diff --git a/parse/train/8PPmLbV6WiQ/8PPmLbV6WiQ.md b/parse/train/8PPmLbV6WiQ/8PPmLbV6WiQ.md new file mode 100644 index 0000000000000000000000000000000000000000..5184ac16cc6fe6022da28152c4cc97e55e5bdd42 --- /dev/null +++ b/parse/train/8PPmLbV6WiQ/8PPmLbV6WiQ.md @@ -0,0 +1,530 @@ +# Normalization of Language Embeddings for Cross-Lingual Alignment + +Anonymous Author(s) +Affiliation +Address +email + +# Abstract + +1 Learning a good transfer function to map the word vectors from two languages +2 into a shared cross-lingual word vector space plays a crucial role in cross-lingual +3 NLP. It is useful in translation tasks and important in allowing complex models +4 built on a high-resource language like English to be directly applied on an aligned +5 low resource language. While Procrustes and other techniques can align language +6 models with some success, it has recently been identified that structural differences +7 (for instance, due to differing word frequency) create different profiles for various +8 monolingual embedding. When these profiles differ across languages, it corre +9 lates with how well languages can align and their performance on cross-lingual +10 downstream tasks. In this work, we develop a very general language embedding +11 normalization procedure, building and subsuming various previous approaches, +12 which removes these structural profiles across languages without destroying their +13 intrinsic meaning. We demonstrate that meaning is retained and alignment is +14 improved on similarity, translation, and cross-language classification tasks. Our +15 proposed normalization clearly outperforms all prior approaches like centering and +16 vector normalization on each task and with each alignment approach. + +# 17 1 Introduction + +18 The best multilingual NLP approaches typically do not jointly learn a single embedding, since words +19 of the same language tend to cluster, and thus are not useful for translation and cross-lingual learning +20 tasks. Rather, after learning individual embeddings, the standard approach is to map word vectors +21 from multiple languages into a shared cross-lingual word vector space [15]. This shared space creates +22 a cross-lingual word embedding (CLWE) [22, 43]. These serve as a valuable tool for transferring +23 data across different languages, understanding cross-linguistic differences, and cross-lingual transfer +24 for downstream tasks, such as direct translation [16, 20, 24], cross-lingual information retrieval [42], +25 cross-lingual document classification [23], and cross-lingual dependency parsing [17, 39]. +26 A common element of almost all CLWE methods is the use of a rigid, orthogonal transformation +27 mapping one embedding onto another so they inhabit a shared linguistic space. An orthogonal +28 transformation is a special class of transformations that can be interpreted as the space of (in our +29 case, high-dimensional) rotations around the origin, and also allowing a mirror flip. This family of +30 transformations preserves (a) linear and (b) angular properties. By linear properties, we mean that +31 the straight-line Euclidean distance between elements is preserved, as are more powerful properties +32 like analogies (e.g., Paris - France $^ +$ Italy $\approx$ Rome). Angular properties refer to measuring angles +33 between pairs of points (from the origin), and as a result, cosine distance is preserved. Given a +34 correspondence between pairs of objects across two embeddings, the classic Procrustes method, +35 provides a closed-form solution which minimizes the sum of Euclidean distances. Moreover, if the +36 vectors are all first made as unit vectors, then this also maximizes the sum of cosine similarities [10]. +37 Under this framework, there has been a flurry of work significantly improving CLWE model per +38 formance along two directions. Semi-supervised and unsupervised models make these approaches +39 require less input, and more amenable to lower-resource languages. For example Bootstrap Procrustes +40 (PROC-B) [15, 41] is semi-supervised in that it starts with a small pairwise correspondence (of +41 500-1000 words), aligns those to infer a larger correspondence, and repeats applying Procrustes +42 alignment. Methods like MUSE [8] are unsupervised, and use a GAN to estimate a correspondence +43 before applying a Procrustes procedure. +44 The second direction is preprocessing the embeddings before applying the Procrustes alignment. +45 These involve methods like removing the mean, removing principal components, and normalization +46 which we will discuss in depth later. In principle, these methods aim to remove the geometry of +47 data intrinsic to particular languages (but not shared across languages) while preserving similarity +48 properties as assured by orthogonal alignment. The space of transformations allowed under orthogonal +49 alignments is quite large, and we make the point that unless this data geometry is “normalized” it +50 inhibits the alignment from optimizing over the entirety of this large space. +51 Finally, we note that methods like Canonical Correlation Analysis (CCA) [13], Discriminative Latent +52 Variable (DLV) [36], and Ranking-based optimization (RCSLS) [21] have also been applied towards +53 finding an orthogonal alignment (or pair of alignments) which minimizes a different optimization +54 function – since the objective function may not align with sums of squared Euclidean of cosine +55 distance [8, 38]. Unlike the others, the RCSLS method notably does not require a rigid transformation. +56 The focus of this paper is on embedding preprocessing, and is agnostic to the method of alignment +57 used afterward, whether it is Procrustes-based, or optimizing something else. +58 Our contribution. This work proposes a new and general approach to preprocessing word embed +59 dings, subsuming many previous approaches. The key is Spectral Normalization which regularizes +60 the spectral properties of monolingual embeddings by setting all of the top singular vectors to have +61 the same singular value. However, it leaves alone the smaller singular value; these capture important +62 information and cannot be zeroed out, but making them the same value as the top singular vectors +63 introduces too much noise. Spectral normalization already performs as well as the best previous +64 approaches on alignment and translation tasks, and since it applies a fairly uniform stretching to the +65 embeddings it does not distort monolingual similarity performance. Moreover, we show layering +66 Spectral Normalization within an iterative sequence with also centering and vector length normaliza +67 tion improves results further. We first demonstrate this improvement on the standard translation task +68 (BLI). We also show that this normalization preserves the core ontological structure of embeddings +69 across languages, and that applying our normalization before aligning a low resource language to +70 English improves performance on topic classification and on a natural language inference task. + +# 71 2 Existing Methods for Orthogonal Vector Spaces Alignment + +72 Given a language $L$ , our starting point is an embedded representation of a set of $n$ words. Indexing +73 words from $i = 1 \dots n$ , each word is associated with a vector $x _ { L } ^ { i } \in \mathbb { R } ^ { d }$ . And let $X _ { L } = \{ x _ { L } ^ { 1 } , . . . , x _ { L } ^ { n } \bar \}$ +74 be the set of $n$ words as their vector representation. These vector representations (derived by methods +75 like word2vec [28], GloVe [33], or FastText [4]) are chosen so words with similar pairwise cosine +76 similarity are found in the similar local context in large text corpora on which they are trained. +77 Higher-level linear structure is shown to emerge, such as concept subspaces and analogies [29]. +78 The focus of this paper is on aligning embeddings of two languages $L _ { 1 }$ and $L _ { 2 }$ . Each embedding +79 $X _ { L _ { 1 } }$ and $X _ { L _ { 2 } }$ , only is designed to ensure pairwise relationships between its word vectors, but the +80 actual coordinates of those vectors do not have any explicit meaning. Yet, previous work has clearly +81 demonstrated that there exists significant overall structural similarity, and alignment seeks to make +82 correspondences between those structures for translation and joint understanding. +83 Most methods start with a known correspondence (or build one) between a set of $K$ words in +84 two languages, wlog let these be the same first $K$ indexed words in those languages, denoted +85 $X _ { L _ { 1 } } ^ { K } ~ = ^ { \bf { \bar { \Phi } } } \{ x _ { L _ { 1 } } ^ { \bf { \bar { 1 } } } \cdot \cdot \cdot x _ { L _ { 1 } } ^ { \tilde { K } } \}$ and $X _ { L _ { 2 } } ^ { K } ~ = ~ \{ x _ { L _ { 2 } } ^ { 1 } \ldots x _ { L _ { 2 } } ^ { K } \}$ . Then the Procrustes Problem solve for an +86 orthogonal matrix $W ^ { * } = \arg \operatorname* { m i n } \cdot$ W $\| X _ { L _ { 1 } } ^ { K } W - X _ { L _ { 2 } } ^ { K } \| _ { 2 } ^ { 2 }$ . There exists a simple solution [2, 38, 45, 10] +87 as $W ^ { * } = U V ^ { \top }$ where $U \Sigma V ^ { \top } = { \mathsf { s v d } } ( X _ { L _ { 1 } } ^ { K } ( X _ { L _ { 2 } } ^ { K } ) ^ { \top } )$ . Dev et.al. [10] also point out that if all vectors +88 are normalized first, then this procedure also maximizes the sum of cosine similarities. +90 It turns out directly aligning embeddings from two languages (even using the “optimal” Procrustes +91 solution) does not provide the best possible joint embedding for translation tasks. While word +92 meaning appears to hold a similar structure, languages have other properties such as differing word +93 frequency, and this for instance leads to more frequent words having longer vectors in embeddings. +94 This extra language-specific structure tends to interfere with alignment. As a result, a number +95 of techniques have been developed to “normalize” the embeddings before Procrustes (or other) +96 alignment. This in some sense allows the word meaning to dominate the optimization tasks without +97 other confounding factors. We review the most common normalization approaches. +98 Mean Centering (C) subtracts the mean of all vectors in an embedding from each vector in that +99 embedding. The result is that the mean of all vectors is 0. This is a rigid transformation, and so does +100 not change the Euclidean distance between any pair of points in an embedding, and also preserves +101 any linear property like analogies (e.g., Paris - France $^ +$ Italy $\approx$ Rome). Dev et. al. [10] points out +102 that this is the first step (followed by the Procrustes orthogonal transformation) to minimize the sum +103 of squared Euclidean distances among paired words, under any rigid transformation. However, this +104 does change the cosine distance between pairs of points. +05 Length Normalization (L) makes each vector have a 2-norm equal to 1, but retains its direction +06 from the origin [2, 45]. This preprocessing step does not change the cosine distance between any pair +07 of points in an embedding. But, it does change the Euclidean distance between pairs of points. +108 Despite these contrasting goals, these two normalizations each turn out to be individually effective in +109 regularizing the geometry of the embeddings, and allow for better CLWE. Wang et. al. [47], realized +110 doing both was even more effective, and showed that iterating these two steps achieves the state +111 of-the-art way to preprocess, we denote as $\mathbf { I } { \mathbf { - C } } { \mathbf { + L } }$ . Iterative Normalization transforms monolingual +112 word embeddings to have unit-length and zero-mean simultaneously (in practice they terminate this +113 iterative process after a few steps before it achieves these two goals exactly). +114 PCA Removal (PR) computes the principal component analysis (PCA) of an embedding, and then +115 projects away from the direction of the top principal component, removing it [31]. Mu et. al. [31] +116 observed that the top singular values typically do not encode essential semantic relationships between +117 words but rather align strongly with word frequency. Also, they notice that the top principal values +118 are much larger than the other values. After eliminating $d / 1 0 0$ principal components, where d is the +119 dimension of a word representation, they achieve better performance on both intrinsic and extrinsic +120 tasks. Sachidananda et. al. [37] applied this simple pre-processing approach to their Filtered Inner +121 Product Projection (FIPP) alignment method, and they significantly improve on the BLI task. + +# 2.2 Spectral Statistics of Embeddings + +123 Dubossarsky et. al. [11] recently documented how cross-lingual alignment is strongly affected by the +124 spectral statistics of monolingual embeddings. We stack the embedded vectors $x _ { L } ^ { \bar { i } } \in \mathbb { R } ^ { d }$ as rows in +125 a $n \times d$ matrix $A \in \mathbb { R } ^ { n \times d }$ . The SVD decomposes $A$ into $U { \boldsymbol { \Sigma } } V ^ { \top }$ where $U$ and $V$ contain the left +126 and right singular vectors, and the singular values $\sigma _ { 1 } \geq \sigma _ { 2 } \geq . . . \geq \sigma _ { d } \geq 0$ are on the diagonal of $\Sigma$ . +127 The effedefined 128 $A$ s a smo where therewith k compone. The effec +$\mathsf { e r } ( A ) = e ^ { H } ( \Sigma )$ $\begin{array} { r } { H ( \Sigma ) = - \sum _ { i = 1 } ^ { d } \bar { \sigma } _ { i } \log \bar { \sigma } _ { i } } \end{array}$ $\bar { \sigma } _ { i } = { \sigma _ { i } } / { \sum _ { i = 1 } ^ { d } { \sigma _ { i } } }$ +condition number $\kappa _ { \mathsf { e f f } } ( A ) = \sigma _ { 1 } / \sigma _ { \mathsf { e r } ( A ) }$ , which replaces the numerator (of condition number, $\sigma _ { d . }$ ) +130 with the more robust singular value at the effective rank. This is desired to be small in stable data +131 sets. The joint effective condition number measures the harmonic mean of the effective condition +132 number across two matrices A, A0 as ECOND-HM(A, A0) = 2κeff(A)κeff(A0)κ (A)+κ (A0) . The singular value gap +133 measures how similar the singular value sequences are between two matrices as $\operatorname { s v g } ( A , A ^ { \prime } ) =$ +134 $\textstyle \sum _ { i = 1 } ^ { d } ( \log \sigma _ { i } - \log \sigma _ { i } ^ { \prime } ) ^ { 2 }$ . These should both be smaller, for more comparable data sets. +135 Dubossarsky et. al. [11] applied these to monolingual embeddings and demonstrated that the +136 performance of several CLWE methods were closely tied to these spectral properties. Basically +137 embeddings align better if they are better jointly conditioned, especially measured via joint effective +138 condition number and the singular value gap. Motivated by this idea, we propose methods that +139 spectrally normalize embeddings improving these statistics while retaining intra-embedding meaning. +141 We introduce more direct and more general techniques to normalize monolingual word embeddings +142 to more effectively prepare them for alignment. The goal is to remove language-specific geometry +143 while maintaining the intrinsic similarity and structure captured within them. + +# 144 3.1 Geometric Median Normalization + +145 Iterative Normalization enforces individual word embeddings to have a unit length and each monolin +146 gual embedding to have a zero mean through an iterative technique. Wang et. al. [47] showed that +147 iterating solutions for these distinct goals will eventually converge to a solution which satisfies both. +148 In this paper, we observe that both goals can be done in one shot without iterating – by solving the +149 Fermat-Weber problem [27, 25]. This dates to the 17th century, and corresponds with identifying the +150 geometric median of a point set. Formally, the goal is a point $x ^ { * } \in \mathbb { R } ^ { d }$ that minimizes the sum of +151 distances from $n$ anchor points $\{ a _ { 1 } , \ldots , \dot { a _ { n } } \} \subset \bar { \mathbb { R } } ^ { d }$ which are not collinear: $\begin{array} { r } { x ^ { * } = \operatorname* { m i n } _ { x } \sum _ { i = 1 } ^ { n } \| x - { } } \end{array}$ +152 $a _ { i } \|$ . Several methods [6, 12, 32] have been proposed; the most popular is the Weiszfeld’s algorithm +153 (Weiszfeld, see Appendix A). It is folklore that the solution $x ^ { * }$ satisfies that $\begin{array} { r } { 0 = \sum _ { i = 1 } ^ { n } { \frac { a _ { i } - x ^ { * } } { \left\| a _ { i } - x ^ { * } \right\| } } } \end{array}$ k we ; +154 do not know of a written proof, so prove this in Appendix A.1 for completeness. +155 Using this characteristic of the geometric median, we can simultaneously enforce monolingual +156 word embeddings to have unit-length and zero-mean in just one step. This can be done using the +157 Geometric Median normalization (GeoMediaN) algorithm (as Algorithm 1). Given a monolingual +158 word embedding $A$ , we compute the geometric median $x ^ { * }$ , and "center" the data on this point, and +159 unit length normalizes the centered embedding. + +# Algorithm 1 Geometric Median Normalization: GeoMediaN(A) + +1: x∗ ← Weiszfeld(A) +2: for all $a _ { i } \in A$ do $\begin{array} { r } { a _ { i } \frac { a _ { i } - x ^ { * } } { \| a _ { i } - x ^ { * } \| } } \end{array}$ +3: return A +60 After these steps, all vectors are unit length, and because of the folklore property (Theorem A.1), the +61 mean of those points is also 0. As a result, we can state the following property. + +162 Theorem 3.1. The output of GeoMediaN $( A )$ is centered and length normalized. + +163 Despite the Geometric Median Normalization algorithm’s ability to enforce unit-length and zero +164 mean in just one step, we will observe that it does not perform especially well on the BLI task. +165 Both GeoMediaN and $\mathrm { I - C + L }$ achieve one of many solutions which achieve these joint goals. We +166 next investigate another one that works better: it preserves meaning and structure, and removes +167 language-specific geometry allowing improved alignment. + +# 168 3.2 Spectral Normalization + +169 The predominant effect of unit-length and zero-mean normalization on monolingual word embeddings +170 is that it makes embedding vectors from a language lie on a hypersphere with the center of the +171 hypersphere centered at the origin. However, this does not take into account how the word embeddings +172 vectors are spread out or clustered on the hypersphere. Approaches like PCA removal and mean +173 centering have the effect of reducing the top principal component or top singular vector. As a result, +174 if the spectral properties are extreme, it can help regularize them. However, this approach can be a +175 bit blunt. PCA removal makes the top singular value exactly 0, so the condition number becomes +176 infinite. Other quantities like the effective condition number, $\kappa _ { \mathsf { e f f } } ( A )$ , do however, tend to decrease. +177 To this effect, we propose a new algorithm Spectral Normalization that more gently regularizes +178 the spectral properties of word embeddings; see Algorithm 2. We will then combine it with other +179 approaches to again ensure the embedding vectors lie on the unit sphere. + +# Algorithm 2 Spectral Normalization $( { \mathsf { S p e c N o r m } } ( A , \beta ) )$ + +1: Compute $\mathsf { s v d } ( A ) = U \Sigma V ^ { \top }$ ; Let $D \in \mathbb { R } ^ { d }$ be a diagonal matrix. +2: Compute $\eta = \sqrt { \| A \| _ { F } ^ { 2 } / d }$ , where $d$ is the dimension of the word embedding +3: for $i = 1 , \ldots , d$ do +4: if $\left( \Sigma _ { i i } > \beta \eta \right)$ then $D _ { i i } \Sigma _ { i i } / ( \beta \eta )$ +5: else $D _ { i i } = 1$ . +6: return $A V D ^ { - 1 }$ +180 Given a monolingual word embedding $A$ it updates part of the spectral properties of $A$ as a whole, +181 using on a parameter $\beta \geq 0$ . Based on an average of singular values $\eta = \sqrt { \| A \| _ { F } ^ { 2 } / d }$ , if a value is +182 above $\beta$ times that average, it adjusts it to $\beta \eta$ . Hence, all of the top directions are given the same +183 singular value. Otherwise, if it is below $\beta \eta$ , it is considered a minor effect (some are quite small, +184 and fairly noisy), and it is left alone. If these small ones are completely zeroed out, the critical +185 information within is destroyed. However, if these small ones are also given the same value (i.e., $\beta \eta )$ +186 then components which do not contribute to the most prevalent aspects of a vectors similarity is given +187 more importance, and we observed (see Section 4.1) that the usefulness of the embedding decreased. +188 Iterative Spectral Normalization. Spectral normalization makes the most sense (see Appendix +189 G) in a setting where the vectors are already centered, and also unit length. While SpecNorm does +190 not change the center of the data, it does not maintain the length of individual vectors. As such, we +191 advocate combining these methods into a single iterative algorithm: $\scriptstyle \mathrm { I - C + S N + L }$ as in Algorithm 3. + +Algorithm 3 Iterative Spectral Normalization with $\mathrm { C + L }$ normalization (I-C+SN+L(A, #Iter)) + +1: for #Iter steps do +2: A ← Center A +3: A ← SpecNorm(A) +4: A ← Unit length normalization of A +5: return A +192 We observe in Figure 1 that this process significantly improves the spectral properties, compare to +193 any other approach. Without preprocessing (None), the languages (EN: English, DE: German, HI: +194 Hindi, JA: Japanese shown) have large effective condition numbers – indicating that there is a large +195 disparity between meaningful singular values. Note the y-axis is in log scale. Hence, aligning these +196 languages without normalization would likely restrict alignment among top singular vectors, not +197 allowing enough degree of freedom to align corresponding words. +198 In contrast, after preprocessing when these values are more uniform, rotations among the dimensions +199 containing the top principal components will not have an influence on the data distribution, and +200 can fully optimize the alignment between words. Moreover, Figure 1 shows that $\mathbf { I - C + S N + L }$ most +201 decreases the effective condition number, joint effective condition number, and singular value gap. +202 Further, these values are fairly uniform across languages, despite great variation beforehand (as +203 shown with None). In fact, $\scriptstyle \mathrm { I - C + S N + L }$ is much more effective than other methods. +205 We provide an evaluation of our proposed preprocessing methods using 8 language embeddings +206 pretrained on Wikipedia [4] of each language: Croatian (HR), English (EN), Finnish (FI), French +207 (FR), German (DE), Italian (IT), Russian (RU), and Turkish (TR). We use the 300-dimensional +208 fastText $[ 4 ] ^ { 1 }$ embeddings, and all vocabularies are trimmed to the 200K most frequent words. +209 Alignment evaluation tasks: BLI We evaluate and compare our proposed preprocessing methods +210 mostly on the Bilingual Lexicon Induction (BLI) task, a word translation task. We discuss two more +211 global applications Cross-lingual document classification (CLDC), and Cross-lingual transfer for +212 natural language inference (XNLI) later. BLI is more direct, and has become the de facto evaluation +213 task for CLWE models. For words in the source language, this task retrieves the nearest neighbors in +214 the target language after alignment to check if it contains the translation. It reports the mean average +215 precision (MAP) [15], which is equivalent to the mean reciprocal rank (MRR), of the translation. +216 Unless stated otherwise, reported values on baseline methods are taken from [15], and use the Google +217 Translate (GTrans) dictionary from $[ 1 5 ] ^ { 2 }$ . We trained (aligned) using 1k, 3k and 5k source words and +218 evaluated (tested) on separate 2k source test queries, unless noted otherwise. +219 Alignment Algorithms. We evaluated and compared the result of several supervised rigid +220 transformation CLWE models on the evaluation benchmarks using our proposed methods. All +221 have publically available code, links are found in the reference citation. These include Canonical +222 Correlation Analysis (CCA) [13], Procrustes (PROC) [2, 38, 45, 10], Bootstrapping Procrustes +223 (PROC-B) [15], and Discriminative Latent-Variable (DLV) [36], as discussed in Section 1. We also +224 consider Ranking-Based Optimization (RCSLS) [21] which is not a rigid alignment. In a few places, +225 we also compare with VECMAP [3] as an example of an unsupervised alignment process. This should +226 only use the geometry of the global embedding structure, e.g., derived from the natural ontology, and +227 our normalization method still helps when using this approach. + +![](images/4457cfc7c0f470ce62ba0a8960c8b44495e221a08fec678f9a5a67f4ed9b61fb.jpg) +Figure 1: Spectral Measures of four (4) monolingual word embeddings, before (None) and after applying various normalization methods. + +# 4.1 Hyperparameter Tuning + +Our main proposed algorithm $\scriptstyle \mathrm { I - C + S N + L }$ has a few simple parameters. To avoid overfitting, we choose these through cross-validation on English (EN) and a held-out set of 5 languages Hindi (HI), Russian (RU), Chinese (ZH), Japanese (JA), Turkish (TR). Ten (10) Language pairs of the form EN-X and X-EN were considered. The hyperparameters $\beta \in \{ 1 , 2 , 3 , 4 , 5 \}$ and #Iter (number of iterations) $\in$ $\{ 1 , 2 , 3 , 4 , 5 \}$ were fine-tuned for $\scriptstyle \mathrm { I - C + S N + L }$ . + +234 We used the publicly available MUSE3 translation dictionary [8] for hyperparameter tuning. The +235 Procrustes alignment algorithm was trained on $5 \mathrm { k }$ source words and evaluated on $1 . 5 \mathrm { k }$ source test +236 queries. We reported the mean average precision (MAP) in Table 1 for $\beta \in \{ 1 , 2 , 3 , 4 , 5 \}$ and with +237 #Iter $\in \{ 1 , 2 , 3 , 4 , 5 \}$ . We observe the value of $\beta = 2$ was consistently the best threshold (although +238 any $\beta \geq 2$ performed similarly). These singular values were normalized, and those below $\beta$ times the +239 average we judged as noise, and left as is. However, the result did not change much with respect to +240 the number of iterations. +241 The tie between the #Iter hyperparameter was broken using their performance on thirteen English +242 word similarity benchmarks; see Appendix C for more details. In Table 2 below, we report the +243 average Spearman rank coefficient score on the word similarity task (None means no normalization). +244 $( \beta , \# \mathrm { { I t e r } } ) \mathbf { \bar { \Psi } } = \mathbf { \Psi } ( 2 , 5 )$ achieved the highest score. So hereafter, we applying $\scriptstyle \mathrm { I - C + S N + L }$ with the +245 hyperparameter $( \beta , \# \mathrm { { I t e r } } ) = ( 2 , 5 )$ . +246 Note that the proposed approach $( \mathrm { I - C + S N + L } )$ only increased this score for these similarity tasks, so +247 showed no signs of distorting inherent information. Although Spectral Normalization does not exactly +248 preserve the linear properties or angular properties (as centering and length normalization do, one each, +249 respectively), it does not suffer ill effects. We hypothesize this is because it is somewhat uniformly +250 stretching words along the major modes of variation, and is effectively removing information not +251 relevant to meaning, like frequency of occurrence. This benign effect is on contrast to other spectral +252 adjustments (removing small singular values, or setting all to the same value) shown in Appendix C. + +Table 1: Cross-Validation for Hyperparameter Tuning: MAP after Procrustes for 10 language pairs. + +
β#Iter=1#Iter=2#Iter=3#Iter=4#Iter=5
10.3630.3400.3280.3220.317
20.3850.3860.3860.3860.386
30.3810.3840.3840.3840.384
40.3810.3820.3820.3820.382
50.3800.3810.3810.3810.381
+ +Table 2: Monolingual word similarity Task; Average Spearman rank coefficient + +
None(β,#Iter)= (2,2)(β,#Iter)= (2,3)(β,#Iter)= (2,4) (β,#Iter)= (2,5)
0.6510.670770.671010.671080.67111
+ +# 4.2 BLI Performance across Normalization and Alignment Algorithms + +254 We compare and evaluate the BLI performance (MAP) of various normalization algorithms from +255 previous works to our proposed algorithms. Using the MUSE translation dictionary, we trained CCA, +256 PROC, PROC-B and RCSLS on $5 \mathrm { k }$ source words and evaluated on $1 . 5 \mathrm { k }$ source test queries. The +257 following normalization algorithms were used in the comparison analysis: PR (PCA Removal) [31], +258 GeoMediaN (Geometric Median Normalization), $\mathrm { C + L }$ (Mean centering and Length normalization, +259 1 round), $\mathrm { I - C + L }$ (Iterative Mean centering and Length normalization, 5 rounds) [47], SN (Spectral +260 Normalization, 1 round), $\mathrm { C + S N + L }$ (Mean centering, Spectral Normalization and Length normal +261 ization, 1 round), and $_ { \mathrm { I - C + S N + L } }$ (Iterative Mean centering, Spectral Normalization and Length +262 normalization, 5 rounds). Specifically, we evaluated 18 language pairs, i.e., English (EN) from/to +263 Bulgarian (BG), Catalan (CA), Czech (CS), German (DE), Spanish (ES), Korean (KO), Thai (TH) and +264 Chinese (ZH), separate from hyperparameter tuning. The average is reported in Table 3, all results +265 are in Appendix G. For almost all algorithms $_ { \mathrm { I - C + S N + L } }$ achieves the best scores (and especially on +266 ${ \bf X } _ { L _ { 2 } } \mathrm { - } { \bf E } { \bf N }$ , often considerably better). The only exceptions are on non-rigid RCSLS when $\mathrm { C + S N + L }$ +267 (with no iteration) or just SN (with $\mathrm { C + L }$ ) performs slightly better. So, Spectral Normalization, and in +268 particular $\scriptstyle \mathrm { I - C + S N + L }$ , is shown as the best way to normalize languages before alignment. +269 We also compute the average BLI MAP score across all 28 language pair for more direct comparison +270 to prior work [15], summarized in Table 4 and Appendix D. All results are in Appendix H. We +271 compare $\scriptstyle \mathrm { I - C + S N + L }$ (denoted with SN) against no normalization on various dictionary sizes: 1k, 3k +272 and $5 \mathrm { k }$ source words and evaluated on 2k source test queries. In all cases, $\scriptstyle \mathbf { I - C + S N + L }$ significantly +273 improves over the baseline. This includes improvement over RCSLS which is non-rigid, so in +274 principle could “learn" adjustments similar to our normalization in the process of alignment. We also +275 tested on VECMAP, an unsupervised approach; $_ { \mathrm { I - C + S N + L } }$ preprocessing also improves this result +276 from 0.375 to 0.410. + +Table 3: BLI performance (MAP) on aligning $\mathbf { E N - X } _ { L _ { 2 } }$ and ${ \bf X } _ { L _ { 2 } } \mathrm { - } { \bf E } { \bf N }$ + +
Methods : EN-XL2Methods : XL_-EN
NormalizationCCAPROCPROC-BRCSLSCCAPROCPROC-BRCSLS
None0.3580.3650.3770.3940.3980.3990.4050.428
PR0.3940.3910.4040.3730.4340.4300.4420.425
GeoMediaN0.3930.3910.4000.3790.4330.4320.4400.429
C+L0.3930.3940.4080.4040.4390.4370.4450.464
I-C+L0.3940.3950.4100.4060.4390.4380.4480.460
SN0.3910.3940.4080.4050.4400.4380.4510.468
C+SN+L0.3950.3960.4130.4070.4440.4440.4580.466
I-C+SN+L0.3960.3980.4140.4060.4450.4460.4610.466
+ +Table 4: Summary of BLI performance (MAP), average scores for all 28 language pairs. No normalization results from [15], against $_ { \mathrm { I - C + S N + L } }$ (denoted SN). + +
CCCCPPRDPPPPRDPPPPREPPPPSRPNsATQPSSSSY PSSSS
1K.289.314.299.326.379.407.289.332.331.331
3K.378.401.384.408.398.415.381.429.415.427
5K.400.423.405.42911.403.452.437.460
+ +# 277 4.3 Downstream Tasks + +We conclude by demonstrating that Spectral Normalization not only improves in direct translation tasks, but also captures an important global structure that generalizes from a high resource language (i.e., English, EN) to lower resource languages. In both examples, a powerful classifier is trained on the EN embedding (after normalization), and then we demonstrate that after a lower resource language (e.g., German, DE) has been normalized and align the analysis task can be directly applied to that language. And in particular, adding the simple process of our normalization $( \mathrm { I - C + S N + L }$ ) dramatically improves the results over not doing that step. + +285 Cross-lingual Document Classification (CLDC). The CLDC task builds a topic classification +286 using a language model on a high resource language (in our case English EN) across 15 topics. +287 The TED CLDC corpus assembled by [19] was used for training and evaluation. Following [15] +288 a simple CNN was used to train. Table 5 summarizes the average F1-score for all topic classifiers +289 on 5 language pairs. The CLWEs induced by $\mathrm { P R O C ^ { S N } }$ , PROC- $\cdot \breve { \mathbf { B } } ^ { \mathsf { S N } }$ , ${ \bf D } { \bf U } ^ { { \sf S } { \sf N } }$ , and $\mathrm { R C S L S ^ { S N } }$ (using +290 $\scriptstyle \mathrm { I - C + S N + L }$ ) outperformed the baseline result (with no normalization) on the CLDC task, significantly +291 improving the best average score from 0.421 to 0.461. Glavas et.al. [15] use only 12 of 15 topics, +292 but could not confirm which, so we re-ran all baselines using all 15 topics. +293 Cross-lingual Natural Language Inference (XNLI). We evaluated the CLWE on a cross-lingual +294 natural language inference (XNLI) task. We used a multi-lingual XNLI corpus created by [9], which +295 is a collection of sentence pairs from the English MultiNLI corpus [44] translated into 14 languages. +296 The MultiNLI corpus contains $4 3 3 \mathrm { k }$ sentence pairs with the labels entailment, contradiction, and +297 neutral. The intersection between XNLI languages and BLI languages result in four XNLI evaluation +298 pairs: EN-DE, EN-FR. EN-TR and EN-RU. We use the training setup in [15] with the Enhanced +299 Sequential Inference Model [7] on English after normalization. First, we aligned normalized versions +300 of each language onto the normalized EN embedding to obtain the shared cross-lingual embedding. +301 Then we used the $5 \mathrm { k }$ test pairs from the XNLI corpus to evaluate each language alignment. Table 6 +302 shows the result for PROC, PROC-B, and RCSLS alignments (DLV and VECMAP transform the +303 EN embedding in the process, so were omitted). We compare against the same procedure without + +Table 5: CLDC performance (micro-averaged $F _ { 1 }$ scores). Cross-lingual transfer EN–X + +
ModelDictEN-DEEN-FREN-ITEN-RUEN-TRAvg
PROC5K.366.258.338.288.278.306
PROCSN5K.436.366.427.517.511.451
PROC-B3K.364.304.299.336.317.324
PROC-BSN3K.448.396.423.522.517.461
DLV5K.419.336.397.493.458.421
DLVSN5K.433.323.406.499.472.427
RCSLS5K.466.397.403.403.406.415
RCSLSSN5K.468.500.443.488.394.459
+ +normalization from Glavas et.al. [15]304 $( \mathrm { I - C + S N + L }$ normalization denoted SN ). As in other experiments, 305 our normalization improves the average test accuracy with each alignment approach. + +Table 6: XNLI performance (test set accuracy) + +
ModelDictEN-DEEN-FREN-TREN-RUAvg
PROC PROCSN5K 5K.607 .611.534 .638.568 .542.585 .596.574 .597
PROC-B PROC-BSN3K 3K.615 .624.532 .638.573 .548.599 .601.580 .603
RCSLS RCSLSSN5K 5K.390 .499.363 .482.387 .504.399 .556.385 .510
+ +# 306 5 Conclusion & Discussion + +307 We introduce a new way to normalize embeddings, based on spectral normalization, for use in creating +308 cross-lingual word embeddings. Our approach generalizes previous approaches, and effectively +309 removes much of the clustering of words based on properties other than the similarity which encodes +310 meaning. When used to individually preprocess monolingual embeddings, our approach allows +311 any alignment procedure to find better alignments: resulting in improved performance on direct +312 translation tasks as well as cross-lingual topic classification and natural language inference tasks. +313 Social impacts. The vast majority of NLP research and cutting-edge advancements are in English. +314 This disadvantages those who primarily operate in other languages, with less developed models, or +315 less data to train models. As large language models are the cornerstone of most NLP research and +316 development in English, one of this work’s main goals is to port these advances to other languages, +317 and those who use them. This will help unlock this technology to many others around the world. As +318 with most models, this accuracy and improvement may vary across tasks and languages. +319 While language models have many positive use cases including improving accessibility, better +320 recommendations, and increased automation, they have some negative effects as well. These include +321 requiring potentially large computational and hence environmental cost, encoding and exacerbating +322 bias, and aiding in automatically generating fake or deceitful content. While this paper is unlikely to +323 change the desire to use embeddings, it aims to reduce the burden of use and increase the effectiveness +324 in lower-resource settings. And in particular to port models trained in English to other languages. +325 This would reduce the cost of retraining in other languages if the English model can be reused, easing +326 environmental costs. We support the maturing efforts in attenuating bias in all such embeddings. +327 And while we acknowledge the possibility of this work aiding in the automated creation of deceitful +328 content and the harm it can cause, we believe the many benefits outweigh the harms. + +Limitations. The overarching goal in this line of work is to port the many advances built on embeddings from high-resource settings (like built on the English language), to lower-resource settings (like Turkish). This work can apply a powerful model built on an English language model (e.g., for natural language inference) and automatically invoke it in Turkish after the embeddings have been aligned. However, the alignment will not be any better than the low-resource embedding. If the embedding is too noisy or limited, then the analysis will likely not be effective. + +335 Also, this work focuses on non-contextual embeddings like FastText, but not contextual ones like +336 RoBERTa which have proven almost universally more effective in NLP on English. While in principle +337 a normalization function and alignment could be applied to contextual settings, we are unaware of +338 any technique for learning these mappings. We believe it could be important future work. + +Data, Code, and Experiments. All existing methods are compared with publicly available code with publicly available data, with links above or in references. The exception is code for CLDC and XNLI is shared by Glavás et.al. [15]. Everything is run with default parameters; the exception is RCSLS were we follow the suggested hyperparameter selection strategy [21] (with learning rate in $\{ 1 , 1 0 , 2 5 , 5 0 \}$ and epoch number in $\{ 1 0 , 2 0 \}$ ). 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Masaryk University, 2006. +[47] Mozhi Zhang, Keyulu Xu, Ken-ichi Kawarabayashi, Stefanie Jegelka, and Jordan Boyd-Graber. Are girls neko or shojo? cross-lingual alignment of non-isomorphic embeddings with iterative ¯ normalization. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 3180–3189, Florence, Italy, July 2019. Association for Computational Linguistics. https://github.com/zhangmozhi/iternorm?utm_source $=$ catalyzex.com. + +# Checklist + +1. For all authors... + +(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] +(b) Did you describe the limitations of your work? [Yes] +(c) Did you discuss any potential negative societal impacts of your work? [Yes] +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] + +3. If you ran experiments... + +(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] +(b) Did you mention the license of the assets? [N/A] +(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] See code in Appendix E. +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] + +534 (b) Did you describe any potential participant risks, with links to Institutional Review +535 Board (IRB) approvals, if applicable? [N/A] +536 (c) Did you include the estimated hourly wage paid to participants and the total amount +537 spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/train/8PPmLbV6WiQ/8PPmLbV6WiQ_content_list.json b/parse/train/8PPmLbV6WiQ/8PPmLbV6WiQ_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..5e0092935ea42f4deff13bc9490b33b6be08d321 --- /dev/null +++ b/parse/train/8PPmLbV6WiQ/8PPmLbV6WiQ_content_list.json @@ -0,0 +1,1073 @@ +[ + { + "type": "text", + "text": "Normalization of Language Embeddings for Cross-Lingual Alignment ", + "text_level": 1, + "bbox": [ + 232, + 122, + 767, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 226, + 580, + 281 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 318, + 535, + 334 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Learning a good transfer function to map the word vectors from two languages \n2 into a shared cross-lingual word vector space plays a crucial role in cross-lingual \n3 NLP. It is useful in translation tasks and important in allowing complex models \n4 built on a high-resource language like English to be directly applied on an aligned \n5 low resource language. While Procrustes and other techniques can align language \n6 models with some success, it has recently been identified that structural differences \n7 (for instance, due to differing word frequency) create different profiles for various \n8 monolingual embedding. When these profiles differ across languages, it corre \n9 lates with how well languages can align and their performance on cross-lingual \n10 downstream tasks. In this work, we develop a very general language embedding \n11 normalization procedure, building and subsuming various previous approaches, \n12 which removes these structural profiles across languages without destroying their \n13 intrinsic meaning. We demonstrate that meaning is retained and alignment is \n14 improved on similarity, translation, and cross-language classification tasks. Our \n15 proposed normalization clearly outperforms all prior approaches like centering and \n16 vector normalization on each task and with each alignment approach. ", + "bbox": [ + 148, + 349, + 766, + 571 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "17 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 598, + 312, + 616 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "18 The best multilingual NLP approaches typically do not jointly learn a single embedding, since words \n19 of the same language tend to cluster, and thus are not useful for translation and cross-lingual learning \n20 tasks. Rather, after learning individual embeddings, the standard approach is to map word vectors \n21 from multiple languages into a shared cross-lingual word vector space [15]. This shared space creates \n22 a cross-lingual word embedding (CLWE) [22, 43]. These serve as a valuable tool for transferring \n23 data across different languages, understanding cross-linguistic differences, and cross-lingual transfer \n24 for downstream tasks, such as direct translation [16, 20, 24], cross-lingual information retrieval [42], \n25 cross-lingual document classification [23], and cross-lingual dependency parsing [17, 39]. \n26 A common element of almost all CLWE methods is the use of a rigid, orthogonal transformation \n27 mapping one embedding onto another so they inhabit a shared linguistic space. An orthogonal \n28 transformation is a special class of transformations that can be interpreted as the space of (in our \n29 case, high-dimensional) rotations around the origin, and also allowing a mirror flip. This family of \n30 transformations preserves (a) linear and (b) angular properties. By linear properties, we mean that \n31 the straight-line Euclidean distance between elements is preserved, as are more powerful properties \n32 like analogies (e.g., Paris - France $^ +$ Italy $\\approx$ Rome). Angular properties refer to measuring angles \n33 between pairs of points (from the origin), and as a result, cosine distance is preserved. Given a \n34 correspondence between pairs of objects across two embeddings, the classic Procrustes method, \n35 provides a closed-form solution which minimizes the sum of Euclidean distances. Moreover, if the \n36 vectors are all first made as unit vectors, then this also maximizes the sum of cosine similarities [10]. \n37 Under this framework, there has been a flurry of work significantly improving CLWE model per \n38 formance along two directions. Semi-supervised and unsupervised models make these approaches \n39 require less input, and more amenable to lower-resource languages. For example Bootstrap Procrustes \n40 (PROC-B) [15, 41] is semi-supervised in that it starts with a small pairwise correspondence (of \n41 500-1000 words), aligns those to infer a larger correspondence, and repeats applying Procrustes \n42 alignment. Methods like MUSE [8] are unsupervised, and use a GAN to estimate a correspondence \n43 before applying a Procrustes procedure. \n44 The second direction is preprocessing the embeddings before applying the Procrustes alignment. \n45 These involve methods like removing the mean, removing principal components, and normalization \n46 which we will discuss in depth later. In principle, these methods aim to remove the geometry of \n47 data intrinsic to particular languages (but not shared across languages) while preserving similarity \n48 properties as assured by orthogonal alignment. The space of transformations allowed under orthogonal \n49 alignments is quite large, and we make the point that unless this data geometry is “normalized” it \n50 inhibits the alignment from optimizing over the entirety of this large space. \n51 Finally, we note that methods like Canonical Correlation Analysis (CCA) [13], Discriminative Latent \n52 Variable (DLV) [36], and Ranking-based optimization (RCSLS) [21] have also been applied towards \n53 finding an orthogonal alignment (or pair of alignments) which minimizes a different optimization \n54 function – since the objective function may not align with sums of squared Euclidean of cosine \n55 distance [8, 38]. Unlike the others, the RCSLS method notably does not require a rigid transformation. \n56 The focus of this paper is on embedding preprocessing, and is agnostic to the method of alignment \n57 used afterward, whether it is Procrustes-based, or optimizing something else. \n58 Our contribution. This work proposes a new and general approach to preprocessing word embed \n59 dings, subsuming many previous approaches. The key is Spectral Normalization which regularizes \n60 the spectral properties of monolingual embeddings by setting all of the top singular vectors to have \n61 the same singular value. However, it leaves alone the smaller singular value; these capture important \n62 information and cannot be zeroed out, but making them the same value as the top singular vectors \n63 introduces too much noise. Spectral normalization already performs as well as the best previous \n64 approaches on alignment and translation tasks, and since it applies a fairly uniform stretching to the \n65 embeddings it does not distort monolingual similarity performance. Moreover, we show layering \n66 Spectral Normalization within an iterative sequence with also centering and vector length normaliza \n67 tion improves results further. We first demonstrate this improvement on the standard translation task \n68 (BLI). We also show that this normalization preserves the core ontological structure of embeddings \n69 across languages, and that applying our normalization before aligning a low resource language to \n70 English improves performance on topic classification and on a natural language inference task. ", + "bbox": [ + 147, + 631, + 825, + 742 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 748, + 825, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 92, + 825, + 189 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 194, + 825, + 292 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 297, + 825, + 367 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 151, + 373, + 825, + 401 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 420, + 826, + 601 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "71 2 Existing Methods for Orthogonal Vector Spaces Alignment ", + "text_level": 1, + "bbox": [ + 147, + 623, + 696, + 641 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "72 Given a language $L$ , our starting point is an embedded representation of a set of $n$ words. Indexing \n73 words from $i = 1 \\dots n$ , each word is associated with a vector $x _ { L } ^ { i } \\in \\mathbb { R } ^ { d }$ . And let $X _ { L } = \\{ x _ { L } ^ { 1 } , . . . , x _ { L } ^ { n } \\bar \\}$ \n74 be the set of $n$ words as their vector representation. These vector representations (derived by methods \n75 like word2vec [28], GloVe [33], or FastText [4]) are chosen so words with similar pairwise cosine \n76 similarity are found in the similar local context in large text corpora on which they are trained. \n77 Higher-level linear structure is shown to emerge, such as concept subspaces and analogies [29]. \n78 The focus of this paper is on aligning embeddings of two languages $L _ { 1 }$ and $L _ { 2 }$ . Each embedding \n79 $X _ { L _ { 1 } }$ and $X _ { L _ { 2 } }$ , only is designed to ensure pairwise relationships between its word vectors, but the \n80 actual coordinates of those vectors do not have any explicit meaning. Yet, previous work has clearly \n81 demonstrated that there exists significant overall structural similarity, and alignment seeks to make \n82 correspondences between those structures for translation and joint understanding. \n83 Most methods start with a known correspondence (or build one) between a set of $K$ words in \n84 two languages, wlog let these be the same first $K$ indexed words in those languages, denoted \n85 $X _ { L _ { 1 } } ^ { K } ~ = ^ { \\bf { \\bar { \\Phi } } } \\{ x _ { L _ { 1 } } ^ { \\bf { \\bar { 1 } } } \\cdot \\cdot \\cdot x _ { L _ { 1 } } ^ { \\tilde { K } } \\}$ and $X _ { L _ { 2 } } ^ { K } ~ = ~ \\{ x _ { L _ { 2 } } ^ { 1 } \\ldots x _ { L _ { 2 } } ^ { K } \\}$ . Then the Procrustes Problem solve for an \n86 orthogonal matrix $W ^ { * } = \\arg \\operatorname* { m i n } \\cdot$ W $\\| X _ { L _ { 1 } } ^ { K } W - X _ { L _ { 2 } } ^ { K } \\| _ { 2 } ^ { 2 }$ . There exists a simple solution [2, 38, 45, 10] \n87 as $W ^ { * } = U V ^ { \\top }$ where $U \\Sigma V ^ { \\top } = { \\mathsf { s v d } } ( X _ { L _ { 1 } } ^ { K } ( X _ { L _ { 2 } } ^ { K } ) ^ { \\top } )$ . Dev et.al. [10] also point out that if all vectors \n88 are normalized first, then this procedure also maximizes the sum of cosine similarities. \n90 It turns out directly aligning embeddings from two languages (even using the “optimal” Procrustes \n91 solution) does not provide the best possible joint embedding for translation tasks. While word \n92 meaning appears to hold a similar structure, languages have other properties such as differing word \n93 frequency, and this for instance leads to more frequent words having longer vectors in embeddings. \n94 This extra language-specific structure tends to interfere with alignment. As a result, a number \n95 of techniques have been developed to “normalize” the embeddings before Procrustes (or other) \n96 alignment. This in some sense allows the word meaning to dominate the optimization tasks without \n97 other confounding factors. We review the most common normalization approaches. \n98 Mean Centering (C) subtracts the mean of all vectors in an embedding from each vector in that \n99 embedding. The result is that the mean of all vectors is 0. This is a rigid transformation, and so does \n100 not change the Euclidean distance between any pair of points in an embedding, and also preserves \n101 any linear property like analogies (e.g., Paris - France $^ +$ Italy $\\approx$ Rome). Dev et. al. [10] points out \n102 that this is the first step (followed by the Procrustes orthogonal transformation) to minimize the sum \n103 of squared Euclidean distances among paired words, under any rigid transformation. However, this \n104 does change the cosine distance between pairs of points. \n05 Length Normalization (L) makes each vector have a 2-norm equal to 1, but retains its direction \n06 from the origin [2, 45]. This preprocessing step does not change the cosine distance between any pair \n07 of points in an embedding. But, it does change the Euclidean distance between pairs of points. \n108 Despite these contrasting goals, these two normalizations each turn out to be individually effective in \n109 regularizing the geometry of the embeddings, and allow for better CLWE. Wang et. al. [47], realized \n110 doing both was even more effective, and showed that iterating these two steps achieves the state \n111 of-the-art way to preprocess, we denote as $\\mathbf { I } { \\mathbf { - C } } { \\mathbf { + L } }$ . Iterative Normalization transforms monolingual \n112 word embeddings to have unit-length and zero-mean simultaneously (in practice they terminate this \n113 iterative process after a few steps before it achieves these two goals exactly). \n114 PCA Removal (PR) computes the principal component analysis (PCA) of an embedding, and then \n115 projects away from the direction of the top principal component, removing it [31]. Mu et. al. [31] \n116 observed that the top singular values typically do not encode essential semantic relationships between \n117 words but rather align strongly with word frequency. Also, they notice that the top principal values \n118 are much larger than the other values. After eliminating $d / 1 0 0$ principal components, where d is the \n119 dimension of a word representation, they achieve better performance on both intrinsic and extrinsic \n120 tasks. Sachidananda et. al. [37] applied this simple pre-processing approach to their Filtered Inner \n121 Product Projection (FIPP) alignment method, and they significantly improve on the BLI task. ", + "bbox": [ + 147, + 656, + 825, + 741 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 746, + 825, + 815 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 821, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 117, + 825, + 228 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 242, + 825, + 340 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 151, + 354, + 825, + 397 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 404, + 825, + 487 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 502, + 825, + 613 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Spectral Statistics of Embeddings ", + "text_level": 1, + "bbox": [ + 165, + 630, + 446, + 645 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "123 Dubossarsky et. al. [11] recently documented how cross-lingual alignment is strongly affected by the \n124 spectral statistics of monolingual embeddings. We stack the embedded vectors $x _ { L } ^ { \\bar { i } } \\in \\mathbb { R } ^ { d }$ as rows in \n125 a $n \\times d$ matrix $A \\in \\mathbb { R } ^ { n \\times d }$ . The SVD decomposes $A$ into $U { \\boldsymbol { \\Sigma } } V ^ { \\top }$ where $U$ and $V$ contain the left \n126 and right singular vectors, and the singular values $\\sigma _ { 1 } \\geq \\sigma _ { 2 } \\geq . . . \\geq \\sigma _ { d } \\geq 0$ are on the diagonal of $\\Sigma$ . \n127 The effedefined 128 $A$ s a smo where therewith k compone. The effec \n$\\mathsf { e r } ( A ) = e ^ { H } ( \\Sigma )$ $\\begin{array} { r } { H ( \\Sigma ) = - \\sum _ { i = 1 } ^ { d } \\bar { \\sigma } _ { i } \\log \\bar { \\sigma } _ { i } } \\end{array}$ $\\bar { \\sigma } _ { i } = { \\sigma _ { i } } / { \\sum _ { i = 1 } ^ { d } { \\sigma _ { i } } }$ \ncondition number $\\kappa _ { \\mathsf { e f f } } ( A ) = \\sigma _ { 1 } / \\sigma _ { \\mathsf { e r } ( A ) }$ , which replaces the numerator (of condition number, $\\sigma _ { d . }$ ) \n130 with the more robust singular value at the effective rank. This is desired to be small in stable data \n131 sets. The joint effective condition number measures the harmonic mean of the effective condition \n132 number across two matrices A, A0 as ECOND-HM(A, A0) = 2κeff(A)κeff(A0)κ (A)+κ (A0) . The singular value gap \n133 measures how similar the singular value sequences are between two matrices as $\\operatorname { s v g } ( A , A ^ { \\prime } ) =$ \n134 $\\textstyle \\sum _ { i = 1 } ^ { d } ( \\log \\sigma _ { i } - \\log \\sigma _ { i } ^ { \\prime } ) ^ { 2 }$ . These should both be smaller, for more comparable data sets. \n135 Dubossarsky et. al. [11] applied these to monolingual embeddings and demonstrated that the \n136 performance of several CLWE methods were closely tied to these spectral properties. Basically \n137 embeddings align better if they are better jointly conditioned, especially measured via joint effective \n138 condition number and the singular value gap. Motivated by this idea, we propose methods that \n139 spectrally normalize embeddings improving these statistics while retaining intra-embedding meaning. \n141 We introduce more direct and more general techniques to normalize monolingual word embeddings \n142 to more effectively prepare them for alignment. The goal is to remove language-specific geometry \n143 while maintaining the intrinsic similarity and structure captured within them. ", + "bbox": [ + 140, + 655, + 826, + 837 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 842, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 125, + 825, + 166 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "144 3.1 Geometric Median Normalization ", + "text_level": 1, + "bbox": [ + 150, + 190, + 447, + 205 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "145 Iterative Normalization enforces individual word embeddings to have a unit length and each monolin \n146 gual embedding to have a zero mean through an iterative technique. Wang et. al. [47] showed that \n147 iterating solutions for these distinct goals will eventually converge to a solution which satisfies both. \n148 In this paper, we observe that both goals can be done in one shot without iterating – by solving the \n149 Fermat-Weber problem [27, 25]. This dates to the 17th century, and corresponds with identifying the \n150 geometric median of a point set. Formally, the goal is a point $x ^ { * } \\in \\mathbb { R } ^ { d }$ that minimizes the sum of \n151 distances from $n$ anchor points $\\{ a _ { 1 } , \\ldots , \\dot { a _ { n } } \\} \\subset \\bar { \\mathbb { R } } ^ { d }$ which are not collinear: $\\begin{array} { r } { x ^ { * } = \\operatorname* { m i n } _ { x } \\sum _ { i = 1 } ^ { n } \\| x - { } } \\end{array}$ \n152 $a _ { i } \\|$ . Several methods [6, 12, 32] have been proposed; the most popular is the Weiszfeld’s algorithm \n153 (Weiszfeld, see Appendix A). It is folklore that the solution $x ^ { * }$ satisfies that $\\begin{array} { r } { 0 = \\sum _ { i = 1 } ^ { n } { \\frac { a _ { i } - x ^ { * } } { \\left\\| a _ { i } - x ^ { * } \\right\\| } } } \\end{array}$ k we ; \n154 do not know of a written proof, so prove this in Appendix A.1 for completeness. \n155 Using this characteristic of the geometric median, we can simultaneously enforce monolingual \n156 word embeddings to have unit-length and zero-mean in just one step. This can be done using the \n157 Geometric Median normalization (GeoMediaN) algorithm (as Algorithm 1). Given a monolingual \n158 word embedding $A$ , we compute the geometric median $x ^ { * }$ , and \"center\" the data on this point, and \n159 unit length normalizes the centered embedding. ", + "bbox": [ + 142, + 218, + 826, + 261 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 266, + 825, + 369 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 375, + 825, + 444 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 1 Geometric Median Normalization: GeoMediaN(A) ", + "text_level": 1, + "bbox": [ + 173, + 468, + 606, + 482 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1: x∗ ← Weiszfeld(A) \n2: for all $a _ { i } \\in A$ do $\\begin{array} { r } { a _ { i } \\frac { a _ { i } - x ^ { * } } { \\| a _ { i } - x ^ { * } \\| } } \\end{array}$ \n3: return A \n60 After these steps, all vectors are unit length, and because of the folklore property (Theorem A.1), the \n61 mean of those points is also 0. As a result, we can state the following property. ", + "bbox": [ + 179, + 486, + 413, + 535 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 155, + 566, + 823, + 594 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "162 Theorem 3.1. The output of GeoMediaN $( A )$ is centered and length normalized. ", + "bbox": [ + 151, + 602, + 710, + 617 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "163 Despite the Geometric Median Normalization algorithm’s ability to enforce unit-length and zero \n164 mean in just one step, we will observe that it does not perform especially well on the BLI task. \n165 Both GeoMediaN and $\\mathrm { I - C + L }$ achieve one of many solutions which achieve these joint goals. We \n166 next investigate another one that works better: it preserves meaning and structure, and removes \n167 language-specific geometry allowing improved alignment. ", + "bbox": [ + 142, + 631, + 825, + 700 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "168 3.2 Spectral Normalization ", + "text_level": 1, + "bbox": [ + 147, + 724, + 375, + 739 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "169 The predominant effect of unit-length and zero-mean normalization on monolingual word embeddings \n170 is that it makes embedding vectors from a language lie on a hypersphere with the center of the \n171 hypersphere centered at the origin. However, this does not take into account how the word embeddings \n172 vectors are spread out or clustered on the hypersphere. Approaches like PCA removal and mean \n173 centering have the effect of reducing the top principal component or top singular vector. As a result, \n174 if the spectral properties are extreme, it can help regularize them. However, this approach can be a \n175 bit blunt. PCA removal makes the top singular value exactly 0, so the condition number becomes \n176 infinite. Other quantities like the effective condition number, $\\kappa _ { \\mathsf { e f f } } ( A )$ , do however, tend to decrease. \n177 To this effect, we propose a new algorithm Spectral Normalization that more gently regularizes \n178 the spectral properties of word embeddings; see Algorithm 2. We will then combine it with other \n179 approaches to again ensure the embedding vectors lie on the unit sphere. ", + "bbox": [ + 140, + 752, + 825, + 863 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 869, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm 2 Spectral Normalization $( { \\mathsf { S p e c N o r m } } ( A , \\beta ) )$ ", + "text_level": 1, + "bbox": [ + 178, + 92, + 540, + 107 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1: Compute $\\mathsf { s v d } ( A ) = U \\Sigma V ^ { \\top }$ ; Let $D \\in \\mathbb { R } ^ { d }$ be a diagonal matrix. \n2: Compute $\\eta = \\sqrt { \\| A \\| _ { F } ^ { 2 } / d }$ , where $d$ is the dimension of the word embedding \n3: for $i = 1 , \\ldots , d$ do \n4: if $\\left( \\Sigma _ { i i } > \\beta \\eta \\right)$ then $D _ { i i } \\Sigma _ { i i } / ( \\beta \\eta )$ \n5: else $D _ { i i } = 1$ . \n6: return $A V D ^ { - 1 }$ \n180 Given a monolingual word embedding $A$ it updates part of the spectral properties of $A$ as a whole, \n181 using on a parameter $\\beta \\geq 0$ . Based on an average of singular values $\\eta = \\sqrt { \\| A \\| _ { F } ^ { 2 } / d }$ , if a value is \n182 above $\\beta$ times that average, it adjusts it to $\\beta \\eta$ . Hence, all of the top directions are given the same \n183 singular value. Otherwise, if it is below $\\beta \\eta$ , it is considered a minor effect (some are quite small, \n184 and fairly noisy), and it is left alone. If these small ones are completely zeroed out, the critical \n185 information within is destroyed. However, if these small ones are also given the same value (i.e., $\\beta \\eta )$ \n186 then components which do not contribute to the most prevalent aspects of a vectors similarity is given \n187 more importance, and we observed (see Section 4.1) that the usefulness of the embedding decreased. \n188 Iterative Spectral Normalization. Spectral normalization makes the most sense (see Appendix \n189 G) in a setting where the vectors are already centered, and also unit length. While SpecNorm does \n190 not change the center of the data, it does not maintain the length of individual vectors. As such, we \n191 advocate combining these methods into a single iterative algorithm: $\\scriptstyle \\mathrm { I - C + S N + L }$ as in Algorithm 3. ", + "bbox": [ + 178, + 109, + 699, + 199 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 226, + 826, + 339 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 356, + 825, + 412 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 3 Iterative Spectral Normalization with $\\mathrm { C + L }$ normalization (I-C+SN+L(A, #Iter)) ", + "bbox": [ + 173, + 431, + 784, + 445 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1: for #Iter steps do \n2: A ← Center A \n3: A ← SpecNorm(A) \n4: A ← Unit length normalization of A \n5: return A \n192 We observe in Figure 1 that this process significantly improves the spectral properties, compare to \n193 any other approach. Without preprocessing (None), the languages (EN: English, DE: German, HI: \n194 Hindi, JA: Japanese shown) have large effective condition numbers – indicating that there is a large \n195 disparity between meaningful singular values. Note the y-axis is in log scale. Hence, aligning these \n196 languages without normalization would likely restrict alignment among top singular vectors, not \n197 allowing enough degree of freedom to align corresponding words. \n198 In contrast, after preprocessing when these values are more uniform, rotations among the dimensions \n199 containing the top principal components will not have an influence on the data distribution, and \n200 can fully optimize the alignment between words. Moreover, Figure 1 shows that $\\mathbf { I - C + S N + L }$ most \n201 decreases the effective condition number, joint effective condition number, and singular value gap. \n202 Further, these values are fairly uniform across languages, despite great variation beforehand (as \n203 shown with None). In fact, $\\scriptstyle \\mathrm { I - C + S N + L }$ is much more effective than other methods. \n205 We provide an evaluation of our proposed preprocessing methods using 8 language embeddings \n206 pretrained on Wikipedia [4] of each language: Croatian (HR), English (EN), Finnish (FI), French \n207 (FR), German (DE), Italian (IT), Russian (RU), and Turkish (TR). We use the 300-dimensional \n208 fastText $[ 4 ] ^ { 1 }$ embeddings, and all vocabularies are trimmed to the 200K most frequent words. \n209 Alignment evaluation tasks: BLI We evaluate and compare our proposed preprocessing methods \n210 mostly on the Bilingual Lexicon Induction (BLI) task, a word translation task. We discuss two more \n211 global applications Cross-lingual document classification (CLDC), and Cross-lingual transfer for \n212 natural language inference (XNLI) later. BLI is more direct, and has become the de facto evaluation \n213 task for CLWE models. For words in the source language, this task retrieves the nearest neighbors in \n214 the target language after alignment to check if it contains the translation. It reports the mean average \n215 precision (MAP) [15], which is equivalent to the mean reciprocal rank (MRR), of the translation. \n216 Unless stated otherwise, reported values on baseline methods are taken from [15], and use the Google \n217 Translate (GTrans) dictionary from $[ 1 5 ] ^ { 2 }$ . We trained (aligned) using 1k, 3k and 5k source words and \n218 evaluated (tested) on separate 2k source test queries, unless noted otherwise. \n219 Alignment Algorithms. We evaluated and compared the result of several supervised rigid \n220 transformation CLWE models on the evaluation benchmarks using our proposed methods. All \n221 have publically available code, links are found in the reference citation. These include Canonical \n222 Correlation Analysis (CCA) [13], Procrustes (PROC) [2, 38, 45, 10], Bootstrapping Procrustes \n223 (PROC-B) [15], and Discriminative Latent-Variable (DLV) [36], as discussed in Section 1. We also \n224 consider Ranking-Based Optimization (RCSLS) [21] which is not a rigid alignment. In a few places, \n225 we also compare with VECMAP [3] as an example of an unsupervised alignment process. This should \n226 only use the geometry of the global embedding structure, e.g., derived from the natural ontology, and \n227 our normalization method still helps when using this approach. ", + "bbox": [ + 178, + 449, + 460, + 518 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 547, + 825, + 631 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 637, + 825, + 720 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/4457cfc7c0f470ce62ba0a8960c8b44495e221a08fec678f9a5a67f4ed9b61fb.jpg", + "image_caption": [ + "Figure 1: Spectral Measures of four (4) monolingual word embeddings, before (None) and after applying various normalization methods. " + ], + "image_footnote": [], + "bbox": [ + 176, + 736, + 820, + 867 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 121, + 825, + 176 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 190, + 825, + 329 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 344, + 825, + 469 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Hyperparameter Tuning ", + "text_level": 1, + "bbox": [ + 158, + 484, + 383, + 501 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Our main proposed algorithm $\\scriptstyle \\mathrm { I - C + S N + L }$ has a few simple parameters. To avoid overfitting, we choose these through cross-validation on English (EN) and a held-out set of 5 languages Hindi (HI), Russian (RU), Chinese (ZH), Japanese (JA), Turkish (TR). Ten (10) Language pairs of the form EN-X and X-EN were considered. The hyperparameters $\\beta \\in \\{ 1 , 2 , 3 , 4 , 5 \\}$ and #Iter (number of iterations) $\\in$ $\\{ 1 , 2 , 3 , 4 , 5 \\}$ were fine-tuned for $\\scriptstyle \\mathrm { I - C + S N + L }$ . ", + "bbox": [ + 171, + 511, + 823, + 580 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "234 We used the publicly available MUSE3 translation dictionary [8] for hyperparameter tuning. The \n235 Procrustes alignment algorithm was trained on $5 \\mathrm { k }$ source words and evaluated on $1 . 5 \\mathrm { k }$ source test \n236 queries. We reported the mean average precision (MAP) in Table 1 for $\\beta \\in \\{ 1 , 2 , 3 , 4 , 5 \\}$ and with \n237 #Iter $\\in \\{ 1 , 2 , 3 , 4 , 5 \\}$ . We observe the value of $\\beta = 2$ was consistently the best threshold (although \n238 any $\\beta \\geq 2$ performed similarly). These singular values were normalized, and those below $\\beta$ times the \n239 average we judged as noise, and left as is. However, the result did not change much with respect to \n240 the number of iterations. \n241 The tie between the #Iter hyperparameter was broken using their performance on thirteen English \n242 word similarity benchmarks; see Appendix C for more details. In Table 2 below, we report the \n243 average Spearman rank coefficient score on the word similarity task (None means no normalization). \n244 $( \\beta , \\# \\mathrm { { I t e r } } ) \\mathbf { \\bar { \\Psi } } = \\mathbf { \\Psi } ( 2 , 5 )$ achieved the highest score. So hereafter, we applying $\\scriptstyle \\mathrm { I - C + S N + L }$ with the \n245 hyperparameter $( \\beta , \\# \\mathrm { { I t e r } } ) = ( 2 , 5 )$ . \n246 Note that the proposed approach $( \\mathrm { I - C + S N + L } )$ only increased this score for these similarity tasks, so \n247 showed no signs of distorting inherent information. Although Spectral Normalization does not exactly \n248 preserve the linear properties or angular properties (as centering and length normalization do, one each, \n249 respectively), it does not suffer ill effects. We hypothesize this is because it is somewhat uniformly \n250 stretching words along the major modes of variation, and is effectively removing information not \n251 relevant to meaning, like frequency of occurrence. This benign effect is on contrast to other spectral \n252 adjustments (removing small singular values, or setting all to the same value) shown in Appendix C. ", + "bbox": [ + 148, + 585, + 825, + 684 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/630b95d9bde48580f67e9ec5174a29782f0f333b8aba7d839fb544c8e0ce6d99.jpg", + "table_caption": [ + "Table 1: Cross-Validation for Hyperparameter Tuning: MAP after Procrustes for 10 language pairs. " + ], + "table_footnote": [], + "table_body": "
β#Iter=1#Iter=2#Iter=3#Iter=4#Iter=5
10.3630.3400.3280.3220.317
20.3850.3860.3860.3860.386
30.3810.3840.3840.3840.384
40.3810.3820.3820.3820.382
50.3800.3810.3810.3810.381
", + "bbox": [ + 313, + 715, + 684, + 815 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 832, + 825, + 861 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 90, + 828, + 133 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/e7f7724a654ae65ad70f0a94d2a2e3a44434b652e6f98652e069d4669d6cb81f.jpg", + "table_caption": [ + "Table 2: Monolingual word similarity Task; Average Spearman rank coefficient " + ], + "table_footnote": [], + "table_body": "
None(β,#Iter)= (2,2)(β,#Iter)= (2,3)(β,#Iter)= (2,4) (β,#Iter)= (2,5)
0.6510.670770.671010.671080.67111
", + "bbox": [ + 187, + 167, + 808, + 213 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 226, + 825, + 324 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 BLI Performance across Normalization and Alignment Algorithms ", + "text_level": 1, + "bbox": [ + 169, + 339, + 676, + 354 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "254 We compare and evaluate the BLI performance (MAP) of various normalization algorithms from \n255 previous works to our proposed algorithms. Using the MUSE translation dictionary, we trained CCA, \n256 PROC, PROC-B and RCSLS on $5 \\mathrm { k }$ source words and evaluated on $1 . 5 \\mathrm { k }$ source test queries. The \n257 following normalization algorithms were used in the comparison analysis: PR (PCA Removal) [31], \n258 GeoMediaN (Geometric Median Normalization), $\\mathrm { C + L }$ (Mean centering and Length normalization, \n259 1 round), $\\mathrm { I - C + L }$ (Iterative Mean centering and Length normalization, 5 rounds) [47], SN (Spectral \n260 Normalization, 1 round), $\\mathrm { C + S N + L }$ (Mean centering, Spectral Normalization and Length normal \n261 ization, 1 round), and $_ { \\mathrm { I - C + S N + L } }$ (Iterative Mean centering, Spectral Normalization and Length \n262 normalization, 5 rounds). Specifically, we evaluated 18 language pairs, i.e., English (EN) from/to \n263 Bulgarian (BG), Catalan (CA), Czech (CS), German (DE), Spanish (ES), Korean (KO), Thai (TH) and \n264 Chinese (ZH), separate from hyperparameter tuning. The average is reported in Table 3, all results \n265 are in Appendix G. For almost all algorithms $_ { \\mathrm { I - C + S N + L } }$ achieves the best scores (and especially on \n266 ${ \\bf X } _ { L _ { 2 } } \\mathrm { - } { \\bf E } { \\bf N }$ , often considerably better). The only exceptions are on non-rigid RCSLS when $\\mathrm { C + S N + L }$ \n267 (with no iteration) or just SN (with $\\mathrm { C + L }$ ) performs slightly better. So, Spectral Normalization, and in \n268 particular $\\scriptstyle \\mathrm { I - C + S N + L }$ , is shown as the best way to normalize languages before alignment. \n269 We also compute the average BLI MAP score across all 28 language pair for more direct comparison \n270 to prior work [15], summarized in Table 4 and Appendix D. All results are in Appendix H. We \n271 compare $\\scriptstyle \\mathrm { I - C + S N + L }$ (denoted with SN) against no normalization on various dictionary sizes: 1k, 3k \n272 and $5 \\mathrm { k }$ source words and evaluated on 2k source test queries. In all cases, $\\scriptstyle \\mathbf { I - C + S N + L }$ significantly \n273 improves over the baseline. This includes improvement over RCSLS which is non-rigid, so in \n274 principle could “learn\" adjustments similar to our normalization in the process of alignment. We also \n275 tested on VECMAP, an unsupervised approach; $_ { \\mathrm { I - C + S N + L } }$ preprocessing also improves this result \n276 from 0.375 to 0.410. ", + "bbox": [ + 143, + 364, + 826, + 573 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/0602ce6555b2015198f7c4bcb3e95f958060e0fdc6e6dc8051fbcd92d7097150.jpg", + "table_caption": [ + "Table 3: BLI performance (MAP) on aligning $\\mathbf { E N - X } _ { L _ { 2 } }$ and ${ \\bf X } _ { L _ { 2 } } \\mathrm { - } { \\bf E } { \\bf N }$ " + ], + "table_footnote": [], + "table_body": "
Methods : EN-XL2Methods : XL_-EN
NormalizationCCAPROCPROC-BRCSLSCCAPROCPROC-BRCSLS
None0.3580.3650.3770.3940.3980.3990.4050.428
PR0.3940.3910.4040.3730.4340.4300.4420.425
GeoMediaN0.3930.3910.4000.3790.4330.4320.4400.429
C+L0.3930.3940.4080.4040.4390.4370.4450.464
I-C+L0.3940.3950.4100.4060.4390.4380.4480.460
SN0.3910.3940.4080.4050.4400.4380.4510.468
C+SN+L0.3950.3960.4130.4070.4440.4440.4580.466
I-C+SN+L0.3960.3980.4140.4060.4450.4460.4610.466
", + "bbox": [ + 174, + 607, + 823, + 767 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 781, + 825, + 893 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/955a25e1dd4929c02f5929dc4fa71fd0dde9bd99c185c1f93635751e24d9a11d.jpg", + "table_caption": [ + "Table 4: Summary of BLI performance (MAP), average scores for all 28 language pairs. No normalization results from [15], against $_ { \\mathrm { I - C + S N + L } }$ (denoted SN). " + ], + "table_footnote": [], + "table_body": "
CCCCPPRDPPPPRDPPPPREPPPPSRPNsATQPSSSSY PSSSS
1K.289.314.299.326.379.407.289.332.331.331
3K.378.401.384.408.398.415.381.429.415.427
5K.400.423.405.42911.403.452.437.460
", + "bbox": [ + 232, + 123, + 764, + 247 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "277 4.3 Downstream Tasks ", + "text_level": 1, + "bbox": [ + 142, + 275, + 346, + 290 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We conclude by demonstrating that Spectral Normalization not only improves in direct translation tasks, but also captures an important global structure that generalizes from a high resource language (i.e., English, EN) to lower resource languages. In both examples, a powerful classifier is trained on the EN embedding (after normalization), and then we demonstrate that after a lower resource language (e.g., German, DE) has been normalized and align the analysis task can be directly applied to that language. And in particular, adding the simple process of our normalization $( \\mathrm { I - C + S N + L }$ ) dramatically improves the results over not doing that step. ", + "bbox": [ + 156, + 301, + 825, + 398 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "285 Cross-lingual Document Classification (CLDC). The CLDC task builds a topic classification \n286 using a language model on a high resource language (in our case English EN) across 15 topics. \n287 The TED CLDC corpus assembled by [19] was used for training and evaluation. Following [15] \n288 a simple CNN was used to train. Table 5 summarizes the average F1-score for all topic classifiers \n289 on 5 language pairs. The CLWEs induced by $\\mathrm { P R O C ^ { S N } }$ , PROC- $\\cdot \\breve { \\mathbf { B } } ^ { \\mathsf { S N } }$ , ${ \\bf D } { \\bf U } ^ { { \\sf S } { \\sf N } }$ , and $\\mathrm { R C S L S ^ { S N } }$ (using \n290 $\\scriptstyle \\mathrm { I - C + S N + L }$ ) outperformed the baseline result (with no normalization) on the CLDC task, significantly \n291 improving the best average score from 0.421 to 0.461. Glavas et.al. [15] use only 12 of 15 topics, \n292 but could not confirm which, so we re-ran all baselines using all 15 topics. \n293 Cross-lingual Natural Language Inference (XNLI). We evaluated the CLWE on a cross-lingual \n294 natural language inference (XNLI) task. We used a multi-lingual XNLI corpus created by [9], which \n295 is a collection of sentence pairs from the English MultiNLI corpus [44] translated into 14 languages. \n296 The MultiNLI corpus contains $4 3 3 \\mathrm { k }$ sentence pairs with the labels entailment, contradiction, and \n297 neutral. The intersection between XNLI languages and BLI languages result in four XNLI evaluation \n298 pairs: EN-DE, EN-FR. EN-TR and EN-RU. We use the training setup in [15] with the Enhanced \n299 Sequential Inference Model [7] on English after normalization. First, we aligned normalized versions \n300 of each language onto the normalized EN embedding to obtain the shared cross-lingual embedding. \n301 Then we used the $5 \\mathrm { k }$ test pairs from the XNLI corpus to evaluate each language alignment. Table 6 \n302 shows the result for PROC, PROC-B, and RCSLS alignments (DLV and VECMAP transform the \n303 EN embedding in the process, so were omitted). We compare against the same procedure without ", + "bbox": [ + 140, + 416, + 826, + 529 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/cba30513c261f5bcf9a52ab5a1b53d88939fa09d89581579a47b0a7bd7db5bf8.jpg", + "table_caption": [ + "Table 5: CLDC performance (micro-averaged $F _ { 1 }$ scores). Cross-lingual transfer EN–X " + ], + "table_footnote": [], + "table_body": "
ModelDictEN-DEEN-FREN-ITEN-RUEN-TRAvg
PROC5K.366.258.338.288.278.306
PROCSN5K.436.366.427.517.511.451
PROC-B3K.364.304.299.336.317.324
PROC-BSN3K.448.396.423.522.517.461
DLV5K.419.336.397.493.458.421
DLVSN5K.433.323.406.499.472.427
RCSLS5K.466.397.403.403.406.415
RCSLSSN5K.468.500.443.488.394.459
", + "bbox": [ + 235, + 564, + 763, + 729 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 758, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "normalization from Glavas et.al. [15]304 $( \\mathrm { I - C + S N + L }$ normalization denoted SN ). As in other experiments, 305 our normalization improves the average test accuracy with each alignment approach. ", + "bbox": [ + 140, + 90, + 825, + 119 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/5a8a627363d1c1f6ae07c0bff29422e55b0bd15cc2bcc05d8274b94a4c9c1a89.jpg", + "table_caption": [ + "Table 6: XNLI performance (test set accuracy) " + ], + "table_footnote": [], + "table_body": "
ModelDictEN-DEEN-FREN-TREN-RUAvg
PROC PROCSN5K 5K.607 .611.534 .638.568 .542.585 .596.574 .597
PROC-B PROC-BSN3K 3K.615 .624.532 .638.573 .548.599 .601.580 .603
RCSLS RCSLSSN5K 5K.390 .499.363 .482.387 .504.399 .556.385 .510
", + "bbox": [ + 266, + 154, + 732, + 284 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "306 5 Conclusion & Discussion ", + "text_level": 1, + "bbox": [ + 145, + 310, + 411, + 327 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "307 We introduce a new way to normalize embeddings, based on spectral normalization, for use in creating \n308 cross-lingual word embeddings. Our approach generalizes previous approaches, and effectively \n309 removes much of the clustering of words based on properties other than the similarity which encodes \n310 meaning. When used to individually preprocess monolingual embeddings, our approach allows \n311 any alignment procedure to find better alignments: resulting in improved performance on direct \n312 translation tasks as well as cross-lingual topic classification and natural language inference tasks. \n313 Social impacts. The vast majority of NLP research and cutting-edge advancements are in English. \n314 This disadvantages those who primarily operate in other languages, with less developed models, or \n315 less data to train models. As large language models are the cornerstone of most NLP research and \n316 development in English, one of this work’s main goals is to port these advances to other languages, \n317 and those who use them. This will help unlock this technology to many others around the world. As \n318 with most models, this accuracy and improvement may vary across tasks and languages. \n319 While language models have many positive use cases including improving accessibility, better \n320 recommendations, and increased automation, they have some negative effects as well. These include \n321 requiring potentially large computational and hence environmental cost, encoding and exacerbating \n322 bias, and aiding in automatically generating fake or deceitful content. While this paper is unlikely to \n323 change the desire to use embeddings, it aims to reduce the burden of use and increase the effectiveness \n324 in lower-resource settings. And in particular to port models trained in English to other languages. \n325 This would reduce the cost of retraining in other languages if the English model can be reused, easing \n326 environmental costs. We support the maturing efforts in attenuating bias in all such embeddings. \n327 And while we acknowledge the possibility of this work aiding in the automated creation of deceitful \n328 content and the harm it can cause, we believe the many benefits outweigh the harms. ", + "bbox": [ + 140, + 342, + 825, + 424 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 439, + 825, + 522 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 529, + 826, + 667 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Limitations. The overarching goal in this line of work is to port the many advances built on embeddings from high-resource settings (like built on the English language), to lower-resource settings (like Turkish). This work can apply a powerful model built on an English language model (e.g., for natural language inference) and automatically invoke it in Turkish after the embeddings have been aligned. However, the alignment will not be any better than the low-resource embedding. If the embedding is too noisy or limited, then the analysis will likely not be effective. ", + "bbox": [ + 171, + 681, + 825, + 765 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "335 Also, this work focuses on non-contextual embeddings like FastText, but not contextual ones like \n336 RoBERTa which have proven almost universally more effective in NLP on English. While in principle \n337 a normalization function and alignment could be applied to contextual settings, we are unaware of \n338 any technique for learning these mappings. We believe it could be important future work. ", + "bbox": [ + 143, + 771, + 825, + 827 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Data, Code, and Experiments. All existing methods are compared with publicly available code with publicly available data, with links above or in references. The exception is code for CLDC and XNLI is shared by Glavás et.al. [15]. Everything is run with default parameters; the exception is RCSLS were we follow the suggested hyperparameter selection strategy [21] (with learning rate in $\\{ 1 , 1 0 , 2 5 , 5 0 \\}$ and epoch number in $\\{ 1 0 , 2 0 \\}$ ). Our new code for SpecNorm is in Appendix E. 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[N/A] \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ", + "bbox": [ + 238, + 597, + 825, + 715 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 215, + 719, + 823, + 734 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] \n(b) Did you mention the license of the assets? [N/A] \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] See code in Appendix E. \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 238, + 738, + 825, + 861 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 215, + 864, + 707, + 878 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] ", + "bbox": [ + 235, + 883, + 826, + 912 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "534 (b) Did you describe any potential participant risks, with links to Institutional Review \n535 Board (IRB) approvals, if applicable? [N/A] \n536 (c) Did you include the estimated hourly wage paid to participants and the total amount \n537 spent on participant compensation? 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This family of", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 88, + 639, + 100, + 649 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "score": 1.0, + "content": "transformations preserves (a) linear and (b) angular properties. By linear properties, we mean that", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 646, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 88, + 649, + 99, + 659 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 105, + 646, + 506, + 661 + ], + "score": 1.0, + "content": "the straight-line Euclidean distance between elements is preserved, as are more powerful properties", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 88, + 660, + 100, + 671 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 105, + 658, + 246, + 672 + ], + "score": 1.0, + "content": "like analogies (e.g., Paris - France", + "type": "text" + }, + { + "bbox": [ + 246, + 660, + 255, + 669 + ], + "score": 0.68, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 658, + 276, + 672 + ], + "score": 1.0, + "content": "Italy", + "type": "text" + }, + { + "bbox": [ + 276, + 659, + 287, + 669 + ], + "score": 0.59, + "content": "\\approx", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 658, + 506, + 672 + ], + "score": 1.0, + "content": "Rome). 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Semi-supervised and unsupervised models make these approaches", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 95, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 89, + 97, + 100, + 106 + ], + "score": 1.0, + "content": "39", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 95, + 505, + 107 + ], + "score": 1.0, + "content": "require less input, and more amenable to lower-resource languages. For example Bootstrap Procrustes", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 89, + 108, + 100, + 117 + ], + "score": 1.0, + "content": "40", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "(PROC-B) [15, 41] is semi-supervised in that it starts with a small pairwise correspondence (of", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 89, + 119, + 99, + 127 + ], + "score": 1.0, + "content": "41", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "500-1000 words), aligns those to infer a larger correspondence, and repeats applying Procrustes", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 89, + 130, + 100, + 138 + ], + "score": 1.0, + "content": "42", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "score": 1.0, + "content": "alignment. Methods like MUSE [8] are unsupervised, and use a GAN to estimate a correspondence", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 138, + 267, + 151 + ], + "spans": [ + { + "bbox": [ + 90, + 140, + 99, + 150 + ], + "score": 1.0, + "content": "43", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 138, + 267, + 151 + ], + "score": 1.0, + "content": "before applying a Procrustes procedure.", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 89, + 157, + 99, + 165 + ], + "score": 1.0, + "content": "44", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "The second direction is preprocessing the embeddings before applying the Procrustes alignment.", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 89, + 167, + 99, + 177 + ], + "score": 1.0, + "content": "45", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "These involve methods like removing the mean, removing principal components, and normalization", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 177, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 89, + 179, + 100, + 188 + ], + "score": 1.0, + "content": "46", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 177, + 506, + 189 + ], + "score": 1.0, + "content": "which we will discuss in depth later. In principle, these methods aim to remove the geometry of", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 188, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 89, + 190, + 99, + 198 + ], + "score": 1.0, + "content": "47", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 188, + 505, + 200 + ], + "score": 1.0, + "content": "data intrinsic to particular languages (but not shared across languages) while preserving similarity", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 89, + 200, + 100, + 210 + ], + "score": 1.0, + "content": "48", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "properties as assured by orthogonal alignment. The space of transformations allowed under orthogonal", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 89, + 212, + 100, + 221 + ], + "score": 1.0, + "content": "49", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "alignments is quite large, and we make the point that unless this data geometry is “normalized” it", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 219, + 408, + 233 + ], + "spans": [ + { + "bbox": [ + 89, + 222, + 100, + 232 + ], + "score": 1.0, + "content": "50", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 219, + 408, + 233 + ], + "score": 1.0, + "content": "inhibits the alignment from optimizing over the entirety of this large space.", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 89, + 238, + 99, + 248 + ], + "score": 1.0, + "content": "51", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "Finally, we note that methods like Canonical Correlation Analysis (CCA) [13], Discriminative Latent", + "type": "text", + "cross_page": true + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 246, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 89, + 249, + 99, + 259 + ], + "score": 1.0, + "content": "52", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 246, + 506, + 261 + ], + "score": 1.0, + "content": "Variable (DLV) [36], and Ranking-based optimization (RCSLS) [21] have also been applied towards", + "type": "text", + "cross_page": true + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 258, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 89, + 261, + 100, + 270 + ], + "score": 1.0, + "content": "53", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 258, + 505, + 270 + ], + "score": 1.0, + "content": "finding an orthogonal alignment (or pair of alignments) which minimizes a different optimization", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 89, + 271, + 100, + 281 + ], + "score": 1.0, + "content": "54", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "function – since the objective function may not align with sums of squared Euclidean of cosine", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 280, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 89, + 282, + 100, + 292 + ], + "score": 1.0, + "content": "55", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 280, + 506, + 293 + ], + "score": 1.0, + "content": "distance [8, 38]. Unlike the others, the RCSLS method notably does not require a rigid transformation.", + "type": "text", + "cross_page": true + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 92, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 92, + 299, + 100, + 306 + ], + "score": 1.0, + "content": "56", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 102, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "The focus of this paper is on embedding preprocessing, and is agnostic to the method of alignment", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 306, + 416, + 320 + ], + "spans": [ + { + "bbox": [ + 91, + 306, + 416, + 320 + ], + "score": 1.0, + "content": "57 used afterward, whether it is Procrustes-based, or optimizing something else.", + "type": "text", + "cross_page": true + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 333, + 507, + 347 + ], + "spans": [ + { + "bbox": [ + 89, + 336, + 100, + 345 + ], + "score": 1.0, + "content": "58", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 333, + 507, + 347 + ], + "score": 1.0, + "content": "Our contribution. This work proposes a new and general approach to preprocessing word embed-", + "type": "text", + "cross_page": true + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 344, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 89, + 346, + 100, + 356 + ], + "score": 1.0, + "content": "59", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 344, + 505, + 357 + ], + "score": 1.0, + "content": "dings, subsuming many previous approaches. The key is Spectral Normalization which regularizes", + "type": "text", + "cross_page": true + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 355, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 89, + 358, + 100, + 367 + ], + "score": 1.0, + "content": "60", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 355, + 506, + 368 + ], + "score": 1.0, + "content": "the spectral properties of monolingual embeddings by setting all of the top singular vectors to have", + "type": "text", + "cross_page": true + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 88, + 369, + 100, + 378 + ], + "score": 1.0, + "content": "61", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "the same singular value. 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In principle, these methods aim to remove the geometry of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 89, + 188, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 89, + 190, + 99, + 198 + ], + "score": 1.0, + "content": "47", + "type": "text" + }, + { + "bbox": [ + 105, + 188, + 505, + 200 + ], + "score": 1.0, + "content": "data intrinsic to particular languages (but not shared across languages) while preserving similarity", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 89, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 89, + 200, + 100, + 210 + ], + "score": 1.0, + "content": "48", + "type": "text" + }, + { + "bbox": [ + 104, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "properties as assured by orthogonal alignment. 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Unlike the others, the RCSLS method notably does not require a rigid transformation.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 93, + 296, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 92, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 92, + 299, + 100, + 306 + ], + "score": 1.0, + "content": "56", + "type": "text" + }, + { + "bbox": [ + 102, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "The focus of this paper is on embedding preprocessing, and is agnostic to the method of alignment", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 91, + 306, + 416, + 320 + ], + "spans": [ + { + "bbox": [ + 91, + 306, + 416, + 320 + ], + "score": 1.0, + "content": "57 used afterward, whether it is Procrustes-based, or optimizing something else.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 89, + 333, + 506, + 476 + ], + "lines": [ + { + "bbox": [ + 89, + 333, + 507, + 347 + ], + "spans": [ + { + "bbox": [ + 89, + 336, + 100, + 345 + ], + "score": 1.0, + "content": "58", + "type": "text" + }, + { + "bbox": [ + 105, + 333, + 507, + 347 + ], + "score": 1.0, + "content": "Our contribution. 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These vector representations (derived by methods", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 89, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 89, + 555, + 100, + 565 + ], + "score": 1.0, + "content": "75", + "type": "text" + }, + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "like word2vec [28], GloVe [33], or FastText [4]) are chosen so words with similar pairwise cosine", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 89, + 564, + 507, + 577 + ], + "spans": [ + { + "bbox": [ + 89, + 566, + 100, + 576 + ], + "score": 1.0, + "content": "76", + "type": "text" + }, + { + "bbox": [ + 105, + 564, + 507, + 577 + ], + "score": 1.0, + "content": "similarity are found in the similar local context in large text corpora on which they are trained.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 89, + 576, + 488, + 587 + ], + "spans": [ + { + "bbox": [ + 89, + 577, + 100, + 587 + ], + "score": 1.0, + "content": "77", + "type": "text" + }, + { + "bbox": [ + 106, + 576, + 488, + 587 + ], + "score": 1.0, + "content": "Higher-level linear structure is shown to emerge, such as concept subspaces and analogies [29].", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 90, + 591, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 89, + 590, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 89, + 593, + 100, + 603 + ], + "score": 1.0, + "content": "78", + "type": "text" + }, + { + "bbox": [ + 105, + 590, + 385, + 605 + ], + "score": 1.0, + "content": "The focus of this paper is on aligning embeddings of two languages", + "type": "text" + }, + { + "bbox": [ + 385, + 592, + 398, + 603 + ], + "score": 0.89, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 590, + 417, + 605 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 417, + 592, + 429, + 603 + ], + "score": 0.88, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 590, + 506, + 605 + ], + "score": 1.0, + "content": ". 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While word", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 88, + 117, + 99, + 127 + ], + "score": 1.0, + "content": "92", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "meaning appears to hold a similar structure, languages have other properties such as differing word", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 88, + 127, + 99, + 138 + ], + "score": 1.0, + "content": "93", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "frequency, and this for instance leads to more frequent words having longer vectors in embeddings.", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 88, + 138, + 100, + 148 + ], + "score": 1.0, + "content": "94", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "This extra language-specific structure tends to interfere with alignment. As a result, a number", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 147, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 88, + 149, + 99, + 159 + ], + "score": 1.0, + "content": "95", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 147, + 506, + 159 + ], + "score": 1.0, + "content": "of techniques have been developed to “normalize” the embeddings before Procrustes (or other)", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 88, + 159, + 100, + 171 + ], + "score": 1.0, + "content": "96", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "alignment. This in some sense allows the word meaning to dominate the optimization tasks without", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 169, + 441, + 182 + ], + "spans": [ + { + "bbox": [ + 89, + 172, + 100, + 181 + ], + "score": 1.0, + "content": "97", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 169, + 441, + 182 + ], + "score": 1.0, + "content": "other confounding factors. We review the most common normalization approaches.", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 193, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 89, + 194, + 100, + 204 + ], + "score": 1.0, + "content": "98", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "score": 1.0, + "content": "Mean Centering (C) subtracts the mean of all vectors in an embedding from each vector in that", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 204, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 89, + 205, + 99, + 215 + ], + "score": 1.0, + "content": "99", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "score": 1.0, + "content": "embedding. The result is that the mean of all vectors is 0. This is a rigid transformation, and so does", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 86, + 217, + 100, + 226 + ], + "score": 1.0, + "content": "100", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "not change the Euclidean distance between any pair of points in an embedding, and also preserves", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 86, + 228, + 99, + 236 + ], + "score": 1.0, + "content": "101", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 226, + 325, + 238 + ], + "score": 1.0, + "content": "any linear property like analogies (e.g., Paris - France", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 325, + 227, + 334, + 236 + ], + "score": 0.7, + "content": "^ +", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 334, + 226, + 354, + 238 + ], + "score": 1.0, + "content": "Italy", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 354, + 226, + 365, + 235 + ], + "score": 0.64, + "content": "\\approx", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 366, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "Rome). Dev et. al. [10] points out", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 237, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 86, + 238, + 100, + 248 + ], + "score": 1.0, + "content": "102", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 237, + 505, + 248 + ], + "score": 1.0, + "content": "that this is the first step (followed by the Procrustes orthogonal transformation) to minimize the sum", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 86, + 249, + 100, + 259 + ], + "score": 1.0, + "content": "103", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "of squared Euclidean distances among paired words, under any rigid transformation. However, this", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 258, + 334, + 271 + ], + "spans": [ + { + "bbox": [ + 86, + 260, + 100, + 270 + ], + "score": 1.0, + "content": "104", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 258, + 334, + 271 + ], + "score": 1.0, + "content": "does change the cosine distance between pairs of points.", + "type": "text", + "cross_page": true + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 93, + 285, + 99, + 291 + ], + "score": 1.0, + "content": "05", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "Length Normalization (L) makes each vector have a 2-norm equal to 1, but retains its direction", + "type": "text", + "cross_page": true + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 93, + 297, + 99, + 303 + ], + "score": 1.0, + "content": "06", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "from the origin [2, 45]. This preprocessing step does not change the cosine distance between any pair", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 93, + 303, + 485, + 316 + ], + "spans": [ + { + "bbox": [ + 93, + 307, + 99, + 313 + ], + "score": 1.0, + "content": "07", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 303, + 485, + 316 + ], + "score": 1.0, + "content": "of points in an embedding. But, it does change the Euclidean distance between pairs of points.", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 86, + 322, + 100, + 331 + ], + "score": 1.0, + "content": "108", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "Despite these contrasting goals, these two normalizations each turn out to be individually effective in", + "type": "text", + "cross_page": true + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 331, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 85, + 333, + 100, + 343 + ], + "score": 1.0, + "content": "109", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "score": 1.0, + "content": "regularizing the geometry of the embeddings, and allow for better CLWE. Wang et. al. [47], realized", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 85, + 343, + 100, + 354 + ], + "score": 1.0, + "content": "110", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "doing both was even more effective, and showed that iterating these two steps achieves the state-", + "type": "text", + "cross_page": true + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 85, + 354, + 100, + 364 + ], + "score": 1.0, + "content": "111", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 351, + 278, + 366 + ], + "score": 1.0, + "content": "of-the-art way to preprocess, we denote as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 278, + 353, + 306, + 363 + ], + "score": 0.81, + "content": "\\mathbf { I } { \\mathbf { - C } } { \\mathbf { + L } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 307, + 351, + 506, + 366 + ], + "score": 1.0, + "content": ". Iterative Normalization transforms monolingual", + "type": "text", + "cross_page": true + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 364, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 85, + 366, + 100, + 375 + ], + "score": 1.0, + "content": "112", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 364, + 505, + 375 + ], + "score": 1.0, + "content": "word embeddings to have unit-length and zero-mean simultaneously (in practice they terminate this", + "type": "text", + "cross_page": true + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 375, + 414, + 387 + ], + "spans": [ + { + "bbox": [ + 85, + 376, + 100, + 387 + ], + "score": 1.0, + "content": "113", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 375, + 414, + 387 + ], + "score": 1.0, + "content": "iterative process after a few steps before it achieves these two goals exactly).", + "type": "text", + "cross_page": true + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 86, + 400, + 99, + 409 + ], + "score": 1.0, + "content": "114", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "PCA Removal (PR) computes the principal component analysis (PCA) of an embedding, and then", + "type": "text", + "cross_page": true + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 86, + 411, + 100, + 421 + ], + "score": 1.0, + "content": "115", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "projects away from the direction of the top principal component, removing it [31]. Mu et. al. [31]", + "type": "text", + "cross_page": true + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 86, + 421, + 100, + 432 + ], + "score": 1.0, + "content": "116", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "observed that the top singular values typically do not encode essential semantic relationships between", + "type": "text", + "cross_page": true + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 86, + 432, + 100, + 443 + ], + "score": 1.0, + "content": "117", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "words but rather align strongly with word frequency. Also, they notice that the top principal values", + "type": "text", + "cross_page": true + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 86, + 444, + 99, + 453 + ], + "score": 1.0, + "content": "118", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 441, + 329, + 455 + ], + "score": 1.0, + "content": "are much larger than the other values. After eliminating", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 329, + 442, + 356, + 453 + ], + "score": 0.85, + "content": "d / 1 0 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 356, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "principal components, where d is the", + "type": "text", + "cross_page": true + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 86, + 455, + 100, + 464 + ], + "score": 1.0, + "content": "119", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "dimension of a word representation, they achieve better performance on both intrinsic and extrinsic", + "type": "text", + "cross_page": true + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 86, + 465, + 100, + 476 + ], + "score": 1.0, + "content": "120", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "tasks. Sachidananda et. al. [37] applied this simple pre-processing approach to their Filtered Inner", + "type": "text", + "cross_page": true + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 474, + 480, + 487 + ], + "spans": [ + { + "bbox": [ + 86, + 476, + 100, + 486 + ], + "score": 1.0, + "content": "121", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 474, + 480, + 487 + ], + "score": 1.0, + "content": "Product Projection (FIPP) alignment method, and they significantly improve on the BLI task.", + "type": "text", + "cross_page": true + } + ], + "index": 31, + "is_list_start_line": true + } + ], + "index": 37.5, + "bbox_fs": [ + 89, + 519, + 507, + 587 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 591, + 505, + 646 + ], + "lines": [], + "index": 43, + "bbox_fs": [ + 89, + 590, + 506, + 649 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 90, + 651, + 505, + 722 + ], + "lines": [], + "index": 48.5, + "bbox_fs": [ + 88, + 651, + 509, + 723 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 89, + 93, + 505, + 181 + ], + "lines": [ + { + "bbox": [ + 89, + 93, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 89, + 95, + 99, + 105 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 105, + 93, + 506, + 105 + ], + "score": 1.0, + "content": "It turns out directly aligning embeddings from two languages (even using the “optimal” Procrustes", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 88, + 104, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 88, + 106, + 99, + 115 + ], + "score": 1.0, + "content": "91", + "type": "text" + }, + { + "bbox": [ + 105, + 104, + 505, + 116 + ], + "score": 1.0, + "content": "solution) does not provide the best possible joint embedding for translation tasks. While word", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 88, + 114, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 88, + 117, + 99, + 127 + ], + "score": 1.0, + "content": "92", + "type": "text" + }, + { + "bbox": [ + 105, + 114, + 506, + 127 + ], + "score": 1.0, + "content": "meaning appears to hold a similar structure, languages have other properties such as differing word", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 88, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 88, + 127, + 99, + 138 + ], + "score": 1.0, + "content": "93", + "type": "text" + }, + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "frequency, and this for instance leads to more frequent words having longer vectors in embeddings.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 88, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 88, + 138, + 100, + 148 + ], + "score": 1.0, + "content": "94", + "type": "text" + }, + { + "bbox": [ + 105, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "This extra language-specific structure tends to interfere with alignment. As a result, a number", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 88, + 147, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 88, + 149, + 99, + 159 + ], + "score": 1.0, + "content": "95", + "type": "text" + }, + { + "bbox": [ + 105, + 147, + 506, + 159 + ], + "score": 1.0, + "content": "of techniques have been developed to “normalize” the embeddings before Procrustes (or other)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 88, + 158, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 88, + 159, + 100, + 171 + ], + "score": 1.0, + "content": "96", + "type": "text" + }, + { + "bbox": [ + 105, + 158, + 506, + 171 + ], + "score": 1.0, + "content": "alignment. This in some sense allows the word meaning to dominate the optimization tasks without", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 89, + 169, + 441, + 182 + ], + "spans": [ + { + "bbox": [ + 89, + 172, + 100, + 181 + ], + "score": 1.0, + "content": "97", + "type": "text" + }, + { + "bbox": [ + 105, + 169, + 441, + 182 + ], + "score": 1.0, + "content": "other confounding factors. We review the most common normalization approaches.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 87, + 192, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 89, + 193, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 89, + 194, + 100, + 204 + ], + "score": 1.0, + "content": "98", + "type": "text" + }, + { + "bbox": [ + 106, + 193, + 505, + 204 + ], + "score": 1.0, + "content": "Mean Centering (C) subtracts the mean of all vectors in an embedding from each vector in that", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 89, + 204, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 89, + 205, + 99, + 215 + ], + "score": 1.0, + "content": "99", + "type": "text" + }, + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "score": 1.0, + "content": "embedding. The result is that the mean of all vectors is 0. This is a rigid transformation, and so does", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 214, + 506, + 228 + ], + "spans": [ + { + "bbox": [ + 86, + 217, + 100, + 226 + ], + "score": 1.0, + "content": "100", + "type": "text" + }, + { + "bbox": [ + 104, + 214, + 506, + 228 + ], + "score": 1.0, + "content": "not change the Euclidean distance between any pair of points in an embedding, and also preserves", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 86, + 228, + 99, + 236 + ], + "score": 1.0, + "content": "101", + "type": "text" + }, + { + "bbox": [ + 105, + 226, + 325, + 238 + ], + "score": 1.0, + "content": "any linear property like analogies (e.g., Paris - France", + "type": "text" + }, + { + "bbox": [ + 325, + 227, + 334, + 236 + ], + "score": 0.7, + "content": "^ +", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 226, + 354, + 238 + ], + "score": 1.0, + "content": "Italy", + "type": "text" + }, + { + "bbox": [ + 354, + 226, + 365, + 235 + ], + "score": 0.64, + "content": "\\approx", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "Rome). Dev et. al. [10] points out", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 237, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 86, + 238, + 100, + 248 + ], + "score": 1.0, + "content": "102", + "type": "text" + }, + { + "bbox": [ + 106, + 237, + 505, + 248 + ], + "score": 1.0, + "content": "that this is the first step (followed by the Procrustes orthogonal transformation) to minimize the sum", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 86, + 249, + 100, + 259 + ], + "score": 1.0, + "content": "103", + "type": "text" + }, + { + "bbox": [ + 105, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "of squared Euclidean distances among paired words, under any rigid transformation. However, this", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 86, + 258, + 334, + 271 + ], + "spans": [ + { + "bbox": [ + 86, + 260, + 100, + 270 + ], + "score": 1.0, + "content": "104", + "type": "text" + }, + { + "bbox": [ + 105, + 258, + 334, + 271 + ], + "score": 1.0, + "content": "does change the cosine distance between pairs of points.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 93, + 281, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 93, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 93, + 285, + 99, + 291 + ], + "score": 1.0, + "content": "05", + "type": "text" + }, + { + "bbox": [ + 104, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "Length Normalization (L) makes each vector have a 2-norm equal to 1, but retains its direction", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 93, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 93, + 297, + 99, + 303 + ], + "score": 1.0, + "content": "06", + "type": "text" + }, + { + "bbox": [ + 104, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "from the origin [2, 45]. This preprocessing step does not change the cosine distance between any pair", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 93, + 303, + 485, + 316 + ], + "spans": [ + { + "bbox": [ + 93, + 307, + 99, + 313 + ], + "score": 1.0, + "content": "07", + "type": "text" + }, + { + "bbox": [ + 104, + 303, + 485, + 316 + ], + "score": 1.0, + "content": "of points in an embedding. But, it does change the Euclidean distance between pairs of points.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 86, + 320, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 86, + 319, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 86, + 322, + 100, + 331 + ], + "score": 1.0, + "content": "108", + "type": "text" + }, + { + "bbox": [ + 105, + 319, + 506, + 332 + ], + "score": 1.0, + "content": "Despite these contrasting goals, these two normalizations each turn out to be individually effective in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 85, + 331, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 85, + 333, + 100, + 343 + ], + "score": 1.0, + "content": "109", + "type": "text" + }, + { + "bbox": [ + 105, + 331, + 506, + 343 + ], + "score": 1.0, + "content": "regularizing the geometry of the embeddings, and allow for better CLWE. Wang et. al. [47], realized", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 85, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 85, + 343, + 100, + 354 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 106, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "doing both was even more effective, and showed that iterating these two steps achieves the state-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 85, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 85, + 354, + 100, + 364 + ], + "score": 1.0, + "content": "111", + "type": "text" + }, + { + "bbox": [ + 104, + 351, + 278, + 366 + ], + "score": 1.0, + "content": "of-the-art way to preprocess, we denote as", + "type": "text" + }, + { + "bbox": [ + 278, + 353, + 306, + 363 + ], + "score": 0.81, + "content": "\\mathbf { I } { \\mathbf { - C } } { \\mathbf { + L } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 351, + 506, + 366 + ], + "score": 1.0, + "content": ". Iterative Normalization transforms monolingual", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 85, + 364, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 85, + 366, + 100, + 375 + ], + "score": 1.0, + "content": "112", + "type": "text" + }, + { + "bbox": [ + 106, + 364, + 505, + 375 + ], + "score": 1.0, + "content": "word embeddings to have unit-length and zero-mean simultaneously (in practice they terminate this", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 85, + 375, + 414, + 387 + ], + "spans": [ + { + "bbox": [ + 85, + 376, + 100, + 387 + ], + "score": 1.0, + "content": "113", + "type": "text" + }, + { + "bbox": [ + 105, + 375, + 414, + 387 + ], + "score": 1.0, + "content": "iterative process after a few steps before it achieves these two goals exactly).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 86, + 398, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 86, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 86, + 400, + 99, + 409 + ], + "score": 1.0, + "content": "114", + "type": "text" + }, + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "PCA Removal (PR) computes the principal component analysis (PCA) of an embedding, and then", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 409, + 505, + 422 + ], + "spans": [ + { + "bbox": [ + 86, + 411, + 100, + 421 + ], + "score": 1.0, + "content": "115", + "type": "text" + }, + { + "bbox": [ + 104, + 409, + 505, + 422 + ], + "score": 1.0, + "content": "projects away from the direction of the top principal component, removing it [31]. Mu et. al. [31]", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 86, + 421, + 100, + 432 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 106, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "observed that the top singular values typically do not encode essential semantic relationships between", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 86, + 432, + 100, + 443 + ], + "score": 1.0, + "content": "117", + "type": "text" + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "words but rather align strongly with word frequency. Also, they notice that the top principal values", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 86, + 444, + 99, + 453 + ], + "score": 1.0, + "content": "118", + "type": "text" + }, + { + "bbox": [ + 105, + 441, + 329, + 455 + ], + "score": 1.0, + "content": "are much larger than the other values. After eliminating", + "type": "text" + }, + { + "bbox": [ + 329, + 442, + 356, + 453 + ], + "score": 0.85, + "content": "d / 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "principal components, where d is the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 86, + 455, + 100, + 464 + ], + "score": 1.0, + "content": "119", + "type": "text" + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "dimension of a word representation, they achieve better performance on both intrinsic and extrinsic", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 86, + 465, + 100, + 476 + ], + "score": 1.0, + "content": "120", + "type": "text" + }, + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "tasks. Sachidananda et. al. [37] applied this simple pre-processing approach to their Filtered Inner", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 474, + 480, + 487 + ], + "spans": [ + { + "bbox": [ + 86, + 476, + 100, + 486 + ], + "score": 1.0, + "content": "121", + "type": "text" + }, + { + "bbox": [ + 105, + 474, + 480, + 487 + ], + "score": 1.0, + "content": "Product Projection (FIPP) alignment method, and they significantly improve on the BLI task.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 101, + 499, + 273, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 274, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 274, + 514 + ], + "score": 1.0, + "content": "2.2 Spectral Statistics of Embeddings", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 86, + 519, + 506, + 663 + ], + "lines": [ + { + "bbox": [ + 86, + 520, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 86, + 521, + 100, + 531 + ], + "score": 1.0, + "content": "123", + "type": "text" + }, + { + "bbox": [ + 105, + 520, + 505, + 531 + ], + "score": 1.0, + "content": "Dubossarsky et. al. [11] recently documented how cross-lingual alignment is strongly affected by the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 86, + 530, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 86, + 532, + 100, + 542 + ], + "score": 1.0, + "content": "124", + "type": "text" + }, + { + "bbox": [ + 105, + 530, + 424, + 542 + ], + "score": 1.0, + "content": "spectral statistics of monolingual embeddings. We stack the embedded vectors", + "type": "text" + }, + { + "bbox": [ + 424, + 530, + 461, + 542 + ], + "score": 0.92, + "content": "x _ { L } ^ { \\bar { i } } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 530, + 506, + 542 + ], + "score": 1.0, + "content": "as rows in", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 540, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 86, + 545, + 99, + 554 + ], + "score": 1.0, + "content": "125", + "type": "text" + }, + { + "bbox": [ + 103, + 540, + 113, + 555 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 543, + 139, + 553 + ], + "score": 0.9, + "content": "n \\times d", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 540, + 169, + 555 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 170, + 542, + 214, + 553 + ], + "score": 0.92, + "content": "A \\in \\mathbb { R } ^ { n \\times d }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 540, + 313, + 555 + ], + "score": 1.0, + "content": ". The SVD decomposes", + "type": "text" + }, + { + "bbox": [ + 314, + 543, + 322, + 552 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 540, + 342, + 555 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 342, + 542, + 373, + 553 + ], + "score": 0.89, + "content": "U { \\boldsymbol { \\Sigma } } V ^ { \\top }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 540, + 402, + 555 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 402, + 543, + 411, + 552 + ], + "score": 0.81, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 540, + 430, + 555 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 430, + 543, + 440, + 553 + ], + "score": 0.77, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 540, + 506, + 555 + ], + "score": 1.0, + "content": "contain the left", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 552, + 507, + 567 + ], + "spans": [ + { + "bbox": [ + 86, + 555, + 100, + 565 + ], + "score": 1.0, + "content": "126", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 303, + 567 + ], + "score": 1.0, + "content": "and right singular vectors, and the singular values", + "type": "text" + }, + { + "bbox": [ + 304, + 554, + 405, + 565 + ], + "score": 0.9, + "content": "\\sigma _ { 1 } \\geq \\sigma _ { 2 } \\geq . . . \\geq \\sigma _ { d } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 552, + 495, + 567 + ], + "score": 1.0, + "content": "are on the diagonal of", + "type": "text" + }, + { + "bbox": [ + 495, + 554, + 503, + 563 + ], + "score": 0.82, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 552, + 507, + 567 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 563, + 489, + 597 + ], + "spans": [ + { + "bbox": [ + 86, + 563, + 139, + 597 + ], + "score": 1.0, + "content": "127 The effedefined 128", + "type": "text" + }, + { + "bbox": [ + 192, + 565, + 201, + 574 + ], + "score": 0.82, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 563, + 236, + 597 + ], + "score": 1.0, + "content": "s a smo where", + "type": "text" + }, + { + "bbox": [ + 346, + 563, + 369, + 597 + ], + "score": 1.0, + "content": "therewith", + "type": "text" + }, + { + "bbox": [ + 444, + 563, + 489, + 597 + ], + "score": 1.0, + "content": "k compone. The effec", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 140, + 575, + 444, + 590 + ], + "spans": [ + { + "bbox": [ + 140, + 576, + 206, + 589 + ], + "score": 0.92, + "content": "\\mathsf { e r } ( A ) = e ^ { H } ( \\Sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 575, + 345, + 590 + ], + "score": 0.93, + "content": "\\begin{array} { r } { H ( \\Sigma ) = - \\sum _ { i = 1 } ^ { d } \\bar { \\sigma } _ { i } \\log \\bar { \\sigma } _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 575, + 444, + 590 + ], + "score": 0.92, + "content": "\\bar { \\sigma } _ { i } = { \\sigma _ { i } } / { \\sum _ { i = 1 } ^ { d } { \\sigma _ { i } } }", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 507, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 182, + 602 + ], + "score": 1.0, + "content": "condition number", + "type": "text" + }, + { + "bbox": [ + 182, + 588, + 268, + 601 + ], + "score": 0.92, + "content": "\\kappa _ { \\mathsf { e f f } } ( A ) = \\sigma _ { 1 } / \\sigma _ { \\mathsf { e r } ( A ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 587, + 489, + 602 + ], + "score": 1.0, + "content": ", which replaces the numerator (of condition number,", + "type": "text" + }, + { + "bbox": [ + 489, + 590, + 501, + 599 + ], + "score": 0.79, + "content": "\\sigma _ { d . }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 587, + 507, + 602 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 86, + 599, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 86, + 601, + 100, + 611 + ], + "score": 1.0, + "content": "130", + "type": "text" + }, + { + "bbox": [ + 105, + 599, + 506, + 611 + ], + "score": 1.0, + "content": "with the more robust singular value at the effective rank. 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The goal is to remove language-specific geometry", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 87, + 120, + 416, + 134 + ], + "spans": [ + { + "bbox": [ + 87, + 123, + 100, + 133 + ], + "score": 1.0, + "content": "143", + "type": "text" + }, + { + "bbox": [ + 104, + 120, + 416, + 134 + ], + "score": 1.0, + "content": "while maintaining the intrinsic similarity and structure captured within them.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 92, + 151, + 274, + 163 + ], + "lines": [ + { + "bbox": [ + 88, + 150, + 275, + 164 + ], + "spans": [ + { + "bbox": [ + 88, + 150, + 275, + 164 + ], + "score": 1.0, + "content": "144 3.1 Geometric Median Normalization", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 87, + 173, + 506, + 207 + ], + "lines": [ + { + "bbox": [ + 86, + 173, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 86, + 175, + 100, + 185 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 105, + 173, + 506, + 185 + ], + "score": 1.0, + "content": "Iterative Normalization enforces individual word embeddings to have a unit length and each monolin-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 86, + 185, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 86, + 187, + 99, + 195 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 105, + 185, + 506, + 197 + ], + "score": 1.0, + "content": "gual embedding to have a zero mean through an iterative technique. Wang et. al. [47] showed that", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 86, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 86, + 198, + 100, + 207 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "iterating solutions for these distinct goals will eventually converge to a solution which satisfies both.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 86, + 211, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 86, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 86, + 213, + 100, + 223 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 105, + 212, + 505, + 223 + ], + "score": 1.0, + "content": "In this paper, we observe that both goals can be done in one shot without iterating – by solving the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 85, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 85, + 224, + 100, + 234 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "Fermat-Weber problem [27, 25]. 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Several methods [6, 12, 32] have been proposed; the most popular is the Weiszfeld’s algorithm", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 263, + 507, + 285 + ], + "spans": [ + { + "bbox": [ + 86, + 270, + 100, + 280 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 103, + 263, + 345, + 285 + ], + "score": 1.0, + "content": "(Weiszfeld, see Appendix A). 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This can be done using the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 86, + 321, + 99, + 330 + ], + "score": 1.0, + "content": "157", + "type": "text" + }, + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "Geometric Median normalization (GeoMediaN) algorithm (as Algorithm 1). Given a monolingual", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 86, + 333, + 100, + 342 + ], + "score": 1.0, + "content": "158", + "type": "text" + }, + { + "bbox": [ + 105, + 330, + 177, + 343 + ], + "score": 1.0, + "content": "word embedding", + "type": "text" + }, + { + "bbox": [ + 177, + 331, + 185, + 340 + ], + "score": 0.65, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 330, + 330, + 343 + ], + "score": 1.0, + "content": ", we compute the geometric median", + "type": "text" + }, + { + "bbox": [ + 331, + 331, + 342, + 340 + ], + "score": 0.85, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 330, + 505, + 343 + ], + "score": 1.0, + "content": ", and \"center\" the data on this point, and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 340, + 298, + 354 + ], + "spans": [ + { + "bbox": [ + 86, + 343, + 99, + 352 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 105, + 340, + 298, + 354 + ], + "score": 1.0, + "content": "unit length normalizes the centered embedding.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 106, + 371, + 371, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 371, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 371, + 385 + ], + "score": 1.0, + "content": "Algorithm 1 Geometric Median Normalization: GeoMediaN(A)", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 110, + 385, + 253, + 424 + ], + "lines": [ + { + "bbox": [ + 111, + 384, + 208, + 397 + ], + "spans": [ + { + "bbox": [ + 111, + 384, + 208, + 397 + ], + "score": 1.0, + "content": "1: x∗ ← Weiszfeld(A)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 110, + 397, + 253, + 413 + ], + "spans": [ + { + "bbox": [ + 110, + 397, + 151, + 413 + ], + "score": 1.0, + "content": "2: for all", + "type": "text" + }, + { + "bbox": [ + 151, + 398, + 181, + 410 + ], + "score": 0.64, + "content": "a _ { i } \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 397, + 195, + 413 + ], + "score": 1.0, + "content": "do", + "type": "text" + }, + { + "bbox": [ + 195, + 397, + 253, + 413 + ], + "score": 0.74, + "content": "\\begin{array} { r } { a _ { i } \\frac { a _ { i } - x ^ { * } } { \\| a _ { i } - x ^ { * } \\| } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 110, + 411, + 163, + 422 + ], + "spans": [ + { + "bbox": [ + 110, + 411, + 163, + 422 + ], + "score": 1.0, + "content": "3: return A", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 95, + 449, + 504, + 471 + ], + "lines": [ + { + "bbox": [ + 91, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 91, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "60 After these steps, all vectors are unit length, and because of the folklore property (Theorem A.1), the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 91, + 459, + 423, + 473 + ], + "spans": [ + { + "bbox": [ + 91, + 459, + 423, + 473 + ], + "score": 1.0, + "content": "61 mean of those points is also 0. As a result, we can state the following property.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 93, + 477, + 435, + 489 + ], + "lines": [ + { + "bbox": [ + 88, + 477, + 435, + 491 + ], + "spans": [ + { + "bbox": [ + 88, + 477, + 278, + 491 + ], + "score": 1.0, + "content": "162 Theorem 3.1. The output of GeoMediaN", + "type": "text" + }, + { + "bbox": [ + 278, + 477, + 294, + 489 + ], + "score": 0.69, + "content": "( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 477, + 435, + 491 + ], + "score": 1.0, + "content": "is centered and length normalized.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 87, + 500, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 86, + 500, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 86, + 502, + 99, + 511 + ], + "score": 1.0, + "content": "163", + "type": "text" + }, + { + "bbox": [ + 105, + 500, + 506, + 512 + ], + "score": 1.0, + "content": "Despite the Geometric Median Normalization algorithm’s ability to enforce unit-length and zero-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 86, + 513, + 99, + 522 + ], + "score": 1.0, + "content": "164", + "type": "text" + }, + { + "bbox": [ + 105, + 511, + 506, + 523 + ], + "score": 1.0, + "content": "mean in just one step, we will observe that it does not perform especially well on the BLI task.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 86, + 524, + 99, + 533 + ], + "score": 1.0, + "content": "165", + "type": "text" + }, + { + "bbox": [ + 105, + 521, + 199, + 535 + ], + "score": 1.0, + "content": "Both GeoMediaN and", + "type": "text" + }, + { + "bbox": [ + 199, + 522, + 227, + 533 + ], + "score": 0.82, + "content": "\\mathrm { I - C + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "achieve one of many solutions which achieve these joint goals. We", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 86, + 535, + 99, + 544 + ], + "score": 1.0, + "content": "166", + "type": "text" + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "next investigate another one that works better: it preserves meaning and structure, and removes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 544, + 341, + 557 + ], + "spans": [ + { + "bbox": [ + 86, + 546, + 99, + 555 + ], + "score": 1.0, + "content": "167", + "type": "text" + }, + { + "bbox": [ + 105, + 544, + 341, + 557 + ], + "score": 1.0, + "content": "language-specific geometry allowing improved alignment.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 90, + 574, + 230, + 586 + ], + "lines": [ + { + "bbox": [ + 87, + 573, + 231, + 587 + ], + "spans": [ + { + "bbox": [ + 87, + 573, + 231, + 587 + ], + "score": 1.0, + "content": "168 3.2 Spectral Normalization", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 86, + 596, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 86, + 595, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 86, + 599, + 99, + 608 + ], + "score": 1.0, + "content": "169", + "type": "text" + }, + { + "bbox": [ + 104, + 595, + 505, + 610 + ], + "score": 1.0, + "content": "The predominant effect of unit-length and zero-mean normalization on monolingual word embeddings", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 86, + 609, + 100, + 618 + ], + "score": 1.0, + "content": "170", + "type": "text" + }, + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "is that it makes embedding vectors from a language lie on a hypersphere with the center of the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 85, + 617, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 85, + 619, + 99, + 630 + ], + "score": 1.0, + "content": "171", + "type": "text" + }, + { + "bbox": [ + 104, + 617, + 505, + 632 + ], + "score": 1.0, + "content": "hypersphere centered at the origin. However, this does not take into account how the word embeddings", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 85, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 85, + 631, + 100, + 641 + ], + "score": 1.0, + "content": "172", + "type": "text" + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "vectors are spread out or clustered on the hypersphere. Approaches like PCA removal and mean", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 85, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 85, + 641, + 100, + 652 + ], + "score": 1.0, + "content": "173", + "type": "text" + }, + { + "bbox": [ + 106, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "centering have the effect of reducing the top principal component or top singular vector. As a result,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 650, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 86, + 653, + 100, + 662 + ], + "score": 1.0, + "content": "174", + "type": "text" + }, + { + "bbox": [ + 104, + 650, + 506, + 664 + ], + "score": 1.0, + "content": "if the spectral properties are extreme, it can help regularize them. However, this approach can be a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 661, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 86, + 664, + 100, + 673 + ], + "score": 1.0, + "content": "175", + "type": "text" + }, + { + "bbox": [ + 105, + 661, + 505, + 675 + ], + "score": 1.0, + "content": "bit blunt. PCA removal makes the top singular value exactly 0, so the condition number becomes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 86, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 86, + 675, + 100, + 684 + ], + "score": 1.0, + "content": "176", + "type": "text" + }, + { + "bbox": [ + 105, + 673, + 351, + 685 + ], + "score": 1.0, + "content": "infinite. Other quantities like the effective condition number,", + "type": "text" + }, + { + "bbox": [ + 351, + 673, + 381, + 685 + ], + "score": 0.91, + "content": "\\kappa _ { \\mathsf { e f f } } ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 673, + 505, + 685 + ], + "score": 1.0, + "content": ", do however, tend to decrease.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 87, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 87, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 87, + 691, + 99, + 700 + ], + "score": 1.0, + "content": "177", + "type": "text" + }, + { + "bbox": [ + 104, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "To this effect, we propose a new algorithm Spectral Normalization that more gently regularizes", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 86, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 86, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "178", + "type": "text" + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "the spectral properties of word embeddings; see Algorithm 2. We will then combine it with other", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 86, + 711, + 397, + 724 + ], + "spans": [ + { + "bbox": [ + 86, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "179", + "type": "text" + }, + { + "bbox": [ + 104, + 711, + 397, + 724 + ], + "score": 1.0, + "content": "approaches to again ensure the embedding vectors lie on the unit sphere.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 88, + 71, + 272, + 84 + ], + "lines": [ + { + "bbox": [ + 84, + 70, + 274, + 86 + ], + "spans": [ + { + "bbox": [ + 84, + 70, + 274, + 86 + ], + "score": 1.0, + "content": "140 3 New Normalization Methods", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 87, + 99, + 505, + 132 + ], + "lines": [], + "index": 1, + "bbox_fs": [ + 87, + 99, + 505, + 134 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 92, + 151, + 274, + 163 + ], + "lines": [ + { + "bbox": [ + 88, + 150, + 275, + 164 + ], + "spans": [ + { + "bbox": [ + 88, + 150, + 275, + 164 + ], + "score": 1.0, + "content": "144 3.1 Geometric Median Normalization", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "index", + "bbox": [ + 87, + 173, + 506, + 207 + ], + "lines": [ + { + "bbox": [ + 86, + 173, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 86, + 175, + 100, + 185 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 105, + 173, + 506, + 185 + ], + "score": 1.0, + "content": "Iterative Normalization enforces individual word embeddings to have a unit length and each monolin-", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 185, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 86, + 187, + 99, + 195 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 105, + 185, + 506, + 197 + ], + "score": 1.0, + "content": "gual embedding to have a zero mean through an iterative technique. Wang et. al. [47] showed that", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 86, + 198, + 100, + 207 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "iterating solutions for these distinct goals will eventually converge to a solution which satisfies both.", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 212, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 86, + 213, + 100, + 223 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 105, + 212, + 505, + 223 + ], + "score": 1.0, + "content": "In this paper, we observe that both goals can be done in one shot without iterating – by solving the", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 85, + 224, + 100, + 234 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "Fermat-Weber problem [27, 25]. This dates to the 17th century, and corresponds with identifying the", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 233, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 85, + 235, + 100, + 245 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 104, + 233, + 358, + 245 + ], + "score": 1.0, + "content": "geometric median of a point set. Formally, the goal is a point", + "type": "text" + }, + { + "bbox": [ + 359, + 233, + 395, + 244 + ], + "score": 0.92, + "content": "x ^ { * } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 233, + 506, + 245 + ], + "score": 1.0, + "content": "that minimizes the sum of", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 241, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 85, + 246, + 100, + 256 + ], + "score": 1.0, + "content": "151", + "type": "text" + }, + { + "bbox": [ + 103, + 241, + 167, + 261 + ], + "score": 1.0, + "content": "distances from", + "type": "text" + }, + { + "bbox": [ + 167, + 246, + 175, + 254 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 241, + 232, + 261 + ], + "score": 1.0, + "content": "anchor points", + "type": "text" + }, + { + "bbox": [ + 233, + 244, + 311, + 256 + ], + "score": 0.94, + "content": "\\{ a _ { 1 } , \\ldots , \\dot { a _ { n } } \\} \\subset \\bar { \\mathbb { R } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 241, + 409, + 261 + ], + "score": 1.0, + "content": "which are not collinear:", + "type": "text" + }, + { + "bbox": [ + 410, + 244, + 506, + 257 + ], + "score": 0.93, + "content": "\\begin{array} { r } { x ^ { * } = \\operatorname* { m i n } _ { x } \\sum _ { i = 1 } ^ { n } \\| x - { } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 255, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 85, + 257, + 100, + 267 + ], + "score": 1.0, + "content": "152", + "type": "text" + }, + { + "bbox": [ + 106, + 255, + 121, + 267 + ], + "score": 0.9, + "content": "a _ { i } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 255, + 506, + 268 + ], + "score": 1.0, + "content": ". Several methods [6, 12, 32] have been proposed; the most popular is the Weiszfeld’s algorithm", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 263, + 507, + 285 + ], + "spans": [ + { + "bbox": [ + 86, + 270, + 100, + 280 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 103, + 263, + 345, + 285 + ], + "score": 1.0, + "content": "(Weiszfeld, see Appendix A). It is folklore that the solution", + "type": "text" + }, + { + "bbox": [ + 345, + 268, + 357, + 279 + ], + "score": 0.87, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 263, + 409, + 285 + ], + "score": 1.0, + "content": "satisfies that", + "type": "text" + }, + { + "bbox": [ + 409, + 266, + 487, + 283 + ], + "score": 0.89, + "content": "\\begin{array} { r } { 0 = \\sum _ { i = 1 } ^ { n } { \\frac { a _ { i } - x ^ { * } } { \\left\\| a _ { i } - x ^ { * } \\right\\| } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 269, + 507, + 281 + ], + "score": 1.0, + "content": "k we", + "type": "text" + }, + { + "bbox": [ + 487, + 263, + 489, + 285 + ], + "score": 1.0, + "content": ";", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 280, + 429, + 294 + ], + "spans": [ + { + "bbox": [ + 86, + 283, + 100, + 292 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 105, + 280, + 429, + 294 + ], + "score": 1.0, + "content": "do not know of a written proof, so prove this in Appendix A.1 for completeness.", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 86, + 300, + 99, + 309 + ], + "score": 1.0, + "content": "155", + "type": "text" + }, + { + "bbox": [ + 105, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "Using this characteristic of the geometric median, we can simultaneously enforce monolingual", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 86, + 311, + 99, + 320 + ], + "score": 1.0, + "content": "156", + "type": "text" + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "word embeddings to have unit-length and zero-mean in just one step. This can be done using the", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 86, + 321, + 99, + 330 + ], + "score": 1.0, + "content": "157", + "type": "text" + }, + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "Geometric Median normalization (GeoMediaN) algorithm (as Algorithm 1). Given a monolingual", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 86, + 333, + 100, + 342 + ], + "score": 1.0, + "content": "158", + "type": "text" + }, + { + "bbox": [ + 105, + 330, + 177, + 343 + ], + "score": 1.0, + "content": "word embedding", + "type": "text" + }, + { + "bbox": [ + 177, + 331, + 185, + 340 + ], + "score": 0.65, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 330, + 330, + 343 + ], + "score": 1.0, + "content": ", we compute the geometric median", + "type": "text" + }, + { + "bbox": [ + 331, + 331, + 342, + 340 + ], + "score": 0.85, + "content": "x ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 330, + 505, + 343 + ], + "score": 1.0, + "content": ", and \"center\" the data on this point, and", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 340, + 298, + 354 + ], + "spans": [ + { + "bbox": [ + 86, + 343, + 99, + 352 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 105, + 340, + 298, + 354 + ], + "score": 1.0, + "content": "unit length normalizes the centered embedding.", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + } + ], + "index": 5, + "bbox_fs": [ + 86, + 173, + 506, + 208 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 211, + 505, + 293 + ], + "lines": [], + "index": 10, + "bbox_fs": [ + 85, + 212, + 507, + 294 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 87, + 297, + 505, + 352 + ], + "lines": [], + "index": 16, + "bbox_fs": [ + 86, + 297, + 505, + 354 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 106, + 371, + 371, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 371, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 371, + 385 + ], + "score": 1.0, + "content": "Algorithm 1 Geometric Median Normalization: GeoMediaN(A)", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "index", + "bbox": [ + 110, + 385, + 253, + 424 + ], + "lines": [ + { + "bbox": [ + 111, + 384, + 208, + 397 + ], + "spans": [ + { + "bbox": [ + 111, + 384, + 208, + 397 + ], + "score": 1.0, + "content": "1: x∗ ← Weiszfeld(A)", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 397, + 253, + 413 + ], + "spans": [ + { + "bbox": [ + 110, + 397, + 151, + 413 + ], + "score": 1.0, + "content": "2: for all", + "type": "text" + }, + { + "bbox": [ + 151, + 398, + 181, + 410 + ], + "score": 0.64, + "content": "a _ { i } \\in A", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 397, + 195, + 413 + ], + "score": 1.0, + "content": "do", + "type": "text" + }, + { + "bbox": [ + 195, + 397, + 253, + 413 + ], + "score": 0.74, + "content": "\\begin{array} { r } { a _ { i } \\frac { a _ { i } - x ^ { * } } { \\| a _ { i } - x ^ { * } \\| } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 411, + 163, + 422 + ], + "spans": [ + { + "bbox": [ + 110, + 411, + 163, + 422 + ], + "score": 1.0, + "content": "3: return A", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 91, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "60 After these steps, all vectors are unit length, and because of the folklore property (Theorem A.1), the", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 91, + 459, + 423, + 473 + ], + "spans": [ + { + "bbox": [ + 91, + 459, + 423, + 473 + ], + "score": 1.0, + "content": "61 mean of those points is also 0. As a result, we can state the following property.", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + } + ], + "index": 21, + "bbox_fs": [ + 110, + 384, + 253, + 422 + ] + }, + { + "type": "index", + "bbox": [ + 95, + 449, + 504, + 471 + ], + "lines": [], + "index": 23.5, + "bbox_fs": [ + 91, + 448, + 505, + 473 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 93, + 477, + 435, + 489 + ], + "lines": [ + { + "bbox": [ + 88, + 477, + 435, + 491 + ], + "spans": [ + { + "bbox": [ + 88, + 477, + 278, + 491 + ], + "score": 1.0, + "content": "162 Theorem 3.1. The output of GeoMediaN", + "type": "text" + }, + { + "bbox": [ + 278, + 477, + 294, + 489 + ], + "score": 0.69, + "content": "( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 477, + 435, + 491 + ], + "score": 1.0, + "content": "is centered and length normalized.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 88, + 477, + 435, + 491 + ] + }, + { + "type": "index", + "bbox": [ + 87, + 500, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 86, + 500, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 86, + 502, + 99, + 511 + ], + "score": 1.0, + "content": "163", + "type": "text" + }, + { + "bbox": [ + 105, + 500, + 506, + 512 + ], + "score": 1.0, + "content": "Despite the Geometric Median Normalization algorithm’s ability to enforce unit-length and zero-", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 86, + 513, + 99, + 522 + ], + "score": 1.0, + "content": "164", + "type": "text" + }, + { + "bbox": [ + 105, + 511, + 506, + 523 + ], + "score": 1.0, + "content": "mean in just one step, we will observe that it does not perform especially well on the BLI task.", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 86, + 524, + 99, + 533 + ], + "score": 1.0, + "content": "165", + "type": "text" + }, + { + "bbox": [ + 105, + 521, + 199, + 535 + ], + "score": 1.0, + "content": "Both GeoMediaN and", + "type": "text" + }, + { + "bbox": [ + 199, + 522, + 227, + 533 + ], + "score": 0.82, + "content": "\\mathrm { I - C + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "achieve one of many solutions which achieve these joint goals. 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Hence, aligning these", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 85, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 85, + 479, + 100, + 489 + ], + "score": 1.0, + "content": "196", + "type": "text" + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "languages without normalization would likely restrict alignment among top singular vectors, not", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 488, + 373, + 502 + ], + "spans": [ + { + "bbox": [ + 86, + 491, + 100, + 500 + ], + "score": 1.0, + "content": "197", + "type": "text" + }, + { + "bbox": [ + 105, + 488, + 373, + 502 + ], + "score": 1.0, + "content": "allowing enough degree of freedom to align corresponding words.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 86, + 505, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 86, + 506, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 86, + 507, + 99, + 516 + ], + "score": 1.0, + "content": "198", + "type": "text" + }, + { + "bbox": [ + 105, + 506, + 505, + 517 + ], + "score": 1.0, + "content": "In contrast, after preprocessing when these values are more uniform, rotations among the dimensions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 86, + 518, + 100, + 528 + ], + "score": 1.0, + "content": "199", + "type": "text" + }, + { + "bbox": [ + 104, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "containing the top principal components will not have an influence on the data distribution, and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 86, + 528, + 100, + 538 + ], + "score": 1.0, + "content": "200", + "type": "text" + }, + { + "bbox": [ + 105, + 527, + 435, + 538 + ], + "score": 1.0, + "content": "can fully optimize the alignment between words. 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Otherwise, if it is below", + "type": "text" + }, + { + "bbox": [ + 272, + 214, + 285, + 225 + ], + "score": 0.88, + "content": "\\beta \\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 213, + 506, + 226 + ], + "score": 1.0, + "content": ", it is considered a minor effect (some are quite small,", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 86, + 227, + 100, + 236 + ], + "score": 1.0, + "content": "184", + "type": "text" + }, + { + "bbox": [ + 106, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "and fairly noisy), and it is left alone. 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Spectral normalization makes the most sense (see Appendix", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 86, + 294, + 100, + 306 + ], + "score": 1.0, + "content": "189", + "type": "text" + }, + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "G) in a setting where the vectors are already centered, and also unit length. While SpecNorm does", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 86, + 306, + 100, + 317 + ], + "score": 1.0, + "content": "190", + "type": "text" + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "not change the center of the data, it does not maintain the length of individual vectors. As such, we", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 315, + 500, + 327 + ], + "spans": [ + { + "bbox": [ + 86, + 317, + 99, + 327 + ], + "score": 1.0, + "content": "191", + "type": "text" + }, + { + "bbox": [ + 105, + 315, + 378, + 327 + ], + "score": 1.0, + "content": "advocate combining these methods into a single iterative algorithm:", + "type": "text" + }, + { + "bbox": [ + 379, + 315, + 424, + 326 + ], + "score": 0.87, + "content": "\\scriptstyle \\mathrm { I - C + S N + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 315, + 500, + 327 + ], + "score": 1.0, + "content": "as in Algorithm 3.", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + } + ], + "index": 3.5, + "bbox_fs": [ + 108, + 86, + 428, + 156 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 179, + 506, + 269 + ], + "lines": [], + "index": 10.5, + "bbox_fs": [ + 85, + 179, + 506, + 270 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 87, + 282, + 505, + 327 + ], + "lines": [], + "index": 16.5, + "bbox_fs": [ + 86, + 282, + 505, + 327 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 342, + 480, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 340, + 482, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 311, + 355 + ], + "score": 1.0, + "content": "Algorithm 3 Iterative Spectral Normalization with", + "type": "text" + }, + { + "bbox": [ + 311, + 343, + 331, + 353 + ], + "score": 0.56, + "content": "\\mathrm { C + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 340, + 482, + 355 + ], + "score": 1.0, + "content": "normalization (I-C+SN+L(A, #Iter))", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 340, + 482, + 355 + ] + }, + { + "type": "index", + "bbox": [ + 109, + 356, + 282, + 411 + ], + "lines": [ + { + "bbox": [ + 110, + 354, + 196, + 368 + ], + "spans": [ + { + "bbox": [ + 110, + 354, + 196, + 368 + ], + "score": 1.0, + "content": "1: for #Iter steps do", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 111, + 366, + 196, + 378 + ], + "spans": [ + { + "bbox": [ + 111, + 367, + 121, + 378 + ], + "score": 1.0, + "content": "2:", + "type": "text" + }, + { + "bbox": [ + 132, + 366, + 196, + 378 + ], + "score": 1.0, + "content": "A ← Center A", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 378, + 216, + 389 + ], + "spans": [ + { + "bbox": [ + 110, + 378, + 121, + 389 + ], + "score": 1.0, + "content": "3:", + "type": "text" + }, + { + "bbox": [ + 133, + 378, + 216, + 389 + ], + "score": 1.0, + "content": "A ← SpecNorm(A)", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 388, + 282, + 400 + ], + "spans": [ + { + "bbox": [ + 110, + 389, + 122, + 399 + ], + "score": 1.0, + "content": "4:", + "type": "text" + }, + { + "bbox": [ + 132, + 388, + 282, + 400 + ], + "score": 1.0, + "content": "A ← Unit length normalization of A", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 110, + 399, + 164, + 412 + ], + "spans": [ + { + "bbox": [ + 110, + 399, + 164, + 412 + ], + "score": 1.0, + "content": "5: return A", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 86, + 436, + 100, + 446 + ], + "score": 1.0, + "content": "192", + "type": "text" + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "We observe in Figure 1 that this process significantly improves the spectral properties, compare to", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 85, + 447, + 100, + 457 + ], + "score": 1.0, + "content": "193", + "type": "text" + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "any other approach. Without preprocessing (None), the languages (EN: English, DE: German, HI:", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 86, + 458, + 100, + 467 + ], + "score": 1.0, + "content": "194", + "type": "text" + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "Hindi, JA: Japanese shown) have large effective condition numbers – indicating that there is a large", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 85, + 469, + 100, + 479 + ], + "score": 1.0, + "content": "195", + "type": "text" + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "disparity between meaningful singular values. Note the y-axis is in log scale. Hence, aligning these", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 85, + 479, + 100, + 489 + ], + "score": 1.0, + "content": "196", + "type": "text" + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "languages without normalization would likely restrict alignment among top singular vectors, not", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 488, + 373, + 502 + ], + "spans": [ + { + "bbox": [ + 86, + 491, + 100, + 500 + ], + "score": 1.0, + "content": "197", + "type": "text" + }, + { + "bbox": [ + 105, + 488, + 373, + 502 + ], + "score": 1.0, + "content": "allowing enough degree of freedom to align corresponding words.", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 506, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 86, + 507, + 99, + 516 + ], + "score": 1.0, + "content": "198", + "type": "text" + }, + { + "bbox": [ + 105, + 506, + 505, + 517 + ], + "score": 1.0, + "content": "In contrast, after preprocessing when these values are more uniform, rotations among the dimensions", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 86, + 518, + 100, + 528 + ], + "score": 1.0, + "content": "199", + "type": "text" + }, + { + "bbox": [ + 104, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "containing the top principal components will not have an influence on the data distribution, and", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 86, + 528, + 100, + 538 + ], + "score": 1.0, + "content": "200", + "type": "text" + }, + { + "bbox": [ + 105, + 527, + 435, + 538 + ], + "score": 1.0, + "content": "can fully optimize the alignment between words. Moreover, Figure 1 shows that", + "type": "text" + }, + { + "bbox": [ + 435, + 527, + 482, + 538 + ], + "score": 0.87, + "content": "\\mathbf { I - C + S N + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "most", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 86, + 540, + 99, + 549 + ], + "score": 1.0, + "content": "201", + "type": "text" + }, + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "decreases the effective condition number, joint effective condition number, and singular value gap.", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 549, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 85, + 550, + 100, + 560 + ], + "score": 1.0, + "content": "202", + "type": "text" + }, + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "score": 1.0, + "content": "Further, these values are fairly uniform across languages, despite great variation beforehand (as", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 560, + 437, + 571 + ], + "spans": [ + { + "bbox": [ + 85, + 561, + 100, + 571 + ], + "score": 1.0, + "content": "203", + "type": "text" + }, + { + "bbox": [ + 105, + 560, + 215, + 571 + ], + "score": 1.0, + "content": "shown with None). In fact,", + "type": "text" + }, + { + "bbox": [ + 216, + 560, + 261, + 570 + ], + "score": 0.89, + "content": "\\scriptstyle \\mathrm { I - C + S N + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 560, + 437, + 571 + ], + "score": 1.0, + "content": "is much more effective than other methods.", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 95, + 506, + 109 + ], + "spans": [ + { + "bbox": [ + 85, + 97, + 100, + 108 + ], + "score": 1.0, + "content": "205", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 95, + 506, + 109 + ], + "score": 1.0, + "content": "We provide an evaluation of our proposed preprocessing methods using 8 language embeddings", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 86, + 109, + 100, + 118 + ], + "score": 1.0, + "content": "206", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "pretrained on Wikipedia [4] of each language: Croatian (HR), English (EN), Finnish (FI), French", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 118, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 85, + 119, + 100, + 130 + ], + "score": 1.0, + "content": "207", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 118, + 505, + 130 + ], + "score": 1.0, + "content": "(FR), German (DE), Italian (IT), Russian (RU), and Turkish (TR). We use the 300-dimensional", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 128, + 480, + 141 + ], + "spans": [ + { + "bbox": [ + 86, + 131, + 100, + 140 + ], + "score": 1.0, + "content": "208", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 128, + 141, + 141 + ], + "score": 1.0, + "content": "fastText", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 141, + 128, + 157, + 140 + ], + "score": 0.57, + "content": "[ 4 ] ^ { 1 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 158, + 128, + 480, + 141 + ], + "score": 1.0, + "content": "embeddings, and all vocabularies are trimmed to the 200K most frequent words.", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 151, + 506, + 164 + ], + "spans": [ + { + "bbox": [ + 85, + 153, + 100, + 163 + ], + "score": 1.0, + "content": "209", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 151, + 506, + 164 + ], + "score": 1.0, + "content": "Alignment evaluation tasks: BLI We evaluate and compare our proposed preprocessing methods", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 162, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 85, + 164, + 100, + 174 + ], + "score": 1.0, + "content": "210", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 162, + 506, + 175 + ], + "score": 1.0, + "content": "mostly on the Bilingual Lexicon Induction (BLI) task, a word translation task. We discuss two more", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 173, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 86, + 176, + 99, + 185 + ], + "score": 1.0, + "content": "211", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 173, + 506, + 186 + ], + "score": 1.0, + "content": "global applications Cross-lingual document classification (CLDC), and Cross-lingual transfer for", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 185, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 86, + 187, + 99, + 196 + ], + "score": 1.0, + "content": "212", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 185, + 505, + 196 + ], + "score": 1.0, + "content": "natural language inference (XNLI) later. BLI is more direct, and has become the de facto evaluation", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 194, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 84, + 197, + 100, + 207 + ], + "score": 1.0, + "content": "213", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "score": 1.0, + "content": "task for CLWE models. For words in the source language, this task retrieves the nearest neighbors in", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 84, + 206, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 84, + 208, + 100, + 218 + ], + "score": 1.0, + "content": "214", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 206, + 506, + 219 + ], + "score": 1.0, + "content": "the target language after alignment to check if it contains the translation. It reports the mean average", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 217, + 507, + 229 + ], + "spans": [ + { + "bbox": [ + 85, + 219, + 100, + 229 + ], + "score": 1.0, + "content": "215", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 217, + 507, + 229 + ], + "score": 1.0, + "content": "precision (MAP) [15], which is equivalent to the mean reciprocal rank (MRR), of the translation.", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 86, + 230, + 99, + 239 + ], + "score": 1.0, + "content": "216", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 228, + 505, + 240 + ], + "score": 1.0, + "content": "Unless stated otherwise, reported values on baseline methods are taken from [15], and use the Google", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 85, + 241, + 100, + 251 + ], + "score": 1.0, + "content": "217", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 239, + 246, + 252 + ], + "score": 1.0, + "content": "Translate (GTrans) dictionary from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 246, + 239, + 267, + 250 + ], + "score": 0.46, + "content": "[ 1 5 ] ^ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 268, + 239, + 506, + 252 + ], + "score": 1.0, + "content": ". We trained (aligned) using 1k, 3k and 5k source words and", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 250, + 413, + 262 + ], + "spans": [ + { + "bbox": [ + 86, + 252, + 99, + 261 + ], + "score": 1.0, + "content": "218", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 250, + 413, + 262 + ], + "score": 1.0, + "content": "evaluated (tested) on separate 2k source test queries, unless noted otherwise.", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 273, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 86, + 275, + 99, + 284 + ], + "score": 1.0, + "content": "219", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 273, + 506, + 285 + ], + "score": 1.0, + "content": "Alignment Algorithms. 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Methods : EN-XL2Methods : XL_-EN
NormalizationCCAPROCPROC-BRCSLSCCAPROCPROC-BRCSLS
None0.3580.3650.3770.3940.3980.3990.4050.428
PR0.3940.3910.4040.3730.4340.4300.4420.425
GeoMediaN0.3930.3910.4000.3790.4330.4320.4400.429
C+L0.3930.3940.4080.4040.4390.4370.4450.464
I-C+L0.3940.3950.4100.4060.4390.4380.4480.460
SN0.3910.3940.4080.4050.4400.4380.4510.468
C+SN+L0.3950.3960.4130.4070.4440.4440.4580.466
I-C+SN+L0.3960.3980.4140.4060.4450.4460.4610.466
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None(β,#Iter)= (2,2)(β,#Iter)= (2,3)(β,#Iter)= (2,4) (β,#Iter)= (2,5)
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So, Spectral Normalization, and in", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 442, + 464, + 455 + ], + "spans": [ + { + "bbox": [ + 86, + 445, + 100, + 453 + ], + "score": 1.0, + "content": "268", + "type": "text" + }, + { + "bbox": [ + 105, + 442, + 146, + 455 + ], + "score": 1.0, + "content": "particular", + "type": "text" + }, + { + "bbox": [ + 147, + 442, + 192, + 453 + ], + "score": 0.89, + "content": "\\scriptstyle \\mathrm { I - C + S N + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 442, + 464, + 455 + ], + "score": 1.0, + "content": ", is shown as the best way to normalize languages before alignment.", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 86, + 622, + 99, + 631 + ], + "score": 1.0, + "content": "269", + "type": "text" + }, + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "We also compute the average BLI MAP score across all 28 language pair for more direct comparison", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 85, + 632, + 100, + 643 + ], + "score": 1.0, + "content": "270", + "type": "text" + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "to prior work [15], summarized in Table 4 and Appendix D. All results are in Appendix H. We", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 641, + 506, + 654 + ], + "spans": [ + { + "bbox": [ + 86, + 644, + 99, + 653 + ], + "score": 1.0, + "content": "271", + "type": "text" + }, + { + "bbox": [ + 105, + 641, + 142, + 654 + ], + "score": 1.0, + "content": "compare", + "type": "text" + }, + { + "bbox": [ + 143, + 642, + 188, + 653 + ], + "score": 0.88, + "content": "\\scriptstyle \\mathrm { I - C + S N + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 641, + 506, + 654 + ], + "score": 1.0, + "content": "(denoted with SN) against no normalization on various dictionary sizes: 1k, 3k", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 652, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 85, + 654, + 100, + 665 + ], + "score": 1.0, + "content": "272", + "type": "text" + }, + { + "bbox": [ + 106, + 652, + 123, + 666 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 653, + 136, + 663 + ], + "score": 0.53, + "content": "5 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 652, + 405, + 666 + ], + "score": 1.0, + "content": "source words and evaluated on 2k source test queries. In all cases,", + "type": "text" + }, + { + "bbox": [ + 406, + 653, + 452, + 664 + ], + "score": 0.88, + "content": "\\scriptstyle \\mathbf { I - C + S N + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 652, + 506, + 666 + ], + "score": 1.0, + "content": "significantly", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 663, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 86, + 666, + 99, + 675 + ], + "score": 1.0, + "content": "273", + "type": "text" + }, + { + "bbox": [ + 105, + 663, + 506, + 677 + ], + "score": 1.0, + "content": "improves over the baseline. This includes improvement over RCSLS which is non-rigid, so in", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 673, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 86, + 677, + 100, + 686 + ], + "score": 1.0, + "content": "274", + "type": "text" + }, + { + "bbox": [ + 104, + 673, + 506, + 688 + ], + "score": 1.0, + "content": "principle could “learn\" adjustments similar to our normalization in the process of alignment. 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Methods : EN-XL2Methods : XL_-EN
NormalizationCCAPROCPROC-BRCSLSCCAPROCPROC-BRCSLS
None0.3580.3650.3770.3940.3980.3990.4050.428
PR0.3940.3910.4040.3730.4340.4300.4420.425
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SN0.3910.3940.4080.4050.4400.4380.4510.468
C+SN+L0.3950.3960.4130.4070.4440.4440.4580.466
I-C+SN+L0.3960.3980.4140.4060.4450.4460.4610.466
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In both examples, a powerful classifier is trained", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 272, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 283 + ], + "score": 1.0, + "content": "on the EN embedding (after normalization), and then we demonstrate that after a lower resource", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 104, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "language (e.g., German, DE) has been normalized and align the analysis task can be directly applied", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 103, + 294, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 103, + 294, + 453, + 306 + ], + "score": 1.0, + "content": "to that language. And in particular, adding the simple process of our normalization", + "type": "text" + }, + { + "bbox": [ + 454, + 294, + 502, + 305 + ], + "score": 0.83, + "content": "( \\mathrm { I - C + S N + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 294, + 506, + 306 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 102, + 303, + 340, + 318 + ], + "spans": [ + { + "bbox": [ + 102, + 303, + 340, + 318 + ], + "score": 1.0, + "content": "dramatically improves the results over not doing that step.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 86, + 330, + 506, + 419 + ], + "lines": [ + { + "bbox": [ + 86, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 86, + 333, + 99, + 342 + ], + "score": 1.0, + "content": "285", + "type": "text" + }, + { + "bbox": [ + 106, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "Cross-lingual Document Classification (CLDC). The CLDC task builds a topic classification", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 86, + 341, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 86, + 344, + 99, + 353 + ], + "score": 1.0, + "content": "286", + "type": "text" + }, + { + "bbox": [ + 105, + 341, + 506, + 355 + ], + "score": 1.0, + "content": "using a language model on a high resource language (in our case English EN) across 15 topics.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 351, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 86, + 355, + 99, + 364 + ], + "score": 1.0, + "content": "287", + "type": "text" + }, + { + "bbox": [ + 105, + 351, + 506, + 366 + ], + "score": 1.0, + "content": "The TED CLDC corpus assembled by [19] was used for training and evaluation. Following [15]", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 86, + 366, + 99, + 375 + ], + "score": 1.0, + "content": "288", + "type": "text" + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "a simple CNN was used to train. Table 5 summarizes the average F1-score for all topic classifiers", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 371, + 507, + 388 + ], + "spans": [ + { + "bbox": [ + 86, + 376, + 99, + 385 + ], + "score": 1.0, + "content": "289", + "type": "text" + }, + { + "bbox": [ + 104, + 371, + 290, + 388 + ], + "score": 1.0, + "content": "on 5 language pairs. The CLWEs induced by", + "type": "text" + }, + { + "bbox": [ + 290, + 373, + 327, + 385 + ], + "score": 0.46, + "content": "\\mathrm { P R O C ^ { S N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 371, + 360, + 388 + ], + "score": 1.0, + "content": ", PROC-", + "type": "text" + }, + { + "bbox": [ + 361, + 373, + 378, + 384 + ], + "score": 0.65, + "content": "\\cdot \\breve { \\mathbf { B } } ^ { \\mathsf { S N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 371, + 381, + 388 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 382, + 373, + 413, + 385 + ], + "score": 0.43, + "content": "{ \\bf D } { \\bf U } ^ { { \\sf S } { \\sf N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 371, + 434, + 388 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 434, + 373, + 476, + 385 + ], + "score": 0.6, + "content": "\\mathrm { R C S L S ^ { S N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 371, + 507, + 388 + ], + "score": 1.0, + "content": "(using", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 383, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 86, + 387, + 99, + 397 + ], + "score": 1.0, + "content": "290", + "type": "text" + }, + { + "bbox": [ + 106, + 385, + 151, + 396 + ], + "score": 0.86, + "content": "\\scriptstyle \\mathrm { I - C + S N + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 383, + 506, + 399 + ], + "score": 1.0, + "content": ") outperformed the baseline result (with no normalization) on the CLDC task, significantly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 85, + 395, + 507, + 410 + ], + "spans": [ + { + "bbox": [ + 85, + 398, + 99, + 408 + ], + "score": 1.0, + "content": "291", + "type": "text" + }, + { + "bbox": [ + 104, + 395, + 507, + 410 + ], + "score": 1.0, + "content": "improving the best average score from 0.421 to 0.461. Glavas et.al. 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ModelDictEN-DEEN-FREN-ITEN-RUEN-TRAvg
PROC5K.366.258.338.288.278.306
PROCSN5K.436.366.427.517.511.451
PROC-B3K.364.304.299.336.317.324
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DLV5K.419.336.397.493.458.421
DLVSN5K.433.323.406.499.472.427
RCSLS5K.466.397.403.403.406.415
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Following [15]", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 86, + 366, + 99, + 375 + ], + "score": 1.0, + "content": "288", + "type": "text" + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "a simple CNN was used to train. Table 5 summarizes the average F1-score for all topic classifiers", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 371, + 507, + 388 + ], + "spans": [ + { + "bbox": [ + 86, + 376, + 99, + 385 + ], + "score": 1.0, + "content": "289", + "type": "text" + }, + { + "bbox": [ + 104, + 371, + 290, + 388 + ], + "score": 1.0, + "content": "on 5 language pairs. The CLWEs induced by", + "type": "text" + }, + { + "bbox": [ + 290, + 373, + 327, + 385 + ], + "score": 0.46, + "content": "\\mathrm { P R O C ^ { S N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 371, + 360, + 388 + ], + "score": 1.0, + "content": ", PROC-", + "type": "text" + }, + { + "bbox": [ + 361, + 373, + 378, + 384 + ], + "score": 0.65, + "content": "\\cdot \\breve { \\mathbf { B } } ^ { \\mathsf { S N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 371, + 381, + 388 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 382, + 373, + 413, + 385 + ], + "score": 0.43, + "content": "{ \\bf D } { \\bf U } ^ { { \\sf S } { \\sf N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 371, + 434, + 388 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 434, + 373, + 476, + 385 + ], + "score": 0.6, + "content": "\\mathrm { R C S L S ^ { S N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 371, + 507, + 388 + ], + "score": 1.0, + "content": "(using", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 383, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 86, + 387, + 99, + 397 + ], + "score": 1.0, + "content": "290", + "type": "text" + }, + { + "bbox": [ + 106, + 385, + 151, + 396 + ], + "score": 0.86, + "content": "\\scriptstyle \\mathrm { I - C + S N + L }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 383, + 506, + 399 + ], + "score": 1.0, + "content": ") outperformed the baseline result (with no normalization) on the CLDC task, significantly", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 395, + 507, + 410 + ], + "spans": [ + { + "bbox": [ + 85, + 398, + 99, + 408 + ], + "score": 1.0, + "content": "291", + "type": "text" + }, + { + "bbox": [ + 104, + 395, + 507, + 410 + ], + "score": 1.0, + "content": "improving the best average score from 0.421 to 0.461. Glavas et.al. [15] use only 12 of 15 topics,", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 407, + 404, + 420 + ], + "spans": [ + { + "bbox": [ + 86, + 409, + 99, + 418 + ], + "score": 1.0, + "content": "292", + "type": "text" + }, + { + "bbox": [ + 106, + 407, + 404, + 420 + ], + "score": 1.0, + "content": "but could not confirm which, so we re-ran all baselines using all 15 topics.", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 86, + 604, + 99, + 613 + ], + "score": 1.0, + "content": "293", + "type": "text" + }, + { + "bbox": [ + 106, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "Cross-lingual Natural Language Inference (XNLI). 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Our approach generalizes previous approaches, and effectively", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 85, + 291, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 85, + 294, + 100, + 304 + ], + "score": 1.0, + "content": "309", + "type": "text" + }, + { + "bbox": [ + 104, + 291, + 506, + 306 + ], + "score": 1.0, + "content": "removes much of the clustering of words based on properties other than the similarity which encodes", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 85, + 304, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 85, + 306, + 100, + 315 + ], + "score": 1.0, + "content": "310", + "type": "text" + }, + { + "bbox": [ + 105, + 304, + 505, + 316 + ], + "score": 1.0, + "content": "meaning. When used to individually preprocess monolingual embeddings, our approach allows", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 85, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 85, + 316, + 100, + 326 + ], + "score": 1.0, + "content": "311", + "type": "text" + }, + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "any alignment procedure to find better alignments: resulting in improved performance on direct", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 325, + 496, + 338 + ], + "spans": [ + { + "bbox": [ + 86, + 327, + 99, + 336 + ], + "score": 1.0, + "content": "312", + "type": "text" + }, + { + "bbox": [ + 105, + 325, + 496, + 338 + ], + "score": 1.0, + "content": "translation tasks as well as cross-lingual topic classification and natural language inference tasks.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 86, + 348, + 505, + 414 + ], + "lines": [ + { + "bbox": [ + 85, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 85, + 350, + 100, + 360 + ], + "score": 1.0, + "content": "313", + "type": "text" + }, + { + "bbox": [ + 106, + 348, + 506, + 360 + ], + "score": 1.0, + "content": "Social impacts. The vast majority of NLP research and cutting-edge advancements are in English.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 85, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 85, + 361, + 100, + 371 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "This disadvantages those who primarily operate in other languages, with less developed models, or", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 85, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 85, + 371, + 100, + 382 + ], + "score": 1.0, + "content": "315", + "type": "text" + }, + { + "bbox": [ + 104, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "less data to train models. As large language models are the cornerstone of most NLP research and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 85, + 380, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 85, + 383, + 100, + 393 + ], + "score": 1.0, + "content": "316", + "type": "text" + }, + { + "bbox": [ + 105, + 380, + 507, + 394 + ], + "score": 1.0, + "content": "development in English, one of this work’s main goals is to port these advances to other languages,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 85, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 85, + 393, + 100, + 403 + ], + "score": 1.0, + "content": "317", + "type": "text" + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "and those who use them. This will help unlock this technology to many others around the world. As", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 85, + 403, + 460, + 416 + ], + "spans": [ + { + "bbox": [ + 85, + 405, + 100, + 415 + ], + "score": 1.0, + "content": "318", + "type": "text" + }, + { + "bbox": [ + 105, + 403, + 460, + 416 + ], + "score": 1.0, + "content": "with most models, this accuracy and improvement may vary across tasks and languages.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 85, + 419, + 506, + 529 + ], + "lines": [ + { + "bbox": [ + 85, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 85, + 421, + 100, + 431 + ], + "score": 1.0, + "content": "319", + "type": "text" + }, + { + "bbox": [ + 105, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "While language models have many positive use cases including improving accessibility, better", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 85, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 85, + 432, + 100, + 442 + ], + "score": 1.0, + "content": "320", + "type": "text" + }, + { + "bbox": [ + 104, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "recommendations, and increased automation, they have some negative effects as well. These include", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 85, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 85, + 443, + 99, + 453 + ], + "score": 1.0, + "content": "321", + "type": "text" + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "requiring potentially large computational and hence environmental cost, encoding and exacerbating", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 85, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 85, + 453, + 100, + 464 + ], + "score": 1.0, + "content": "322", + "type": "text" + }, + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "bias, and aiding in automatically generating fake or deceitful content. While this paper is unlikely to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 85, + 463, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 85, + 464, + 100, + 474 + ], + "score": 1.0, + "content": "323", + "type": "text" + }, + { + "bbox": [ + 106, + 463, + 506, + 474 + ], + "score": 1.0, + "content": "change the desire to use embeddings, it aims to reduce the burden of use and increase the effectiveness", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 85, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 85, + 475, + 100, + 486 + ], + "score": 1.0, + "content": "324", + "type": "text" + }, + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "in lower-resource settings. And in particular to port models trained in English to other languages.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 85, + 483, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 85, + 486, + 100, + 496 + ], + "score": 1.0, + "content": "325", + "type": "text" + }, + { + "bbox": [ + 104, + 483, + 506, + 498 + ], + "score": 1.0, + "content": "This would reduce the cost of retraining in other languages if the English model can be reused, easing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 85, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 85, + 497, + 100, + 507 + ], + "score": 1.0, + "content": "326", + "type": "text" + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "environmental costs. We support the maturing efforts in attenuating bias in all such embeddings.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 85, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 85, + 508, + 100, + 518 + ], + "score": 1.0, + "content": "327", + "type": "text" + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "And while we acknowledge the possibility of this work aiding in the automated creation of deceitful", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 518, + 445, + 529 + ], + "spans": [ + { + "bbox": [ + 85, + 519, + 100, + 529 + ], + "score": 1.0, + "content": "328", + "type": "text" + }, + { + "bbox": [ + 106, + 518, + 445, + 529 + ], + "score": 1.0, + "content": "content and the harm it can cause, we believe the many benefits outweigh the harms.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 105, + 540, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "Limitations. The overarching goal in this line of work is to port the many advances built on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "embeddings from high-resource settings (like built on the English language), to lower-resource", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "settings (like Turkish). 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While in principle", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 86, + 634, + 100, + 645 + ], + "score": 1.0, + "content": "337", + "type": "text" + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "a normalization function and alignment could be applied to contextual settings, we are unaware of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 644, + 465, + 657 + ], + "spans": [ + { + "bbox": [ + 86, + 646, + 99, + 656 + ], + "score": 1.0, + "content": "338", + "type": "text" + }, + { + "bbox": [ + 105, + 644, + 465, + 657 + ], + "score": 1.0, + "content": "any technique for learning these mappings. 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The vast majority of NLP research and cutting-edge advancements are in English.", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 85, + 361, + 100, + 371 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "This disadvantages those who primarily operate in other languages, with less developed models, or", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 369, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 85, + 371, + 100, + 382 + ], + "score": 1.0, + "content": "315", + "type": "text" + }, + { + "bbox": [ + 104, + 369, + 506, + 383 + ], + "score": 1.0, + "content": "less data to train models. As large language models are the cornerstone of most NLP research and", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 380, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 85, + 383, + 100, + 393 + ], + "score": 1.0, + "content": "316", + "type": "text" + }, + { + "bbox": [ + 105, + 380, + 507, + 394 + ], + "score": 1.0, + "content": "development in English, one of this work’s main goals is to port these advances to other languages,", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 391, + 506, + 404 + ], + "spans": [ + { + "bbox": [ + 85, + 393, + 100, + 403 + ], + "score": 1.0, + "content": "317", + "type": "text" + }, + { + "bbox": [ + 105, + 391, + 506, + 404 + ], + "score": 1.0, + "content": "and those who use them. This will help unlock this technology to many others around the world. As", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 403, + 460, + 416 + ], + "spans": [ + { + "bbox": [ + 85, + 405, + 100, + 415 + ], + "score": 1.0, + "content": "318", + "type": "text" + }, + { + "bbox": [ + 105, + 403, + 460, + 416 + ], + "score": 1.0, + "content": "with most models, this accuracy and improvement may vary across tasks and languages.", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 419, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 85, + 421, + 100, + 431 + ], + "score": 1.0, + "content": "319", + "type": "text" + }, + { + "bbox": [ + 105, + 419, + 505, + 430 + ], + "score": 1.0, + "content": "While language models have many positive use cases including improving accessibility, better", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 85, + 432, + 100, + 442 + ], + "score": 1.0, + "content": "320", + "type": "text" + }, + { + "bbox": [ + 104, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "recommendations, and increased automation, they have some negative effects as well. These include", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 441, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 85, + 443, + 99, + 453 + ], + "score": 1.0, + "content": "321", + "type": "text" + }, + { + "bbox": [ + 105, + 441, + 506, + 454 + ], + "score": 1.0, + "content": "requiring potentially large computational and hence environmental cost, encoding and exacerbating", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 85, + 453, + 100, + 464 + ], + "score": 1.0, + "content": "322", + "type": "text" + }, + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "bias, and aiding in automatically generating fake or deceitful content. While this paper is unlikely to", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 463, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 85, + 464, + 100, + 474 + ], + "score": 1.0, + "content": "323", + "type": "text" + }, + { + "bbox": [ + 106, + 463, + 506, + 474 + ], + "score": 1.0, + "content": "change the desire to use embeddings, it aims to reduce the burden of use and increase the effectiveness", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 85, + 475, + 100, + 486 + ], + "score": 1.0, + "content": "324", + "type": "text" + }, + { + "bbox": [ + 105, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "in lower-resource settings. And in particular to port models trained in English to other languages.", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 483, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 85, + 486, + 100, + 496 + ], + "score": 1.0, + "content": "325", + "type": "text" + }, + { + "bbox": [ + 104, + 483, + 506, + 498 + ], + "score": 1.0, + "content": "This would reduce the cost of retraining in other languages if the English model can be reused, easing", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 85, + 497, + 100, + 507 + ], + "score": 1.0, + "content": "326", + "type": "text" + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "environmental costs. We support the maturing efforts in attenuating bias in all such embeddings.", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 85, + 508, + 100, + 518 + ], + "score": 1.0, + "content": "327", + "type": "text" + }, + { + "bbox": [ + 105, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "And while we acknowledge the possibility of this work aiding in the automated creation of deceitful", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 518, + 445, + 529 + ], + "spans": [ + { + "bbox": [ + 85, + 519, + 100, + 529 + ], + "score": 1.0, + "content": "328", + "type": "text" + }, + { + "bbox": [ + 106, + 518, + 445, + 529 + ], + "score": 1.0, + "content": "content and the harm it can cause, we believe the many benefits outweigh the harms.", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + } + ], + "index": 9.5, + "bbox_fs": [ + 85, + 269, + 506, + 338 + ] + }, + { + "type": "index", + "bbox": [ + 86, + 348, + 505, + 414 + ], + "lines": [], + "index": 15.5, + "bbox_fs": [ + 85, + 348, + 507, + 416 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 85, + 419, + 506, + 529 + ], + "lines": [], + "index": 23.5, + "bbox_fs": [ + 85, + 419, + 506, + 529 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 105, + 540, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 553 + ], + "score": 1.0, + "content": "Limitations. 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In", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 127, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 127, + 113, + 506, + 127 + ], + "score": 1.0, + "content": "Proceedings of Human Language Technologies: The 2009 Annual Conference of the North", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 124, + 121, + 508, + 139 + ], + "spans": [ + { + "bbox": [ + 124, + 121, + 508, + 139 + ], + "score": 1.0, + "content": "American Chapter of the Association for Computational Linguistics, pages 19–27, Boulder,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 127, + 136, + 390, + 147 + ], + "spans": [ + { + "bbox": [ + 127, + 136, + 390, + 147 + ], + "score": 1.0, + "content": "Colorado, June 2009. 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for #Iter stepsdo
1: 2:A←CenterA
3:A ← SpecNorm(A)
4:A ←Unit length normalization of A
5:
return A
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Methods : EN-XL2Methods : XL_-EN
NormalizationCCAPROCPROC-BRCSLSCCAPROCPROC-BRCSLS
None0.3580.3650.3770.3940.3980.3990.4050.428
PR0.3940.3910.4040.3730.4340.4300.4420.425
GeoMediaN0.3930.3910.4000.3790.4330.4320.4400.429
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I-C+L0.3940.3950.4100.4060.4390.4380.4480.460
SN0.3910.3940.4080.4050.4400.4380.4510.468
C+SN+L0.3950.3960.4130.4070.4440.4440.4580.466
I-C+SN+L0.3960.3980.4140.4060.4450.4460.4610.466
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None(β,#Iter)= (2,2)(β,#Iter)= (2,3)(β,#Iter)= (2,4) (β,#Iter)= (2,5)
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+University of California, Los Angeles +{jianwen, yanglv, ruiqigao}@ucla.edu, {sczhu, ywu}@stat.ucla.edu + +# ABSTRACT + +This paper studies the cooperative training of two probabilistic models of signals such as images. Both models are parametrized by convolutional neural networks (ConvNets). The first network is a descriptor network, which is an exponential family model or an energy-based model, whose feature statistics or energy function are defined by a bottom-up ConvNet, which maps the observed signal to the feature statistics. The second network is a generator network, which is a nonlinear version of factor analysis. It is defined by a top-down ConvNet, which maps the latent factors to the observed signal. The maximum likelihood training algorithms of both the descriptor net and the generator net are in the form of alternating back-propagation, and both algorithms involve Langevin sampling. We observe that the two training algorithms can cooperate with each other by jumpstarting each other’s Langevin sampling, and they can be seamlessly interwoven into a CoopNets algorithm that can train both nets simultaneously. + +# 1 INTRODUCTION + +# 1.1 TWO CONVNETS OF OPPOSITE DIRECTIONS + +We begin with a story that the reader of this paper can readily relate to. A student writes up an initial draft of a paper. His advisor then revises it. After that they submit the revised paper for review. The student then learns from his advisor’s revision, while the advisor learns from the outside review. In this story, the advisor guides the student, but the student does most of the work. + +This paper is about two probabilistic models of signals such as images, and they play the roles of student and advisor as described above. Both models are parametrized by convolutional neural networks (ConvNets or CNNs) (LeCun et al., 1998; Krizhevsky et al., 2012). The two nets take two opposite directions. One is bottom-up, and the other is top-down, as illustrate by the following diagram: + +
Bottom-up ConvNet featuresTop-down ConvNet latent variables
个←
signal (a) Descriptor Netsignal (b) Generator Net
+ +The simultaneous training of such two nets was first studied by the recent work of Kim & Bengio (2016). These two nets belong to two major classes of probabilistic models. (a) The exponential family models or the energy-based models (LeCun et al., 2006) or the Markov random field models (Zhu et al., 1997), where the probability distribution is defined by feature statistics or energy function computed from the signal by a bottom-up process. (b) The latent variable models or the directed graphical models, where the signal is assumed to be a transformation of the latent factors that follow a known prior distribution. The latent factors generate the signal by a top-down process. A classical example is factor analysis. + +The two classes of models have been contrasted by Zhu (2003); Teh et al. (2003); Ngiam et al. (2011). Zhu (2003) called the two classes of models the descriptive models and the generative models respectively. Both classes of models can benefit from the high capacity of the multi-layer ConvNets. (a) In the exponential family models or the energy-based models, the feature statistics or the energy function can be defined by a bottom-up ConvNet that maps the signal to the features and the energy function (Ngiam et al., 2011; Xie et al., 2016). We call the resulting model a descriptive network or a descriptor net following Zhu (2003), because it is built on descriptive feature statistics. (b) In the latent variable models or the directed graphical models, the transformation from the latent factors to the signal can be defined by a top-down ConvNet (Dosovitskiy et al., 2015), which maps the latent factors to the signal. We call the resulting model a generative network or generator net following Goodfellow et al. (2014), who proposed such a model in their work on the generative adversarial networks (GAN). + +![](images/799da87ccece60114024503ad08a2702e46d67f851a2c657025c4fc10d8cf9f5.jpg) +Figure 1: (a) Algorithm D involves sampling from the current model by Langevin dynamics. (b) Algorithm G involves sampling from the posterior distribution of the latent factors by Langevin dynamics. (c) CoopNets algorithm. The part of the flowchart for training the descriptor is similar to Algorithm D, except that the D1 Langevin sampling is initialized from the initial synthesized examples supplied by the generator. The part of the flowchart for training the generator can also be mapped to Algorithm G, except that the revised synthesized examples play the role of the observed data, and the known generated latent factors can be used as inferred latent factors (or be used to initialize the G1 Langevin sampling of the latent factors). + +# 1.2 TWO TRAINING ALGORITHMS AND THEIR COOPERATION + +Fig. 1(a) and (b) display the flowcharts of the maximum likelihood learning algorithms for training the descriptor and generator nets . We call the two algorithms Algorithm D and Algorithm G respectively. Algorithm D (Xie et al., 2016) iterates two steps: Step D1 synthesizes examples by sampling from the current model by Langevin dynamics. Step D2 updates the parameters to shift the density from the synthesized examples towards the observed examples. Algorithm G (Han et al., 2017) also iterates two steps. Step G1 infers latent factors for each observed example by sampling from their posterior distribution by Langevin dynamics. Step G2 updates the parameters by a non-linear regression of the observed examples on their corresponding latent factors. We use Langevin dynamics for Markov chain Monte Carlo (MCMC) sampling because the gradient term of Langevin dynamics can be readily computed via back-propagation. Thus all the steps D1, D2 and G1, G2 are powered by back-propagation, and both Algorithms D and G are alternating back-propagation algorithms. + +In this article, we propose to couple Algorithms D and G into a cooperative training algorithm that interweaves the steps of the two algorithms seamlessly. We call the resulting algorithm the CoopNets algorithm, and we show that it can train both nets simultaneously. + +Figure 1(c) displays the flowchart of the CoopNets algorithm. The generator is like the student. It generates the initial draft of the synthesized examples. The descriptor is like the advisor. It revises the initial draft by initializing its Langevin dynamics from the initial draft in Step D1, which produces the revised draft of the synthesized examples. The descriptor learns from the outside review in Step D2, which is in the form of the difference between the observed examples and the revised synthesized examples. The generator learns from how the descriptor revises the initial draft by reconstructing the revised draft in Step G2. For each synthesized example, the generator knows the latent factors that generate the initial draft, so that Step G1 can infer the latent factors by initializing its Langevin dynamics from their known values. + +In the CoopNets algorithm, the generator fuels the MCMC of the descriptor by supplying initial synthesized examples, which can be obtained by direct ancestral sampling. The generator then learns from the revised synthesized examples with virtually known latent factors. The cooperation is thus beneficial to both nets. + +# 2 RELATED WORK + +Our work is inspired by the generative adversarial networks (GAN) (Goodfellow et al., 2014; Denton et al., 2015; Radford et al., 2015). In GAN, the generator net is paired with a discriminator net. The two nets play adversarial roles. In our work, the generator net and the descriptor net play cooperative roles, and they feed each other the initial, revised and reconstructed synthesized data. The learning of both nets is based on maximum likelihood, and the learning process is quite stable because of the cooperative nature and the consistent directions of the two maximum likelihood training algorithms. + +Another method to train the generator network is variational auto-encoder (VAE) (Kingma & Welling, 2014; Rezende et al., 2014; Mnih & Gregor, 2014), which learns an inferential or recognition network to approximate the posterior distribution of the latent factors. + +The connection between the descriptor net and the discriminator net has been explored by Xie et al. +(2016), where the descriptor can be derived from the discriminator. + +Our work is most similar to the recent work of Kim & Bengio (2016). In fact, the settings of the two nets are the same. In their work, the generator learns from the descriptor by minimizing the Kullback-Leibler divergence from the generator to the descriptor, which can be decomposed into an energy term and an entropy term. In our work, the two nets interact with each other via synthesized data, and the generator learns from the descriptor by reconstructing the revised draft of synthesized examples. Our method does not need to approximate the intractable entropy term. + +Our work is related to the contrastive divergence algorithm (Hinton, 2002) for training the descriptor net. The contrastive divergence initializes the MCMC sampling from the observed examples. The CoopNets algorithm initializes the MCMC sampling from the examples supplied by the generator. + +# 3 TWO NETS AND TWO TRAINING ALGORITHMS + +# 3.1 DESCRIPTOR NET AND TRAINING ALGORITHM + +Let $Y$ be the $D$ -dimensional signal, such as an image. The descriptor model is in the form of exponential tilting of a reference distribution (Xie et al., 2016): + +$$ +P _ { \mathcal { D } } ( Y ; W _ { \mathcal { D } } ) = \frac { 1 } { Z ( W _ { \mathcal { D } } ) } \exp \left[ f ( Y ; W _ { \mathcal { D } } ) \right] q ( Y ) , +$$ + +where $q ( Y )$ is the reference distribution such as Gaussian white noise $q ( Y ) \propto \exp \left( - \| Y \| ^ { 2 } / 2 s ^ { 2 } \right) ,$ $f ( Y ; W _ { \mathcal { D } } )$ $( f$ stands for features) is the feature statistics or energy function, defined by a ConvNet whose parameters are denoted by $W _ { \mathcal { D } }$ . This ConvNet is bottom-up because it maps the signal $Y$ to a number. See the diagram in (1). $\begin{array} { r } { Z ( W _ { \mathcal { D } } ) = \int \exp { [ f ( Y ; W _ { \mathcal { D } } ) ] } \bar { q } ( Y ) d Y = \mathrm { E } _ { q } \bar { \left\{ \exp [ f ( \bar { Y } ; W _ { \mathcal { D } } ) ] \right\} } } \end{array}$ is the normalizing constant, where $\operatorname { E } _ { q }$ is the expectation with respect to $q$ . + +Suppose we observe training examples $\{ Y _ { i } , i ~ = ~ 1 , . . . , n \}$ from an unknown data distribution $P _ { \mathrm { d a t a } } ( Y )$ . The maximum likelihood training seeks to maximize the log-likelihood function $\begin{array} { r } { L _ { \mathcal { D } } ( W _ { \mathcal { D } } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \log P _ { \mathcal { D } } ( Y _ { i } ; W _ { \mathcal { D } } ) } \end{array}$ . If the sample size Kullback-Leibler di $n$ is large, the maximum likelihorgence from the data distribution $\mathrm { K L } ( P _ { \mathrm { d a t a } } | P _ { \mathcal { D } } )$ $P _ { \mathrm { d a t a } }$ to the model distribution $P _ { D }$ . The gradient of the $L _ { \mathcal { D } } ( W _ { \mathcal { D } } )$ is + +$$ +\begin{array} { r c l } { { { \cal L } _ { \mathcal D } ^ { \prime } ( W _ { \mathcal D } ) } } & { { = } } & { { \displaystyle { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \partial } { \partial W _ { \mathcal D } } f ( Y _ { i } ; W _ { \mathcal D } ) - \mathrm { E } _ { W _ { \mathcal D } } \left[ \frac { \partial } { \partial W _ { \mathcal D } } f ( Y ; W _ { \mathcal D } ) \right] } , } } \end{array} +$$ + +where $\operatorname { E } _ { W _ { \mathcal { D } } }$ denotes the expectation with respect to $P _ { \mathcal { D } } ( Y ; W _ { \mathcal { D } } )$ + +The expectation in equation (3) is analytically intractable and has to be approximated by MCMC, such as Langevin dynamics, which iterates the following step: + +$$ +Y _ { \tau + 1 } = Y _ { \tau } - \frac { \delta ^ { 2 } } { 2 } \left[ \frac { Y _ { \tau } } { s ^ { 2 } } - \frac { \partial } { \partial Y } f ( Y _ { \tau } ; W _ { \mathcal { D } } ) \right] + \delta U _ { \tau } , +$$ + +where $\tau$ indexes the time steps of the Langevin dynamics, $\delta$ is the step size, and $U _ { \tau } \sim \mathrm { N } ( 0 , I _ { D } )$ is the Gaussian white noise term. + +We can run $\tilde { n }$ parallel chains of Langevin dynamics according to (4) to obtain the synthesized examples $\{ \tilde { Y } _ { i } , i = 1 , . . . , \tilde { n } \}$ . The Monte Carlo approximation to $L _ { \mathcal { D } } ^ { \prime } ( W _ { \mathcal { D } } )$ is + +$$ +L _ { \mathcal { D } } ^ { \prime } ( W _ { \mathcal { D } } ) \quad \approx \quad \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \partial } { \partial W _ { \mathcal { D } } } f ( Y _ { i } ; W _ { \mathcal { D } } ) - \frac { 1 } { \tilde { n } } \sum _ { i = 1 } ^ { \tilde { n } } \frac { \partial } { \partial W _ { \mathcal { D } } } f ( \tilde { Y } _ { i } ; W _ { \mathcal { D } } ) , +$$ + +which is used to update $W _ { \mathcal { D } }$ + +Algorithm D (Xie et al., 2016) iterates the following two steps after initializing $W _ { \mathcal { D } }$ and $\{ \tilde { Y } _ { i } , i =$ $1 , . . . , \tilde { n } \}$ . Step $D l$ : Run $l _ { \mathcal { D } }$ steps of Langevin from the current $\{ \tilde { Y } \}$ according according to (4). Step $D 2$ : update $\boldsymbol { W _ { D } ^ { ( t + 1 ) } } = \boldsymbol { W _ { D } ^ { ( t ) } } + \gamma _ { t } \boldsymbol { L _ { D } ^ { \prime } } ( \boldsymbol { W _ { D } ^ { ( t ) } } )$ with learning rate $\gamma _ { t }$ . The convergence of such an algorithm follows Younes (1999). + +# 3.2 GENERATOR NET AND TRAINING ALGORITHM + +The generator net (Goodfellow et al., 2014) seeks to explain the signal $Y$ of dimension $D$ by a vector of latent factors $X$ of dimension $d$ , and usually $d \ll D$ . The model is of the following form: + +$$ +X \sim \mathrm { N } ( 0 , I _ { d } ) , Y = g ( X ; W _ { \mathcal { G } } ) + \epsilon , \epsilon \sim \mathrm { N } ( 0 , \sigma ^ { 2 } I _ { D } ) . +$$ + +$g ( X ; W _ { \mathcal { G } } )$ $( g$ stands for generator) is a top-down ConvNet defined by the parameters $W _ { \mathcal { G } }$ . The ConvNet $g$ maps the latent factors $X$ to the signal $Y$ . See the diagram in (1). + +The joint density of model (6) is $P _ { \mathcal { G } } ( X , Y ; W _ { \mathcal { G } } ) = P _ { \mathcal { G } } ( X ) P _ { \mathcal { G } } ( Y | X ; W _ { \mathcal { G } } )$ , and + +$$ +\log P _ { \mathcal { G } } ( X , Y ; W _ { \mathcal { G } } ) = - \frac { 1 } { 2 \sigma ^ { 2 } } \| Y - g ( X ; W _ { \mathcal { G } } ) \| ^ { 2 } - \frac { 1 } { 2 } \| X \| ^ { 2 } + \mathrm { c o n s t a n t } , +$$ + +where the constant term is independent of $X , Y$ and $W _ { \mathcal { G } }$ . The marginal density is obtained by integrating out the latent factors $X$ , i.e., $\begin{array} { r } { P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ) = \int P _ { \mathcal { G } } ( X , Y ; W _ { \mathcal { G } } ) d X } \end{array}$ . The inference of $X$ given $Y$ is based on the posterior density $P _ { \mathcal { G } } ( X | Y ; W _ { \mathcal { G } } ) \stackrel { \smile } { = } P _ { \mathcal { G } } ( X , Y ; W _ { \mathcal { G } } ) / P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ) \propto P _ { \mathcal { G } } ( X , Y ; W _ { \mathcal { G } } )$ as a function of $X$ . + +For the training data $\{ Y _ { i } , i = 1 , . . . , n \}$ , the generator net can be trained by maximizing the loglikelihood $\begin{array} { r } { L _ { \mathcal { G } } ( W _ { \mathcal { G } } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \log P _ { \mathcal { G } } ( Y _ { i } ; W _ { \mathcal { G } } ) } \end{array}$ . For large sample, the learned $W _ { \mathcal { G } }$ minimizes the Kullback-Leibler divergence $\mathrm { K L } ( P _ { \mathrm { d a t a } } | P _ { \mathcal { G } } )$ from the data distribution $P _ { \mathrm { d a t a } }$ to the model distribution $P _ { \mathcal { G } }$ . The gradient of $L _ { \mathcal { G } } ( W _ { \mathcal { G } } )$ is obtained according to the following identity + +$$ +\begin{array} { r c l } { \displaystyle \frac { \partial } { \partial W _ { \mathcal { G } } } \log P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ) } & { = } & { \displaystyle \frac { 1 } { P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ) } \frac { \partial } { \partial W _ { \mathcal { G } } } \int P _ { \mathcal { G } } ( Y , X ; W _ { \mathcal { G } } ) d X } \\ & { = } & { \displaystyle \frac { 1 } { P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ) } \int \left[ \frac { \partial } { \partial W _ { \mathcal { G } } } \log P _ { \mathcal { G } } ( Y , X ; W _ { \mathcal { G } } ) \right] P _ { \mathcal { G } } ( Y , X ; W _ { \mathcal { G } } ) d X } \\ & { = } & { \displaystyle \int \left[ \frac { \partial } { \partial W _ { \mathcal { G } } } \log P _ { \mathcal { G } } ( Y , X ; W _ { \mathcal { G } } ) \right] \frac { P _ { \mathcal { G } } ( Y , X ; W _ { \mathcal { G } } ) } { P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ) } d X } \\ & { = } & { \displaystyle \mathrm { E } _ { P _ { \mathcal { G } } ( X \mid Y ; W _ { \mathcal { G } } ) } \left[ \frac { \partial } { \partial W _ { \mathcal { G } } } \log P _ { \mathcal { G } } ( X , Y ; W _ { \mathcal { G } } ) \right] , } \end{array} +$$ + +which underlies the EM algorithm. In general, the expectation in (8) is analytically intractable, and has to be approximated by MCMC that samples from the posterior $P _ { G } ( X | Y ; W _ { \mathcal { G } } )$ , such as Langevin dynamics, which iterates + +$$ +X _ { \tau + 1 } = X _ { \tau } + \frac { \delta ^ { 2 } } { 2 } \frac { \partial } { \partial X } \log P _ { \mathcal { G } } ( X _ { \tau } , Y ; W _ { \mathcal { G } } ) + \delta U _ { \tau } , +$$ + +where $U _ { \tau } \sim \mathrm { N } ( 0 , I _ { d } )$ . With $X _ { i }$ sampled from $P g ( X _ { i } \mid Y _ { i } , W g )$ for each observation $Y _ { i }$ , the Monte Carlo approximation to $L _ { \mathcal { G } } ^ { \prime } ( W _ { \mathcal { G } } )$ is + +$$ +L _ { \mathcal { G } } ^ { \prime } ( W _ { \mathcal { G } } ) \approx \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { \partial } { \partial W _ { \mathcal { G } } } \log P _ { \mathcal { G } } ( X _ { i } , Y _ { i } ; W _ { \mathcal { G } } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { 1 } { \sigma ^ { 2 } } \left( Y _ { i } - g ( X _ { i } ; W _ { \mathcal { G } } ) \right) \frac { \partial } { \partial W _ { \mathcal { G } } } g ( X _ { i } ; W _ { \mathcal { G } } ) . +$$ + +Algorithm G (Han et al., 2017) iterates the following two steps after initializing $W _ { \mathcal { G } }$ and $\{ X _ { i } , i =$ $1 , . . . , n \}$ . Step $G I$ : run $l _ { \mathcal { G } }$ steps of Langevin from the current $\{ X _ { i } \}$ according to (9). Step $G 2$ : update $\boldsymbol { W _ { \mathcal { G } } ^ { ( t + 1 ) } } = \boldsymbol { W } _ { \mathcal { G } } ^ { ( t ) } + \gamma _ { t } \boldsymbol { L } _ { \mathcal { G } } ^ { \prime } ( \boldsymbol { W } _ { \mathcal { G } } ^ { ( t ) } )$ with learning rate $\gamma _ { t }$ . The convergence of such an algorithm follows Younes (1999). + +# 4 COOPNETS ALGORITHM: RECONSTRUCTING THE REVISION + +In Algorithms $\mathrm { D }$ and $\mathbf { G }$ , both steps D1 and G1 are Langevin dynamics, which may be slow to converge. An interesting observation is that the two algorithms can cooperate with each other by jumpstarting each other’s Langevin sampling. + +Specifically, in Step D1, we can initialize the synthesized examples by generating examples from the generator net. We first generate $\hat { X } _ { i } \sim \mathrm { N } ( 0 , I _ { d } )$ , and then generate $\bar { \hat { Y } } _ { i } = g ( \bar { \hat { X } } _ { i } ; W _ { \mathcal { G } } ) \bar { + } \epsilon _ { i }$ , for $i = 1 , . . . , \tilde { n }$ . If the current generator $P _ { \mathcal { G } }$ is close to the current descriptor $P _ { \mathcal { D } }$ , then the generated $\{ \hat { Y } _ { i } \}$ should be a good initialization for sampling from the descriptor net, i.e., starting from the $\{ \hat { Y } _ { i } , i = 1 , . . . , \tilde { n } \}$ , we run Langevin dynamics in Step D1 for $l _ { \mathcal { D } }$ steps to get $\{ \tilde { Y } _ { i } , i = 1 , . . . , \tilde { n } \}$ , which are revised versions of $\{ \hat { Y } _ { i } \}$ . These $\{ \tilde { Y } _ { i } \}$ can be used as the synthesized examples from the descriptor net. We can then update $W _ { \mathcal { D } }$ according to Step D2 of Algorithm D. + +In order to update $W _ { \mathcal { G } }$ of the generator net, we treat the $\{ \tilde { Y } _ { i } , i = 1 , . . . , \tilde { n } \}$ produced by the above Step D1 as the training data for the generator. Since these $\{ \tilde { Y } _ { i } \}$ are obtained by the Langevin dynamics initialized from the $\{ \hat { Y } _ { i } , i = 1 , . . . , \tilde { n } \}$ produced by the generator net with known latent factors $\{ \hat { X } _ { i } , i = 1 , . . . , \tilde { n } \}$ , we can update $W _ { \mathcal { G } }$ by learning from $\{ ( \tilde { Y } _ { i } , \hat { X } _ { i } ) , i = 1 , . . . , \tilde { n } \}$ , which is a supervised learning problem, or more specifically, a non-linear regression of $\tilde { Y } _ { i }$ on ${ \hat { X } } _ { i }$ . At $W _ { \mathcal { G } } ^ { ( t ) }$ , the latent factors ${ \hat { X } } _ { i }$ generates and thus reconstructs the initial example $\hat { Y } _ { i }$ . After updating $W _ { \mathcal { G } }$ , we want ${ \hat { X } } _ { i }$ to reconstruct the revised example $\tilde { Y _ { i } }$ . That is, we revise $W _ { \mathcal { G } }$ to absorb the revision from ${ \hat { Y } } _ { i }$ to $\tilde { Y _ { i } }$ , so that the generator shifts its density from $\{ \hat { Y } _ { i } \}$ to $\{ \tilde { Y } _ { i } \}$ . The reconstruction error can tell us whether the generator has caught up with the descriptor by fully absorbing the revision. + +The left diagram in (11) illustrates the basic idea. + +$$ + \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & \end{array} +$$ + +In the two diagrams in (11), the double-line arrows indicate generation and reconstruction by the generator net, while the dashed-line arrows indicate Langevin dynamics for revision and inference in the two nets. The diagram on the right in (11) illustrates a more rigorous method, where we initialize the Langevin inference of $\{ X _ { i } , i = 1 , . . . , \tilde { n } \}$ in Step G1 from $\{ \hat { X } _ { i } \}$ , and then update $W _ { \mathcal { G } }$ in Step G2 based on $\{ ( \tilde { Y } _ { i } , X _ { i } ) , i = \bar { 1 } , . . . , \tilde { n } \}$ . The diagram on the right shows how the two nets jumpstart each other’s Langevin dynamics. + +Algorithm 1 describes the cooperative training that interweaves Algorithm D and Algorithm G. See Figure 1(c) for the flowchart of the CoopNets algorithm. In our experiments, we set $l _ { \mathcal { G } } = 0$ and infer $X _ { i } = \hat { X } _ { i }$ for simplicity, i.e., we follow the left diagram in (11). + +See Appendix for a theoretical understanding of the convergence of the CoopNets algorithm. + +# Algorithm 1 CoopNets Algorithm + +# + +(1) training examples $\{ Y _ { i } , i = 1 , . . . , n \}$ (2) numbers of Langevin steps $l _ { \mathcal { D } }$ ad $l _ { \mathcal { G } }$ (3) number of learning iterations $T$ + +# + +(1) estimated parameters $W _ { \mathcal { D } }$ and $W _ { \mathcal { G } }$ (2) synthesized examples $\{ \hat { Y } _ { i } , \tilde { Y } _ { i } , i = 1 , . . . , \tilde { n } \}$ + +1: Let $t \gets 0$ , initialize $W _ { \mathcal { D } }$ and $W _ { \mathcal { G } }$ + +2: repeat + +3: Step G0: For $i = 1 , . . . , \tilde { n }$ , generate $\hat { X } _ { i } \sim \mathrm { N } ( 0 , I _ { d } )$ , and generate $\hat { Y } _ { i } = g ( \hat { X } _ { i } ; W _ { \mathcal { G } } ^ { ( t ) } ) + \epsilon _ { i }$ + +4: Step D1: For $i = 1 , . . . , \tilde { n }$ , starting from $\hat { Y _ { i } }$ , Run $l _ { \mathcal { D } }$ steps of Langevin dynamics to obtain $\tilde { Y _ { i } }$ , each step following equation (4). + +5: Step G1: Treat the current $\{ \tilde { Y } _ { i } , i = 1 , . . . , \tilde { n } \}$ as the training data, for each $i$ , infer $X _ { i } = { \hat { X } } _ { i }$ . Or more rigorously, starting from $X _ { i } = { \hat { X } } _ { i }$ , run $l _ { \mathcal { G } }$ steps of Langevin dynamics to update $X _ { i }$ , each step following equation (9). + +6: Step D2: Update $\begin{array} { r } { \dot { W } _ { \mathcal { D } } ^ { ( \bar { t } + 1 ) } = W _ { \mathcal { D } } ^ { ( t ) } + \gamma _ { t } L _ { \mathcal { D } } ^ { \prime } ( W _ { \mathcal { D } } ^ { ( t ) } ) } \end{array}$ , where $L _ { \mathcal { D } } ^ { \prime } ( W _ { \mathcal { D } } ^ { ( t ) } )$ is computed according + +7: Step G2: Update $\boldsymbol { W _ { \mathcal { G } } ^ { ( t + 1 ) } } = \boldsymbol { W _ { \mathcal { G } } ^ { ( t ) } } + \gamma _ { t } \boldsymbol { L _ { \mathcal { G } } } ^ { \prime } ( \boldsymbol { W _ { \mathcal { G } } ^ { ( t ) } } )$ where $L _ { \mathcal { G } } ^ { \prime } ( W _ { G } )$ is computed according $Y _ { i }$ is replaced by $\tilde { Y _ { i } }$ , and $n$ by $\tilde { n }$ + +8: Let $t \gets t + 1$ +9: until $t = T$ + +# 5 EXPERIMENTS + +We use the MatConvNet of Vedaldi & Lenc (2015) for coding. For the descriptor net, we adopt the structure of Xie et al. (2016), where the bottom-up network consists of multiple layers of convolution by linear filtering, ReLU non-linearity, and down-sampling. We adopt the structure of the generator network of Radford et al. (2015); Dosovitskiy et al. (2015), where the top-down network consists of multiple layers of deconvolution by linear superposition, ReLU non-linearity, and up-sampling, with tanh non-linearity at the bottom-layer (Radford et al., 2015) to make the signals fall within $[ - 1 , 1 ]$ . + +# 5.1 QUANTITATIVE EXPERIMENT ON FACE COMPLETION + +We conduct an experiment on learning from complete training images of human faces, and then testing the learned model on completing the occluded testing images. The structure of the generator network is the same as in (Radford et al., 2015; Dosovitskiy et al., 2015). We adopt a 4-layer descriptor net. The first layer has $9 6 5 \times 5$ filters with sub-sampling of 2, the second layers has $1 2 8 \mathrm { ~ 5 ~ } \times \mathrm { ~ 5 ~ }$ filters with sub-sampling of 2, the third layer has $2 5 6 ~ 5 ~ \times ~ 5$ filters with sub-sampling of 2, and the final layer is a fully connected layer with 50 channels as output. We use $L { = } 1 0$ steps of Langevin revision dynamics within each learning iteration, and the Langevin step size is set at 0.002. The learning rate is 0.07. The training data are 10, 000 human faces randomly selected from CelebA dataset (Liu et al., 2015). We run 600 cooperative learning iterations. Figure 2 displays 144 synthesized human faces by the descriptor net. + +To quantitatively test whether we have learned a good generator net $g ( X ; W _ { \mathcal { G } } )$ even though it has never seen the training images directly in the training stage, we apply it to the task of recovering the occluded pixels of testing images. For each occluded testing image $Y$ , we use Step G1 of Algorithm G to infer the latent factors $X$ . The only change is with respect to the term $\| Y - g ( X ; W _ { \mathcal { G } } ) \| ^ { 2 }$ , where the sum of squares is over all the observed pixels of $Y$ in back-propagation computation. We run 1000 Langevin steps, initializing $X$ from $\mathrm { N } ( 0 , I _ { d } )$ . After inferring $X$ , the completed image $g ( X ; W _ { \mathcal { G } } )$ is automatically obtained. We design 3 experiments, where we randomly place a $2 0 \times 2 0$ , $3 0 \times 3 0$ , or $4 0 \times 4 0$ mask on each $6 4 \times 6 4$ testing image. These 3 experiments are denoted by $\mathbf { M } 2 0 \ \mathbf { M } 3 0$ , and M40 respectively (M for mask). We report the recovery errors and compare our method with 8 different image inpainting methods as well as the DCGAN of Radford et al. (2015). + +![](images/00dee8a7ae0f21424076c4f3c1068bafffe2b10b9e9f036ee0ad57ae66f5d096.jpg) +Figure 2: Generating human face pattern. The synthesized images are generated by the CoopNets algorithm that learns from 10, 000 images. + +For DCGAN, we use the parameter setting in Radford et al. (2015) except changing the number of learning iterations to 600. We use the same 10, 000 training images to learn DCGAN. After the model is learned, we keep the generator and use the same method as ours to infer latent factors $X$ , and recover the unobserved pixels. In 8 inpainting methods, Methods 1 and 2 are based on Markov random field prior where the nearest neighbor potential terms are $\ell _ { 2 }$ and $\ell _ { 1 }$ differences respectively. Methods 3 to 8 are interpolation methods. Please refer to D’Errico (2004) for more details. Table 1 displays the recovery errors of the 3 experiments, where the error is measured by per pixel difference (relative to the range of pixel values) between the original image and the recovered image on the occluded region, averaged over 100 testing images. Fig. 3 displays some recovery results by our method. The first row shows the original images as the ground truth. The second row displays the testing images with occluded pixels. The third row displays the recovered images by the generator net trained by the CoopNets algorithm on the 10,000 training images. + +Table 1: Comparison of recovery errors among different inpainting methods in 3 experiments + +
ExpOursGAN12345678
M20.0966.2535.1545.1506.1277.1123.2493.1123.1126.1277
M30.1112.2606.1820.1792.1679.1321.3367.1310.1312.1679
M40.1184.2618.2055.2032.1894.1544.3809.1525.1526.1894
+ +![](images/0f85ee69c8caee33ddcb82bc8eacc824192883782a133aa5e14a7314a197032a.jpg) +Figure 3: Row 1: ground-truth images. Row 2: testing images with occluded pixels. Row 3: recovered images by our method. + +# 5.2 QUALITATIVE EXPERIMENT ON SYNTHESIS + +We conduct an experiment on synthesizing images of categories from Imagenet ILSVRC2012 dataset (Deng et al., 2009) and MIT places205 dataset (Zhou et al., 2014). We adopt a 4-layer descriptor net. The first layer has $6 4 ~ 5 \times 5$ filters with sub-sampling of 2, the second layers has $1 2 8 3 \times 3$ filters with sub-sampling of 2, the third layer has $2 5 6 3 \times 3$ filters with sub-sampling of 1, and the final layer is a fully connected layer with 100 channels as output. We set the number of Langevin dynamics steps in each learning iteration to 10 and the step size to 0.002. The learning rate is 0.07. For each category, we randomly choose 1,000 images as training data and resize the images to $6 4 \times 6 4$ . We run 1, 000 cooperative learning iterations to train the model. Figures 4 and 5 display the results for two categories, where for each category, we show 144 original images sampled from the training set, and 144 synthesized images generated by our method. The appendix contains more synthesis results. + +As a comparison, we apply the Algorithm G alone and GAN code on the same 1,000 hotel room training images to learn the generator of the same structure as in CoopNets. Figure 6 displays the synthesis results. + +We also try to synthesize images at high resolution $( 2 2 4 \times 2 2 4 )$ ). We adopt a 4-layer descriptor net. The first layer has $1 2 8 1 5 \times 1 5$ filters with sub-sampling of 3, the second layer has $2 5 6 3 \times 3$ filters with sub-sampling of 2, the third layer has $5 1 2 3 \times 3$ filters with sub-sampling of 1, and the final layer is a fully connected layer with 100 channels as output. We enlarge the filters of the final layer of generator net to $1 4 \times 1 4$ to generate $2 2 4 \times 2 2 4$ images. The learning rate is 0.05. We run 1000 cooperative learning iterations to train the model. Figures 7 and 8 show the synthesized images of two categories from MIT places205 dataset. + +# 6 CONCLUSION + +The most unique feature of our work is that the two networks feed each other the synthesized data in the learning process, including initial, revised, and reconstructed synthesized data. + +Another unique feature of our work is that the learning process interweaves the existing maximum likelihood learning algorithms for the two networks. + +A third unique feature of our work is that the MCMC for the descriptor keeps rejuvenating the chains by refreshing the samples by independent replacements supplied by the generator, so that a single chain effectively amounts to an infinite number of chains or the evolution of the whole marginal distribution modeled by the generator. + +# CODE AND DATA + +http://www.stat.ucla.edu/˜ywu/CoopNets/main.html + +# 7 APPENDIX: CONVERGENCE + +# 7.1 GENERATOR OF INFINITE CAPACITY + +In the CoopNets algorithm, the descriptor learns from the observed examples, while the generator learns from the descriptor through the synthesized examples. Therefore, the descriptor is the driving force in terms of learning, although the generator is the driving force in terms of synthesis. In order to understand the convergence of learning, we can start from Algorithm D for learning the descriptor. + +Algorithm D is a stochastic approximation algorithm (Robbins & Monro, 1951), except that the samples are generated by finite step MCMC transitions. According to Younes (1999), Algorithm D converges to the maximum likelihood estimate under suitable regularity conditions on the mixing of the transition kernel of the MCMC and the schedule of the learning rate $\gamma _ { t }$ , even if the number of Langevin steps $l _ { \mathcal { D } }$ is finite or small (e.g., $l _ { \mathcal { D } } = 1 .$ ), and even if the number of parallel chains $\tilde { n }$ is finite or small (e.g., $\tilde { n } = 1$ ). The reason is that the random fluctuations caused by the finite number of chains, $\tilde { n }$ , and the limited mixing caused by the finite steps of MCMC, $l _ { \mathcal { D } }$ , are mitigated if the learning rate is sufficiently small. At learning iteration $t$ , let $W _ { \mathcal { D } } ^ { ( t ) }$ be the estimated parameter of D i D because lD is finite (P (t+1)D = PD(Y ; W (t)D ) if lD → ∞), we still have W (t)D the descriptor. Let $P _ { \mathcal { D } } ^ { ( t + 1 ) }$ be the marginal distribution of $\{ \tilde { Y } _ { i } \}$ D . Even though $W _ { \mathcal { D } } ^ { ( t ) } \hat { W } _ { \mathcal { D } }$ $P _ { \mathcal { D } } ^ { ( t + 1 ) } \neq P _ { \mathcal { D } } ( Y ; W _ { \mathcal { D } } ^ { ( t ) } )$ in probability according to Younes (1999), where $\hat { W } _ { \mathcal { D } }$ is the maximum likelihood estimate of $W _ { \mathcal { D } }$ . + +![](images/d4240e8ec86be0e776562781d3937712ce9947ec86ce6287e18cf6909f78c0b8.jpg) +(a) Original images + +![](images/27375fd3fc42fded0524ee180cc0241c3cddcd2cd4adbe32a593ef4295b0b3a4.jpg) +Figure 4: Generating forest road images. The category is from MIT places205 dataset. + +![](images/3b814fea1df0f26d03e177841edda77251a006c1d205523349743b71c3f8a821.jpg) +(a) Original images + +![](images/92c52da281e005d4ec4e61edf5d120b84b1ec7988226271a673f3d42cc16f074.jpg) +Figure 5: Generating hotel room images. The category is from MIT places205 dataset. + +![](images/6fb76193e697b5c26e40eed9a8fe339173923b9204009b88d0ee14072fa89502.jpg) +(a) Generated by Algorithm G alone. + +![](images/5e1e7b4b4e9b634c5fd2e2c24a30124a39d196894b57ce504d0eb3868ce4b020.jpg) +Figure 6: Generating hotel room images by Algorithm G alone and by GAN. + +![](images/9da43a4b287c88e80fbcf6b14085c1dd16285443159635e7c180a39e11c150b9.jpg) +Figure 7: Generating forest road images at high resolution $( 2 2 4 \times 2 2 4 )$ . + +![](images/66128d06a3eef868dcfe60224334a982f3e70707b3795c8172556dafa82e917e.jpg) +Figure 8: Generating hotel room images at high resolution $( 2 2 4 \times 2 2 4 )$ . + +The efficiency of Algorithm D increases if the number of parallel chains $\tilde { n }$ is large because it leads to more accurate estimation of the expectation in the gradient $L _ { \mathcal { D } } ^ { \prime } ( W _ { \mathcal { D } } )$ of equation (3), so that we can afford to use larger learning rate $\gamma _ { t }$ for faster convergence. + +Now let us come back to the CoopNets algorithm. In order to understand how the descriptor net helps the training of the generator net, let us consider the idealized scenario where the number of parallel chains $\tilde { n } \infty$ , and the generator has infinite capacity, and in each iteration it estimates $W _ { \mathcal { G } }$ by maximum likelihood using the synthesized data from P (t+1)D . In this idealized scenario, the learned generator $P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ^ { ( t + 1 ) } )$ will reproduce $P _ { \mathcal { D } } ^ { ( t + 1 ) }$ by minimizing $\operatorname { K L } ( P _ { \mathcal { D } } ^ { ( t + 1 ) } ( Y ) | P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ) )$ , with $P _ { \mathcal { D } } ^ { ( t + 1 ) }$ serving as its data distribution. Then eventually the learned generator $P _ { \mathcal { G } } ( Y , \hat { W } _ { \mathcal { G } } )$ will reproduce $P _ { \mathcal { D } } ( Y ; \hat { W } _ { \mathcal { D } } )$ . Thus the cooperative training helps the learning of the generator. Note that the learned generator $P _ { \mathcal { G } } ( Y , \hat { W } _ { \mathcal { G } } )$ will not reproduce the distribution of the observed data $P _ { \mathrm { d a t a } }$ , unless the descriptor is of infinite capacity too. + +Conversely, the generator net also helps the learning of the descriptor net in the CoopNets algorithm. In Algorithm D, it is impractical to make the number of parallel chains $\tilde { n }$ too large. On the other hand, it would be difficult for a small number of chains $\{ \bar { \tilde { Y } _ { i } } , i = 1 , . . . , \tilde { n } \}$ to explore the state space. In the CoopNets algorithm, because $P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ^ { ( t ) } )$ reproduces $P _ { \mathcal { D } } ^ { ( t ) }$ , we can generate a completely new batch of independent samples $\{ \hat { Y } _ { i } \}$ from $\overset { \cdot } { P _ { \mathcal { G } } } ( Y ; W _ { \mathcal { G } } ^ { ( t ) } )$ , and revise $\{ \hat { Y } _ { i } \}$ to $\{ \tilde { Y } _ { i } \}$ by Langevin dynamics, instead of running Langevin dynamics from the same old batch of $\{ \tilde { Y } _ { i } \}$ as in the original Algorithm D. This is like implementing an infinite number of parallel chains, because each iteration evolves a fresh batch of examples, as if each iteration evolves a new set of chains. By updating the generator $W _ { \mathcal { G } }$ , it is like we are updating the infinite number of parallel chains, because $W _ { \mathcal { G } }$ memorizes the whole distribution. Even if $\tilde { n }$ in the CoopNets algorithm is small, e.g., $\tilde { n } \ = \ 1$ , viewed from the perspective of Algorithm D, it is as if $\tilde { n } \to \infty$ . Thus the above idealization $\tilde { n } \to \infty$ is sound. + +# 7.2 GENERATOR OF FINITE CAPACITY + +From an information geometry point of view, let $\mathcal { D } = \{ P _ { \mathcal { D } } ( Y ; W _ { \mathcal { D } } ) , \forall W _ { \mathcal { D } } \}$ be the manifold of the descriptor models, where each distribution $P _ { \mathcal { D } } ( Y ; W _ { \mathcal { D } } )$ is a point on this manifold. Then the maximum likelihood estimate of $W _ { \mathcal { D } }$ is a projection of the data distribution $P _ { \mathrm { d a t a } }$ onto the manifold $\mathcal { D }$ . Let $\mathcal { G } = \{ P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ) , \forall W _ { \mathcal { G } } \}$ be the manifold of the generator models, where each distribution $P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } )$ is a point on this manifold. Then the maximum likelihood estimate of $W _ { \mathcal { G } }$ is a projection of the data distribution $P _ { \mathrm { d a t a } }$ onto the manifold $\mathcal { G }$ . + +From now on, for notational simplicity and with a slight abuse of notation, we use $W _ { \mathcal { D } }$ to denote the descriptor distribution $P _ { \mathcal { D } } ( Y ; W _ { \mathcal { D } } )$ , and use $W _ { \mathcal { G } }$ to denote the generator distribution $P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } )$ . + +We assume both the observed data size $n$ and the synthesized data size $\tilde { n }$ are large enough so that we shall work on distributions or populations instead of finite samples. As explained above, assuming $\tilde { n } \to \infty$ is sound because the generator net can supply unlimited number of examples. + +The Langevin revision dynamics runs a Markov chain from $W _ { \mathcal { G } } ^ { ( t ) }$ ) towards W (t)D . Let $\mathbf { L } _ { W _ { \mathcal { D } } }$ be the Markov transition kernel of $l _ { \mathcal { D } }$ steps of Langevin revisions towards $W _ { \mathcal { D } }$ . The distribution of the revised synthesized data is + +$$ +P _ { \mathcal { D } } ^ { ( t + 1 ) } = \mathbf { L } _ { W _ { \mathcal { D } } ^ { ( t ) } } \cdot W _ { \mathcal { G } } ^ { ( t ) } , +$$ + +where the notation $\mathbf { L } \cdot P$ denotes the marginal distribution obtained by running the Markov transition L from $P$ . The distribution $P _ { \mathcal { D } } ^ { ( t + 1 ) }$ is in the middle between the two nets $W _ { \mathcal { G } } ^ { ( t ) }$ ) and W (t)D , and it serves as the data distribution to train the generator, i.e., we project this distribution onto the manifold $\mathcal { G } = \{ P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ) , \forall W _ { \mathcal { G } } \} = \{ W _ { \mathcal { G } } \}$ (recall we use $W _ { \mathcal { G } }$ to denote the distribution $P _ { \mathcal { G } } ( Y ; W _ { \mathcal { G } } ) )$ in the + +information geometry picture, so that + +$$ +W _ { \mathcal { G } } ^ { ( t + 1 ) } = \arg \operatorname* { m i n } _ { \mathcal { G } } \mathrm { K L } ( P _ { \mathcal { D } } ^ { ( t + 1 ) } | W _ { \mathcal { G } } ) . +$$ + +The learning process alternates between Markov transition in (12) and projection in (13), as illustrated by Figure 9. + +![](images/b43b09ce1302d71cd85c8097eff4f0ee917cfe70c93d31247f6b45de0963750b.jpg) +Figure 9: The learning of the generator alternates between Markov transition and projection. The family of the generator models $\mathcal { G }$ is illustrated by the black curve. Each distribution is illustrated by a point. + +In the case of $l _ { \mathcal { D } } \infty$ + +$$ +\begin{array} { r l } & { W _ { \mathcal { D } } ^ { ( t ) } \to \hat { W } _ { \mathcal { D } } = \arg \underset { \mathcal { D } } { \operatorname* { m i n } } \mathrm { K L } ( P _ { \mathrm { d a t a } } | W _ { \mathcal { D } } ) , } \\ & { W _ { \mathcal { G } } ^ { ( t ) } \to \hat { W } _ { \mathcal { G } } = \arg \underset { \mathcal { G } } { \operatorname* { m i n } } \mathrm { K L } ( \hat { W } _ { \mathcal { D } } | W _ { \mathcal { G } } ) . } \end{array} +$$ + +That is, we first project $P _ { \mathrm { d a t a } }$ onto $\mathcal { D }$ , and from there continue to project onto $\mathcal { G }$ . Therefore, $W _ { \mathcal { D } }$ converges to the maximum likelihood estimate with $P _ { \mathrm { d a t a } }$ being the data distribution, while $W _ { \mathcal { G } }$ converges to the maximum likelihood estimate with $\hat { W _ { \mathcal { D } } }$ serving as the data distribution. + +For finite $l _ { \mathcal { D } }$ , the algorithm may converge to the following fixed points. The fixed point for the generator satisfies + +$$ +\hat { W } _ { \mathcal { G } } = \arg \operatorname* { m i n } _ { \mathcal { G } } \mathrm { K L } ( L _ { \hat { W } _ { \mathcal { D } } } \cdot \hat { W } _ { \mathcal { G } } | W _ { \mathcal { G } } ) . +$$ + +The fixed point for the descriptor satisfies + +$$ +\begin{array} { r l r } { \hat { W } _ { \mathcal D } } & { = } & { \arg \underset { \mathcal D } { \operatorname* { m i n } } \left[ \mathrm { K L } ( P _ { \mathrm { d a t a } } | W _ { \mathcal D } ) - \mathrm { K L } ( \mathbf L _ { \hat { W } _ { \mathcal D } } \cdot \hat { W } _ { \mathcal G } | W _ { \mathcal D } ) \right] , } \end{array} +$$ + +which is similar to contrastive divergence (Hinton, 2002), except that $\hat { W } _ { G }$ takes the place of $P _ { \mathrm { d a t a } }$ in the second Kullback-Leibler divergence. Because $\hat { W } _ { \mathcal G }$ is supposed to be close to $\hat { W } _ { \mathcal { D } }$ , the second Kullback-Leibler divergence is supposed to be small, hence our algorithm is closer to maximum likelihood learning than contrastive divergence. + +Kim & Bengio (2016) learned the generator by gradient descent on $\mathrm { K L } ( W _ { \mathcal { G } } | W _ { \mathcal { D } } ^ { ( t ) } )$ over $\mathcal { G }$ . The objective function is $\operatorname { K L } ( W _ { \mathcal { G } } | W _ { \mathcal { D } } ^ { ( t ) } ) = \operatorname { E } _ { W _ { \mathcal { G } } } [ \log P _ { G } ( Y ; W _ { \mathcal { G } } ) ] - \operatorname { E } _ { W _ { \mathcal { G } } } [ \log P _ { \mathcal { D } } ( Y ; W _ { \mathcal { D } } ^ { ( t ) } ) ]$ , where the first term is the negative entropy that is intractable, and the second term is the expected energy that is tractable. Our learning method for the generator is consistent with the learning objective $\mathrm { K L } ( W _ { \mathcal { G } } | W _ { \mathcal { D } } ^ { ( t ) } )$ , because + +$$ +\mathrm { K L } ( P _ { \mathcal { D } } ^ { ( t + 1 ) } | W _ { \mathcal { D } } ^ { ( t ) } ) \leq \mathrm { K L } ( W _ { \mathcal { G } } ^ { ( t ) } | W _ { \mathcal { D } } ^ { ( t ) } ) . +$$ + +In fact, $\mathrm { K L } ( P _ { \mathcal { D } } ^ { ( t + 1 ) } | W _ { \mathcal { D } } ^ { ( t ) } ) \to 0$ monotonically as $l _ { \mathcal { D } } \infty$ due to the second law of thermodynamics. The reduction of the Kullback-Leibler divergence in (18) and the projection in (13) in our learning of the generator are consistent with the learning objective of reducing $\mathrm { K L } ( W _ { \mathcal { G } } | W _ { \mathcal { D } } ^ { ( t ) } )$ in $\operatorname { K i m } \&$ Bengio (2016). But the Monte Carlo implementation of in our work avoids the need to approximate the intractable entropy term. + +# 7.3 MORE SYNTHESIS RESULTS + +We display more synthesis results at the resolution of $6 4 \times 6 4$ . + +![](images/fa57b10313cd350d46c64e652a218e577e9c5d82f42bf54e6cbda4051968c44b.jpg) +(a) Original images + +![](images/4344d87e0bc5dba658a5951713b369d4ebd3961a2be7cc7e3bef6fb9b8b0da35.jpg) +Figure 10: Generating swimming pool images. The category is from MIT places205 dataset. + +![](images/4f930503433b04f0f4f3a8d878fdf742a9e39069207fa6e40cbb81dd51708443.jpg) +(a) Original images + +![](images/643e35f770dfe5dc178ae9f9c9a07ba34b55669747d4d354cb29229f83ebe2cf.jpg) +Figure 11: Generating volcano images. The category is from MIT places205 dataset. + +![](images/c9545a3708bd3fb1fe08e41242a58e0180a661542fad4b2bdcd92a881bdb4175.jpg) +(a) Original images + +![](images/3e53647e608ce0c4acea2f61b7245b65ae4faaf262144fad822b943eda9ad142.jpg) +Figure 12: Generating rock images. The category is from MIT places205 dataset. + +![](images/b12cace848d065b0fa8a488392ab77b34a9746b62f0664058f2c3712139e8447.jpg) +(a) Original images + +![](images/5ee1f47ee99d367d33c8750e5f6afc134f8e3eb3ab8afa1c6349f1a4f3dcedae.jpg) +Figure 13: Generating desert images. The category is from MIT places205 dataset. + +![](images/c240c83a2ecf8e50df6b5c62bed9bc301a8fa3ed14365d75d5d059ea8c66eede.jpg) +Figure 14: Generating schoolbus images. The category is from Imagenet ILSVRC2012 1000 object categories. + +![](images/84da1ea2ce2aa8bb875b9475ae90c024adb01c0be4582788db870e1abbee7bca.jpg) +(a) Original images + +![](images/a8d36572a8199b6a19dcb1243b1e8da604cc228c9da6312c998be9f6ed69efbb.jpg) +Figure 15: Generating lifeboat images. The category is from Imagenet ILSVRC2012 1000 object categories. + +![](images/46d20f827362e767e77661eededb4fe10435504996bd25a3b41fa459497b2123.jpg) +(a) Original images + +![](images/e648eb571b9779541d94d339c142f7e168d3c07e852fd01745814cf2fbbbb81b.jpg) +Figure 16: Generating zebra images. The category is from Imagenet ILSVRC2012 1000 object categories. + +![](images/27711f6e1fdb6eb29d18e2f1525701702e0482d73d2d5267a3d0ec86c16756d5.jpg) +(a) Original images + +![](images/16832906f2ab1dce4cda5308ebb29a4000dea39e6fc33a2f320c55c689537a8c.jpg) +Figure 17: Generating strawberry images. The category is from Imagenet ILSVRC2012 1000 object categories. + +![](images/6d5cf44133ea0452bf4dfa5b37dabb9272e680f425069fb07a5732da64e15174.jpg) +Figure 18: Generating lemon images. The category is from Imagenet ILSVRC2012 1000 object categories. + +![](images/922628eee4c815b3f90fe271603a9b61b34ae36d5547fd0c779439a40bfa6723.jpg) +(a) Original images + +![](images/c07c2e08ef357ea34cea8c5bd5c0f147c0e503a97915274b1c7941023530fefd.jpg) +Figure 19: Generating apartment building images. The category is from Imagenet ILSVRC2012 1000 object categories. + +![](images/7217d8f4633ecbedd3c65d26e0b7c52899f72d7542e691c35e0c04a25cf095ca.jpg) +Figure 20: Generating dinning table images. The category is from Imagenet ILSVRC2012 1000 object categories. + +![](images/43d047ffdb94f8574ee56bff737c033f91dd1af046867e25f5900be4d091b51d.jpg) +Figure 21: Generating balloon images. The category is from Imagenet ILSVRC2012 1000 object categories. + +# ACKNOWLEDGEMENT + +We thank Hansheng Jiang for her work on this project as a summer visiting student. We thank Tian Han for sharing the code on learning the generator network, and for helpful discussions. + +The work is supported by NSF DMS 1310391, DARPA SIMPLEX N66001-15-C-4035, ONR MURI N00014-16-1-2007, and DARPA ARO W911NF-16-1-0579. + +# REFERENCES + +Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pp. 248–255. IEEE, 2009. + +Emily L Denton, Soumith Chintala, Rob Fergus, et al. Deep generative image models using a laplacian pyramid of adversarial networks. In Advances in Neural Information Processing Systems, pp. 1486–1494, 2015. + +John D’Errico. Interpolation inpainting, 2004. URL https://www.mathworks.com/ matlabcentral/fileexchange/4551-inpaint-nans. + +E Dosovitskiy, J. T. Springenberg, and T Brox. Learning to generate chairs with convolutional neural networks. In IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), 2015. + +Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems, pp. 2672–2680, 2014. + +Tian Han, Yang Lu, Song-Chun Zhu, and Ying Nian Wu. Alternating back-propagation for generator network. In 31st AAAI Conference on Artificial Intelligence, 2017. + +Geoffrey E Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14(8):1771–1800, 2002. + +Taesup Kim and Yoshua Bengio. Deep directed generative models with energy-based probability estimation. arXiv preprint arXiv:1606.03439, 2016. + +Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. ICLR, 2014. + +Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In NIPS, pp. 1097–1105, 2012. + +Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to ´ document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. + +Yann LeCun, Sumit Chopra, Rata Hadsell, Mare’Aurelio Ranzato, and Fu Jie Huang. A tutorial on energy-based learning. In Predicting Structured Data. MIT Press, 2006. + +Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In Proceedings of the IEEE International Conference on Computer Vision, pp. 3730–3738, 2015. + +Andriy Mnih and Karol Gregor. Neural variational inference and learning in belief networks. In ICML, 2014. + +Jiquan Ngiam, Zhenghao Chen, Pang Wei Koh, and Andrew Y. Ng. Learning deep energy models. In International Conference on Machine Learning, 2011. + +Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015. + +Danilo J. Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Tony Jebara and Eric P. Xing (eds.), ICML, pp. 1278–1286. JMLR Workshop and Conference Proceedings, 2014. + +Herbert Robbins and Sutton Monro. A stochastic approximation method. The annals of mathematical statistics, pp. 400–407, 1951. + +Yee Whye Teh, Max Welling, Simon Osindero, and Geoffrey E Hinton. Energy-based models for sparse overcomplete representations. Journal of Machine Learning Research, 4(Dec):1235–1260, 2003. + +A. Vedaldi and K. Lenc. Matconvnet – convolutional neural networks for matlab. In Proceeding of the ACM Int. Conf. on Multimedia, 2015. +Jianwen Xie, Yang Lu, Song-Chun Zhu, and Ying Nian Wu. A theory of generative convnet. In ICML, 2016. +Laurent Younes. On the convergence of markovian stochastic algorithms with rapidly decreasing ergodicity rates. Stochastics: An International Journal of Probability and Stochastic Processes, 65(3-4):177–228, 1999. +Bolei Zhou, Agata Lapedriza, Jianxiong Xiao, Antonio Torralba, and Aude Oliva. Learning deep features for scene recognition using places database. In Advances in neural information processing systems, pp. 487–495, 2014. +Song-Chun Zhu. Statistical modeling and conceptualization of visual patterns. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(6):691–712, 2003. +Song-Chun Zhu, Ying Nian Wu, and David Mumford. Minimax entropy principle and its application to texture modeling. Neural Computation, 9(8):1627–1660, 1997. \ No newline at end of file diff --git a/parse/train/B1E7Pwqgl/B1E7Pwqgl_content_list.json b/parse/train/B1E7Pwqgl/B1E7Pwqgl_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..9eef692793fc966220fa309bce17c41dbd20eb08 --- /dev/null +++ b/parse/train/B1E7Pwqgl/B1E7Pwqgl_content_list.json @@ -0,0 +1,2174 @@ +[ + { + "type": "text", + "text": "COOPERATIVE TRAINING OF DESCRIPTOR AND GEN-ERATOR NETWORKS", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jianwen Xie, Yang Lu, Ruiqi Gao, Song-Chun Zhu & Ying Nian Wu ", + "bbox": [ + 186, + 170, + 660, + 185 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Statistics \nUniversity of California, Los Angeles \n{jianwen, yanglv, ruiqigao}@ucla.edu, {sczhu, ywu}@stat.ucla.edu ", + "bbox": [ + 184, + 186, + 805, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 262, + 544, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This paper studies the cooperative training of two probabilistic models of signals such as images. Both models are parametrized by convolutional neural networks (ConvNets). The first network is a descriptor network, which is an exponential family model or an energy-based model, whose feature statistics or energy function are defined by a bottom-up ConvNet, which maps the observed signal to the feature statistics. The second network is a generator network, which is a nonlinear version of factor analysis. It is defined by a top-down ConvNet, which maps the latent factors to the observed signal. The maximum likelihood training algorithms of both the descriptor net and the generator net are in the form of alternating back-propagation, and both algorithms involve Langevin sampling. We observe that the two training algorithms can cooperate with each other by jumpstarting each other’s Langevin sampling, and they can be seamlessly interwoven into a CoopNets algorithm that can train both nets simultaneously. ", + "bbox": [ + 233, + 292, + 764, + 473 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 498, + 336, + 515 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1.1 TWO CONVNETS OF OPPOSITE DIRECTIONS ", + "text_level": 1, + "bbox": [ + 179, + 529, + 514, + 544 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We begin with a story that the reader of this paper can readily relate to. A student writes up an initial draft of a paper. His advisor then revises it. After that they submit the revised paper for review. The student then learns from his advisor’s revision, while the advisor learns from the outside review. In this story, the advisor guides the student, but the student does most of the work. ", + "bbox": [ + 174, + 555, + 823, + 611 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This paper is about two probabilistic models of signals such as images, and they play the roles of student and advisor as described above. Both models are parametrized by convolutional neural networks (ConvNets or CNNs) (LeCun et al., 1998; Krizhevsky et al., 2012). The two nets take two opposite directions. One is bottom-up, and the other is top-down, as illustrate by the following diagram: ", + "bbox": [ + 174, + 618, + 825, + 688 + ], + "page_idx": 0 + }, + { + "type": "table", + "img_path": "images/24b6527b1d0278282a4b710a40b45f0b9e043a5d9a2e104e0df3fcdb234effa0.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Bottom-up ConvNet featuresTop-down ConvNet latent variables
个←
signal (a) Descriptor Netsignal (b) Generator Net
", + "bbox": [ + 334, + 694, + 635, + 765 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The simultaneous training of such two nets was first studied by the recent work of Kim & Bengio (2016). These two nets belong to two major classes of probabilistic models. (a) The exponential family models or the energy-based models (LeCun et al., 2006) or the Markov random field models (Zhu et al., 1997), where the probability distribution is defined by feature statistics or energy function computed from the signal by a bottom-up process. (b) The latent variable models or the directed graphical models, where the signal is assumed to be a transformation of the latent factors that follow a known prior distribution. The latent factors generate the signal by a top-down process. A classical example is factor analysis. ", + "bbox": [ + 173, + 776, + 825, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The two classes of models have been contrasted by Zhu (2003); Teh et al. (2003); Ngiam et al. (2011). Zhu (2003) called the two classes of models the descriptive models and the generative models respectively. Both classes of models can benefit from the high capacity of the multi-layer ConvNets. (a) In the exponential family models or the energy-based models, the feature statistics or the energy function can be defined by a bottom-up ConvNet that maps the signal to the features and the energy function (Ngiam et al., 2011; Xie et al., 2016). We call the resulting model a descriptive network or a descriptor net following Zhu (2003), because it is built on descriptive feature statistics. (b) In the latent variable models or the directed graphical models, the transformation from the latent factors to the signal can be defined by a top-down ConvNet (Dosovitskiy et al., 2015), which maps the latent factors to the signal. We call the resulting model a generative network or generator net following Goodfellow et al. (2014), who proposed such a model in their work on the generative adversarial networks (GAN). ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/799da87ccece60114024503ad08a2702e46d67f851a2c657025c4fc10d8cf9f5.jpg", + "image_caption": [ + "Figure 1: (a) Algorithm D involves sampling from the current model by Langevin dynamics. (b) Algorithm G involves sampling from the posterior distribution of the latent factors by Langevin dynamics. (c) CoopNets algorithm. The part of the flowchart for training the descriptor is similar to Algorithm D, except that the D1 Langevin sampling is initialized from the initial synthesized examples supplied by the generator. The part of the flowchart for training the generator can also be mapped to Algorithm G, except that the revised synthesized examples play the role of the observed data, and the known generated latent factors can be used as inferred latent factors (or be used to initialize the G1 Langevin sampling of the latent factors). " + ], + "image_footnote": [], + "bbox": [ + 196, + 101, + 802, + 303 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 460, + 825, + 599 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.2 TWO TRAINING ALGORITHMS AND THEIR COOPERATION", + "text_level": 1, + "bbox": [ + 176, + 619, + 604, + 632 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Fig. 1(a) and (b) display the flowcharts of the maximum likelihood learning algorithms for training the descriptor and generator nets . We call the two algorithms Algorithm D and Algorithm G respectively. Algorithm D (Xie et al., 2016) iterates two steps: Step D1 synthesizes examples by sampling from the current model by Langevin dynamics. Step D2 updates the parameters to shift the density from the synthesized examples towards the observed examples. Algorithm G (Han et al., 2017) also iterates two steps. Step G1 infers latent factors for each observed example by sampling from their posterior distribution by Langevin dynamics. Step G2 updates the parameters by a non-linear regression of the observed examples on their corresponding latent factors. We use Langevin dynamics for Markov chain Monte Carlo (MCMC) sampling because the gradient term of Langevin dynamics can be readily computed via back-propagation. Thus all the steps D1, D2 and G1, G2 are powered by back-propagation, and both Algorithms D and G are alternating back-propagation algorithms. ", + "bbox": [ + 173, + 645, + 825, + 797 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this article, we propose to couple Algorithms D and G into a cooperative training algorithm that interweaves the steps of the two algorithms seamlessly. We call the resulting algorithm the CoopNets algorithm, and we show that it can train both nets simultaneously. ", + "bbox": [ + 176, + 805, + 820, + 847 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Figure 1(c) displays the flowchart of the CoopNets algorithm. The generator is like the student. It generates the initial draft of the synthesized examples. The descriptor is like the advisor. It revises the initial draft by initializing its Langevin dynamics from the initial draft in Step D1, which produces the revised draft of the synthesized examples. The descriptor learns from the outside review in Step D2, which is in the form of the difference between the observed examples and the revised synthesized examples. The generator learns from how the descriptor revises the initial draft by reconstructing the revised draft in Step G2. For each synthesized example, the generator knows the latent factors that generate the initial draft, so that Step G1 can infer the latent factors by initializing its Langevin dynamics from their known values. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the CoopNets algorithm, the generator fuels the MCMC of the descriptor by supplying initial synthesized examples, which can be obtained by direct ancestral sampling. The generator then learns from the revised synthesized examples with virtually known latent factors. The cooperation is thus beneficial to both nets. ", + "bbox": [ + 174, + 166, + 825, + 222 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 242, + 341, + 258 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our work is inspired by the generative adversarial networks (GAN) (Goodfellow et al., 2014; Denton et al., 2015; Radford et al., 2015). In GAN, the generator net is paired with a discriminator net. The two nets play adversarial roles. In our work, the generator net and the descriptor net play cooperative roles, and they feed each other the initial, revised and reconstructed synthesized data. The learning of both nets is based on maximum likelihood, and the learning process is quite stable because of the cooperative nature and the consistent directions of the two maximum likelihood training algorithms. ", + "bbox": [ + 174, + 273, + 825, + 358 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Another method to train the generator network is variational auto-encoder (VAE) (Kingma & Welling, 2014; Rezende et al., 2014; Mnih & Gregor, 2014), which learns an inferential or recognition network to approximate the posterior distribution of the latent factors. ", + "bbox": [ + 176, + 364, + 821, + 406 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The connection between the descriptor net and the discriminator net has been explored by Xie et al. \n(2016), where the descriptor can be derived from the discriminator. ", + "bbox": [ + 173, + 412, + 821, + 441 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our work is most similar to the recent work of Kim & Bengio (2016). In fact, the settings of the two nets are the same. In their work, the generator learns from the descriptor by minimizing the Kullback-Leibler divergence from the generator to the descriptor, which can be decomposed into an energy term and an entropy term. In our work, the two nets interact with each other via synthesized data, and the generator learns from the descriptor by reconstructing the revised draft of synthesized examples. Our method does not need to approximate the intractable entropy term. ", + "bbox": [ + 174, + 449, + 825, + 534 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our work is related to the contrastive divergence algorithm (Hinton, 2002) for training the descriptor net. The contrastive divergence initializes the MCMC sampling from the observed examples. The CoopNets algorithm initializes the MCMC sampling from the examples supplied by the generator. ", + "bbox": [ + 174, + 539, + 823, + 582 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 TWO NETS AND TWO TRAINING ALGORITHMS ", + "text_level": 1, + "bbox": [ + 174, + 602, + 584, + 618 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 DESCRIPTOR NET AND TRAINING ALGORITHM ", + "text_level": 1, + "bbox": [ + 174, + 633, + 535, + 647 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Let $Y$ be the $D$ -dimensional signal, such as an image. The descriptor model is in the form of exponential tilting of a reference distribution (Xie et al., 2016): ", + "bbox": [ + 174, + 659, + 823, + 688 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/ab13d106e7c44c34db04ec5b4fe0cea1fd15f71b3dcd0b1d0e3658c03ca790cd.jpg", + "text": "$$\nP _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } ) = \\frac { 1 } { Z ( W _ { \\mathcal { D } } ) } \\exp \\left[ f ( Y ; W _ { \\mathcal { D } } ) \\right] q ( Y ) ,\n$$", + "text_format": "latex", + "bbox": [ + 325, + 693, + 638, + 727 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $q ( Y )$ is the reference distribution such as Gaussian white noise $q ( Y ) \\propto \\exp \\left( - \\| Y \\| ^ { 2 } / 2 s ^ { 2 } \\right) ,$ $f ( Y ; W _ { \\mathcal { D } } )$ $( f$ stands for features) is the feature statistics or energy function, defined by a ConvNet whose parameters are denoted by $W _ { \\mathcal { D } }$ . This ConvNet is bottom-up because it maps the signal $Y$ to a number. See the diagram in (1). $\\begin{array} { r } { Z ( W _ { \\mathcal { D } } ) = \\int \\exp { [ f ( Y ; W _ { \\mathcal { D } } ) ] } \\bar { q } ( Y ) d Y = \\mathrm { E } _ { q } \\bar { \\left\\{ \\exp [ f ( \\bar { Y } ; W _ { \\mathcal { D } } ) ] \\right\\} } } \\end{array}$ is the normalizing constant, where $\\operatorname { E } _ { q }$ is the expectation with respect to $q$ . ", + "bbox": [ + 173, + 732, + 825, + 805 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Suppose we observe training examples $\\{ Y _ { i } , i ~ = ~ 1 , . . . , n \\}$ from an unknown data distribution $P _ { \\mathrm { d a t a } } ( Y )$ . The maximum likelihood training seeks to maximize the log-likelihood function $\\begin{array} { r } { L _ { \\mathcal { D } } ( W _ { \\mathcal { D } } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\log P _ { \\mathcal { D } } ( Y _ { i } ; W _ { \\mathcal { D } } ) } \\end{array}$ . If the sample size Kullback-Leibler di $n$ is large, the maximum likelihorgence from the data distribution $\\mathrm { K L } ( P _ { \\mathrm { d a t a } } | P _ { \\mathcal { D } } )$ $P _ { \\mathrm { d a t a } }$ to the model distribution $P _ { D }$ . The gradient of the $L _ { \\mathcal { D } } ( W _ { \\mathcal { D } } )$ is ", + "bbox": [ + 174, + 810, + 823, + 882 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8808962764fa6d5bc575b3a744e97d49b5950fdf30a4f2b1d0adb7896e2fa4fb.jpg", + "text": "$$\n\\begin{array} { r c l } { { { \\cal L } _ { \\mathcal D } ^ { \\prime } ( W _ { \\mathcal D } ) } } & { { = } } & { { \\displaystyle { \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\partial } { \\partial W _ { \\mathcal D } } f ( Y _ { i } ; W _ { \\mathcal D } ) - \\mathrm { E } _ { W _ { \\mathcal D } } \\left[ \\frac { \\partial } { \\partial W _ { \\mathcal D } } f ( Y ; W _ { \\mathcal D } ) \\right] } , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 887, + 725, + 928 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\operatorname { E } _ { W _ { \\mathcal { D } } }$ denotes the expectation with respect to $P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } )$ ", + "bbox": [ + 173, + 103, + 599, + 119 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The expectation in equation (3) is analytically intractable and has to be approximated by MCMC, such as Langevin dynamics, which iterates the following step: ", + "bbox": [ + 174, + 125, + 825, + 154 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/371b5514ff286662a87c852430bd000284424468e77d98454a0bd0e0ca1cbcf0.jpg", + "text": "$$\nY _ { \\tau + 1 } = Y _ { \\tau } - \\frac { \\delta ^ { 2 } } { 2 } \\left[ \\frac { Y _ { \\tau } } { s ^ { 2 } } - \\frac { \\partial } { \\partial Y } f ( Y _ { \\tau } ; W _ { \\mathcal { D } } ) \\right] + \\delta U _ { \\tau } ,\n$$", + "text_format": "latex", + "bbox": [ + 318, + 157, + 647, + 193 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\tau$ indexes the time steps of the Langevin dynamics, $\\delta$ is the step size, and $U _ { \\tau } \\sim \\mathrm { N } ( 0 , I _ { D } )$ is the Gaussian white noise term. ", + "bbox": [ + 174, + 196, + 825, + 226 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We can run $\\tilde { n }$ parallel chains of Langevin dynamics according to (4) to obtain the synthesized examples $\\{ \\tilde { Y } _ { i } , i = 1 , . . . , \\tilde { n } \\}$ . The Monte Carlo approximation to $L _ { \\mathcal { D } } ^ { \\prime } ( W _ { \\mathcal { D } } )$ is ", + "bbox": [ + 174, + 231, + 823, + 263 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/3559e4f1ac9e6f22a69b16183c211172921c8622827b8fb281b0ff7198f01eb2.jpg", + "text": "$$\nL _ { \\mathcal { D } } ^ { \\prime } ( W _ { \\mathcal { D } } ) \\quad \\approx \\quad \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\partial } { \\partial W _ { \\mathcal { D } } } f ( Y _ { i } ; W _ { \\mathcal { D } } ) - \\frac { 1 } { \\tilde { n } } \\sum _ { i = 1 } ^ { \\tilde { n } } \\frac { \\partial } { \\partial W _ { \\mathcal { D } } } f ( \\tilde { Y } _ { i } ; W _ { \\mathcal { D } } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 277, + 268, + 720, + 310 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "which is used to update $W _ { \\mathcal { D } }$ ", + "bbox": [ + 174, + 314, + 362, + 329 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Algorithm D (Xie et al., 2016) iterates the following two steps after initializing $W _ { \\mathcal { D } }$ and $\\{ \\tilde { Y } _ { i } , i =$ $1 , . . . , \\tilde { n } \\}$ . Step $D l$ : Run $l _ { \\mathcal { D } }$ steps of Langevin from the current $\\{ \\tilde { Y } \\}$ according according to (4). Step $D 2$ : update $\\boldsymbol { W _ { D } ^ { ( t + 1 ) } } = \\boldsymbol { W _ { D } ^ { ( t ) } } + \\gamma _ { t } \\boldsymbol { L _ { D } ^ { \\prime } } ( \\boldsymbol { W _ { D } ^ { ( t ) } } )$ with learning rate $\\gamma _ { t }$ . The convergence of such an algorithm follows Younes (1999). ", + "bbox": [ + 173, + 335, + 825, + 400 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 GENERATOR NET AND TRAINING ALGORITHM ", + "text_level": 1, + "bbox": [ + 178, + 416, + 529, + 430 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The generator net (Goodfellow et al., 2014) seeks to explain the signal $Y$ of dimension $D$ by a vector of latent factors $X$ of dimension $d$ , and usually $d \\ll D$ . The model is of the following form: ", + "bbox": [ + 174, + 440, + 823, + 470 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f1f58b077c53e4969220a720956cb7d4b8741c568776f15d8be5e2db5f968619.jpg", + "text": "$$\nX \\sim \\mathrm { N } ( 0 , I _ { d } ) , Y = g ( X ; W _ { \\mathcal { G } } ) + \\epsilon , \\epsilon \\sim \\mathrm { N } ( 0 , \\sigma ^ { 2 } I _ { D } ) .\n$$", + "text_format": "latex", + "bbox": [ + 338, + 474, + 692, + 492 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "$g ( X ; W _ { \\mathcal { G } } )$ $( g$ stands for generator) is a top-down ConvNet defined by the parameters $W _ { \\mathcal { G } }$ . The ConvNet $g$ maps the latent factors $X$ to the signal $Y$ . See the diagram in (1). ", + "bbox": [ + 173, + 496, + 831, + 525 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The joint density of model (6) is $P _ { \\mathcal { G } } ( X , Y ; W _ { \\mathcal { G } } ) = P _ { \\mathcal { G } } ( X ) P _ { \\mathcal { G } } ( Y | X ; W _ { \\mathcal { G } } )$ , and ", + "bbox": [ + 174, + 530, + 691, + 547 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/fb8fa4aaa81be9f3ac895c6a4c06f21fb08d5c80f1201153e1b7ce6be5477100.jpg", + "text": "$$\n\\log P _ { \\mathcal { G } } ( X , Y ; W _ { \\mathcal { G } } ) = - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\| Y - g ( X ; W _ { \\mathcal { G } } ) \\| ^ { 2 } - \\frac { 1 } { 2 } \\| X \\| ^ { 2 } + \\mathrm { c o n s t a n t } ,\n$$", + "text_format": "latex", + "bbox": [ + 251, + 551, + 746, + 582 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the constant term is independent of $X , Y$ and $W _ { \\mathcal { G } }$ . The marginal density is obtained by integrating out the latent factors $X$ , i.e., $\\begin{array} { r } { P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) = \\int P _ { \\mathcal { G } } ( X , Y ; W _ { \\mathcal { G } } ) d X } \\end{array}$ . The inference of $X$ given $Y$ is based on the posterior density $P _ { \\mathcal { G } } ( X | Y ; W _ { \\mathcal { G } } ) \\stackrel { \\smile } { = } P _ { \\mathcal { G } } ( X , Y ; W _ { \\mathcal { G } } ) / P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) \\propto P _ { \\mathcal { G } } ( X , Y ; W _ { \\mathcal { G } } )$ as a function of $X$ . ", + "bbox": [ + 173, + 583, + 825, + 640 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For the training data $\\{ Y _ { i } , i = 1 , . . . , n \\}$ , the generator net can be trained by maximizing the loglikelihood $\\begin{array} { r } { L _ { \\mathcal { G } } ( W _ { \\mathcal { G } } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\log P _ { \\mathcal { G } } ( Y _ { i } ; W _ { \\mathcal { G } } ) } \\end{array}$ . For large sample, the learned $W _ { \\mathcal { G } }$ minimizes the Kullback-Leibler divergence $\\mathrm { K L } ( P _ { \\mathrm { d a t a } } | P _ { \\mathcal { G } } )$ from the data distribution $P _ { \\mathrm { d a t a } }$ to the model distribution $P _ { \\mathcal { G } }$ . The gradient of $L _ { \\mathcal { G } } ( W _ { \\mathcal { G } } )$ is obtained according to the following identity ", + "bbox": [ + 173, + 646, + 825, + 704 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/4e65911487f910d916ecc7c522c8bc808f5430f398734f87f5b97fecd7e93be1.jpg", + "text": "$$\n\\begin{array} { r c l } { \\displaystyle \\frac { \\partial } { \\partial W _ { \\mathcal { G } } } \\log P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) } & { = } & { \\displaystyle \\frac { 1 } { P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) } \\frac { \\partial } { \\partial W _ { \\mathcal { G } } } \\int P _ { \\mathcal { G } } ( Y , X ; W _ { \\mathcal { G } } ) d X } \\\\ & { = } & { \\displaystyle \\frac { 1 } { P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) } \\int \\left[ \\frac { \\partial } { \\partial W _ { \\mathcal { G } } } \\log P _ { \\mathcal { G } } ( Y , X ; W _ { \\mathcal { G } } ) \\right] P _ { \\mathcal { G } } ( Y , X ; W _ { \\mathcal { G } } ) d X } \\\\ & { = } & { \\displaystyle \\int \\left[ \\frac { \\partial } { \\partial W _ { \\mathcal { G } } } \\log P _ { \\mathcal { G } } ( Y , X ; W _ { \\mathcal { G } } ) \\right] \\frac { P _ { \\mathcal { G } } ( Y , X ; W _ { \\mathcal { G } } ) } { P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) } d X } \\\\ & { = } & { \\displaystyle \\mathrm { E } _ { P _ { \\mathcal { G } } ( X \\mid Y ; W _ { \\mathcal { G } } ) } \\left[ \\frac { \\partial } { \\partial W _ { \\mathcal { G } } } \\log P _ { \\mathcal { G } } ( X , Y ; W _ { \\mathcal { G } } ) \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 707, + 797, + 848 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "which underlies the EM algorithm. In general, the expectation in (8) is analytically intractable, and has to be approximated by MCMC that samples from the posterior $P _ { G } ( X | Y ; W _ { \\mathcal { G } } )$ , such as Langevin dynamics, which iterates ", + "bbox": [ + 174, + 849, + 825, + 892 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/93c2f45595400fad814f9758ed16673f1eeac09f1a40d5dda5ebdb14b945b147.jpg", + "text": "$$\nX _ { \\tau + 1 } = X _ { \\tau } + \\frac { \\delta ^ { 2 } } { 2 } \\frac { \\partial } { \\partial X } \\log P _ { \\mathcal { G } } ( X _ { \\tau } , Y ; W _ { \\mathcal { G } } ) + \\delta U _ { \\tau } ,\n$$", + "text_format": "latex", + "bbox": [ + 313, + 896, + 650, + 929 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $U _ { \\tau } \\sim \\mathrm { N } ( 0 , I _ { d } )$ . With $X _ { i }$ sampled from $P g ( X _ { i } \\mid Y _ { i } , W g )$ for each observation $Y _ { i }$ , the Monte Carlo approximation to $L _ { \\mathcal { G } } ^ { \\prime } ( W _ { \\mathcal { G } } )$ is ", + "bbox": [ + 173, + 102, + 825, + 133 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/925aae53d9a306f25ca0bc7ac07f51e298b5d77c2eb5d4bd9718845cc5d25cae.jpg", + "text": "$$\nL _ { \\mathcal { G } } ^ { \\prime } ( W _ { \\mathcal { G } } ) \\approx \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { \\partial } { \\partial W _ { \\mathcal { G } } } \\log P _ { \\mathcal { G } } ( X _ { i } , Y _ { i } ; W _ { \\mathcal { G } } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { 1 } { \\sigma ^ { 2 } } \\left( Y _ { i } - g ( X _ { i } ; W _ { \\mathcal { G } } ) \\right) \\frac { \\partial } { \\partial W _ { \\mathcal { G } } } g ( X _ { i } ; W _ { \\mathcal { G } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 169, + 138, + 795, + 181 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm G (Han et al., 2017) iterates the following two steps after initializing $W _ { \\mathcal { G } }$ and $\\{ X _ { i } , i =$ $1 , . . . , n \\}$ . Step $G I$ : run $l _ { \\mathcal { G } }$ steps of Langevin from the current $\\{ X _ { i } \\}$ according to (9). Step $G 2$ : update $\\boldsymbol { W _ { \\mathcal { G } } ^ { ( t + 1 ) } } = \\boldsymbol { W } _ { \\mathcal { G } } ^ { ( t ) } + \\gamma _ { t } \\boldsymbol { L } _ { \\mathcal { G } } ^ { \\prime } ( \\boldsymbol { W } _ { \\mathcal { G } } ^ { ( t ) } )$ with learning rate $\\gamma _ { t }$ . The convergence of such an algorithm follows Younes (1999). ", + "bbox": [ + 173, + 193, + 825, + 253 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 COOPNETS ALGORITHM: RECONSTRUCTING THE REVISION ", + "text_level": 1, + "bbox": [ + 174, + 273, + 697, + 290 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In Algorithms $\\mathrm { D }$ and $\\mathbf { G }$ , both steps D1 and G1 are Langevin dynamics, which may be slow to converge. An interesting observation is that the two algorithms can cooperate with each other by jumpstarting each other’s Langevin sampling. ", + "bbox": [ + 176, + 303, + 825, + 347 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Specifically, in Step D1, we can initialize the synthesized examples by generating examples from the generator net. We first generate $\\hat { X } _ { i } \\sim \\mathrm { N } ( 0 , I _ { d } )$ , and then generate $\\bar { \\hat { Y } } _ { i } = g ( \\bar { \\hat { X } } _ { i } ; W _ { \\mathcal { G } } ) \\bar { + } \\epsilon _ { i }$ , for $i = 1 , . . . , \\tilde { n }$ . If the current generator $P _ { \\mathcal { G } }$ is close to the current descriptor $P _ { \\mathcal { D } }$ , then the generated $\\{ \\hat { Y } _ { i } \\}$ should be a good initialization for sampling from the descriptor net, i.e., starting from the $\\{ \\hat { Y } _ { i } , i = 1 , . . . , \\tilde { n } \\}$ , we run Langevin dynamics in Step D1 for $l _ { \\mathcal { D } }$ steps to get $\\{ \\tilde { Y } _ { i } , i = 1 , . . . , \\tilde { n } \\}$ , which are revised versions of $\\{ \\hat { Y } _ { i } \\}$ . These $\\{ \\tilde { Y } _ { i } \\}$ can be used as the synthesized examples from the descriptor net. We can then update $W _ { \\mathcal { D } }$ according to Step D2 of Algorithm D. ", + "bbox": [ + 173, + 352, + 825, + 460 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In order to update $W _ { \\mathcal { G } }$ of the generator net, we treat the $\\{ \\tilde { Y } _ { i } , i = 1 , . . . , \\tilde { n } \\}$ produced by the above Step D1 as the training data for the generator. Since these $\\{ \\tilde { Y } _ { i } \\}$ are obtained by the Langevin dynamics initialized from the $\\{ \\hat { Y } _ { i } , i = 1 , . . . , \\tilde { n } \\}$ produced by the generator net with known latent factors $\\{ \\hat { X } _ { i } , i = 1 , . . . , \\tilde { n } \\}$ , we can update $W _ { \\mathcal { G } }$ by learning from $\\{ ( \\tilde { Y } _ { i } , \\hat { X } _ { i } ) , i = 1 , . . . , \\tilde { n } \\}$ , which is a supervised learning problem, or more specifically, a non-linear regression of $\\tilde { Y } _ { i }$ on ${ \\hat { X } } _ { i }$ . At $W _ { \\mathcal { G } } ^ { ( t ) }$ , the latent factors ${ \\hat { X } } _ { i }$ generates and thus reconstructs the initial example $\\hat { Y } _ { i }$ . After updating $W _ { \\mathcal { G } }$ , we want ${ \\hat { X } } _ { i }$ to reconstruct the revised example $\\tilde { Y _ { i } }$ . That is, we revise $W _ { \\mathcal { G } }$ to absorb the revision from ${ \\hat { Y } } _ { i }$ to $\\tilde { Y _ { i } }$ , so that the generator shifts its density from $\\{ \\hat { Y } _ { i } \\}$ to $\\{ \\tilde { Y } _ { i } \\}$ . The reconstruction error can tell us whether the generator has caught up with the descriptor by fully absorbing the revision. ", + "bbox": [ + 173, + 467, + 826, + 613 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The left diagram in (11) illustrates the basic idea. ", + "bbox": [ + 174, + 619, + 496, + 635 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/8b8f6dc3118c57c451e86597a2fed5636fdc270f871726a561a4ab737ac2a702.jpg", + "text": "$$\n \\begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 642, + 665, + 752 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In the two diagrams in (11), the double-line arrows indicate generation and reconstruction by the generator net, while the dashed-line arrows indicate Langevin dynamics for revision and inference in the two nets. The diagram on the right in (11) illustrates a more rigorous method, where we initialize the Langevin inference of $\\{ X _ { i } , i = 1 , . . . , \\tilde { n } \\}$ in Step G1 from $\\{ \\hat { X } _ { i } \\}$ , and then update $W _ { \\mathcal { G } }$ in Step G2 based on $\\{ ( \\tilde { Y } _ { i } , X _ { i } ) , i = \\bar { 1 } , . . . , \\tilde { n } \\}$ . The diagram on the right shows how the two nets jumpstart each other’s Langevin dynamics. ", + "bbox": [ + 173, + 762, + 825, + 852 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1 describes the cooperative training that interweaves Algorithm D and Algorithm G. See Figure 1(c) for the flowchart of the CoopNets algorithm. In our experiments, we set $l _ { \\mathcal { G } } = 0$ and infer $X _ { i } = \\hat { X } _ { i }$ for simplicity, i.e., we follow the left diagram in (11). ", + "bbox": [ + 174, + 858, + 825, + 904 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "See Appendix for a theoretical understanding of the convergence of the CoopNets algorithm. ", + "bbox": [ + 168, + 909, + 779, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1 CoopNets Algorithm ", + "text_level": 1, + "bbox": [ + 176, + 103, + 401, + 118 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "text_level": 1, + "bbox": [ + 174, + 122, + 220, + 133 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "(1) training examples $\\{ Y _ { i } , i = 1 , . . . , n \\}$ (2) numbers of Langevin steps $l _ { \\mathcal { D } }$ ad $l _ { \\mathcal { G } }$ (3) number of learning iterations $T$ ", + "bbox": [ + 200, + 135, + 460, + 175 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "text_level": 1, + "bbox": [ + 173, + 176, + 232, + 189 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "(1) estimated parameters $W _ { \\mathcal { D } }$ and $W _ { \\mathcal { G } }$ (2) synthesized examples $\\{ \\hat { Y } _ { i } , \\tilde { Y } _ { i } , i = 1 , . . . , \\tilde { n } \\}$ ", + "bbox": [ + 200, + 190, + 506, + 220 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1: Let $t \\gets 0$ , initialize $W _ { \\mathcal { D } }$ and $W _ { \\mathcal { G } }$ ", + "bbox": [ + 183, + 233, + 424, + 247 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2: repeat ", + "bbox": [ + 183, + 248, + 250, + 261 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3: Step G0: For $i = 1 , . . . , \\tilde { n }$ , generate $\\hat { X } _ { i } \\sim \\mathrm { N } ( 0 , I _ { d } )$ , and generate $\\hat { Y } _ { i } = g ( \\hat { X } _ { i } ; W _ { \\mathcal { G } } ^ { ( t ) } ) + \\epsilon _ { i }$ ", + "bbox": [ + 181, + 262, + 797, + 279 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4: Step D1: For $i = 1 , . . . , \\tilde { n }$ , starting from $\\hat { Y _ { i } }$ , Run $l _ { \\mathcal { D } }$ steps of Langevin dynamics to obtain $\\tilde { Y _ { i } }$ , each step following equation (4). ", + "bbox": [ + 178, + 281, + 823, + 309 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5: Step G1: Treat the current $\\{ \\tilde { Y } _ { i } , i = 1 , . . . , \\tilde { n } \\}$ as the training data, for each $i$ , infer $X _ { i } = { \\hat { X } } _ { i }$ . Or more rigorously, starting from $X _ { i } = { \\hat { X } } _ { i }$ , run $l _ { \\mathcal { G } }$ steps of Langevin dynamics to update $X _ { i }$ , each step following equation (9). ", + "bbox": [ + 178, + 310, + 823, + 354 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "6: Step D2: Update $\\begin{array} { r } { \\dot { W } _ { \\mathcal { D } } ^ { ( \\bar { t } + 1 ) } = W _ { \\mathcal { D } } ^ { ( t ) } + \\gamma _ { t } L _ { \\mathcal { D } } ^ { \\prime } ( W _ { \\mathcal { D } } ^ { ( t ) } ) } \\end{array}$ , where $L _ { \\mathcal { D } } ^ { \\prime } ( W _ { \\mathcal { D } } ^ { ( t ) } )$ is computed according ", + "bbox": [ + 178, + 356, + 820, + 386 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "7: Step G2: Update $\\boldsymbol { W _ { \\mathcal { G } } ^ { ( t + 1 ) } } = \\boldsymbol { W _ { \\mathcal { G } } ^ { ( t ) } } + \\gamma _ { t } \\boldsymbol { L _ { \\mathcal { G } } } ^ { \\prime } ( \\boldsymbol { W _ { \\mathcal { G } } ^ { ( t ) } } )$ where $L _ { \\mathcal { G } } ^ { \\prime } ( W _ { G } )$ is computed according $Y _ { i }$ is replaced by $\\tilde { Y _ { i } }$ , and $n$ by $\\tilde { n }$ ", + "bbox": [ + 187, + 386, + 825, + 421 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "8: Let $t \\gets t + 1$ \n9: until $t = T$ ", + "bbox": [ + 179, + 421, + 310, + 448 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 477, + 326, + 493 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We use the MatConvNet of Vedaldi & Lenc (2015) for coding. For the descriptor net, we adopt the structure of Xie et al. (2016), where the bottom-up network consists of multiple layers of convolution by linear filtering, ReLU non-linearity, and down-sampling. We adopt the structure of the generator network of Radford et al. (2015); Dosovitskiy et al. (2015), where the top-down network consists of multiple layers of deconvolution by linear superposition, ReLU non-linearity, and up-sampling, with tanh non-linearity at the bottom-layer (Radford et al., 2015) to make the signals fall within $[ - 1 , 1 ]$ . ", + "bbox": [ + 174, + 508, + 825, + 593 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 QUANTITATIVE EXPERIMENT ON FACE COMPLETION", + "text_level": 1, + "bbox": [ + 174, + 613, + 575, + 626 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We conduct an experiment on learning from complete training images of human faces, and then testing the learned model on completing the occluded testing images. The structure of the generator network is the same as in (Radford et al., 2015; Dosovitskiy et al., 2015). We adopt a 4-layer descriptor net. The first layer has $9 6 5 \\times 5$ filters with sub-sampling of 2, the second layers has $1 2 8 \\mathrm { ~ 5 ~ } \\times \\mathrm { ~ 5 ~ }$ filters with sub-sampling of 2, the third layer has $2 5 6 ~ 5 ~ \\times ~ 5$ filters with sub-sampling of 2, and the final layer is a fully connected layer with 50 channels as output. We use $L { = } 1 0$ steps of Langevin revision dynamics within each learning iteration, and the Langevin step size is set at 0.002. The learning rate is 0.07. The training data are 10, 000 human faces randomly selected from CelebA dataset (Liu et al., 2015). We run 600 cooperative learning iterations. Figure 2 displays 144 synthesized human faces by the descriptor net. ", + "bbox": [ + 173, + 638, + 825, + 777 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To quantitatively test whether we have learned a good generator net $g ( X ; W _ { \\mathcal { G } } )$ even though it has never seen the training images directly in the training stage, we apply it to the task of recovering the occluded pixels of testing images. For each occluded testing image $Y$ , we use Step G1 of Algorithm G to infer the latent factors $X$ . The only change is with respect to the term $\\| Y - g ( X ; W _ { \\mathcal { G } } ) \\| ^ { 2 }$ , where the sum of squares is over all the observed pixels of $Y$ in back-propagation computation. We run 1000 Langevin steps, initializing $X$ from $\\mathrm { N } ( 0 , I _ { d } )$ . After inferring $X$ , the completed image $g ( X ; W _ { \\mathcal { G } } )$ is automatically obtained. We design 3 experiments, where we randomly place a $2 0 \\times 2 0$ , $3 0 \\times 3 0$ , or $4 0 \\times 4 0$ mask on each $6 4 \\times 6 4$ testing image. These 3 experiments are denoted by $\\mathbf { M } 2 0 \\ \\mathbf { M } 3 0$ , and M40 respectively (M for mask). We report the recovery errors and compare our method with 8 different image inpainting methods as well as the DCGAN of Radford et al. (2015). ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/00dee8a7ae0f21424076c4f3c1068bafffe2b10b9e9f036ee0ad57ae66f5d096.jpg", + "image_caption": [ + "Figure 2: Generating human face pattern. The synthesized images are generated by the CoopNets algorithm that learns from 10, 000 images. " + ], + "image_footnote": [], + "bbox": [ + 303, + 101, + 692, + 402 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For DCGAN, we use the parameter setting in Radford et al. (2015) except changing the number of learning iterations to 600. We use the same 10, 000 training images to learn DCGAN. After the model is learned, we keep the generator and use the same method as ours to infer latent factors $X$ , and recover the unobserved pixels. In 8 inpainting methods, Methods 1 and 2 are based on Markov random field prior where the nearest neighbor potential terms are $\\ell _ { 2 }$ and $\\ell _ { 1 }$ differences respectively. Methods 3 to 8 are interpolation methods. Please refer to D’Errico (2004) for more details. Table 1 displays the recovery errors of the 3 experiments, where the error is measured by per pixel difference (relative to the range of pixel values) between the original image and the recovered image on the occluded region, averaged over 100 testing images. Fig. 3 displays some recovery results by our method. The first row shows the original images as the ground truth. The second row displays the testing images with occluded pixels. The third row displays the recovered images by the generator net trained by the CoopNets algorithm on the 10,000 training images. ", + "bbox": [ + 173, + 474, + 825, + 641 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/ac9842e2fbe3369eb19f0e69e5a97e8ec7be8de531d48cabba6485068b461fea.jpg", + "table_caption": [ + "Table 1: Comparison of recovery errors among different inpainting methods in 3 experiments " + ], + "table_footnote": [], + "table_body": "
ExpOursGAN12345678
M20.0966.2535.1545.1506.1277.1123.2493.1123.1126.1277
M30.1112.2606.1820.1792.1679.1321.3367.1310.1312.1679
M40.1184.2618.2055.2032.1894.1544.3809.1525.1526.1894
", + "bbox": [ + 186, + 683, + 810, + 741 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/0f85ee69c8caee33ddcb82bc8eacc824192883782a133aa5e14a7314a197032a.jpg", + "image_caption": [ + "Figure 3: Row 1: ground-truth images. Row 2: testing images with occluded pixels. Row 3: recovered images by our method. " + ], + "image_footnote": [], + "bbox": [ + 305, + 772, + 691, + 868 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 QUALITATIVE EXPERIMENT ON SYNTHESIS ", + "text_level": 1, + "bbox": [ + 178, + 104, + 509, + 117 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We conduct an experiment on synthesizing images of categories from Imagenet ILSVRC2012 dataset (Deng et al., 2009) and MIT places205 dataset (Zhou et al., 2014). We adopt a 4-layer descriptor net. The first layer has $6 4 ~ 5 \\times 5$ filters with sub-sampling of 2, the second layers has $1 2 8 3 \\times 3$ filters with sub-sampling of 2, the third layer has $2 5 6 3 \\times 3$ filters with sub-sampling of 1, and the final layer is a fully connected layer with 100 channels as output. We set the number of Langevin dynamics steps in each learning iteration to 10 and the step size to 0.002. The learning rate is 0.07. For each category, we randomly choose 1,000 images as training data and resize the images to $6 4 \\times 6 4$ . We run 1, 000 cooperative learning iterations to train the model. Figures 4 and 5 display the results for two categories, where for each category, we show 144 original images sampled from the training set, and 144 synthesized images generated by our method. The appendix contains more synthesis results. ", + "bbox": [ + 173, + 130, + 825, + 282 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As a comparison, we apply the Algorithm G alone and GAN code on the same 1,000 hotel room training images to learn the generator of the same structure as in CoopNets. Figure 6 displays the synthesis results. ", + "bbox": [ + 176, + 289, + 821, + 332 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also try to synthesize images at high resolution $( 2 2 4 \\times 2 2 4 )$ ). We adopt a 4-layer descriptor net. The first layer has $1 2 8 1 5 \\times 1 5$ filters with sub-sampling of 3, the second layer has $2 5 6 3 \\times 3$ filters with sub-sampling of 2, the third layer has $5 1 2 3 \\times 3$ filters with sub-sampling of 1, and the final layer is a fully connected layer with 100 channels as output. We enlarge the filters of the final layer of generator net to $1 4 \\times 1 4$ to generate $2 2 4 \\times 2 2 4$ images. The learning rate is 0.05. We run 1000 cooperative learning iterations to train the model. Figures 7 and 8 show the synthesized images of two categories from MIT places205 dataset. ", + "bbox": [ + 174, + 338, + 825, + 436 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 457, + 318, + 473 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The most unique feature of our work is that the two networks feed each other the synthesized data in the learning process, including initial, revised, and reconstructed synthesized data. ", + "bbox": [ + 174, + 489, + 821, + 517 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Another unique feature of our work is that the learning process interweaves the existing maximum likelihood learning algorithms for the two networks. ", + "bbox": [ + 174, + 525, + 823, + 553 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "A third unique feature of our work is that the MCMC for the descriptor keeps rejuvenating the chains by refreshing the samples by independent replacements supplied by the generator, so that a single chain effectively amounts to an infinite number of chains or the evolution of the whole marginal distribution modeled by the generator. ", + "bbox": [ + 176, + 559, + 825, + 616 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "CODE AND DATA ", + "text_level": 1, + "bbox": [ + 176, + 637, + 313, + 654 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "http://www.stat.ucla.edu/˜ywu/CoopNets/main.html ", + "bbox": [ + 174, + 670, + 643, + 683 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 APPENDIX: CONVERGENCE ", + "text_level": 1, + "bbox": [ + 176, + 704, + 436, + 722 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7.1 GENERATOR OF INFINITE CAPACITY ", + "text_level": 1, + "bbox": [ + 176, + 737, + 462, + 751 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In the CoopNets algorithm, the descriptor learns from the observed examples, while the generator learns from the descriptor through the synthesized examples. Therefore, the descriptor is the driving force in terms of learning, although the generator is the driving force in terms of synthesis. In order to understand the convergence of learning, we can start from Algorithm D for learning the descriptor. ", + "bbox": [ + 174, + 763, + 823, + 819 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Algorithm D is a stochastic approximation algorithm (Robbins & Monro, 1951), except that the samples are generated by finite step MCMC transitions. According to Younes (1999), Algorithm D converges to the maximum likelihood estimate under suitable regularity conditions on the mixing of the transition kernel of the MCMC and the schedule of the learning rate $\\gamma _ { t }$ , even if the number of Langevin steps $l _ { \\mathcal { D } }$ is finite or small (e.g., $l _ { \\mathcal { D } } = 1 .$ ), and even if the number of parallel chains $\\tilde { n }$ is finite or small (e.g., $\\tilde { n } = 1$ ). The reason is that the random fluctuations caused by the finite number of chains, $\\tilde { n }$ , and the limited mixing caused by the finite steps of MCMC, $l _ { \\mathcal { D } }$ , are mitigated if the learning rate is sufficiently small. At learning iteration $t$ , let $W _ { \\mathcal { D } } ^ { ( t ) }$ be the estimated parameter of D i D because lD is finite (P (t+1)D = PD(Y ; W (t)D ) if lD → ∞), we still have W (t)D the descriptor. Let $P _ { \\mathcal { D } } ^ { ( t + 1 ) }$ be the marginal distribution of $\\{ \\tilde { Y } _ { i } \\}$ D . Even though $W _ { \\mathcal { D } } ^ { ( t ) } \\hat { W } _ { \\mathcal { D } }$ $P _ { \\mathcal { D } } ^ { ( t + 1 ) } \\neq P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } ^ { ( t ) } )$ in probability according to Younes (1999), where $\\hat { W } _ { \\mathcal { D } }$ is the maximum likelihood estimate of $W _ { \\mathcal { D } }$ . ", + "bbox": [ + 174, + 825, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/d4240e8ec86be0e776562781d3937712ce9947ec86ce6287e18cf6909f78c0b8.jpg", + "image_caption": [ + "(a) Original images " + ], + "image_footnote": [], + "bbox": [ + 271, + 121, + 725, + 473 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/27375fd3fc42fded0524ee180cc0241c3cddcd2cd4adbe32a593ef4295b0b3a4.jpg", + "image_caption": [ + "Figure 4: Generating forest road images. The category is from MIT places205 dataset. " + ], + "image_footnote": [], + "bbox": [ + 267, + 502, + 723, + 869 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/3b814fea1df0f26d03e177841edda77251a006c1d205523349743b71c3f8a821.jpg", + "image_caption": [ + "(a) Original images " + ], + "image_footnote": [], + "bbox": [ + 269, + 121, + 725, + 473 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/92c52da281e005d4ec4e61edf5d120b84b1ec7988226271a673f3d42cc16f074.jpg", + "image_caption": [ + "Figure 5: Generating hotel room images. The category is from MIT places205 dataset. " + ], + "image_footnote": [], + "bbox": [ + 267, + 502, + 723, + 869 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/6fb76193e697b5c26e40eed9a8fe339173923b9204009b88d0ee14072fa89502.jpg", + "image_caption": [ + "(a) Generated by Algorithm G alone. " + ], + "image_footnote": [], + "bbox": [ + 271, + 122, + 725, + 472 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/5e1e7b4b4e9b634c5fd2e2c24a30124a39d196894b57ce504d0eb3868ce4b020.jpg", + "image_caption": [ + "Figure 6: Generating hotel room images by Algorithm G alone and by GAN. " + ], + "image_footnote": [], + "bbox": [ + 267, + 501, + 723, + 869 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/9da43a4b287c88e80fbcf6b14085c1dd16285443159635e7c180a39e11c150b9.jpg", + "image_caption": [ + "Figure 7: Generating forest road images at high resolution $( 2 2 4 \\times 2 2 4 )$ . " + ], + "image_footnote": [], + "bbox": [ + 174, + 244, + 823, + 747 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/66128d06a3eef868dcfe60224334a982f3e70707b3795c8172556dafa82e917e.jpg", + "image_caption": [ + "Figure 8: Generating hotel room images at high resolution $( 2 2 4 \\times 2 2 4 )$ . " + ], + "image_footnote": [], + "bbox": [ + 174, + 244, + 823, + 748 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 101, + 825, + 171 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The efficiency of Algorithm D increases if the number of parallel chains $\\tilde { n }$ is large because it leads to more accurate estimation of the expectation in the gradient $L _ { \\mathcal { D } } ^ { \\prime } ( W _ { \\mathcal { D } } )$ of equation (3), so that we can afford to use larger learning rate $\\gamma _ { t }$ for faster convergence. ", + "bbox": [ + 176, + 178, + 823, + 222 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Now let us come back to the CoopNets algorithm. In order to understand how the descriptor net helps the training of the generator net, let us consider the idealized scenario where the number of parallel chains $\\tilde { n } \\infty$ , and the generator has infinite capacity, and in each iteration it estimates $W _ { \\mathcal { G } }$ by maximum likelihood using the synthesized data from P (t+1)D . In this idealized scenario, the learned generator $P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ^ { ( t + 1 ) } )$ will reproduce $P _ { \\mathcal { D } } ^ { ( t + 1 ) }$ by minimizing $\\operatorname { K L } ( P _ { \\mathcal { D } } ^ { ( t + 1 ) } ( Y ) | P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) )$ , with $P _ { \\mathcal { D } } ^ { ( t + 1 ) }$ serving as its data distribution. Then eventually the learned generator $P _ { \\mathcal { G } } ( Y , \\hat { W } _ { \\mathcal { G } } )$ will reproduce $P _ { \\mathcal { D } } ( Y ; \\hat { W } _ { \\mathcal { D } } )$ . Thus the cooperative training helps the learning of the generator. Note that the learned generator $P _ { \\mathcal { G } } ( Y , \\hat { W } _ { \\mathcal { G } } )$ will not reproduce the distribution of the observed data $P _ { \\mathrm { d a t a } }$ , unless the descriptor is of infinite capacity too. ", + "bbox": [ + 173, + 227, + 825, + 372 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Conversely, the generator net also helps the learning of the descriptor net in the CoopNets algorithm. In Algorithm D, it is impractical to make the number of parallel chains $\\tilde { n }$ too large. On the other hand, it would be difficult for a small number of chains $\\{ \\bar { \\tilde { Y } _ { i } } , i = 1 , . . . , \\tilde { n } \\}$ to explore the state space. In the CoopNets algorithm, because $P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ^ { ( t ) } )$ reproduces $P _ { \\mathcal { D } } ^ { ( t ) }$ , we can generate a completely new batch of independent samples $\\{ \\hat { Y } _ { i } \\}$ from $\\overset { \\cdot } { P _ { \\mathcal { G } } } ( Y ; W _ { \\mathcal { G } } ^ { ( t ) } )$ , and revise $\\{ \\hat { Y } _ { i } \\}$ to $\\{ \\tilde { Y } _ { i } \\}$ by Langevin dynamics, instead of running Langevin dynamics from the same old batch of $\\{ \\tilde { Y } _ { i } \\}$ as in the original Algorithm D. This is like implementing an infinite number of parallel chains, because each iteration evolves a fresh batch of examples, as if each iteration evolves a new set of chains. By updating the generator $W _ { \\mathcal { G } }$ , it is like we are updating the infinite number of parallel chains, because $W _ { \\mathcal { G } }$ memorizes the whole distribution. Even if $\\tilde { n }$ in the CoopNets algorithm is small, e.g., $\\tilde { n } \\ = \\ 1$ , viewed from the perspective of Algorithm D, it is as if $\\tilde { n } \\to \\infty$ . Thus the above idealization $\\tilde { n } \\to \\infty$ is sound. ", + "bbox": [ + 173, + 377, + 826, + 559 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "7.2 GENERATOR OF FINITE CAPACITY ", + "text_level": 1, + "bbox": [ + 178, + 577, + 444, + 592 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "From an information geometry point of view, let $\\mathcal { D } = \\{ P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } ) , \\forall W _ { \\mathcal { D } } \\}$ be the manifold of the descriptor models, where each distribution $P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } )$ is a point on this manifold. Then the maximum likelihood estimate of $W _ { \\mathcal { D } }$ is a projection of the data distribution $P _ { \\mathrm { d a t a } }$ onto the manifold $\\mathcal { D }$ . Let $\\mathcal { G } = \\{ P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) , \\forall W _ { \\mathcal { G } } \\}$ be the manifold of the generator models, where each distribution $P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } )$ is a point on this manifold. Then the maximum likelihood estimate of $W _ { \\mathcal { G } }$ is a projection of the data distribution $P _ { \\mathrm { d a t a } }$ onto the manifold $\\mathcal { G }$ . ", + "bbox": [ + 174, + 602, + 825, + 688 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "From now on, for notational simplicity and with a slight abuse of notation, we use $W _ { \\mathcal { D } }$ to denote the descriptor distribution $P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } )$ , and use $W _ { \\mathcal { G } }$ to denote the generator distribution $P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } )$ . ", + "bbox": [ + 174, + 694, + 820, + 723 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We assume both the observed data size $n$ and the synthesized data size $\\tilde { n }$ are large enough so that we shall work on distributions or populations instead of finite samples. As explained above, assuming $\\tilde { n } \\to \\infty$ is sound because the generator net can supply unlimited number of examples. ", + "bbox": [ + 174, + 729, + 823, + 772 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The Langevin revision dynamics runs a Markov chain from $W _ { \\mathcal { G } } ^ { ( t ) }$ ) towards W (t)D . Let $\\mathbf { L } _ { W _ { \\mathcal { D } } }$ be the Markov transition kernel of $l _ { \\mathcal { D } }$ steps of Langevin revisions towards $W _ { \\mathcal { D } }$ . The distribution of the revised synthesized data is ", + "bbox": [ + 174, + 779, + 825, + 824 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/0b1abe29c1fadb9f40422aae387da813f496fc83989ecce3a77752bf6fb3e65f.jpg", + "text": "$$\nP _ { \\mathcal { D } } ^ { ( t + 1 ) } = \\mathbf { L } _ { W _ { \\mathcal { D } } ^ { ( t ) } } \\cdot W _ { \\mathcal { G } } ^ { ( t ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 401, + 832, + 562, + 857 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where the notation $\\mathbf { L } \\cdot P$ denotes the marginal distribution obtained by running the Markov transition L from $P$ . The distribution $P _ { \\mathcal { D } } ^ { ( t + 1 ) }$ is in the middle between the two nets $W _ { \\mathcal { G } } ^ { ( t ) }$ ) and W (t)D , and it serves as the data distribution to train the generator, i.e., we project this distribution onto the manifold $\\mathcal { G } = \\{ P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) , \\forall W _ { \\mathcal { G } } \\} = \\{ W _ { \\mathcal { G } } \\}$ (recall we use $W _ { \\mathcal { G } }$ to denote the distribution $P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) )$ in the ", + "bbox": [ + 174, + 864, + 823, + 925 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "information geometry picture, so that ", + "bbox": [ + 174, + 103, + 421, + 118 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/afe4900ba6e434247d51b0560471e6694b9e60f7e550ef0ef48f1c63b6fa5943.jpg", + "text": "$$\nW _ { \\mathcal { G } } ^ { ( t + 1 ) } = \\arg \\operatorname* { m i n } _ { \\mathcal { G } } \\mathrm { K L } ( P _ { \\mathcal { D } } ^ { ( t + 1 ) } | W _ { \\mathcal { G } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 123, + 606, + 151 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The learning process alternates between Markov transition in (12) and projection in (13), as illustrated by Figure 9. ", + "bbox": [ + 176, + 156, + 825, + 185 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/b43b09ce1302d71cd85c8097eff4f0ee917cfe70c93d31247f6b45de0963750b.jpg", + "image_caption": [ + "Figure 9: The learning of the generator alternates between Markov transition and projection. The family of the generator models $\\mathcal { G }$ is illustrated by the black curve. Each distribution is illustrated by a point. " + ], + "image_footnote": [], + "bbox": [ + 413, + 199, + 580, + 276 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In the case of $l _ { \\mathcal { D } } \\infty$ ", + "bbox": [ + 174, + 354, + 330, + 369 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/9badcab69da5dbb27a2bc3df598aceb47872a1fbbc4a6cc9d1eda912a9f24278.jpg", + "text": "$$\n\\begin{array} { r l } & { W _ { \\mathcal { D } } ^ { ( t ) } \\to \\hat { W } _ { \\mathcal { D } } = \\arg \\underset { \\mathcal { D } } { \\operatorname* { m i n } } \\mathrm { K L } ( P _ { \\mathrm { d a t a } } | W _ { \\mathcal { D } } ) , } \\\\ & { W _ { \\mathcal { G } } ^ { ( t ) } \\to \\hat { W } _ { \\mathcal { G } } = \\arg \\underset { \\mathcal { G } } { \\operatorname* { m i n } } \\mathrm { K L } ( \\hat { W } _ { \\mathcal { D } } | W _ { \\mathcal { G } } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 375, + 373, + 650, + 430 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "That is, we first project $P _ { \\mathrm { d a t a } }$ onto $\\mathcal { D }$ , and from there continue to project onto $\\mathcal { G }$ . Therefore, $W _ { \\mathcal { D } }$ converges to the maximum likelihood estimate with $P _ { \\mathrm { d a t a } }$ being the data distribution, while $W _ { \\mathcal { G } }$ converges to the maximum likelihood estimate with $\\hat { W _ { \\mathcal { D } } }$ serving as the data distribution. ", + "bbox": [ + 173, + 434, + 823, + 479 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For finite $l _ { \\mathcal { D } }$ , the algorithm may converge to the following fixed points. The fixed point for the generator satisfies ", + "bbox": [ + 176, + 486, + 823, + 515 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/2b65595fae96734d8ddcc606cd96fe5f874b6310fe69bf8357d69b381fafbe32.jpg", + "text": "$$\n\\hat { W } _ { \\mathcal { G } } = \\arg \\operatorname* { m i n } _ { \\mathcal { G } } \\mathrm { K L } ( L _ { \\hat { W } _ { \\mathcal { D } } } \\cdot \\hat { W } _ { \\mathcal { G } } | W _ { \\mathcal { G } } ) .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 520, + 606, + 546 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The fixed point for the descriptor satisfies ", + "bbox": [ + 176, + 551, + 447, + 566 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/2d5a66740267290a6ebf4e03c26337d2a1fb27772dce027f9b9c497d5e63ef92.jpg", + "text": "$$\n\\begin{array} { r l r } { \\hat { W } _ { \\mathcal D } } & { = } & { \\arg \\underset { \\mathcal D } { \\operatorname* { m i n } } \\left[ \\mathrm { K L } ( P _ { \\mathrm { d a t a } } | W _ { \\mathcal D } ) - \\mathrm { K L } ( \\mathbf L _ { \\hat { W } _ { \\mathcal D } } \\cdot \\hat { W } _ { \\mathcal G } | W _ { \\mathcal D } ) \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 571, + 702, + 597 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "which is similar to contrastive divergence (Hinton, 2002), except that $\\hat { W } _ { G }$ takes the place of $P _ { \\mathrm { d a t a } }$ in the second Kullback-Leibler divergence. Because $\\hat { W } _ { \\mathcal G }$ is supposed to be close to $\\hat { W } _ { \\mathcal { D } }$ , the second Kullback-Leibler divergence is supposed to be small, hence our algorithm is closer to maximum likelihood learning than contrastive divergence. ", + "bbox": [ + 174, + 604, + 825, + 665 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Kim & Bengio (2016) learned the generator by gradient descent on $\\mathrm { K L } ( W _ { \\mathcal { G } } | W _ { \\mathcal { D } } ^ { ( t ) } )$ over $\\mathcal { G }$ . The objective function is $\\operatorname { K L } ( W _ { \\mathcal { G } } | W _ { \\mathcal { D } } ^ { ( t ) } ) = \\operatorname { E } _ { W _ { \\mathcal { G } } } [ \\log P _ { G } ( Y ; W _ { \\mathcal { G } } ) ] - \\operatorname { E } _ { W _ { \\mathcal { G } } } [ \\log P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } ^ { ( t ) } ) ]$ , where the first term is the negative entropy that is intractable, and the second term is the expected energy that is tractable. Our learning method for the generator is consistent with the learning objective $\\mathrm { K L } ( W _ { \\mathcal { G } } | W _ { \\mathcal { D } } ^ { ( t ) } )$ , because ", + "bbox": [ + 173, + 672, + 825, + 752 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f3859cb16c2fa37ed242d00338f1fbc3822e04ed888078ff5d98f0fcf852a790.jpg", + "text": "$$\n\\mathrm { K L } ( P _ { \\mathcal { D } } ^ { ( t + 1 ) } | W _ { \\mathcal { D } } ^ { ( t ) } ) \\leq \\mathrm { K L } ( W _ { \\mathcal { G } } ^ { ( t ) } | W _ { \\mathcal { D } } ^ { ( t ) } ) .\n$$", + "text_format": "latex", + "bbox": [ + 352, + 758, + 611, + 781 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In fact, $\\mathrm { K L } ( P _ { \\mathcal { D } } ^ { ( t + 1 ) } | W _ { \\mathcal { D } } ^ { ( t ) } ) \\to 0$ monotonically as $l _ { \\mathcal { D } } \\infty$ due to the second law of thermodynamics. The reduction of the Kullback-Leibler divergence in (18) and the projection in (13) in our learning of the generator are consistent with the learning objective of reducing $\\mathrm { K L } ( W _ { \\mathcal { G } } | W _ { \\mathcal { D } } ^ { ( t ) } )$ in $\\operatorname { K i m } \\&$ Bengio (2016). But the Monte Carlo implementation of in our work avoids the need to approximate the intractable entropy term. ", + "bbox": [ + 173, + 787, + 825, + 864 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "7.3 MORE SYNTHESIS RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 880, + 405, + 895 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We display more synthesis results at the resolution of $6 4 \\times 6 4$ . ", + "bbox": [ + 173, + 906, + 584, + 921 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/fa57b10313cd350d46c64e652a218e577e9c5d82f42bf54e6cbda4051968c44b.jpg", + "image_caption": [ + "(a) Original images " + ], + "image_footnote": [], + "bbox": [ + 271, + 122, + 727, + 473 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/4344d87e0bc5dba658a5951713b369d4ebd3961a2be7cc7e3bef6fb9b8b0da35.jpg", + "image_caption": [ + "Figure 10: Generating swimming pool images. The category is from MIT places205 dataset. " + ], + "image_footnote": [], + "bbox": [ + 267, + 502, + 723, + 869 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/4f930503433b04f0f4f3a8d878fdf742a9e39069207fa6e40cbb81dd51708443.jpg", + "image_caption": [ + "(a) Original images " + ], + "image_footnote": [], + "bbox": [ + 269, + 122, + 725, + 473 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/643e35f770dfe5dc178ae9f9c9a07ba34b55669747d4d354cb29229f83ebe2cf.jpg", + "image_caption": [ + "Figure 11: Generating volcano images. The category is from MIT places205 dataset. " + ], + "image_footnote": [], + "bbox": [ + 267, + 501, + 723, + 869 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/c9545a3708bd3fb1fe08e41242a58e0180a661542fad4b2bdcd92a881bdb4175.jpg", + "image_caption": [ + "(a) Original images " + ], + "image_footnote": [], + "bbox": [ + 271, + 122, + 725, + 473 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/3e53647e608ce0c4acea2f61b7245b65ae4faaf262144fad822b943eda9ad142.jpg", + "image_caption": [ + "Figure 12: Generating rock images. The category is from MIT places205 dataset. " + ], + "image_footnote": [], + "bbox": [ + 267, + 502, + 723, + 869 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/b12cace848d065b0fa8a488392ab77b34a9746b62f0664058f2c3712139e8447.jpg", + "image_caption": [ + "(a) Original images " + ], + "image_footnote": [], + "bbox": [ + 269, + 122, + 727, + 472 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/5ee1f47ee99d367d33c8750e5f6afc134f8e3eb3ab8afa1c6349f1a4f3dcedae.jpg", + "image_caption": [ + "Figure 13: Generating desert images. The category is from MIT places205 dataset. " + ], + "image_footnote": [], + "bbox": [ + 267, + 501, + 723, + 869 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/c240c83a2ecf8e50df6b5c62bed9bc301a8fa3ed14365d75d5d059ea8c66eede.jpg", + "image_caption": [ + "Figure 14: Generating schoolbus images. The category is from Imagenet ILSVRC2012 1000 object categories. " + ], + "image_footnote": [], + "bbox": [ + 267, + 109, + 727, + 878 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/84da1ea2ce2aa8bb875b9475ae90c024adb01c0be4582788db870e1abbee7bca.jpg", + "image_caption": [ + "(a) Original images " + ], + "image_footnote": [], + "bbox": [ + 269, + 114, + 727, + 467 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/a8d36572a8199b6a19dcb1243b1e8da604cc228c9da6312c998be9f6ed69efbb.jpg", + "image_caption": [ + "Figure 15: Generating lifeboat images. The category is from Imagenet ILSVRC2012 1000 object categories. " + ], + "image_footnote": [], + "bbox": [ + 267, + 494, + 723, + 862 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/46d20f827362e767e77661eededb4fe10435504996bd25a3b41fa459497b2123.jpg", + "image_caption": [ + "(a) Original images " + ], + "image_footnote": [], + "bbox": [ + 271, + 116, + 725, + 467 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/e648eb571b9779541d94d339c142f7e168d3c07e852fd01745814cf2fbbbb81b.jpg", + "image_caption": [ + "Figure 16: Generating zebra images. The category is from Imagenet ILSVRC2012 1000 object categories. " + ], + "image_footnote": [], + "bbox": [ + 267, + 494, + 723, + 862 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/27711f6e1fdb6eb29d18e2f1525701702e0482d73d2d5267a3d0ec86c16756d5.jpg", + "image_caption": [ + "(a) Original images " + ], + "image_footnote": [], + "bbox": [ + 269, + 116, + 727, + 467 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/16832906f2ab1dce4cda5308ebb29a4000dea39e6fc33a2f320c55c689537a8c.jpg", + "image_caption": [ + "Figure 17: Generating strawberry images. The category is from Imagenet ILSVRC2012 1000 object categories. " + ], + "image_footnote": [], + "bbox": [ + 267, + 494, + 723, + 863 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/6d5cf44133ea0452bf4dfa5b37dabb9272e680f425069fb07a5732da64e15174.jpg", + "image_caption": [ + "Figure 18: Generating lemon images. The category is from Imagenet ILSVRC2012 1000 object categories. " + ], + "image_footnote": [], + "bbox": [ + 267, + 109, + 728, + 881 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/922628eee4c815b3f90fe271603a9b61b34ae36d5547fd0c779439a40bfa6723.jpg", + "image_caption": [ + "(a) Original images " + ], + "image_footnote": [], + "bbox": [ + 269, + 114, + 725, + 467 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/c07c2e08ef357ea34cea8c5bd5c0f147c0e503a97915274b1c7941023530fefd.jpg", + "image_caption": [ + "Figure 19: Generating apartment building images. The category is from Imagenet ILSVRC2012 1000 object categories. " + ], + "image_footnote": [], + "bbox": [ + 267, + 494, + 723, + 863 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/7217d8f4633ecbedd3c65d26e0b7c52899f72d7542e691c35e0c04a25cf095ca.jpg", + "image_caption": [ + "Figure 20: Generating dinning table images. The category is from Imagenet ILSVRC2012 1000 object categories. " + ], + "image_footnote": [], + "bbox": [ + 267, + 113, + 728, + 881 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/43d047ffdb94f8574ee56bff737c033f91dd1af046867e25f5900be4d091b51d.jpg", + "image_caption": [ + "Figure 21: Generating balloon images. The category is from Imagenet ILSVRC2012 1000 object categories. " + ], + "image_footnote": [], + "bbox": [ + 267, + 114, + 728, + 881 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENT ", + "text_level": 1, + "bbox": [ + 176, + 104, + 328, + 117 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "We thank Hansheng Jiang for her work on this project as a summer visiting student. We thank Tian Han for sharing the code on learning the generator network, and for helpful discussions. ", + "bbox": [ + 176, + 127, + 823, + 156 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "The work is supported by NSF DMS 1310391, DARPA SIMPLEX N66001-15-C-4035, ONR MURI N00014-16-1-2007, and DARPA ARO W911NF-16-1-0579. ", + "bbox": [ + 174, + 162, + 823, + 190 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 287, + 117 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pp. 248–255. IEEE, 2009. ", + "bbox": [ + 173, + 126, + 823, + 169 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Emily L Denton, Soumith Chintala, Rob Fergus, et al. Deep generative image models using a laplacian pyramid of adversarial networks. 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We", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 340, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 340, + 470, + 356 + ], + "score": 1.0, + "content": "observe that the two training algorithms can cooperate with each other by jump-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 469, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 365 + ], + "score": 1.0, + "content": "starting each other’s Langevin sampling, and they can be seamlessly interwoven", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 364, + 409, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 364, + 409, + 376 + ], + "score": 1.0, + "content": "into a CoopNets algorithm that can train both nets simultaneously.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13, + "bbox_fs": [ + 141, + 233, + 470, + 376 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 395, + 206, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 208, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 208, + 411 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 110, + 419, + 315, + 431 + ], + "lines": [ + { + "bbox": [ + 107, + 419, + 317, + 432 + ], + "spans": [ + { + "bbox": [ + 107, + 419, + 317, + 432 + ], + "score": 1.0, + "content": "1.1 TWO CONVNETS OF OPPOSITE DIRECTIONS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 504, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "We begin with a story that the reader of this paper can readily relate to. A student writes up an initial", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 452, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 506, + 463 + ], + "score": 1.0, + "content": "draft of a paper. His advisor then revises it. After that they submit the revised paper for review. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 474 + ], + "score": 1.0, + "content": "student then learns from his advisor’s revision, while the advisor learns from the outside review. In", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 474, + 426, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 426, + 485 + ], + "score": 1.0, + "content": "this story, the advisor guides the student, but the student does most of the work.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 440, + 506, + 485 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 490, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 503 + ], + "score": 1.0, + "content": "This paper is about two probabilistic models of signals such as images, and they play the roles", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "of student and advisor as described above. Both models are parametrized by convolutional neural", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 524 + ], + "score": 1.0, + "content": "networks (ConvNets or CNNs) (LeCun et al., 1998; Krizhevsky et al., 2012). The two nets take", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 538 + ], + "score": 1.0, + "content": "two opposite directions. One is bottom-up, and the other is top-down, as illustrate by the following", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 533, + 145, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 145, + 549 + ], + "score": 1.0, + "content": "diagram:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 490, + 505, + 549 + ] + }, + { + "type": "table", + "bbox": [ + 205, + 550, + 389, + 606 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 205, + 550, + 389, + 606 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 205, + 550, + 389, + 606 + ], + "spans": [ + { + "bbox": [ + 205, + 550, + 389, + 606 + ], + "score": 0.933, + "html": "
Bottom-up ConvNet featuresTop-down ConvNet latent variables
个←
signal (a) Descriptor Netsignal (b) Generator Net
", + "type": "table", + "image_path": "24b6527b1d0278282a4b710a40b45f0b9e043a5d9a2e104e0df3fcdb234effa0.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 550, + 389, + 564.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 205, + 564.0, + 389, + 578.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 205, + 578.0, + 389, + 592.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 205, + 592.0, + 389, + 606.0 + ], + "spans": [], + "index": 34 + } + ] + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 615, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "The simultaneous training of such two nets was first studied by the recent work of Kim & Bengio", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "(2016). These two nets belong to two major classes of probabilistic models. (a) The exponential", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 651 + ], + "score": 1.0, + "content": "family models or the energy-based models (LeCun et al., 2006) or the Markov random field models", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "(Zhu et al., 1997), where the probability distribution is defined by feature statistics or energy function", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "computed from the signal by a bottom-up process. (b) The latent variable models or the directed", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "graphical models, where the signal is assumed to be a transformation of the latent factors that follow", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "a known prior distribution. The latent factors generate the signal by a top-down process. A classical", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 693, + 214, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 214, + 705 + ], + "score": 1.0, + "content": "example is factor analysis.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 616, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "The two classes of models have been contrasted by Zhu (2003); Teh et al. (2003); Ngiam et al.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "(2011). Zhu (2003) called the two classes of models the descriptive models and the generative", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "models respectively. Both classes of models can benefit from the high capacity of the multi-layer", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "ConvNets. (a) In the exponential family models or the energy-based models, the feature statistics or", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "the energy function can be defined by a bottom-up ConvNet that maps the signal to the features and", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "the energy function (Ngiam et al., 2011; Xie et al., 2016). We call the resulting model a descriptive", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "network or a descriptor net following Zhu (2003), because it is built on descriptive feature statistics.", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "(b) In the latent variable models or the directed graphical models, the transformation from the latent", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "factors to the signal can be defined by a top-down ConvNet (Dosovitskiy et al., 2015), which maps", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "the latent factors to the signal. We call the resulting model a generative network or generator net", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "following Goodfellow et al. (2014), who proposed such a model in their work on the generative", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 464, + 224, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 224, + 475 + ], + "score": 1.0, + "content": "adversarial networks (GAN).", + "type": "text", + "cross_page": true + } + ], + "index": 20 + } + ], + "index": 43.5, + "bbox_fs": [ + 106, + 708, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 120, + 80, + 491, + 240 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 120, + 80, + 491, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 80, + 491, + 240 + ], + "spans": [ + { + "bbox": [ + 120, + 80, + 491, + 240 + ], + "score": 0.957, + "type": "image", + "image_path": "799da87ccece60114024503ad08a2702e46d67f851a2c657025c4fc10d8cf9f5.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 120, + 80, + 491, + 133.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 120, + 133.33333333333334, + 491, + 186.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 120, + 186.66666666666669, + 491, + 240.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 250, + 505, + 339 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "Figure 1: (a) Algorithm D involves sampling from the current model by Langevin dynamics. (b)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "score": 1.0, + "content": "Algorithm G involves sampling from the posterior distribution of the latent factors by Langevin", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "dynamics. (c) CoopNets algorithm. The part of the flowchart for training the descriptor is similar", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "to Algorithm D, except that the D1 Langevin sampling is initialized from the initial synthesized", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "examples supplied by the generator. The part of the flowchart for training the generator can also be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "mapped to Algorithm G, except that the revised synthesized examples play the role of the observed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "data, and the known generated latent factors can be used as inferred latent factors (or be used to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 328, + 338, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 338, + 340 + ], + "score": 1.0, + "content": "initialize the G1 Langevin sampling of the latent factors).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "models respectively. Both classes of models can benefit from the high capacity of the multi-layer", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "ConvNets. (a) In the exponential family models or the energy-based models, the feature statistics or", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "the energy function can be defined by a bottom-up ConvNet that maps the signal to the features and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "the energy function (Ngiam et al., 2011; Xie et al., 2016). We call the resulting model a descriptive", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "network or a descriptor net following Zhu (2003), because it is built on descriptive feature statistics.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "(b) In the latent variable models or the directed graphical models, the transformation from the latent", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 444 + ], + "score": 1.0, + "content": "factors to the signal can be defined by a top-down ConvNet (Dosovitskiy et al., 2015), which maps", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "the latent factors to the signal. We call the resulting model a generative network or generator net", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "following Goodfellow et al. (2014), who proposed such a model in their work on the generative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 464, + 224, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 224, + 475 + ], + "score": 1.0, + "content": "adversarial networks (GAN).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 491, + 370, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 371, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 371, + 503 + ], + "score": 1.0, + "content": "1.2 TWO TRAINING ALGORITHMS AND THEIR COOPERATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "score": 1.0, + "content": "Fig. 1(a) and (b) display the flowcharts of the maximum likelihood learning algorithms for training", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "the descriptor and generator nets . We call the two algorithms Algorithm D and Algorithm G respec-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "tively. Algorithm D (Xie et al., 2016) iterates two steps: Step D1 synthesizes examples by sampling", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "from the current model by Langevin dynamics. Step D2 updates the parameters to shift the density", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "from the synthesized examples towards the observed examples. Algorithm G (Han et al., 2017) also", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "iterates two steps. Step G1 infers latent factors for each observed example by sampling from their", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "posterior distribution by Langevin dynamics. Step G2 updates the parameters by a non-linear re-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "gression of the observed examples on their corresponding latent factors. We use Langevin dynamics", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "for Markov chain Monte Carlo (MCMC) sampling because the gradient term of Langevin dynamics", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "can be readily computed via back-propagation. Thus all the steps D1, D2 and G1, G2 are powered", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 621, + 493, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 493, + 633 + ], + "score": 1.0, + "content": "by back-propagation, and both Algorithms D and G are alternating back-propagation algorithms.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 638, + 502, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 504, + 650 + ], + "score": 1.0, + "content": "In this article, we propose to couple Algorithms D and G into a cooperative training algorithm that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 649, + 504, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 504, + 661 + ], + "score": 1.0, + "content": "interweaves the steps of the two algorithms seamlessly. We call the resulting algorithm the CoopNets", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 660, + 370, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 370, + 672 + ], + "score": 1.0, + "content": "algorithm, and we show that it can train both nets simultaneously.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "Figure 1(c) displays the flowchart of the CoopNets algorithm. The generator is like the student.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "It generates the initial draft of the synthesized examples. The descriptor is like the advisor. It", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "revises the initial draft by initializing its Langevin dynamics from the initial draft in Step D1, which", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "produces the revised draft of the synthesized examples. The descriptor learns from the outside review", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "in Step D2, which is in the form of the difference between the observed examples and the revised", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "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": [ + 120, + 80, + 491, + 240 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 120, + 80, + 491, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 80, + 491, + 240 + ], + "spans": [ + { + "bbox": [ + 120, + 80, + 491, + 240 + ], + "score": 0.957, + "type": "image", + "image_path": "799da87ccece60114024503ad08a2702e46d67f851a2c657025c4fc10d8cf9f5.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 120, + 80, + 491, + 133.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 120, + 133.33333333333334, + 491, + 186.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 120, + 186.66666666666669, + 491, + 240.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 250, + 505, + 339 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "Figure 1: (a) Algorithm D involves sampling from the current model by Langevin dynamics. (b)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "score": 1.0, + "content": "Algorithm G involves sampling from the posterior distribution of the latent factors by Langevin", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "dynamics. (c) CoopNets algorithm. The part of the flowchart for training the descriptor is similar", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "to Algorithm D, except that the D1 Langevin sampling is initialized from the initial synthesized", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 307 + ], + "score": 1.0, + "content": "examples supplied by the generator. The part of the flowchart for training the generator can also be", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "mapped to Algorithm G, except that the revised synthesized examples play the role of the observed", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "data, and the known generated latent factors can be used as inferred latent factors (or be used to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 328, + 338, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 338, + 340 + ], + "score": 1.0, + "content": "initialize the G1 Langevin sampling of the latent factors).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + } + ], + "index": 3.75 + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 505, + 475 + ], + "lines": [], + "index": 15.5, + "bbox_fs": [ + 105, + 365, + 506, + 475 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 491, + 370, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 371, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 371, + 503 + ], + "score": 1.0, + "content": "1.2 TWO TRAINING ALGORITHMS AND THEIR COOPERATION", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 511, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "score": 1.0, + "content": "Fig. 1(a) and (b) display the flowcharts of the maximum likelihood learning algorithms for training", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "the descriptor and generator nets . We call the two algorithms Algorithm D and Algorithm G respec-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "tively. Algorithm D (Xie et al., 2016) iterates two steps: Step D1 synthesizes examples by sampling", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "from the current model by Langevin dynamics. Step D2 updates the parameters to shift the density", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "from the synthesized examples towards the observed examples. Algorithm G (Han et al., 2017) also", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 578 + ], + "score": 1.0, + "content": "iterates two steps. Step G1 infers latent factors for each observed example by sampling from their", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "posterior distribution by Langevin dynamics. Step G2 updates the parameters by a non-linear re-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "gression of the observed examples on their corresponding latent factors. We use Langevin dynamics", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "for Markov chain Monte Carlo (MCMC) sampling because the gradient term of Langevin dynamics", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "can be readily computed via back-propagation. Thus all the steps D1, D2 and G1, G2 are powered", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 621, + 493, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 493, + 633 + ], + "score": 1.0, + "content": "by back-propagation, and both Algorithms D and G are alternating back-propagation algorithms.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 510, + 506, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 638, + 502, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 504, + 650 + ], + "score": 1.0, + "content": "In this article, we propose to couple Algorithms D and G into a cooperative training algorithm that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 649, + 504, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 504, + 661 + ], + "score": 1.0, + "content": "interweaves the steps of the two algorithms seamlessly. 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The descriptor is like the advisor. It", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "revises the initial draft by initializing its Langevin dynamics from the initial draft in Step D1, which", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "produces the revised draft of the synthesized examples. The descriptor learns from the outside review", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "in Step D2, which is in the form of the difference between the observed examples and the revised", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "synthesized examples. The generator learns from how the descriptor revises the initial draft by", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "reconstructing the revised draft in Step G2. For each synthesized example, the generator knows the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "latent factors that generate the initial draft, so that Step G1 can infer the latent factors by initializing", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 301, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 301, + 127 + ], + "score": 1.0, + "content": "its Langevin dynamics from their known values.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 676, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "synthesized examples. The generator learns from how the descriptor revises the initial draft by", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "reconstructing the revised draft in Step G2. For each synthesized example, the generator knows the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "latent factors that generate the initial draft, so that Step G1 can infer the latent factors by initializing", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 301, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 301, + 127 + ], + "score": 1.0, + "content": "its Langevin dynamics from their known values.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "In the CoopNets algorithm, the generator fuels the MCMC of the descriptor by supplying initial", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "synthesized examples, which can be obtained by direct ancestral sampling. The generator then", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "learns from the revised synthesized examples with virtually known latent factors. The cooperation", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 227, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 227, + 177 + ], + "score": 1.0, + "content": "is thus beneficial to both nets.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 192, + 209, + 205 + ], + "lines": [ + { + "bbox": [ + 104, + 191, + 210, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 210, + 207 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 106, + 218, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 229 + ], + "score": 1.0, + "content": "Our work is inspired by the generative adversarial networks (GAN) (Goodfellow et al., 2014; Denton", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 241 + ], + "score": 1.0, + "content": "et al., 2015; Radford et al., 2015). In GAN, the generator net is paired with a discriminator net. The", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "two nets play adversarial roles. In our work, the generator net and the descriptor net play cooperative", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "score": 1.0, + "content": "roles, and they feed each other the initial, revised and reconstructed synthesized data. The learning", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "score": 1.0, + "content": "of both nets is based on maximum likelihood, and the learning process is quite stable because of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 272, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 285 + ], + "score": 1.0, + "content": "cooperative nature and the consistent directions of the two maximum likelihood training algorithms.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 108, + 289, + 503, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "Another method to train the generator network is variational auto-encoder (VAE) (Kingma &", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 299, + 504, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 504, + 312 + ], + "score": 1.0, + "content": "Welling, 2014; Rezende et al., 2014; Mnih & Gregor, 2014), which learns an inferential or recogni-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 311, + 406, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 406, + 324 + ], + "score": 1.0, + "content": "tion network to approximate the posterior distribution of the latent factors.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 503, + 350 + ], + "lines": [ + { + "bbox": [ + 106, + 328, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 505, + 341 + ], + "score": 1.0, + "content": "The connection between the descriptor net and the discriminator net has been explored by Xie et al.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 339, + 376, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 376, + 352 + ], + "score": 1.0, + "content": "(2016), where the descriptor can be derived from the discriminator.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 505, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "Our work is most similar to the recent work of Kim & Bengio (2016). In fact, the settings of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "two nets are the same. In their work, the generator learns from the descriptor by minimizing the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "Kullback-Leibler divergence from the generator to the descriptor, which can be decomposed into an", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "energy term and an entropy term. 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The", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 449, + 501, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 501, + 463 + ], + "score": 1.0, + "content": "CoopNets algorithm initializes the MCMC sampling from the examples supplied by the generator.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 477, + 358, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 476, + 359, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 359, + 492 + ], + "score": 1.0, + "content": "3 TWO NETS AND TWO TRAINING ALGORITHMS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 502, + 328, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 329, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 329, + 514 + ], + "score": 1.0, + "content": "3.1 DESCRIPTOR NET AND TRAINING ALGORITHM", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 123, + 534 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 523, + 133, + 532 + ], + "score": 0.79, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 522, + 164, + 534 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 165, + 523, + 174, + 532 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "-dimensional signal, such as an image. 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In fact, the settings of the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "two nets are the same. In their work, the generator learns from the descriptor by minimizing the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 377, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 505, + 391 + ], + "score": 1.0, + "content": "Kullback-Leibler divergence from the generator to the descriptor, which can be decomposed into an", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "energy term and an entropy term. 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For the descriptor net, we adopt the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "structure of Xie et al. (2016), where the bottom-up network consists of multiple layers of convolution", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "by linear filtering, ReLU non-linearity, and down-sampling. We adopt the structure of the generator", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "network of Radford et al. (2015); Dosovitskiy et al. (2015), where the top-down network consists of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "multiple layers of deconvolution by linear superposition, ReLU non-linearity, and up-sampling, with", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 459, + 502, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 470, + 472 + ], + "score": 1.0, + "content": "tanh non-linearity at the bottom-layer (Radford et al., 2015) to make the signals fall within", + "type": "text" + }, + { + "bbox": [ + 470, + 459, + 498, + 471 + ], + "score": 0.86, + "content": "[ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 459, + 502, + 472 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 403, + 506, + 472 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 486, + 352, + 496 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 353, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 353, + 498 + ], + "score": 1.0, + "content": "5.1 QUANTITATIVE EXPERIMENT ON FACE COMPLETION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 519 + ], + "score": 1.0, + "content": "We conduct an experiment on learning from complete training images of human faces, and then", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 518, + 504, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 504, + 529 + ], + "score": 1.0, + "content": "testing the learned model on completing the occluded testing images. The structure of the generator", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "network is the same as in (Radford et al., 2015; Dosovitskiy et al., 2015). We adopt a 4-layer", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 248, + 551 + ], + "score": 1.0, + "content": "descriptor net. The first layer has", + "type": "text" + }, + { + "bbox": [ + 248, + 540, + 286, + 550 + ], + "score": 0.86, + "content": "9 6 5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "filters with sub-sampling of 2, the second layers has", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 107, + 550, + 150, + 561 + ], + "score": 0.88, + "content": "1 2 8 \\mathrm { ~ 5 ~ } \\times \\mathrm { ~ 5 ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 549, + 356, + 564 + ], + "score": 1.0, + "content": "filters with sub-sampling of 2, the third layer has", + "type": "text" + }, + { + "bbox": [ + 356, + 550, + 400, + 561 + ], + "score": 0.87, + "content": "2 5 6 ~ 5 ~ \\times ~ 5", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "filters with sub-sampling", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 457, + 574 + ], + "score": 1.0, + "content": "of 2, and the final layer is a fully connected layer with 50 channels as output. We use", + "type": "text" + }, + { + "bbox": [ + 457, + 561, + 481, + 572 + ], + "score": 0.86, + "content": "L { = } 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "steps", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "of Langevin revision dynamics within each learning iteration, and the Langevin step size is set at", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "0.002. The learning rate is 0.07. The training data are 10, 000 human faces randomly selected from", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 592, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 608 + ], + "score": 1.0, + "content": "CelebA dataset (Liu et al., 2015). We run 600 cooperative learning iterations. Figure 2 displays 144", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 605, + 294, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 294, + 618 + ], + "score": 1.0, + "content": "synthesized human faces by the descriptor net.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 506, + 506, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 384, + 635 + ], + "score": 1.0, + "content": "To quantitatively test whether we have learned a good generator net", + "type": "text" + }, + { + "bbox": [ + 384, + 622, + 426, + 634 + ], + "score": 0.92, + "content": "g ( X ; W _ { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "even though it has", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "never seen the training images directly in the training stage, we apply it to the task of recovering the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 374, + 657 + ], + "score": 1.0, + "content": "occluded pixels of testing images. For each occluded testing image", + "type": "text" + }, + { + "bbox": [ + 374, + 644, + 383, + 654 + ], + "score": 0.78, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 643, + 505, + 657 + ], + "score": 1.0, + "content": ", we use Step G1 of Algorithm", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 653, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 222, + 669 + ], + "score": 1.0, + "content": "G to infer the latent factors", + "type": "text" + }, + { + "bbox": [ + 222, + 655, + 232, + 665 + ], + "score": 0.81, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 653, + 423, + 669 + ], + "score": 1.0, + "content": ". The only change is with respect to the term", + "type": "text" + }, + { + "bbox": [ + 423, + 655, + 501, + 667 + ], + "score": 0.91, + "content": "\\| Y - g ( X ; W _ { \\mathcal { G } } ) \\| ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 653, + 505, + 669 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 354, + 678 + ], + "score": 1.0, + "content": "where the sum of squares is over all the observed pixels of", + "type": "text" + }, + { + "bbox": [ + 354, + 666, + 363, + 676 + ], + "score": 0.83, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "in back-propagation computation.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 271, + 690 + ], + "score": 1.0, + "content": "We run 1000 Langevin steps, initializing", + "type": "text" + }, + { + "bbox": [ + 271, + 677, + 281, + 687 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 676, + 304, + 690 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 305, + 677, + 339, + 689 + ], + "score": 0.93, + "content": "\\mathrm { N } ( 0 , I _ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 676, + 405, + 690 + ], + "score": 1.0, + "content": ". After inferring", + "type": "text" + }, + { + "bbox": [ + 405, + 677, + 415, + 687 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 676, + 505, + 690 + ], + "score": 1.0, + "content": ", the completed image", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 149, + 700 + ], + "score": 0.92, + "content": "g ( X ; W _ { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 687, + 470, + 700 + ], + "score": 1.0, + "content": "is automatically obtained. 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These 3 experiments are denoted by", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 149, + 720 + ], + "score": 0.52, + "content": "\\mathbf { M } 2 0 \\ \\mathbf { M } 3 0", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 709, + 506, + 723 + ], + "score": 1.0, + "content": ", and M40 respectively (M for mask). We report the recovery errors and compare our", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "method with 8 different image inpainting methods as well as the DCGAN of Radford et al. 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The synthesized images are generated by the CoopNets", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 342, + 278, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 278, + 355 + ], + "score": 1.0, + "content": "algorithm that learns from 10, 000 images.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 376, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "For DCGAN, we use the parameter setting in Radford et al. (2015) except changing the number", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "of learning iterations to 600. We use the same 10, 000 training images to learn DCGAN. After the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 491, + 410 + ], + "score": 1.0, + "content": "model is learned, we keep the generator and use the same method as ours to infer latent factors", + "type": "text" + }, + { + "bbox": [ + 491, + 398, + 501, + 408 + ], + "score": 0.78, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 398, + 505, + 410 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "and recover the unobserved pixels. In 8 inpainting methods, Methods 1 and 2 are based on Markov", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 367, + 433 + ], + "score": 1.0, + "content": "random field prior where the nearest neighbor potential terms are", + "type": "text" + }, + { + "bbox": [ + 367, + 420, + 378, + 431 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 420, + 395, + 433 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 395, + 420, + 406, + 431 + ], + "score": 0.87, + "content": "\\ell _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "differences respectively.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 505, + 443 + ], + "score": 1.0, + "content": "Methods 3 to 8 are interpolation methods. 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ExpOursGAN12345678
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Row 2: testing images with occluded pixels. Row 3:", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 711, + 241, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 241, + 722 + ], + "score": 1.0, + "content": "recovered images by our method.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5 + } + ], + "index": 39.75 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 109, + 83, + 312, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 314, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 314, + 95 + ], + "score": 1.0, + "content": "5.2 QUALITATIVE EXPERIMENT ON SYNTHESIS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 224 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 116 + ], + "score": 1.0, + "content": "We conduct an experiment on synthesizing images of categories from Imagenet ILSVRC2012", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 127 + ], + "score": 1.0, + "content": "dataset (Deng et al., 2009) and MIT places205 dataset (Zhou et al., 2014). We adopt a 4-layer", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 125, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 248, + 138 + ], + "score": 1.0, + "content": "descriptor net. The first layer has", + "type": "text" + }, + { + "bbox": [ + 248, + 126, + 286, + 136 + ], + "score": 0.9, + "content": "6 4 ~ 5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 125, + 505, + 138 + ], + "score": 1.0, + "content": "filters with sub-sampling of 2, the second layers has", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 136, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 107, + 136, + 149, + 147 + ], + "score": 0.87, + "content": "1 2 8 3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 136, + 348, + 149 + ], + "score": 1.0, + "content": "filters with sub-sampling of 2, the third layer has", + "type": "text" + }, + { + "bbox": [ + 348, + 136, + 391, + 147 + ], + "score": 0.77, + "content": "2 5 6 3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 136, + 506, + 149 + ], + "score": 1.0, + "content": "filters with sub-sampling of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 506, + 159 + ], + "score": 1.0, + "content": "1, and the final layer is a fully connected layer with 100 channels as output. 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For each category, we randomly choose 1,000 images as training data and resize the images", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 178, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 116, + 194 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 180, + 149, + 191 + ], + "score": 0.89, + "content": "6 4 \\times 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 178, + 505, + 194 + ], + "score": 1.0, + "content": ". We run 1, 000 cooperative learning iterations to train the model. Figures 4 and 5 display", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "the results for two categories, where for each category, we show 144 original images sampled from", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "score": 1.0, + "content": "the training set, and 144 synthesized images generated by our method. The appendix contains more", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 177, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 177, + 225 + ], + "score": 1.0, + "content": "synthesis results.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 108, + 229, + 503, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 243 + ], + "score": 1.0, + "content": "As a comparison, we apply the Algorithm G alone and GAN code on the same 1,000 hotel room", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "training images to learn the generator of the same structure as in CoopNets. Figure 6 displays the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 252, + 177, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 177, + 263 + ], + "score": 1.0, + "content": "synthesis results.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 268, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 314, + 282 + ], + "score": 1.0, + "content": "We also try to synthesize images at high resolution", + "type": "text" + }, + { + "bbox": [ + 315, + 269, + 361, + 280 + ], + "score": 0.84, + "content": "( 2 2 4 \\times 2 2 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "). We adopt a 4-layer descriptor net.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 181, + 292 + ], + "score": 1.0, + "content": "The first layer has", + "type": "text" + }, + { + "bbox": [ + 181, + 280, + 232, + 290 + ], + "score": 0.89, + "content": "1 2 8 1 5 \\times 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 280, + 437, + 292 + ], + "score": 1.0, + "content": "filters with sub-sampling of 3, the second layer has", + "type": "text" + }, + { + "bbox": [ + 438, + 280, + 479, + 290 + ], + "score": 0.7, + "content": "2 5 6 3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "filters", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 282, + 303 + ], + "score": 1.0, + "content": "with sub-sampling of 2, the third layer has", + "type": "text" + }, + { + "bbox": [ + 282, + 291, + 325, + 301 + ], + "score": 0.89, + "content": "5 1 2 3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "filters with sub-sampling of 1, and the final", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "layer is a fully connected layer with 100 channels as output. We enlarge the filters of the final layer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 183, + 325 + ], + "score": 1.0, + "content": "of generator net to", + "type": "text" + }, + { + "bbox": [ + 183, + 313, + 218, + 323 + ], + "score": 0.91, + "content": "1 4 \\times 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 312, + 266, + 325 + ], + "score": 1.0, + "content": "to generate", + "type": "text" + }, + { + "bbox": [ + 266, + 313, + 310, + 323 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "images. The learning rate is 0.05. We run 1000", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "cooperative learning iterations to train the model. Figures 7 and 8 show the synthesized images of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 335, + 284, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 284, + 347 + ], + "score": 1.0, + "content": "two categories from MIT places205 dataset.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 362, + 195, + 375 + ], + "lines": [ + { + "bbox": [ + 104, + 360, + 197, + 378 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 197, + 378 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 503, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "The most unique feature of our work is that the two networks feed each other the synthesized data", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 399, + 447, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 447, + 412 + ], + "score": 1.0, + "content": "in the learning process, including initial, revised, and reconstructed synthesized data.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 504, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "Another unique feature of our work is that the learning process interweaves the existing maximum", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 427, + 316, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 316, + 438 + ], + "score": 1.0, + "content": "likelihood learning algorithms for the two networks.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 108, + 443, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "A third unique feature of our work is that the MCMC for the descriptor keeps rejuvenating the chains", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "by refreshing the samples by independent replacements supplied by the generator, so that a single", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "chain effectively amounts to an infinite number of chains or the evolution of the whole marginal", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 477, + 261, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 261, + 489 + ], + "score": 1.0, + "content": "distribution modeled by the generator.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 108, + 505, + 192, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 194, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 194, + 520 + ], + "score": 1.0, + "content": "CODE AND DATA", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 394, + 541 + ], + "lines": [ + { + "bbox": [ + 106, + 530, + 396, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 396, + 543 + ], + "score": 1.0, + "content": "http://www.stat.ucla.edu/˜ywu/CoopNets/main.html", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 108, + 558, + 267, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 268, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 268, + 574 + ], + "score": 1.0, + "content": "7 APPENDIX: CONVERGENCE", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 108, + 584, + 283, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 584, + 285, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 285, + 597 + ], + "score": 1.0, + "content": "7.1 GENERATOR OF INFINITE CAPACITY", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 504, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "In the CoopNets algorithm, the descriptor learns from the observed examples, while the generator", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "learns from the descriptor through the synthesized examples. Therefore, the descriptor is the driving", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "force in terms of learning, although the generator is the driving force in terms of synthesis. In order", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "to understand the convergence of learning, we can start from Algorithm D for learning the descriptor.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "Algorithm D is a stochastic approximation algorithm (Robbins & Monro, 1951), except that the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "samples are generated by finite step MCMC transitions. According to Younes (1999), Algorithm D", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "converges to the maximum likelihood estimate under suitable regularity conditions on the mixing", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 412, + 700 + ], + "score": 1.0, + "content": "of the transition kernel of the MCMC and the schedule of the learning rate", + "type": "text" + }, + { + "bbox": [ + 412, + 690, + 422, + 699 + ], + "score": 0.85, + "content": "\\gamma _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ", even if the number", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 180, + 711 + ], + "score": 1.0, + "content": "of Langevin steps", + "type": "text" + }, + { + "bbox": [ + 180, + 699, + 191, + 710 + ], + "score": 0.87, + "content": "l _ { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 699, + 283, + 711 + ], + "score": 1.0, + "content": "is finite or small (e.g.,", + "type": "text" + }, + { + "bbox": [ + 284, + 699, + 314, + 710 + ], + "score": 0.89, + "content": "l _ { \\mathcal { D } } = 1 .", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 699, + 487, + 711 + ], + "score": 1.0, + "content": "), and even if the number of parallel chains", + "type": "text" + }, + { + "bbox": [ + 487, + 699, + 495, + 709 + ], + "score": 0.82, + "content": "\\tilde { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 187, + 723 + ], + "score": 1.0, + "content": "finite or small (e.g.,", + "type": "text" + }, + { + "bbox": [ + 187, + 710, + 214, + 720 + ], + "score": 0.88, + "content": "\\tilde { n } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "). The reason is that the random fluctuations caused by the finite number", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 149, + 733 + ], + "score": 1.0, + "content": "of chains,", + "type": "text" + }, + { + "bbox": [ + 149, + 721, + 156, + 731 + ], + "score": 0.77, + "content": "\\tilde { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 721, + 409, + 733 + ], + "score": 1.0, + "content": ", and the limited mixing caused by the finite steps of MCMC,", + "type": "text" + }, + { + "bbox": [ + 410, + 721, + 420, + 732 + ], + "score": 0.87, + "content": "l _ { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 721, + 506, + 733 + ], + "score": 1.0, + "content": ", are mitigated if the", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "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": "title", + "bbox": [ + 109, + 83, + 312, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 314, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 314, + 95 + ], + "score": 1.0, + "content": "5.2 QUALITATIVE EXPERIMENT ON SYNTHESIS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 224 + ], + "lines": [ + { + "bbox": [ + 106, + 104, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 116 + ], + "score": 1.0, + "content": "We conduct an experiment on synthesizing images of categories from Imagenet ILSVRC2012", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 113, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 127 + ], + "score": 1.0, + "content": "dataset (Deng et al., 2009) and MIT places205 dataset (Zhou et al., 2014). We adopt a 4-layer", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 125, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 248, + 138 + ], + "score": 1.0, + "content": "descriptor net. The first layer has", + "type": "text" + }, + { + "bbox": [ + 248, + 126, + 286, + 136 + ], + "score": 0.9, + "content": "6 4 ~ 5 \\times 5", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 125, + 505, + 138 + ], + "score": 1.0, + "content": "filters with sub-sampling of 2, the second layers has", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 136, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 107, + 136, + 149, + 147 + ], + "score": 0.87, + "content": "1 2 8 3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 136, + 348, + 149 + ], + "score": 1.0, + "content": "filters with sub-sampling of 2, the third layer has", + "type": "text" + }, + { + "bbox": [ + 348, + 136, + 391, + 147 + ], + "score": 0.77, + "content": "2 5 6 3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 136, + 506, + 149 + ], + "score": 1.0, + "content": "filters with sub-sampling of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 506, + 159 + ], + "score": 1.0, + "content": "1, and the final layer is a fully connected layer with 100 channels as output. We set the number of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 171 + ], + "score": 1.0, + "content": "Langevin dynamics steps in each learning iteration to 10 and the step size to 0.002. The learning rate", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 182 + ], + "score": 1.0, + "content": "is 0.07. For each category, we randomly choose 1,000 images as training data and resize the images", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 178, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 104, + 178, + 116, + 194 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 180, + 149, + 191 + ], + "score": 0.89, + "content": "6 4 \\times 6 4", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 178, + 505, + 194 + ], + "score": 1.0, + "content": ". We run 1, 000 cooperative learning iterations to train the model. Figures 4 and 5 display", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "the results for two categories, where for each category, we show 144 original images sampled from", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 216 + ], + "score": 1.0, + "content": "the training set, and 144 synthesized images generated by our method. The appendix contains more", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 177, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 177, + 225 + ], + "score": 1.0, + "content": "synthesis results.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 104, + 506, + 225 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 229, + 503, + 263 + ], + "lines": [ + { + "bbox": [ + 106, + 229, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 505, + 243 + ], + "score": 1.0, + "content": "As a comparison, we apply the Algorithm G alone and GAN code on the same 1,000 hotel room", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "training images to learn the generator of the same structure as in CoopNets. Figure 6 displays the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 252, + 177, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 177, + 263 + ], + "score": 1.0, + "content": "synthesis results.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 229, + 505, + 263 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 268, + 505, + 346 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 314, + 282 + ], + "score": 1.0, + "content": "We also try to synthesize images at high resolution", + "type": "text" + }, + { + "bbox": [ + 315, + 269, + 361, + 280 + ], + "score": 0.84, + "content": "( 2 2 4 \\times 2 2 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "). We adopt a 4-layer descriptor net.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 181, + 292 + ], + "score": 1.0, + "content": "The first layer has", + "type": "text" + }, + { + "bbox": [ + 181, + 280, + 232, + 290 + ], + "score": 0.89, + "content": "1 2 8 1 5 \\times 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 280, + 437, + 292 + ], + "score": 1.0, + "content": "filters with sub-sampling of 3, the second layer has", + "type": "text" + }, + { + "bbox": [ + 438, + 280, + 479, + 290 + ], + "score": 0.7, + "content": "2 5 6 3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "filters", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 282, + 303 + ], + "score": 1.0, + "content": "with sub-sampling of 2, the third layer has", + "type": "text" + }, + { + "bbox": [ + 282, + 291, + 325, + 301 + ], + "score": 0.89, + "content": "5 1 2 3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "filters with sub-sampling of 1, and the final", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "layer is a fully connected layer with 100 channels as output. We enlarge the filters of the final layer", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 183, + 325 + ], + "score": 1.0, + "content": "of generator net to", + "type": "text" + }, + { + "bbox": [ + 183, + 313, + 218, + 323 + ], + "score": 0.91, + "content": "1 4 \\times 1 4", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 312, + 266, + 325 + ], + "score": 1.0, + "content": "to generate", + "type": "text" + }, + { + "bbox": [ + 266, + 313, + 310, + 323 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "images. The learning rate is 0.05. We run 1000", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "cooperative learning iterations to train the model. Figures 7 and 8 show the synthesized images of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 335, + 284, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 284, + 347 + ], + "score": 1.0, + "content": "two categories from MIT places205 dataset.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 268, + 506, + 347 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 362, + 195, + 375 + ], + "lines": [ + { + "bbox": [ + 104, + 360, + 197, + 378 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 197, + 378 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 503, + 410 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "The most unique feature of our work is that the two networks feed each other the synthesized data", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 399, + 447, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 447, + 412 + ], + "score": 1.0, + "content": "in the learning process, including initial, revised, and reconstructed synthesized data.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 388, + 505, + 412 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 416, + 504, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 429 + ], + "score": 1.0, + "content": "Another unique feature of our work is that the learning process interweaves the existing maximum", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 427, + 316, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 316, + 438 + ], + "score": 1.0, + "content": "likelihood learning algorithms for the two networks.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 106, + 415, + 505, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 443, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "A third unique feature of our work is that the MCMC for the descriptor keeps rejuvenating the chains", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "by refreshing the samples by independent replacements supplied by the generator, so that a single", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "chain effectively amounts to an infinite number of chains or the evolution of the whole marginal", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 477, + 261, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 261, + 489 + ], + "score": 1.0, + "content": "distribution modeled by the generator.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 443, + 505, + 489 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 505, + 192, + 518 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 194, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 194, + 520 + ], + "score": 1.0, + "content": "CODE AND DATA", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 531, + 394, + 541 + ], + "lines": [ + { + "bbox": [ + 106, + 530, + 396, + 543 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 396, + 543 + ], + "score": 1.0, + "content": "http://www.stat.ucla.edu/˜ywu/CoopNets/main.html", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 530, + 396, + 543 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 558, + 267, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 268, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 268, + 574 + ], + "score": 1.0, + "content": "7 APPENDIX: CONVERGENCE", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "title", + "bbox": [ + 108, + 584, + 283, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 584, + 285, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 285, + 597 + ], + "score": 1.0, + "content": "7.1 GENERATOR OF INFINITE CAPACITY", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 504, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "In the CoopNets algorithm, the descriptor learns from the observed examples, while the generator", + "type": "text" + } + ], + "index": 35 + 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Therefore, the descriptor is the driving", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "force in terms of learning, although the generator is the driving force in terms of synthesis. In order", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "to understand the convergence of learning, we can start from Algorithm D for learning the descriptor.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 604, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "Algorithm D is a stochastic approximation algorithm (Robbins & Monro, 1951), except that the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "samples are generated by finite step MCMC transitions. 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In this idealized scenario, the learned", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 101, + 223, + 508, + 249 + ], + "spans": [ + { + "bbox": [ + 101, + 223, + 147, + 249 + ], + "score": 1.0, + "content": "generator", + "type": "text" + }, + { + "bbox": [ + 147, + 228, + 210, + 244 + ], + "score": 0.94, + "content": "P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ^ { ( t + 1 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 223, + 273, + 249 + ], + "score": 1.0, + "content": "will reproduce", + "type": "text" + }, + { + "bbox": [ + 273, + 228, + 301, + 243 + ], + "score": 0.94, + "content": "P _ { \\mathcal { D } } ^ { ( t + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 223, + 365, + 249 + ], + "score": 1.0, + "content": "by minimizing", + "type": "text" + }, + { + "bbox": [ + 365, + 228, + 480, + 243 + ], + "score": 0.92, + "content": "\\operatorname { K L } ( P _ { \\mathcal { D } } ^ { ( t + 1 ) } ( Y ) | P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 223, + 508, + 249 + ], + "score": 1.0, + "content": ", with", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 237, + 509, + 263 + ], + "spans": [ + { + "bbox": [ + 107, + 243, + 135, + 258 + ], + "score": 0.92, + "content": "P _ { \\mathcal { D } } ^ { ( t + 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 237, + 412, + 263 + ], + "score": 1.0, + "content": "serving as its data distribution. Then eventually the learned generator", + "type": "text" + }, + { + "bbox": [ + 413, + 244, + 459, + 258 + ], + "score": 0.93, + "content": "P _ { \\mathcal { G } } ( Y , \\hat { W } _ { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 237, + 509, + 263 + ], + "score": 1.0, + "content": "will repro-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 128, + 271 + ], + "score": 1.0, + "content": "duce", + "type": "text" + }, + { + "bbox": [ + 128, + 258, + 179, + 271 + ], + "score": 0.93, + "content": "P _ { \\mathcal { D } } ( Y ; \\hat { W } _ { \\mathcal { D } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 258, + 505, + 271 + ], + "score": 1.0, + "content": ". Thus the cooperative training helps the learning of the generator. Note that the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 270, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 179, + 285 + ], + "score": 1.0, + "content": "learned generator", + "type": "text" + }, + { + "bbox": [ + 179, + 270, + 225, + 284 + ], + "score": 0.93, + "content": "P _ { \\mathcal { G } } ( Y , \\hat { W } _ { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 271, + 450, + 285 + ], + "score": 1.0, + "content": "will not reproduce the distribution of the observed data", + "type": "text" + }, + { + "bbox": [ + 450, + 272, + 473, + 283 + ], + "score": 0.9, + "content": "P _ { \\mathrm { d a t a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 271, + 505, + 285 + ], + "score": 1.0, + "content": ", unless", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 283, + 267, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 267, + 295 + ], + "score": 1.0, + "content": "the descriptor is of infinite capacity too.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 299, + 506, + 443 + ], + "lines": [ + { + "bbox": [ + 107, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "Conversely, the generator net also helps the learning of the descriptor net in the CoopNets algorithm.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 399, + 324 + ], + "score": 1.0, + "content": "In Algorithm D, it is impractical to make the number of parallel chains", + "type": "text" + }, + { + "bbox": [ + 399, + 311, + 407, + 321 + ], + "score": 0.8, + "content": "\\tilde { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "too large. On the other", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 321, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 329, + 336 + ], + "score": 1.0, + "content": "hand, it would be difficult for a small number of chains", + "type": "text" + }, + { + "bbox": [ + 329, + 321, + 398, + 335 + ], + "score": 0.92, + "content": "\\{ \\bar { \\tilde { Y } _ { i } } , i = 1 , . . . , \\tilde { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "to explore the state space.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 101, + 330, + 509, + 356 + ], + "spans": [ + { + "bbox": [ + 101, + 330, + 255, + 356 + ], + "score": 1.0, + "content": "In the CoopNets algorithm, because", + "type": "text" + }, + { + "bbox": [ + 256, + 334, + 309, + 350 + ], + "score": 0.94, + "content": "P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ^ { ( t ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 330, + 358, + 356 + ], + "score": 1.0, + "content": "reproduces", + "type": "text" + }, + { + "bbox": [ + 359, + 335, + 377, + 349 + ], + "score": 0.91, + "content": "P _ { \\mathcal { D } } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 330, + 509, + 356 + ], + "score": 1.0, + "content": ", we can generate a completely", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 101, + 348, + 509, + 368 + ], + "spans": [ + { + "bbox": [ + 101, + 348, + 249, + 368 + ], + "score": 1.0, + "content": "new batch of independent samples", + "type": "text" + }, + { + "bbox": [ + 249, + 350, + 270, + 365 + ], + "score": 0.92, + "content": "\\{ \\hat { Y } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 348, + 294, + 368 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 294, + 349, + 347, + 365 + ], + "score": 0.94, + "content": "\\overset { \\cdot } { P _ { \\mathcal { G } } } ( Y ; W _ { \\mathcal { G } } ^ { ( t ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 348, + 397, + 368 + ], + "score": 1.0, + "content": ", and revise", + "type": "text" + }, + { + "bbox": [ + 397, + 350, + 417, + 364 + ], + "score": 0.92, + "content": "\\{ \\hat { Y } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 348, + 430, + 368 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 430, + 350, + 450, + 364 + ], + "score": 0.92, + "content": "\\{ \\tilde { Y } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 348, + 509, + 368 + ], + "score": 1.0, + "content": "by Langevin", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 415, + 378 + ], + "score": 1.0, + "content": "dynamics, instead of running Langevin dynamics from the same old batch of", + "type": "text" + }, + { + "bbox": [ + 415, + 365, + 435, + 378 + ], + "score": 0.93, + "content": "\\{ \\tilde { Y } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "as in the original", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "Algorithm D. This is like implementing an infinite number of parallel chains, because each iteration", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 387, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 401 + ], + "score": 1.0, + "content": "evolves a fresh batch of examples, as if each iteration evolves a new set of chains. By updating", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 397, + 504, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 163, + 412 + ], + "score": 1.0, + "content": "the generator", + "type": "text" + }, + { + "bbox": [ + 164, + 399, + 180, + 410 + ], + "score": 0.9, + "content": "W _ { \\mathcal { G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 397, + 487, + 412 + ], + "score": 1.0, + "content": ", it is like we are updating the infinite number of parallel chains, because", + "type": "text" + }, + { + "bbox": [ + 487, + 399, + 504, + 410 + ], + "score": 0.89, + "content": "W _ { \\mathcal { G } }", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 287, + 423 + ], + "score": 1.0, + "content": "memorizes the whole distribution. Even if", + "type": "text" + }, + { + "bbox": [ + 288, + 410, + 295, + 420 + ], + "score": 0.82, + "content": "\\tilde { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 409, + 470, + 423 + ], + "score": 1.0, + "content": "in the CoopNets algorithm is small, e.g.,", + "type": "text" + }, + { + "bbox": [ + 471, + 410, + 501, + 420 + ], + "score": 0.88, + "content": "\\tilde { n } \\ = \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 409, + 506, + 423 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 420, + 504, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 322, + 433 + ], + "score": 1.0, + "content": "viewed from the perspective of Algorithm D, it is as if", + "type": "text" + }, + { + "bbox": [ + 323, + 421, + 356, + 431 + ], + "score": 0.9, + "content": "\\tilde { n } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 420, + 471, + 433 + ], + "score": 1.0, + "content": ". Thus the above idealization", + "type": "text" + }, + { + "bbox": [ + 471, + 421, + 504, + 431 + ], + "score": 0.88, + "content": "\\tilde { n } \\to \\infty", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 432, + 144, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 144, + 442 + ], + "score": 1.0, + "content": "is sound.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 109, + 457, + 272, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 275, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 275, + 470 + ], + "score": 1.0, + "content": "7.2 GENERATOR OF FINITE CAPACITY", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 311, + 492 + ], + "score": 1.0, + "content": "From an information geometry point of view, let", + "type": "text" + }, + { + "bbox": [ + 311, + 478, + 424, + 491 + ], + "score": 0.92, + "content": "\\mathcal { D } = \\{ P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } ) , \\forall W _ { \\mathcal { D } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "be the manifold of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 298, + 502 + ], + "score": 1.0, + "content": "the descriptor models, where each distribution", + "type": "text" + }, + { + "bbox": [ + 298, + 490, + 348, + 502 + ], + "score": 0.92, + "content": "P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "is a point on this manifold. Then the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 238, + 513 + ], + "score": 1.0, + "content": "maximum likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 239, + 501, + 255, + 511 + ], + "score": 0.9, + "content": "W _ { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 500, + 407, + 513 + ], + "score": 1.0, + "content": "is a projection of the data distribution", + "type": "text" + }, + { + "bbox": [ + 407, + 501, + 431, + 511 + ], + "score": 0.91, + "content": "P _ { \\mathrm { d a t a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "onto the manifold", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 107, + 512, + 115, + 521 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 511, + 137, + 524 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 137, + 511, + 241, + 523 + ], + "score": 0.9, + "content": "\\mathcal { G } = \\{ P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) , \\forall W _ { \\mathcal { G } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "be the manifold of the generator models, where each distribution", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 107, + 523, + 155, + 534 + ], + "score": 0.92, + "content": "P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 522, + 430, + 536 + ], + "score": 1.0, + "content": "is a point on this manifold. Then the maximum likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 430, + 523, + 446, + 534 + ], + "score": 0.9, + "content": "W _ { \\mathcal { G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "is a projection", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 309, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 199, + 546 + ], + "score": 1.0, + "content": "of the data distribution", + "type": "text" + }, + { + "bbox": [ + 200, + 534, + 223, + 545 + ], + "score": 0.9, + "content": "P _ { \\mathrm { d a t a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 533, + 297, + 546 + ], + "score": 1.0, + "content": "onto the manifold", + "type": "text" + }, + { + "bbox": [ + 298, + 534, + 305, + 544 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 533, + 309, + 546 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 502, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 433, + 563 + ], + "score": 1.0, + "content": "From now on, for notational simplicity and with a slight abuse of notation, we use", + "type": "text" + }, + { + "bbox": [ + 433, + 550, + 450, + 561 + ], + "score": 0.89, + "content": "W _ { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 550, + 504, + 563 + ], + "score": 1.0, + "content": "to denote the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 560, + 497, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 197, + 575 + ], + "score": 1.0, + "content": "descriptor distribution", + "type": "text" + }, + { + "bbox": [ + 198, + 561, + 248, + 573 + ], + "score": 0.92, + "content": "P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 560, + 284, + 575 + ], + "score": 1.0, + "content": ", and use", + "type": "text" + }, + { + "bbox": [ + 285, + 561, + 300, + 573 + ], + "score": 0.92, + "content": "W _ { \\mathcal { G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 560, + 444, + 575 + ], + "score": 1.0, + "content": "to denote the generator distribution", + "type": "text" + }, + { + "bbox": [ + 444, + 561, + 493, + 573 + ], + "score": 0.92, + "content": "P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 560, + 497, + 575 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 504, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 262, + 591 + ], + "score": 1.0, + "content": "We assume both the observed data size", + "type": "text" + }, + { + "bbox": [ + 262, + 581, + 270, + 588 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 577, + 386, + 591 + ], + "score": 1.0, + "content": "and the synthesized data size", + "type": "text" + }, + { + "bbox": [ + 387, + 579, + 394, + 588 + ], + "score": 0.82, + "content": "\\tilde { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "are large enough so that we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "score": 1.0, + "content": "shall work on distributions or populations instead of finite samples. As explained above, assuming", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 600, + 451, + 613 + ], + "spans": [ + { + "bbox": [ + 107, + 601, + 139, + 610 + ], + "score": 0.86, + "content": "\\tilde { n } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 600, + 451, + 613 + ], + "score": 1.0, + "content": "is sound because the generator net can supply unlimited number of examples.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 617, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 101, + 615, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 101, + 615, + 351, + 638 + ], + "score": 1.0, + "content": "The Langevin revision dynamics runs a Markov chain from", + "type": "text" + }, + { + "bbox": [ + 352, + 617, + 373, + 633 + ], + "score": 0.92, + "content": "W _ { \\mathcal { G } } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 616, + 436, + 633 + ], + "score": 1.0, + "content": ") towards W (t)D .", + "type": "text" + }, + { + "bbox": [ + 431, + 617, + 453, + 634 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 453, + 619, + 475, + 632 + ], + "score": 0.9, + "content": "\\mathbf { L } _ { W _ { \\mathcal { D } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 617, + 506, + 634 + ], + "score": 1.0, + "content": "be the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 630, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 222, + 643 + ], + "score": 1.0, + "content": "Markov transition kernel of", + "type": "text" + }, + { + "bbox": [ + 222, + 631, + 234, + 642 + ], + "score": 0.89, + "content": "l _ { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 630, + 385, + 643 + ], + "score": 1.0, + "content": "steps of Langevin revisions towards", + "type": "text" + }, + { + "bbox": [ + 386, + 631, + 403, + 642 + ], + "score": 0.89, + "content": "W _ { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 630, + 505, + 643 + ], + "score": 1.0, + "content": ". 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Then eventually the learned generator", + "type": "text" + }, + { + "bbox": [ + 413, + 244, + 459, + 258 + ], + "score": 0.93, + "content": "P _ { \\mathcal { G } } ( Y , \\hat { W } _ { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 237, + 509, + 263 + ], + "score": 1.0, + "content": "will repro-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 128, + 271 + ], + "score": 1.0, + "content": "duce", + "type": "text" + }, + { + "bbox": [ + 128, + 258, + 179, + 271 + ], + "score": 0.93, + "content": "P _ { \\mathcal { D } } ( Y ; \\hat { W } _ { \\mathcal { D } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 258, + 505, + 271 + ], + "score": 1.0, + "content": ". Thus the cooperative training helps the learning of the generator. Note that the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 270, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 179, + 285 + ], + "score": 1.0, + "content": "learned generator", + "type": "text" + }, + { + "bbox": [ + 179, + 270, + 225, + 284 + ], + "score": 0.93, + "content": "P _ { \\mathcal { G } } ( Y , \\hat { W } _ { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 271, + 450, + 285 + ], + "score": 1.0, + "content": "will not reproduce the distribution of the observed data", + "type": "text" + }, + { + "bbox": [ + 450, + 272, + 473, + 283 + ], + "score": 0.9, + "content": "P _ { \\mathrm { d a t a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 271, + 505, + 285 + ], + "score": 1.0, + "content": ", unless", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 283, + 267, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 267, + 295 + ], + "score": 1.0, + "content": "the descriptor is of infinite capacity too.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 101, + 181, + 510, + 295 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 299, + 506, + 443 + ], + "lines": [ + { + "bbox": [ + 107, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "Conversely, the generator net also helps the learning of the descriptor net in the CoopNets algorithm.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 399, + 324 + ], + "score": 1.0, + "content": "In Algorithm D, it is impractical to make the number of parallel chains", + "type": "text" + }, + { + "bbox": [ + 399, + 311, + 407, + 321 + ], + "score": 0.8, + "content": "\\tilde { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "too large. On the other", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 321, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 329, + 336 + ], + "score": 1.0, + "content": "hand, it would be difficult for a small number of chains", + "type": "text" + }, + { + "bbox": [ + 329, + 321, + 398, + 335 + ], + "score": 0.92, + "content": "\\{ \\bar { \\tilde { Y } _ { i } } , i = 1 , . . . , \\tilde { n } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 322, + 505, + 336 + ], + "score": 1.0, + "content": "to explore the state space.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 101, + 330, + 509, + 356 + ], + "spans": [ + { + "bbox": [ + 101, + 330, + 255, + 356 + ], + "score": 1.0, + "content": "In the CoopNets algorithm, because", + "type": "text" + }, + { + "bbox": [ + 256, + 334, + 309, + 350 + ], + "score": 0.94, + "content": "P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ^ { ( t ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 330, + 358, + 356 + ], + "score": 1.0, + "content": "reproduces", + "type": "text" + }, + { + "bbox": [ + 359, + 335, + 377, + 349 + ], + "score": 0.91, + "content": "P _ { \\mathcal { D } } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 330, + 509, + 356 + ], + "score": 1.0, + "content": ", we can generate a completely", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 101, + 348, + 509, + 368 + ], + "spans": [ + { + "bbox": [ + 101, + 348, + 249, + 368 + ], + "score": 1.0, + "content": "new batch of independent samples", + "type": "text" + }, + { + "bbox": [ + 249, + 350, + 270, + 365 + ], + "score": 0.92, + "content": "\\{ \\hat { Y } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 348, + 294, + 368 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 294, + 349, + 347, + 365 + ], + "score": 0.94, + "content": "\\overset { \\cdot } { P _ { \\mathcal { G } } } ( Y ; W _ { \\mathcal { G } } ^ { ( t ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 348, + 397, + 368 + ], + "score": 1.0, + "content": ", and revise", + "type": "text" + }, + { + "bbox": [ + 397, + 350, + 417, + 364 + ], + "score": 0.92, + "content": "\\{ \\hat { Y } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 348, + 430, + 368 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 430, + 350, + 450, + 364 + ], + "score": 0.92, + "content": "\\{ \\tilde { Y } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 348, + 509, + 368 + ], + "score": 1.0, + "content": "by Langevin", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 415, + 378 + ], + "score": 1.0, + "content": "dynamics, instead of running Langevin dynamics from the same old batch of", + "type": "text" + }, + { + "bbox": [ + 415, + 365, + 435, + 378 + ], + "score": 0.93, + "content": "\\{ \\tilde { Y } _ { i } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "as in the original", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "Algorithm D. This is like implementing an infinite number of parallel chains, because each iteration", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 387, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 401 + ], + "score": 1.0, + "content": "evolves a fresh batch of examples, as if each iteration evolves a new set of chains. By updating", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 397, + 504, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 163, + 412 + ], + "score": 1.0, + "content": "the generator", + "type": "text" + }, + { + "bbox": [ + 164, + 399, + 180, + 410 + ], + "score": 0.9, + "content": "W _ { \\mathcal { G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 397, + 487, + 412 + ], + "score": 1.0, + "content": ", it is like we are updating the infinite number of parallel chains, because", + "type": "text" + }, + { + "bbox": [ + 487, + 399, + 504, + 410 + ], + "score": 0.89, + "content": "W _ { \\mathcal { G } }", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 287, + 423 + ], + "score": 1.0, + "content": "memorizes the whole distribution. Even if", + "type": "text" + }, + { + "bbox": [ + 288, + 410, + 295, + 420 + ], + "score": 0.82, + "content": "\\tilde { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 409, + 470, + 423 + ], + "score": 1.0, + "content": "in the CoopNets algorithm is small, e.g.,", + "type": "text" + }, + { + "bbox": [ + 471, + 410, + 501, + 420 + ], + "score": 0.88, + "content": "\\tilde { n } \\ = \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 409, + 506, + 423 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 420, + 504, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 322, + 433 + ], + "score": 1.0, + "content": "viewed from the perspective of Algorithm D, it is as if", + "type": "text" + }, + { + "bbox": [ + 323, + 421, + 356, + 431 + ], + "score": 0.9, + "content": "\\tilde { n } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 420, + 471, + 433 + ], + "score": 1.0, + "content": ". Thus the above idealization", + "type": "text" + }, + { + "bbox": [ + 471, + 421, + 504, + 431 + ], + "score": 0.88, + "content": "\\tilde { n } \\to \\infty", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 432, + 144, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 144, + 442 + ], + "score": 1.0, + "content": "is sound.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5, + "bbox_fs": [ + 101, + 300, + 509, + 442 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 457, + 272, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 275, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 275, + 470 + ], + "score": 1.0, + "content": "7.2 GENERATOR OF FINITE CAPACITY", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 477, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 311, + 492 + ], + "score": 1.0, + "content": "From an information geometry point of view, let", + "type": "text" + }, + { + "bbox": [ + 311, + 478, + 424, + 491 + ], + "score": 0.92, + "content": "\\mathcal { D } = \\{ P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } ) , \\forall W _ { \\mathcal { D } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "be the manifold of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 298, + 502 + ], + "score": 1.0, + "content": "the descriptor models, where each distribution", + "type": "text" + }, + { + "bbox": [ + 298, + 490, + 348, + 502 + ], + "score": 0.92, + "content": "P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "is a point on this manifold. Then the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 238, + 513 + ], + "score": 1.0, + "content": "maximum likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 239, + 501, + 255, + 511 + ], + "score": 0.9, + "content": "W _ { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 500, + 407, + 513 + ], + "score": 1.0, + "content": "is a projection of the data distribution", + "type": "text" + }, + { + "bbox": [ + 407, + 501, + 431, + 511 + ], + "score": 0.91, + "content": "P _ { \\mathrm { d a t a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "onto the manifold", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 107, + 512, + 115, + 521 + ], + "score": 0.79, + "content": "\\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 511, + 137, + 524 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 137, + 511, + 241, + 523 + ], + "score": 0.9, + "content": "\\mathcal { G } = \\{ P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } ) , \\forall W _ { \\mathcal { G } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "be the manifold of the generator models, where each distribution", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 522, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 107, + 523, + 155, + 534 + ], + "score": 0.92, + "content": "P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 522, + 430, + 536 + ], + "score": 1.0, + "content": "is a point on this manifold. Then the maximum likelihood estimate of", + "type": "text" + }, + { + "bbox": [ + 430, + 523, + 446, + 534 + ], + "score": 0.9, + "content": "W _ { \\mathcal { G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 522, + 505, + 536 + ], + "score": 1.0, + "content": "is a projection", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 533, + 309, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 199, + 546 + ], + "score": 1.0, + "content": "of the data distribution", + "type": "text" + }, + { + "bbox": [ + 200, + 534, + 223, + 545 + ], + "score": 0.9, + "content": "P _ { \\mathrm { d a t a } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 533, + 297, + 546 + ], + "score": 1.0, + "content": "onto the manifold", + "type": "text" + }, + { + "bbox": [ + 298, + 534, + 305, + 544 + ], + "score": 0.84, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 533, + 309, + 546 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 477, + 506, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 502, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 504, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 433, + 563 + ], + "score": 1.0, + "content": "From now on, for notational simplicity and with a slight abuse of notation, we use", + "type": "text" + }, + { + "bbox": [ + 433, + 550, + 450, + 561 + ], + "score": 0.89, + "content": "W _ { \\mathcal { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 550, + 504, + 563 + ], + "score": 1.0, + "content": "to denote the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 560, + 497, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 197, + 575 + ], + "score": 1.0, + "content": "descriptor distribution", + "type": "text" + }, + { + "bbox": [ + 198, + 561, + 248, + 573 + ], + "score": 0.92, + "content": "P _ { \\mathcal { D } } ( Y ; W _ { \\mathcal { D } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 560, + 284, + 575 + ], + "score": 1.0, + "content": ", and use", + "type": "text" + }, + { + "bbox": [ + 285, + 561, + 300, + 573 + ], + "score": 0.92, + "content": "W _ { \\mathcal { G } }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 560, + 444, + 575 + ], + "score": 1.0, + "content": "to denote the generator distribution", + "type": "text" + }, + { + "bbox": [ + 444, + 561, + 493, + 573 + ], + "score": 0.92, + "content": "P _ { \\mathcal { G } } ( Y ; W _ { \\mathcal { G } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 560, + 497, + 575 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 550, + 504, + 575 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 504, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 262, + 591 + ], + "score": 1.0, + "content": "We assume both the observed data size", + "type": "text" + }, + { + "bbox": [ + 262, + 581, + 270, + 588 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 577, + 386, + 591 + ], + "score": 1.0, + "content": "and the synthesized data size", + "type": "text" + }, + { + "bbox": [ + 387, + 579, + 394, + 588 + ], + "score": 0.82, + "content": "\\tilde { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "are large enough so that we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "score": 1.0, + "content": "shall work on distributions or populations instead of finite samples. 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computer vision workloads like image classification and object detection. The performant systems, however, typically involve big models with numerous parameters. Once trained, a challenging aspect for such top performing models is deployment on resource constrained inference systems — the models (often deep networks or wide networks or both) are compute and memory intensive. Low-precision numerics and model compression using knowledge distillation are popular techniques to lower both the compute requirements and memory footprint of these deployed models. In this paper, we study combination of these two techniques and show that the performance of low-precision networks can be significantly improved by using knowledge distillation techniques. Our approach, $\mathcal { A }$ pprentice, achieves state-of-the-art accuracies using ternary precision and 4-bit precision for variants of ResNet architecture on ImageNet dataset. We present three schemes using which one can apply knowledge distillation techniques to various stages of the train-and-deploy pipeline. + +# 1 INTRODUCTION + +Background: Today’s high performing deep neural networks (DNNs) for computer vision applications comprise of multiple layers and involve numerous parameters. These networks have ${ \mathcal { O } } ( { \mathrm { G i g a - } }$ FLOPS) compute requirements and generate models which are $\mathcal { O }$ (Mega-Bytes) in storage (Canziani et al., 2016). Further, the memory and compute requirements during training and inference are quite different (Mishra et al., 2017). Training is performed on big datasets with large batch-sizes where memory footprint of activations dominates the model memory footprint. On the other hand, batchsize during inference is typically small and the model’s memory footprint dominates the runtime memory requirements. + +Because of complexity in compute, memory and storage requirements, training phase of the networks is performed on CPU and/or GPU clusters in a distributed computing environment. Once trained, a challenging aspect is deployment of trained models on resource constrained inference systems such as portable devices or sensor networks, and for applications in which real-time predictions are required. Performing inference on edge-devices comes with severe constraints on memory, compute and power. Additionally, ensemble based methods, which one can potentially use to get improved accuracy predictions, become prohibitive in resource constrained systems. + +Quantization using low-precision numerics (Vanhoucke et al., 2011; Zhou et al., 2016; Lin et al., 2015; Miyashita et al., 2016; Gupta et al., 2015; Zhu et al., 2016; Rastegari et al., 2016; Courbariaux et al., 2015; Umuroglu et al., 2016; Mishra et al., 2017) and model compression (Bucilua et al. ˇ , 2006; Hinton et al., 2015; Romero et al., 2014) have emerged as popular solutions for resource constrained deployment scenarios. With quantization, a low-precision version of network model is generated and deployed on the device. Operating in lower precision mode reduces compute as well as data movement and storage requirements. However, majority of existing works in low-precision DNNs sacrifice accuracy over baseline full-precision networks. With model compression, a smaller low memory footprint network is trained to mimic the behaviour of the original complex network. During this training, a process called, knowledge distillation is used to “transfer knowledge” from the complex network to the smaller network. Work by Hinton et al. (2015) shows that the knowledge distillation scheme can yield networks at comparable or slightly better accuracy than the original complex model. However, to the best of our knowledge, all prior works using model compression techniques target compression at full-precision. + +Our proposal: In this paper, we study the combination of network quantization with model compression and show that accuracies of low-precision networks can be significantly improved by using knowledge distillation techniques. Previous studies on model compression use a large network as the teacher network and a small network as the student network. The small student network learns from teacher network using distillation process. The network architecture of the student network is typically different from that of the teacher network – for e.g. Hinton et al. (2015) investigate a student network that has fewer number of neurons in the hidden layers compared to the teacher network. In our work, the student network has similar topology as that of teacher network, except that the student network has low-precision neurons compared to the teacher network which has neurons operating at full-precision. + +We call our approach Apprentice1 and study three schemes which produce low-precision networks using knowledge distillation techniques. Each of these three schemes produce state-of-the-art ternary precision and 4-bit precision models. + +In the first scheme, a low-precision network and a full-precision network are jointly trained from scratch using knowledge distillation scheme. Later in the paper we describe the rationale behind this approach. Using this scheme, a new state-of-the-art accuracy is obtained for ternary and 4-bit precision for ResNet-18, ResNet-34 and ResNet-50 on ImageNet dataset. In fact, using this scheme the accuracy of the full-precision model also slightly improves. This scheme then serves as the new baseline for the other two schemes we investigate. + +In the second scheme, we start with a full-precision trained network and transfer knowledge from this trained network continuously to train a low-precision network from scratch. We find that the low-precision network converges faster (albeit to similar accuracies as the first scheme) when a trained complex network guides its training. + +In the third scheme, we start with a trained full-precision large network and an apprentice network that has been initialised with full-precision weights. The apprentice network’s precision is lowered and is fine-tuned using knowledge distillation techniques. We find that the low-precision network’s accuracy marginally improves and surpasses the accuracy obtained via the first scheme. This scheme then sets the new state-of-the-art accuracies for the ResNet models at ternary and 4-bit precision. + +Overall, the contributions of this paper are the techniques to obtain low-precision DNNs using knowledge distillation technique. Each of our scheme produces a low-precision model that surpasses the accuracy of the equivalent low-precision model published to date. One of our schemes also helps a low-precision model converge faster. We envision these accurate low-precision models to simplify the inference deployment process on resource constrained systems and even otherwise on cloud-based deployment systems. + +# 2 MOTIVATION FOR LOW-PRECISION MODEL PARAMETERS + +Lowering precision of model parameters: Resource constrained inference systems impose significant restrictions on memory, compute and power budget. With regard to storage, model (or weight) parameters and activation maps occupy memory during the inference phase of DNNs. During this phase memory is allocated for input (IFM) and output feature maps (OFM) required by a single layer in the DNN, and these dynamic memory allocations are reused for other layers. The total memory allocation during inference is then the maximum of IFM and maximum of OFM memory required across all the layers plus the sum of all weight tensors (Mishra et al., 2017). When inference phase for DNNs is performed with a small batch size, the memory footprint of the weights exceeds the footprint of the activation maps. This aspect is shown in Figure 1 for 4 different networks (AlexNet (Krizhevsky et al., 2012), Inception-Resnet-v2 (Szegedy et al., 2016), ResNet-50 and ResNet-101 (He et al., 2015)) running $2 2 4 \mathrm { x } 2 2 4$ image patches. Thus lowering the precision of the weight tensors helps lower the memory requirements during deployment. One other aspect of lowering memory footprint is that the working set size of the workload starts to fit on chip and by reducing accesses to DRAM (off-chip) memory, the compute core starts to see better performance and energy savings (DRAM accesses are expensive in latency and energy). + +Benefit of low-precision compute: Low-precision compute simplifies hardware implementation. For example, the compute unit to perform the convolution operation (multiplication of two operands) involves a floating-point multiplier when using fullprecision weights and activations. The floatingpoint multiplier can be replaced with a much simpler circuitry (xnor and popcount logic elements) when using binary precision for weights and activations (Courbariaux & Bengio, 2016; Rastegari et al., 2016; Courbariaux et al., 2015). Similarly, when using ternary precision for weights and full-precision for activations, the multiplier unit can be replaced with a sign comparator unit. Simpler hardware also helps lower the inference latency and energy budget. Thus, operating in lower precision mode reduces compute as well as data movement and storage requirements. + +![](images/9891559225e3a0edb31812399103f038b11fe71bbdf203533e28c2c2d622f4f3.jpg) +Figure 1: Memory footprint of activations (ACTs) and weights (W) during inference for mini-batch sizes 1 and 8. + +The drawback of low-precision models, however, is degraded accuracy. We discuss later in the paper the network accuracies obtained using methods proposed in literature. These accuracies serve as the starting point and baselines we compare to in our work. + +# 3 RELATED WORK + +Low-precision networks: Low-precision DNNs are an active area of research. Most low-precision networks acknowledge the over parameterization aspect of today’s DNN architectures and/or the aspect that lowering the precision of neurons post-training often does not impact the final performance. Reducing precision of weights for efficient inference pipeline has been very well studied. Works like Binary connect (BC) (Courbariaux et al., 2015), Ternary-weight networks (TWN) (Li & Liu, 2016), fine-grained ternary quantization (Mellempudi et al., 2017) and INQ (Zhou et al., 2017) target precision reduction of network weights. Accuracy is almost always affected when quantizing the weights significantly below 8-bits of precision. For AlexNet on ImageNet, TWN loses $5 \%$ Top-1 accuracy. Schemes like INQ, work in Sung et al. (2015) and Mellempudi et al. (2017) do fine-tuning to quantize the network weights. + +Work in XNOR-NET (Rastegari et al., 2016), binary neural networks (Courbariaux & Bengio, 2016), DoReFa (Zhou et al., 2016) and trained ternary quantization (TTQ) (Zhu et al., 2016) target training pipeline. While TTQ targets weight quantization, most works targeting activation quantization show that quantizing activations always hurt accuracy. XNOR-NET approach degrades Top-1 accuracy by $12 \%$ and DoReFa by $8 \%$ when quantizing both weights and activations to 1-bit (for AlexNet on ImageNet). Work by Gupta et al. (2015) advocates for low-precision fixed-point numbers for training. They show 16-bits to be sufficient for training on CIFAR10 dataset. Work by Seide et al. (2014) quantizes gradients in a distributed computing system. + +Knowledge distillation methods: The general technique in distillation based methods involves using a teacher-student strategy, where a large deep network trained for a given task teaches shallower student network(s) on the same task. The core concepts behind knowledge distillation or transfer technique have been around for a while. Bucilua et al. ˇ (2006) show that one can compress the information in an ensemble into a single network. Ba & Caurana (2013) extend this approach to study shallow, but wide, fully connected topologies by mimicking deep neural networks. To facilitate learning, the authors introduce the concepts of learning on logits rather than the probability distribution. + +Hinton et al. (2015) propose a framework to transfer knowledge by introducing the concept of temperature. The key idea is to divide the logits by a temperature factor before performing a Softmax function. By using a higher temperature factor the activations of incorrect classes are boosted. This then facilitates more information flowing to the model parameters during back-propagation operation. FitNets (Romero et al., 2014) extend this work by using intermediate hidden layer outputs as target values for training a deeper, but thinner, student model. Net2Net (Chen et al., 2015a) also uses a teacher-student network system with a function-preserving transformation approach to initialize the parameters of the student network. The goal in Net2Net approach is to accelerate the training of a larger student network. Zagoruyko & Komodakis (2016) use attention as a mechanism for transferring knowledge from one network to another. In a similar theme, Yim et al. (2017) propose an information metric using which a teacher DNN can transfer the distilled knowledge to other student DNNs. In N2N learning work, Ashok et al. (2017) propose a reinforcement learning based approach for compressing a teacher network into an equally capable student network. They achieve a compression factor of 10x for ResNet-34 on CIFAR datasets. + +Sparsity and hashing: Few other popular techniques for model compression are pruning (LeCun et al., 1990; Han et al., 2015a; Wen et al., 2016; Han et al., 2015b), hashing (Weinberger et al., 2009) and weight sharing (Chen et al., 2015b; Denil et al., 2013). Pruning leads to removing neurons entirely from the final trained model making the model a sparse structure. With hashing and weight sharing schemes a hash function is used to alias several weight parameters into few hash buckets, effectively lowering the parameter memory footprint. To realize benefits of sparsity and hashing schemes during runtime, efficient hardware support is required (e.g. support for irregular memory accesses (Han et al., 2016; Venkatesh et al., 2016; Parashar et al., 2017)). + +# 4 KNOWLEDGE DISTILLATION + +We introduce the concept of knowledge distillation in this section. Bucilua et al. ˇ (2006), Hinton et al. (2015) and Urban et al. (2016) analyze this topic in great detail. + +Figure 2 shows the schematic of the knowledge distillation setup. Given an input image $x$ , a teacher DNN maps this image to predictions $p ^ { T }$ . The $C$ class predictions are obtained by applying Softmax function on the un-normalized log probability values $z$ (the logits), i.e. $\begin{array} { r } { p ^ { T } = e ^ { z _ { k } ^ { T } } / \sum _ { j } ^ { C } e ^ { z _ { j } ^ { T } } } \end{array}$ . The same image is fed to the student network and it predicts $p ^ { A } = e ^ { z _ { k } ^ { A } } / { \sum _ { j } ^ { C } e ^ { z _ { j } ^ { A } } }$ . During training, the cost function, , is given as: + +$$ +\mathcal { L } ( x ; W _ { T } , W _ { A } ) = \alpha \mathcal { H } ( y , p ^ { T } ) + \beta \mathcal { H } ( y , p ^ { A } ) + \gamma \mathcal { H } ( z ^ { T } , p ^ { A } ) +$$ + +where, $W _ { T }$ and $W _ { A }$ are the parameters of the teacher and the student (apprentice) network, respectively, $y$ is the ground truth, $\mathcal { H } ( \cdot )$ denotes a loss function and, $\alpha , \beta$ and $\gamma$ are weighting factors to prioritize the output of a certain loss function over the other. + +In equation 1, lowering the first term of the cost function gives a better teacher network and lowering the second term gives a better student network. The third term is the knowledge distillation term whereby the student network attempts to mimic the knowledge in the teacher network. In Hinton et al. (2015), the logits of the teacher network are divided by a temperature factor $\tau$ . Using a higher value for $\tau$ produces a softer probability distribution when taking the Softmax of the logits. In our studies, we use cross-entropy function for $\mathcal { H } ( \cdot )$ , set $\alpha = 1$ , $\beta = 0 . 5$ and $\gamma = 0 . 5$ and, perform the transfer learning process using the logits (inputs to the Softmax function) of the teacher network. In our experiments we study the effect of varying the depth of the teacher and the student network, and the precision of the neurons in the student network. + +# 5 OUR APPROACH - APPRENTICE NETWORK + +Low-precision DNNs target the storage and compute efficiency aspects of the network. Model compression targets the same efficiency parameters from the point of view of network architecture. With + +![](images/de4bb554a8cf5d3c54e0cec30d054c2e4a524e80b0ca5c77ec209cd41c0c4ecd.jpg) +Figure 2: Schematic of the knowledge distillation setup. The teacher network is a high precision network and the apprentice network is a low-precision network. + +Apprentice we combine both these techniques to improve the network accuracy as well as the runtime efficiency of DNNs. Using the teacher-student setup described in the last section, we investigate three schemes using which one can obtain a low-precision model for the student network. The first scheme (scheme-A) jointly trains both the networks - full-precision teacher and low-precision student network. The second scheme (scheme-B) trains only the low-precision student network but distills knowledge from a trained full-precision teacher network throughout the training process. The third scheme (scheme-C) starts with a trained full-precision teacher and a full-precision student network but fine-tunes the student network after lowering its precision. Before we get into the details of each of these schemes, we discuss the accuracy numbers obtained using low-precision schemes described in literature. These accuracy figures serve as the baseline for comparative analysis. + +# 5.1 TOP-1 ERROR WITH PRIOR PROPOSALS FOR LOW-PRECISION NETWORKS + +We focus on sub 8-bits precision for inference deployments, specifically ternary and 4-bits precision. We found TTQ (Zhu et al., 2016) scheme achieving the state-of-the-art accuracy with ternary precision for weights and full-precision (32-bits floating-point) for activations. On Imagenet-1K (Russakovsky et al., 2015), TTQ achieves $3 3 . 4 \%$ Top-1 error rate with a ResNet-18 model. We implemented TTQ scheme for ResNet-34 and ResNet-50 models trained on Imagenet-1K and achieved $2 8 . 3 \%$ and $2 5 . 6 \%$ Top-1 error rates, respectively. This scheme is our baseline for 2-bits weight and full-precision activations. For 2-bits weight and 8-bits activation, we find work by Mellempudi et al. (2017) to achieve the best accuracies reported in literature. For ResNet-50, Mellempudi et al. (2017) obtain $2 9 . 2 4 \%$ Top-1 error. We consider this work to be our baseline for 2-bits weight and 8-bits activation models. + +For 4-bits precision, we find WRPN scheme (Mishra et al., 2017) to report the highest accuracy. We implemented this scheme for 4-bits weight and 8-bits activations. For ResNet-34 and ResNet-50 models trained on Imagenet-1K, we achieve $2 9 . 7 \%$ and $2 8 . 4 \%$ Top-1 error rates, respectively. + +# 5.2 SCHEME-A: JOINT TRAINING OF TEACHER-STUDENT NETWORKS + +In the first scheme that we investigate, a full-precision teacher network is jointly trained with a lowprecision student network. Figure 2 shows the overall training framework. We use ResNet topology for both the teacher and student network. When using a certain depth for the student network, we pick the teacher network to have either the same or larger depth. + +In Bucilua et al. ˇ (2006) and Hinton et al. (2015), only the student network trains while distilling knowledge from the teacher network. In our case, we jointly train with the rationale that the teacher network would continuously guide the student network not only with the final trained logits, but also on what path the teacher takes towards generating those final higher accuracy logits. + +We implement pre-activation version of ResNet (He et al., 2016) in TensorFlow (Abadi et al., 2015). The training process closely follows the recipe mentioned in Torch implementation of ResNet - we use a batch size of 256 and no hyper-parameters are changed from what is mentioned in the recipe. For the teacher network, we experiment with ResNet-34, ResNet-50 and ResNet-101 as options. For the student network, we experiment with low-precision variants of ResNet-18, ResNet-34 and ResNet-50. + +For low-precision numerics, when using ternary precision we use the ternary weight network scheme (Li & Liu, 2016) where the weight tensors are quantized into $\{ - 1 , 0 , 1 \}$ with a per-layer scaling coefficient computed based on the mean of the positive terms in the weight tensor. We use the WRPN scheme (Mishra et al., 2017) to quantize weights and activations to 4-bits or 8-bits. We do not lower the precision of the first layer and the final layer in the apprentice network. This is based on the observation in almost all prior works that lowering the precision of these layers degrades the accuracy dramatically. While training and during fine-tuning, the gradients are still maintained at full-precision. + +Table 1: Top-1 validation set error rate $( \% )$ on ImageNet-1K for ResNet-18 student network as precision of activations (A) and weight (W) changes. The last three columns show error rate when the student ResNet-18 is paired with ResNet-34, ResNet-50 and ResNet-101. + +
ResNet-18 BaselineResNet-18 with ResNet-34ResNet-18 with ResNet-50ResNet-18 with ResNet-101
32A,32W30.430.230.130.1
32A,2W33.431.731.531.8
8A,4W33.629.629.629.9
8A,2W33.932.032.232.4
+ +Results with ResNet-18: Table 1 shows the effect of lowering precision on the accuracy (Top1 error) of ResNet-18 with baseline (no teacher) and with ResNet-34, ResNet-50 and ResNet-101 as teachers. In the table, $A$ denotes the precision of the activation maps (in bits) and $W$ denotes the precision of the weights. The baseline Top-1 error for full-precision ResNet-18 is $3 0 . 4 \%$ . By lowering the precision without using any help from a teacher network, the accuracy drops by $3 . 5 \%$ when using ternary and 4-bits precision (the column corresponding to “Res-18 Baseline” in the table). With distillation based technique, the accuracy of low-precision configurations improves significantly. In fact, the accuracy of the fullprecision ResNet-18 also improves when paired with a larger full-precision ResNet model (the row corresponding to $\mathrm { ^ { 6 6 } 3 2 A }$ , 32W” in Table 1). The best full-precision accuracy was achieved with a student ResNet-18 and ResNet-101 as the teacher (improvement by $0 . 3 5 \%$ over the baseline). The gap between full-precision ResNet-18 and the best + +![](images/f42e8f643c1b43106b46bcc4e066bcaba5b3b48a073be588f9ad1aea0c269326.jpg) +Difference (!) in Top-1 error for Res-18 from baseline +Figure 3: Difference in Top-1 error rate for lowprecision variants of ResNet-18 with (blue bars) and without (red bars) distillation scheme. The difference is calculated from the accuracy of ResNet-18 with full-precision numerics. Higher $\%$ difference denotes a better network configuration. + +achieved ternary weight ResNet-18 is only $1 \%$ (improvement of $2 \%$ over previous best). With “8A, 4W”, we find the accuracy of the student ResNet-18 model to beat the baseline accuracy. We hypothesize regularization with low-precision (and distillation) to be the reason for this. “8A, 4W” improving the accuracy beyond baseline figure is only seen for ResNet-18. + +Figure 3 shows the difference in Top-1 error rate achieved by our best low-precision student networks (when trained under the guidance of a teacher network) versus not using any help from a teacher network. For this figure, the difference in Top-1 error of the best low-precision student network is calculated from the baseline full-precision network (i.e. ResNet-18 with $3 0 . 4 \%$ Top-1 error), i.e. we want to see how close a low-precision student network can come to a full-precision baseline model. + +We find our low-precision network accuracies to significantly close the gap between full-precision accuracy (and for some configurations even beat the baseline accuracy). + +Hinton et al. (2015) mention improving the baseline full-precision accuracy when a student network is paired with a teacher network. They mention improving the accuracy of a small model on MNIST dataset. We show the efficacy of distillation based techniques on a much bigger model (ResNet) with much larger dataset (ImageNet). + +Table 2: Top-1 validation set error rate $( \% )$ on ImageNet-1K for ResNet-34 student network as precision of activations (A) and weight (W) changes. The last three columns show error rate when the student ResNet-34 is paired with ResNet-34, ResNet-50 and ResNet-101. + +
ResNet-34 BaselineResNet-34 with ResNet-34ResNet-34 with ResNet-50ResNet-34 with ResNet-101
32A,32W26.426.326.126.1
32A,2W28.327.627.227.2
8A,4W29.727.026.926.9
8A,2W30.828.828.828.5
+ +Table 3: Top-1 validation set error rate $( \% )$ on ImageNet-1K for ResNet-50 student network as precision of activations (A) and weight (W) changes. The final two columns show error rate when the student ResNet-50 is paired with ResNet-50 and ResNet-101. + +
ResNet-50 BaselineResNet-50 with ResNet-50ResNet-50 with ResNet-101
32A,32W23.823.723.5
32A,2W26.125.425.3
8A,4W28.525.525.3
8A,2W29.227.327.2
+ +![](images/47c2437bfd32d6774412562b4a57e97c472a8cd6b2d82e1de69fef711f7fda78.jpg) +Difference (!) in Top-1 error for Res-34 from baseline +Figure 4: Difference in Top-1 error rate for low-precision variants of ResNet-34 and ResNet-50 with (blue bars) and without (red bars) distillation scheme. The difference is calculated from the accuracy of the baseline network (ResNet-34 for (a) and ResNet-50 for (b)) operating at full-precision. Higher $\%$ difference denotes a better network configuration. + +![](images/2e0db05bfecee700dbb72b7cb8a8f0fceab579a6d7c2d9898a51b6829c269068.jpg) +Difference (!) in Top-1 error for Res-50 from baseline +(a) Apprentice versus baseline accuracy for ResNet-34. (b) Apprentice versus baseline accuracy for ResNet-50. + +Results with ResNet-34 and ResNet-50: Table 2 and Table 3 show the effect of lowering precision on the accuracy of ResNet-34 and ResNet-50, respectively, with distillation based technique. With a student ResNet-34 network, we use ResNet-34, ResNet-50 and ResNet-101 as teachers. With a student ResNet-50 network, we use ResNet-50 and ResNet-101 as teachers. The Top-1 error for full-precision ResNet-34 is $2 6 . 4 \%$ . Our best 4-bits weight and 8-bits activation ResNet-34 is within $0 . 5 \%$ of this number ( $2 6 . 9 \%$ error rate with ResNet-34 student and ResNet-50 teacher). This significantly improves upon the previously reported error rate of $2 9 . 7 \%$ . 4-bits weight and 8-bits activation for ResNet-50 gives us a model that is within $1 . 5 \%$ of full-precision model accuracy $2 5 . 3 \%$ vs. $2 3 . 8 \%$ ). Figure 4a and Figure 4b show the difference in Top-1 error achieved by our best low-precision ResNet-34 and ResNet-50 student networks, respectively, and compares with results obtained using methods proposed in literature. Our $\mathcal { A }$ pprentice scheme significantly closes the gap between full-precision baseline networks and low-precision variants of the same networks. In most cases we see our scheme to better the previously reported accuracy numbers by $1 . 5 \% - 3 \%$ . + +Discussion: In scheme-A, we use a teacher network that is always as large or larger in number of parameters than the student network. We experimented with a ternary ResNet-34 student network which was paired with a full-precision ResNet-18. The ternary model for ResNet-34 is about $8 . 5 \mathrm { x }$ smaller in size compared to the full-precision ResNet-18 model. The final trained accuracy of the ResNet-34 ternary model with this setup is $2 . 7 \%$ worse than that obtained by pairing the ternary ResNet-34 network with a ResNet-50 teacher network. This suggests that the distillation scheme works only when the teacher network is higher in accuracy than the student network (and not necessarily bigger in capacity). Further, the benefit from using a larger teacher network saturates at some point. This can be seen by picking up a precision point, say “32A, 2W” and looking at the error rates along the row in Table 1, 2 and 3. + +One concern, we had in the early stages of our investigation, with joint training of a low-precision small network and a high precision large network was the influence of the small network’s accuracy on the accuracy of the large network. When using the joint cost function, the smaller network’s probability scores are matched with the predictions from the teacher network. The joint cost is added as a term to the total loss function (equation 1). This led us to posit that the larger network’s learning capability will be affected by the inherent impairment in the smaller low-precision network. Further, since the smaller student network learns form the larger teacher network, a vicious cycle might form where the student network’s accuracy will further drop because the teacher network’s learning capability is being impeded. However, in practice, we did not see this phenomenon occurring - in each case where the teacher network was jointly trained with a student network, the accuracy of the teacher network was always within $0 . 1 \%$ to $0 . 2 \%$ of the accuracy of the teacher network without it jointly supervising a student network. This could be because of our choice of $\alpha$ , $\beta$ and $\gamma$ values. + +In Section 4, we mentioned about temperature, $\tau$ , for Softmax function and hyper-parameters $\alpha = 1$ , $\beta = 0 . 5$ and $\gamma = 0 . 5$ . Since, we train directly on the logits of the teacher network, we did not have to experiment with the appropriate value of $\tau , \tau$ is required when training on the soft targets produced by the teacher network. Although we did not do extensive studies experimenting with training on soft targets as opposed to logits, we find that $\tau = 1$ gives us best results when training on soft targets. Hinton et al. (2015) mention that when the student network is significantly smaller than the teacher network, small values of $\tau$ are more effective than large values. For few of the low-precision configurations, we experimented with $\alpha = \beta = \gamma = 1$ , and, $\alpha = 0 . 9$ , $\beta = 1$ and $\gamma = 0 . 1$ or 0.3. Each of these configurations, yielded a lower performance model compared to our original choice for these parameters. + +For the third term in equation 1, we experimented with a mean-squared error loss function and also a loss function with logits from both the student and the teacher network (i.e. $\mathcal { H } ( z ^ { T } , z ^ { A } ) )$ . We did not find any improvement in accuracy compared to our original choice of the cost function formulation. A thorough investigation of the behavior of the networks with other values of hyper-parameters and different loss functions is an agenda for our future work. + +Overall, we find the distillation process to be quite effective in getting us high accuracy low-precision models. All our low-precision models surpass previously reported low-precision accuracy figures. For example, TTQ scheme achieves $3 3 . 4 \%$ Top-1 error rate for ResNet-18 with 2-bits weight. Our best ResNet-18 model, using scheme-A, with 2-bits weight achieves ${ \sim } 3 1 . 5 \%$ error rate, improving the model accuracy by ${ \sim } 2 \%$ over TTQ. Similarly, the scheme in Mellempudi et al. (2017) achieves $2 9 . 2 \%$ Top-1 error with 2-bits weight and 8-bits activation. The best performing $A$ pprentice network at this precision achieves $2 7 . 2 \%$ Top-1 error. For Scheme-B and Scheme-C, which we describe next, Scheme-A serves as the new baseline. + +# 5.3 SCHEME-B: DISTILLING KNOWLEDGE FROM A TEACHER + +In this scheme, we start with a trained teacher network. Referring back to Figure 2, the input image is passed to both the teacher and the student network, except that the learning with back-propagation happens only in the low precision student network which is trained from scratch. This is the scheme used by Bucilua et al. ˇ (2006) and Hinton et al. (2015) for training their student networks. In this scheme, the first term in equation 1 zeroes out and only the last two terms in the equation contribute toward the loss function. + +![](images/776aa02a37e8a7359b0d5d75f2da39efde532341c5e092bd7590bc5fbde6419b.jpg) +Figure 5: Top-1 error rate versus epochs of four student networks using scheme-A and scheme-B. + +With scheme-B, one can pre-compute and store the logit values for the input images on disk and access them during training the student network. This saves the forward pass computations in the teacher network. Scheme-B might also help the scenario where a student network attempts to learn the “dark knowledge” from a teacher network that has already been trained on some private or sensitive data (in addition to the data the student network is interested in training on). + +With scheme-A, we had the hypothesis that the student network would be influenced by not only the “dark knowledge” in the teacher network but also the path the teacher adopts to learn the knowledge. With scheme-B we find, that the student network gets to similar accuracy numbers as the teacher network albeit at fewer number of epochs. + +With this scheme, the training accuracies are similar to that reported in Table 1, 2 and 3. The low-precision student networks, however, learn in fewer number of epochs. Figure 5 plots the Top-1 error rates for few of the configurations from our experiment suite. In each of these plots, the student network in scheme-B converges around 80th-85th epoch compared to about 105 epochs in schemeA. In general, we find the student networks with scheme-B to learn in about $10 \% { - } 2 0 \%$ fewer epochs than the student networks trained using scheme-A. + +# 5.4 SCHEME-C: FINE-TUNING THE STUDENT MODEL + +Scheme-C is very similar to scheme-B, except that the student network is primed with full precision training weights before the start of the training process. At the beginning of the training process, the weights and activations are lowered and the student network is sort of fine-tuned on the dataset. + +Similar to scheme-B, only the final two terms in equation 1 comprise the loss function and the lowprecision student network is trained with back-propagation algorithm. Since, the network starts from a good initial point, comparatively low learning rate is used throughout the training process. There is no clear recipe for learning rates (and change of learning rate with epochs) which works across all the configurations. In general, we find training with a learning rate of 1e-3 for 10 to 15 epochs, followed by 1e-4 for another 5 to 10 epochs, followed by 1e-5 for another 5 epochs to give us the best accuracy. Some configurations run for about 40 to 50 epochs before stabilizing. For these configurations, we find training using scheme-B with warm startup (train the student network at full-precision for about 25-30 epochs before lowering the precision) to be equally good. Wu (2016) investigate a similar scheme for binary precision on AlexNet. Our experiments show that distillation is an overkill for AlexNet and one can get comparable accuracies using techniques proposed in (Tang et al., 2017; Mishra et al., 2017). Further, Wu (2016) hypothesize that distillation scheme will work on larger networks, we show in this paper how to make it work. Tann et al. (2017) use a similar scheme for AlexNet and mention starting from a non-global optimal checkpoint gives better accuracy, though we did not find this observation to hold in our experiments. + +We find the final accuracy of the models obtained using scheme-C to be (marginally) better than those obtained using scheme-A or scheme-B. Table 4 shows error rates of few configurations of low-precision student network obtained using scheme-A (or scheme-B) and scheme-C. For ResNet50 student network, the accuracy with ternary weights is further improved by $0 . 6 \%$ compared to that obtained using scheme-A. Note that the performance of ternary networks obtained using schemeA are already state-of-the-art. Hence, for ResNet-50 ternary networks, $2 4 . 7 \%$ Top-1 error rate is the new state-of-the-art. With this, ternary ResNet-50 is within $0 . 9 \%$ of baseline accuracy $( 2 3 . 8 \%$ vs. $2 4 . 7 \%$ ). Similarly, with 4-bits weight and 8-bits activations, ResNet-50 model obtained using scheme-C is $0 . 4 \%$ better than that obtained with scheme-A (closing the gap to be within $1 . 3 \%$ of full-precision ResNet-50 accuracy). + +Table 4: Top-1 ImageNet-1K validation set error rate $( \% )$ with scheme-A and scheme-C for ResNet-34 and ResNet-50 student networks with ternary and 4-bits precision. + +
ResNet-34 student with ResNet-50 teacher32A,2W with scheme-A or B32A,2W with scheme-C
ResNet-50 student with ResNet101 teacher27.2 25.326.9 24.7
ResNet-34 student with ResNet-50 teacher8A,4W with scheme-A or B8A,4W with scheme-C
26.926.8
ResNet-50 student with ResNet101 teacher25.525.1
+ +Scheme-C is useful when one already has a trained network which can be fine-tuned using knowledge distillation schemes to produce a low-precision variant of the trained network. + +# 5.5 DISCUSSION - TERNARY PRECISION VERSUS SPARSITY + +As mentioned earlier, low-precision is a form of model compression. There are many works which target network sparsification and pruning techniques to compress a model. With ternary precision models, the model size reduces by a factor of $2 / 3 2$ compared to full-precision models. With $\mathcal { A }$ pprentice, we show how one can get a performant model with ternary precision. Many works targeting network pruning and sparsification target a full-precision model to implement their scheme. To be comparable in model size to ternary networks, a full-precision model needs to be sparsified by $9 3 . 7 5 \%$ . Further, to be effective, a sparse model needs to store a key for every non-zero value denoting the position of the value in the weight tensor. This adds storage overhead and a sparse model needs to be about $9 5 \%$ sparse to be at-par in memory size as a 2-bit model. Note that ternary precision also has inherent sparsity (zero is a term in the ternary symbol dictionary) – we find our ternary models to be about $50 \%$ sparse. In work by Wen et al. (2016) and Han et al. (2015b), sparsification of full-precision networks is proposed but the sparsity achieved is less than $9 3 . 7 5 \%$ . Further, the network accuracy using techniques in both these works lead to larger degradation in accuracy compared to our ternary models. Overall, we believe, our ternary precision models to be state-of-the-art not only in accuracy (we better the accuracy compared to prior ternary precision models) but also when one considers the size of the model at the accuracy level achieved by low-precision or sparse networks. + +# 6 CONCLUSIONS + +While low-precision networks have system-level benefits, the drawback of such models is degraded accuracy when compared to full-precision models. We present three schemes based on knowledge distillation concept to improve the accuracy of low-precision networks and close the gap between the accuracy of these models and full-precision models. Each of the three schemes improve the accuracy of the low-precision network configuration compared to prior proposals. We motivate the need for a smaller model size in low batch, real-time and resource constrained inference deployment systems. 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In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017. + +Sergey Zagoruyko and Nikos Komodakis. Paying more attention to attention: Improving the performance of convolutional neural networks via attention transfer. CoRR, abs/1612.03928, 2016. URL http://arxiv.org/abs/1612.03928. + +Aojun Zhou, Anbang Yao, Yiwen Guo, Lin Xu, and Yurong Chen. Incremental network quantization: Towards lossless cnns with low-precision weights. CoRR, abs/1702.03044, 2017. URL http://arxiv.org/abs/1702.03044. + +Shuchang Zhou, Zekun Ni, Xinyu Zhou, He Wen, Yuxin Wu, and Yuheng Zou. Dorefa-net: Training low bitwidth convolutional neural networks with low bitwidth gradients. CoRR, abs/1606.06160, 2016. URL http://arxiv.org/abs/1606.06160. + +Chenzhuo Zhu, Song Han, Huizi Mao, and William J. Dally. Trained ternary quantization. CoRR, abs/1612.01064, 2016. URL http://arxiv.org/abs/1612.01064. + +![](images/7647475b497c5ab56acf137b91a8eb2c58c5ffb1c66f3938057b2bfbff0237c1.jpg) +7 APPENDIX: ANALYSIS WITH RESNET ON CIFAR-10 DATASET +Figure 6: Comparison of various configurations of ResNet on CIFAR-10 with and without $\mathcal { A }$ pprentice scheme. + +In addition to ImageNet dataset, we also experiment with Apprentice scheme on CIFAR-10 dataset. CIFAR-10 dataset (Krizhevsky, 2009) consists of 50K training images and 10K testing images in 10 classes. We use various depths of ResNet topology for this study. Our implementation of ResNet for CIFAR-10 closely follows the configuration in He et al. (2015). The network inputs are $3 2 \times 3 2$ images. The first layer is a $3 \times 3$ convolutional layer followed by a stack of $6 n$ layers with $3 \times 3$ convolutions on feature map sizes 32, 16 and 8; with $2 n$ layers for each feature map size. The numbers of filters are 16, 32 and 64 in each set of $2 n$ layers. This is followed by a global average pooling, a 10-way fully connected layer and a softmax layer. Thus, in total there are $6 n { + 2 }$ weight layers. + +Figure 6a shows the impact of lowering precision as the depth of ResNet varies. As the network becomes larger in size, the impact of lowering precision is diminished (relative to the accuracy of the network at that depth when using full-precision). For example, with ResNet-110, full-precision Top-1 error rate is $6 . 1 9 \%$ . At the same depth, ternarizing the model gives similar error rate $( 6 . 2 4 \% )$ . Comparing this with ResNet-20, the gap between full-precision and ternary model (2-bits weight and 32-bits activations) is $0 . 8 \%$ $7 . 9 \%$ vs. $8 . 7 \%$ Top-1 error). Overall, we find that ternarizing a model closely follows accuracy of baseline full-precision model. However, lowering both weights and activations almost always leads to large accuracy degradation. Accuracy of 2-bits weight and 8-bits activation network is $0 . 8 \% 1 . 6 \%$ worse than full-precision model. Using $A$ pprentice scheme this gap is considerably lowered. + +Figure 6b shows the impact of lowering precision when a low-precision (student) network is paired with a full-precision (teacher) network. For this analysis we use scheme-A where we jointly train both the teacher and student network. The mix of ResNet depths we used for this study are ResNet20, 32, 44, 56, 110 and 182. ResNet-20 student network was paired with deeper ResNets from this mix, i.e. ResNet-32, 44, 56, 110 and 182 (as five separate experiments). Similarly, ResNet-44 student network was paired with deeper ResNet-56 and 110 (as two different set of experiments). ResNet-110 student network used ResNet-182 as its teacher network. For a particular ResNet depth, the figure plots the minimum error rate across each of the experiments. + +We find $A$ pprentice scheme to improve the baseline full-precision accuracy. The scheme also helps close the gap between the new improved baseline accuracy and the accuracy when lowering the precision of the weights and activations. The gap between 2-bits weight and 8-bits activation network is now $0 . 4 \% - 0 . 8 \%$ worse than full-precision model. With ImageNet dataset, the accuracy of full-precision networks also improves but very marginally (by $0 . 3 \%$ ). However, the impact of distillation technique on CIFAR-10 is quite pronounced - for example, top-1 error lowers by $1 . 1 \%$ for ResNet-110. + +![](images/58b5bef731307b1943dcb500f0772a4d88af24ac4034b21f19ae656eba91abc5.jpg) +Figure 7: Distillation followed by quantization. + +We experimented with a variation of scheme-C where a network is first compressed using distillation scheme (using a deeper ResNet as the teacher network) followed by lowering the precision and finetuning. The fine-tuning is done for 35-40 epochs with a very low learning-rate without the influence of any teacher network (no distillation). For this experiment, the student network starts from a higher accuracy compressed model compared to scheme-C (since distillation improves accuracy of student network at full-precision as well). Figure 7 shows the results with this experimental setting. For each configuration, we find the error-rate to lie in between the error-rates shown in Figure 6a and Figure 6b for the corresponding configuration, i.e. this scheme is better than low-precision training from scratch but not as good as training with methodology described in scheme-A. On an average, we find scheme-A to give $0 . 7 \%$ better accuracy at low-precision configurations compared to the scheme mentioned here highlighting the benefits of “joint” low-precision training from scratch with distillation (Apprentice scheme). Many works proposing low-precision knobs advocate for training from scratch or training (for a significant number of epochs) with warm-startup – the conclusions from this experiment are in line with the observations in these papers. + +# 8 FUTURE RESEARCH + +Some works proposing low-precision networks advocate for making the layers wider (or the model larger) to recover accuracy at low-precision. These works propose making the layers wider by $2 \mathrm { x }$ or $3 \mathrm { x }$ . While these works show the benefits of low-precision, making the model larger increases the number of raw computations. Future work could investigate low-precision and less layer widening factor (say $1 . 1 0 \mathrm { x }$ or $1 . 2 5 \mathrm { x }$ ). This would help inference latency while maintaining accuracy at-par with baseline full-precision networks. + +As mentioned in section 5.5, sparsifying a model more than a certain percentage leads to accuracy loss. Investigating hyper-sparse network models without accuracy loss using distillation based schemes is another interesting avenue of further research. \ No newline at end of file diff --git a/parse/train/B1ae1lZRb/B1ae1lZRb_content_list.json b/parse/train/B1ae1lZRb/B1ae1lZRb_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..2cb2f8e9e15815dab231f6208be45e9745eee83c --- /dev/null +++ b/parse/train/B1ae1lZRb/B1ae1lZRb_content_list.json @@ -0,0 +1,1615 @@ +[ + { + "type": "text", + "text": "APPRENTICE: USING KNOWLEDGE DISTILLATION TECHNIQUES TO IMPROVE LOW-PRECISION NETWORK ACCURACY ", + "text_level": 1, + "bbox": [ + 174, + 98, + 821, + 171 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Asit Mishra & Debbie Marr \nAccelerator Architecture Lab \nIntel Labs \n{asit.k.mishra,debbie.marr}@intel.com ", + "bbox": [ + 183, + 195, + 545, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 287, + 544, + 304 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep learning networks have achieved state-of-the-art accuracies on computer vision workloads like image classification and object detection. The performant systems, however, typically involve big models with numerous parameters. Once trained, a challenging aspect for such top performing models is deployment on resource constrained inference systems — the models (often deep networks or wide networks or both) are compute and memory intensive. Low-precision numerics and model compression using knowledge distillation are popular techniques to lower both the compute requirements and memory footprint of these deployed models. In this paper, we study combination of these two techniques and show that the performance of low-precision networks can be significantly improved by using knowledge distillation techniques. Our approach, $\\mathcal { A }$ pprentice, achieves state-of-the-art accuracies using ternary precision and 4-bit precision for variants of ResNet architecture on ImageNet dataset. We present three schemes using which one can apply knowledge distillation techniques to various stages of the train-and-deploy pipeline. ", + "bbox": [ + 233, + 320, + 764, + 529 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 558, + 336, + 573 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Background: Today’s high performing deep neural networks (DNNs) for computer vision applications comprise of multiple layers and involve numerous parameters. These networks have ${ \\mathcal { O } } ( { \\mathrm { G i g a - } }$ FLOPS) compute requirements and generate models which are $\\mathcal { O }$ (Mega-Bytes) in storage (Canziani et al., 2016). Further, the memory and compute requirements during training and inference are quite different (Mishra et al., 2017). Training is performed on big datasets with large batch-sizes where memory footprint of activations dominates the model memory footprint. On the other hand, batchsize during inference is typically small and the model’s memory footprint dominates the runtime memory requirements. ", + "bbox": [ + 174, + 589, + 823, + 700 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Because of complexity in compute, memory and storage requirements, training phase of the networks is performed on CPU and/or GPU clusters in a distributed computing environment. Once trained, a challenging aspect is deployment of trained models on resource constrained inference systems such as portable devices or sensor networks, and for applications in which real-time predictions are required. Performing inference on edge-devices comes with severe constraints on memory, compute and power. Additionally, ensemble based methods, which one can potentially use to get improved accuracy predictions, become prohibitive in resource constrained systems. ", + "bbox": [ + 174, + 708, + 823, + 805 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Quantization using low-precision numerics (Vanhoucke et al., 2011; Zhou et al., 2016; Lin et al., 2015; Miyashita et al., 2016; Gupta et al., 2015; Zhu et al., 2016; Rastegari et al., 2016; Courbariaux et al., 2015; Umuroglu et al., 2016; Mishra et al., 2017) and model compression (Bucilua et al. ˇ , 2006; Hinton et al., 2015; Romero et al., 2014) have emerged as popular solutions for resource constrained deployment scenarios. With quantization, a low-precision version of network model is generated and deployed on the device. Operating in lower precision mode reduces compute as well as data movement and storage requirements. However, majority of existing works in low-precision DNNs sacrifice accuracy over baseline full-precision networks. With model compression, a smaller low memory footprint network is trained to mimic the behaviour of the original complex network. During this training, a process called, knowledge distillation is used to “transfer knowledge” from the complex network to the smaller network. Work by Hinton et al. (2015) shows that the knowledge distillation scheme can yield networks at comparable or slightly better accuracy than the original complex model. However, to the best of our knowledge, all prior works using model compression techniques target compression at full-precision. ", + "bbox": [ + 173, + 811, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our proposal: In this paper, we study the combination of network quantization with model compression and show that accuracies of low-precision networks can be significantly improved by using knowledge distillation techniques. Previous studies on model compression use a large network as the teacher network and a small network as the student network. The small student network learns from teacher network using distillation process. The network architecture of the student network is typically different from that of the teacher network – for e.g. Hinton et al. (2015) investigate a student network that has fewer number of neurons in the hidden layers compared to the teacher network. In our work, the student network has similar topology as that of teacher network, except that the student network has low-precision neurons compared to the teacher network which has neurons operating at full-precision. ", + "bbox": [ + 174, + 194, + 825, + 333 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We call our approach Apprentice1 and study three schemes which produce low-precision networks using knowledge distillation techniques. Each of these three schemes produce state-of-the-art ternary precision and 4-bit precision models. ", + "bbox": [ + 176, + 340, + 820, + 382 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the first scheme, a low-precision network and a full-precision network are jointly trained from scratch using knowledge distillation scheme. Later in the paper we describe the rationale behind this approach. Using this scheme, a new state-of-the-art accuracy is obtained for ternary and 4-bit precision for ResNet-18, ResNet-34 and ResNet-50 on ImageNet dataset. In fact, using this scheme the accuracy of the full-precision model also slightly improves. This scheme then serves as the new baseline for the other two schemes we investigate. ", + "bbox": [ + 174, + 388, + 823, + 472 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the second scheme, we start with a full-precision trained network and transfer knowledge from this trained network continuously to train a low-precision network from scratch. We find that the low-precision network converges faster (albeit to similar accuracies as the first scheme) when a trained complex network guides its training. ", + "bbox": [ + 174, + 479, + 823, + 536 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the third scheme, we start with a trained full-precision large network and an apprentice network that has been initialised with full-precision weights. The apprentice network’s precision is lowered and is fine-tuned using knowledge distillation techniques. We find that the low-precision network’s accuracy marginally improves and surpasses the accuracy obtained via the first scheme. This scheme then sets the new state-of-the-art accuracies for the ResNet models at ternary and 4-bit precision. ", + "bbox": [ + 174, + 542, + 823, + 612 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Overall, the contributions of this paper are the techniques to obtain low-precision DNNs using knowledge distillation technique. Each of our scheme produces a low-precision model that surpasses the accuracy of the equivalent low-precision model published to date. One of our schemes also helps a low-precision model converge faster. We envision these accurate low-precision models to simplify the inference deployment process on resource constrained systems and even otherwise on cloud-based deployment systems. ", + "bbox": [ + 174, + 619, + 825, + 703 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 MOTIVATION FOR LOW-PRECISION MODEL PARAMETERS ", + "text_level": 1, + "bbox": [ + 176, + 727, + 673, + 742 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Lowering precision of model parameters: Resource constrained inference systems impose significant restrictions on memory, compute and power budget. With regard to storage, model (or weight) parameters and activation maps occupy memory during the inference phase of DNNs. During this phase memory is allocated for input (IFM) and output feature maps (OFM) required by a single layer in the DNN, and these dynamic memory allocations are reused for other layers. The total memory allocation during inference is then the maximum of IFM and maximum of OFM memory required across all the layers plus the sum of all weight tensors (Mishra et al., 2017). When inference phase for DNNs is performed with a small batch size, the memory footprint of the weights exceeds the footprint of the activation maps. This aspect is shown in Figure 1 for 4 different networks (AlexNet (Krizhevsky et al., 2012), Inception-Resnet-v2 (Szegedy et al., 2016), ResNet-50 and ResNet-101 (He et al., 2015)) running $2 2 4 \\mathrm { x } 2 2 4$ image patches. Thus lowering the precision of the weight tensors helps lower the memory requirements during deployment. One other aspect of lowering memory footprint is that the working set size of the workload starts to fit on chip and by reducing accesses to DRAM (off-chip) memory, the compute core starts to see better performance and energy savings (DRAM accesses are expensive in latency and energy). ", + "bbox": [ + 173, + 758, + 825, + 871 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Benefit of low-precision compute: Low-precision compute simplifies hardware implementation. For example, the compute unit to perform the convolution operation (multiplication of two operands) involves a floating-point multiplier when using fullprecision weights and activations. The floatingpoint multiplier can be replaced with a much simpler circuitry (xnor and popcount logic elements) when using binary precision for weights and activations (Courbariaux & Bengio, 2016; Rastegari et al., 2016; Courbariaux et al., 2015). Similarly, when using ternary precision for weights and full-precision for activations, the multiplier unit can be replaced with a sign comparator unit. Simpler hardware also helps lower the inference latency and energy budget. Thus, operating in lower precision mode reduces compute as well as data movement and storage requirements. ", + "bbox": [ + 174, + 208, + 516, + 457 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/9891559225e3a0edb31812399103f038b11fe71bbdf203533e28c2c2d622f4f3.jpg", + "image_caption": [ + "Figure 1: Memory footprint of activations (ACTs) and weights (W) during inference for mini-batch sizes 1 and 8. " + ], + "image_footnote": [], + "bbox": [ + 540, + 234, + 813, + 371 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The drawback of low-precision models, however, is degraded accuracy. We discuss later in the paper the network accuracies obtained using methods proposed in literature. These accuracies serve as the starting point and baselines we compare to in our work. ", + "bbox": [ + 174, + 464, + 823, + 507 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 537, + 339, + 554 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Low-precision networks: Low-precision DNNs are an active area of research. Most low-precision networks acknowledge the over parameterization aspect of today’s DNN architectures and/or the aspect that lowering the precision of neurons post-training often does not impact the final performance. Reducing precision of weights for efficient inference pipeline has been very well studied. Works like Binary connect (BC) (Courbariaux et al., 2015), Ternary-weight networks (TWN) (Li & Liu, 2016), fine-grained ternary quantization (Mellempudi et al., 2017) and INQ (Zhou et al., 2017) target precision reduction of network weights. Accuracy is almost always affected when quantizing the weights significantly below 8-bits of precision. For AlexNet on ImageNet, TWN loses $5 \\%$ Top-1 accuracy. Schemes like INQ, work in Sung et al. (2015) and Mellempudi et al. (2017) do fine-tuning to quantize the network weights. ", + "bbox": [ + 173, + 575, + 825, + 715 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Work in XNOR-NET (Rastegari et al., 2016), binary neural networks (Courbariaux & Bengio, 2016), DoReFa (Zhou et al., 2016) and trained ternary quantization (TTQ) (Zhu et al., 2016) target training pipeline. While TTQ targets weight quantization, most works targeting activation quantization show that quantizing activations always hurt accuracy. XNOR-NET approach degrades Top-1 accuracy by $12 \\%$ and DoReFa by $8 \\%$ when quantizing both weights and activations to 1-bit (for AlexNet on ImageNet). Work by Gupta et al. (2015) advocates for low-precision fixed-point numbers for training. They show 16-bits to be sufficient for training on CIFAR10 dataset. Work by Seide et al. (2014) quantizes gradients in a distributed computing system. ", + "bbox": [ + 174, + 722, + 825, + 833 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Knowledge distillation methods: The general technique in distillation based methods involves using a teacher-student strategy, where a large deep network trained for a given task teaches shallower student network(s) on the same task. The core concepts behind knowledge distillation or transfer technique have been around for a while. Bucilua et al. ˇ (2006) show that one can compress the information in an ensemble into a single network. Ba & Caurana (2013) extend this approach to study shallow, but wide, fully connected topologies by mimicking deep neural networks. To facilitate learning, the authors introduce the concepts of learning on logits rather than the probability distribution. ", + "bbox": [ + 174, + 840, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Hinton et al. (2015) propose a framework to transfer knowledge by introducing the concept of temperature. The key idea is to divide the logits by a temperature factor before performing a Softmax function. By using a higher temperature factor the activations of incorrect classes are boosted. This then facilitates more information flowing to the model parameters during back-propagation operation. FitNets (Romero et al., 2014) extend this work by using intermediate hidden layer outputs as target values for training a deeper, but thinner, student model. Net2Net (Chen et al., 2015a) also uses a teacher-student network system with a function-preserving transformation approach to initialize the parameters of the student network. The goal in Net2Net approach is to accelerate the training of a larger student network. Zagoruyko & Komodakis (2016) use attention as a mechanism for transferring knowledge from one network to another. In a similar theme, Yim et al. (2017) propose an information metric using which a teacher DNN can transfer the distilled knowledge to other student DNNs. In N2N learning work, Ashok et al. (2017) propose a reinforcement learning based approach for compressing a teacher network into an equally capable student network. They achieve a compression factor of 10x for ResNet-34 on CIFAR datasets. ", + "bbox": [ + 174, + 138, + 825, + 333 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Sparsity and hashing: Few other popular techniques for model compression are pruning (LeCun et al., 1990; Han et al., 2015a; Wen et al., 2016; Han et al., 2015b), hashing (Weinberger et al., 2009) and weight sharing (Chen et al., 2015b; Denil et al., 2013). Pruning leads to removing neurons entirely from the final trained model making the model a sparse structure. With hashing and weight sharing schemes a hash function is used to alias several weight parameters into few hash buckets, effectively lowering the parameter memory footprint. To realize benefits of sparsity and hashing schemes during runtime, efficient hardware support is required (e.g. support for irregular memory accesses (Han et al., 2016; Venkatesh et al., 2016; Parashar et al., 2017)). ", + "bbox": [ + 174, + 340, + 825, + 452 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 KNOWLEDGE DISTILLATION ", + "text_level": 1, + "bbox": [ + 176, + 472, + 441, + 488 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We introduce the concept of knowledge distillation in this section. Bucilua et al. ˇ (2006), Hinton et al. (2015) and Urban et al. (2016) analyze this topic in great detail. ", + "bbox": [ + 173, + 503, + 823, + 532 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Figure 2 shows the schematic of the knowledge distillation setup. Given an input image $x$ , a teacher DNN maps this image to predictions $p ^ { T }$ . The $C$ class predictions are obtained by applying Softmax function on the un-normalized log probability values $z$ (the logits), i.e. $\\begin{array} { r } { p ^ { T } = e ^ { z _ { k } ^ { T } } / \\sum _ { j } ^ { C } e ^ { z _ { j } ^ { T } } } \\end{array}$ . The same image is fed to the student network and it predicts $p ^ { A } = e ^ { z _ { k } ^ { A } } / { \\sum _ { j } ^ { C } e ^ { z _ { j } ^ { A } } }$ . During training, the cost function, , is given as: ", + "bbox": [ + 173, + 539, + 825, + 621 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/3ac18263d0d4d839bbd70e21c7f26661af108096a975645568d9a127ce3055a1.jpg", + "text": "$$\n\\mathcal { L } ( x ; W _ { T } , W _ { A } ) = \\alpha \\mathcal { H } ( y , p ^ { T } ) + \\beta \\mathcal { H } ( y , p ^ { A } ) + \\gamma \\mathcal { H } ( z ^ { T } , p ^ { A } )\n$$", + "text_format": "latex", + "bbox": [ + 303, + 638, + 694, + 657 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where, $W _ { T }$ and $W _ { A }$ are the parameters of the teacher and the student (apprentice) network, respectively, $y$ is the ground truth, $\\mathcal { H } ( \\cdot )$ denotes a loss function and, $\\alpha , \\beta$ and $\\gamma$ are weighting factors to prioritize the output of a certain loss function over the other. ", + "bbox": [ + 174, + 667, + 825, + 710 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In equation 1, lowering the first term of the cost function gives a better teacher network and lowering the second term gives a better student network. The third term is the knowledge distillation term whereby the student network attempts to mimic the knowledge in the teacher network. In Hinton et al. (2015), the logits of the teacher network are divided by a temperature factor $\\tau$ . Using a higher value for $\\tau$ produces a softer probability distribution when taking the Softmax of the logits. In our studies, we use cross-entropy function for $\\mathcal { H } ( \\cdot )$ , set $\\alpha = 1$ , $\\beta = 0 . 5$ and $\\gamma = 0 . 5$ and, perform the transfer learning process using the logits (inputs to the Softmax function) of the teacher network. In our experiments we study the effect of varying the depth of the teacher and the student network, and the precision of the neurons in the student network. ", + "bbox": [ + 173, + 717, + 825, + 843 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "5 OUR APPROACH - APPRENTICE NETWORK", + "text_level": 1, + "bbox": [ + 174, + 864, + 553, + 880 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Low-precision DNNs target the storage and compute efficiency aspects of the network. Model compression targets the same efficiency parameters from the point of view of network architecture. With ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/de4bb554a8cf5d3c54e0cec30d054c2e4a524e80b0ca5c77ec209cd41c0c4ecd.jpg", + "image_caption": [ + "Figure 2: Schematic of the knowledge distillation setup. The teacher network is a high precision network and the apprentice network is a low-precision network. " + ], + "image_footnote": [], + "bbox": [ + 253, + 104, + 745, + 291 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Apprentice we combine both these techniques to improve the network accuracy as well as the runtime efficiency of DNNs. Using the teacher-student setup described in the last section, we investigate three schemes using which one can obtain a low-precision model for the student network. The first scheme (scheme-A) jointly trains both the networks - full-precision teacher and low-precision student network. The second scheme (scheme-B) trains only the low-precision student network but distills knowledge from a trained full-precision teacher network throughout the training process. The third scheme (scheme-C) starts with a trained full-precision teacher and a full-precision student network but fine-tunes the student network after lowering its precision. Before we get into the details of each of these schemes, we discuss the accuracy numbers obtained using low-precision schemes described in literature. These accuracy figures serve as the baseline for comparative analysis. ", + "bbox": [ + 174, + 367, + 825, + 507 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.1 TOP-1 ERROR WITH PRIOR PROPOSALS FOR LOW-PRECISION NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 534, + 710, + 546 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We focus on sub 8-bits precision for inference deployments, specifically ternary and 4-bits precision. We found TTQ (Zhu et al., 2016) scheme achieving the state-of-the-art accuracy with ternary precision for weights and full-precision (32-bits floating-point) for activations. On Imagenet-1K (Russakovsky et al., 2015), TTQ achieves $3 3 . 4 \\%$ Top-1 error rate with a ResNet-18 model. We implemented TTQ scheme for ResNet-34 and ResNet-50 models trained on Imagenet-1K and achieved $2 8 . 3 \\%$ and $2 5 . 6 \\%$ Top-1 error rates, respectively. This scheme is our baseline for 2-bits weight and full-precision activations. For 2-bits weight and 8-bits activation, we find work by Mellempudi et al. (2017) to achieve the best accuracies reported in literature. For ResNet-50, Mellempudi et al. (2017) obtain $2 9 . 2 4 \\%$ Top-1 error. We consider this work to be our baseline for 2-bits weight and 8-bits activation models. ", + "bbox": [ + 174, + 563, + 825, + 700 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For 4-bits precision, we find WRPN scheme (Mishra et al., 2017) to report the highest accuracy. We implemented this scheme for 4-bits weight and 8-bits activations. For ResNet-34 and ResNet-50 models trained on Imagenet-1K, we achieve $2 9 . 7 \\%$ and $2 8 . 4 \\%$ Top-1 error rates, respectively. ", + "bbox": [ + 176, + 708, + 825, + 750 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5.2 SCHEME-A: JOINT TRAINING OF TEACHER-STUDENT NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 776, + 663, + 790 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In the first scheme that we investigate, a full-precision teacher network is jointly trained with a lowprecision student network. Figure 2 shows the overall training framework. We use ResNet topology for both the teacher and student network. When using a certain depth for the student network, we pick the teacher network to have either the same or larger depth. ", + "bbox": [ + 174, + 805, + 825, + 861 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In Bucilua et al. ˇ (2006) and Hinton et al. (2015), only the student network trains while distilling knowledge from the teacher network. In our case, we jointly train with the rationale that the teacher network would continuously guide the student network not only with the final trained logits, but also on what path the teacher takes towards generating those final higher accuracy logits. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We implement pre-activation version of ResNet (He et al., 2016) in TensorFlow (Abadi et al., 2015). The training process closely follows the recipe mentioned in Torch implementation of ResNet - we use a batch size of 256 and no hyper-parameters are changed from what is mentioned in the recipe. For the teacher network, we experiment with ResNet-34, ResNet-50 and ResNet-101 as options. For the student network, we experiment with low-precision variants of ResNet-18, ResNet-34 and ResNet-50. ", + "bbox": [ + 174, + 103, + 825, + 186 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For low-precision numerics, when using ternary precision we use the ternary weight network scheme (Li & Liu, 2016) where the weight tensors are quantized into $\\{ - 1 , 0 , 1 \\}$ with a per-layer scaling coefficient computed based on the mean of the positive terms in the weight tensor. We use the WRPN scheme (Mishra et al., 2017) to quantize weights and activations to 4-bits or 8-bits. We do not lower the precision of the first layer and the final layer in the apprentice network. This is based on the observation in almost all prior works that lowering the precision of these layers degrades the accuracy dramatically. While training and during fine-tuning, the gradients are still maintained at full-precision. ", + "bbox": [ + 173, + 194, + 826, + 305 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/2b2c77fcab7029bc37b02b14a28a349a56aad9253e59af87feb0e25d744748c2.jpg", + "table_caption": [ + "Table 1: Top-1 validation set error rate $( \\% )$ on ImageNet-1K for ResNet-18 student network as precision of activations (A) and weight (W) changes. The last three columns show error rate when the student ResNet-18 is paired with ResNet-34, ResNet-50 and ResNet-101. " + ], + "table_footnote": [], + "table_body": "
ResNet-18 BaselineResNet-18 with ResNet-34ResNet-18 with ResNet-50ResNet-18 with ResNet-101
32A,32W30.430.230.130.1
32A,2W33.431.731.531.8
8A,4W33.629.629.629.9
8A,2W33.932.032.232.4
", + "bbox": [ + 222, + 375, + 776, + 474 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Results with ResNet-18: Table 1 shows the effect of lowering precision on the accuracy (Top1 error) of ResNet-18 with baseline (no teacher) and with ResNet-34, ResNet-50 and ResNet-101 as teachers. In the table, $A$ denotes the precision of the activation maps (in bits) and $W$ denotes the precision of the weights. The baseline Top-1 error for full-precision ResNet-18 is $3 0 . 4 \\%$ . By lowering the precision without using any help from a teacher network, the accuracy drops by $3 . 5 \\%$ when using ternary and 4-bits precision (the column corresponding to “Res-18 Baseline” in the table). With distillation based technique, the accuracy of low-precision configurations improves significantly. In fact, the accuracy of the fullprecision ResNet-18 also improves when paired with a larger full-precision ResNet model (the row corresponding to $\\mathrm { ^ { 6 6 } 3 2 A }$ , 32W” in Table 1). The best full-precision accuracy was achieved with a student ResNet-18 and ResNet-101 as the teacher (improvement by $0 . 3 5 \\%$ over the baseline). The gap between full-precision ResNet-18 and the best ", + "bbox": [ + 174, + 487, + 500, + 791 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/f42e8f643c1b43106b46bcc4e066bcaba5b3b48a073be588f9ad1aea0c269326.jpg", + "image_caption": [ + "Difference\t(!)\tin\tTop-1\terror\tfor\tRes-18\tfrom\tbaseline ", + "Figure 3: Difference in Top-1 error rate for lowprecision variants of ResNet-18 with (blue bars) and without (red bars) distillation scheme. The difference is calculated from the accuracy of ResNet-18 with full-precision numerics. Higher $\\%$ difference denotes a better network configuration. " + ], + "image_footnote": [], + "bbox": [ + 519, + 532, + 812, + 669 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "achieved ternary weight ResNet-18 is only $1 \\%$ (improvement of $2 \\%$ over previous best). With “8A, 4W”, we find the accuracy of the student ResNet-18 model to beat the baseline accuracy. We hypothesize regularization with low-precision (and distillation) to be the reason for this. “8A, 4W” improving the accuracy beyond baseline figure is only seen for ResNet-18. ", + "bbox": [ + 174, + 791, + 825, + 847 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 3 shows the difference in Top-1 error rate achieved by our best low-precision student networks (when trained under the guidance of a teacher network) versus not using any help from a teacher network. For this figure, the difference in Top-1 error of the best low-precision student network is calculated from the baseline full-precision network (i.e. ResNet-18 with $3 0 . 4 \\%$ Top-1 error), i.e. we want to see how close a low-precision student network can come to a full-precision baseline model. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We find our low-precision network accuracies to significantly close the gap between full-precision accuracy (and for some configurations even beat the baseline accuracy). ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Hinton et al. (2015) mention improving the baseline full-precision accuracy when a student network is paired with a teacher network. They mention improving the accuracy of a small model on MNIST dataset. We show the efficacy of distillation based techniques on a much bigger model (ResNet) with much larger dataset (ImageNet). ", + "bbox": [ + 174, + 138, + 825, + 194 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/340bd5a135b902040861bd2b06f736a26543d9c1e4f60098d12a35fa789f70c6.jpg", + "table_caption": [ + "Table 2: Top-1 validation set error rate $( \\% )$ on ImageNet-1K for ResNet-34 student network as precision of activations (A) and weight (W) changes. The last three columns show error rate when the student ResNet-34 is paired with ResNet-34, ResNet-50 and ResNet-101. " + ], + "table_footnote": [], + "table_body": "
ResNet-34 BaselineResNet-34 with ResNet-34ResNet-34 with ResNet-50ResNet-34 with ResNet-101
32A,32W26.426.326.126.1
32A,2W28.327.627.227.2
8A,4W29.727.026.926.9
8A,2W30.828.828.828.5
", + "bbox": [ + 222, + 267, + 776, + 366 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/53c3197f8992bcff31fb7bf64bf943b9e8952b915bdcfd1f23377cc7718f311f.jpg", + "table_caption": [ + "Table 3: Top-1 validation set error rate $( \\% )$ on ImageNet-1K for ResNet-50 student network as precision of activations (A) and weight (W) changes. The final two columns show error rate when the student ResNet-50 is paired with ResNet-50 and ResNet-101. " + ], + "table_footnote": [], + "table_body": "
ResNet-50 BaselineResNet-50 with ResNet-50ResNet-50 with ResNet-101
32A,32W23.823.723.5
32A,2W26.125.425.3
8A,4W28.525.525.3
8A,2W29.227.327.2
", + "bbox": [ + 281, + 446, + 715, + 546 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/47c2437bfd32d6774412562b4a57e97c472a8cd6b2d82e1de69fef711f7fda78.jpg", + "image_caption": [ + "Difference\t(!)\tin\tTop-1\terror\tfor\tRes-34\tfrom\tbaseline ", + "Figure 4: Difference in Top-1 error rate for low-precision variants of ResNet-34 and ResNet-50 with (blue bars) and without (red bars) distillation scheme. The difference is calculated from the accuracy of the baseline network (ResNet-34 for (a) and ResNet-50 for (b)) operating at full-precision. Higher $\\%$ difference denotes a better network configuration. " + ], + "image_footnote": [], + "bbox": [ + 179, + 592, + 486, + 734 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/2e0db05bfecee700dbb72b7cb8a8f0fceab579a6d7c2d9898a51b6829c269068.jpg", + "image_caption": [ + "Difference\t(!)\tin\tTop-1\terror\tfor\tRes-50\tfrom\tbaseline ", + "(a) Apprentice versus baseline accuracy for ResNet-34. (b) Apprentice versus baseline accuracy for ResNet-50. " + ], + "image_footnote": [], + "bbox": [ + 506, + 592, + 812, + 733 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results with ResNet-34 and ResNet-50: Table 2 and Table 3 show the effect of lowering precision on the accuracy of ResNet-34 and ResNet-50, respectively, with distillation based technique. With a student ResNet-34 network, we use ResNet-34, ResNet-50 and ResNet-101 as teachers. With a student ResNet-50 network, we use ResNet-50 and ResNet-101 as teachers. The Top-1 error for full-precision ResNet-34 is $2 6 . 4 \\%$ . Our best 4-bits weight and 8-bits activation ResNet-34 is within $0 . 5 \\%$ of this number ( $2 6 . 9 \\%$ error rate with ResNet-34 student and ResNet-50 teacher). This significantly improves upon the previously reported error rate of $2 9 . 7 \\%$ . 4-bits weight and 8-bits activation for ResNet-50 gives us a model that is within $1 . 5 \\%$ of full-precision model accuracy $2 5 . 3 \\%$ vs. $2 3 . 8 \\%$ ). Figure 4a and Figure 4b show the difference in Top-1 error achieved by our best low-precision ResNet-34 and ResNet-50 student networks, respectively, and compares with results obtained using methods proposed in literature. Our $\\mathcal { A }$ pprentice scheme significantly closes the gap between full-precision baseline networks and low-precision variants of the same networks. In most cases we see our scheme to better the previously reported accuracy numbers by $1 . 5 \\% - 3 \\%$ . ", + "bbox": [ + 173, + 839, + 825, + 922 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Discussion: In scheme-A, we use a teacher network that is always as large or larger in number of parameters than the student network. We experimented with a ternary ResNet-34 student network which was paired with a full-precision ResNet-18. The ternary model for ResNet-34 is about $8 . 5 \\mathrm { x }$ smaller in size compared to the full-precision ResNet-18 model. The final trained accuracy of the ResNet-34 ternary model with this setup is $2 . 7 \\%$ worse than that obtained by pairing the ternary ResNet-34 network with a ResNet-50 teacher network. This suggests that the distillation scheme works only when the teacher network is higher in accuracy than the student network (and not necessarily bigger in capacity). Further, the benefit from using a larger teacher network saturates at some point. This can be seen by picking up a precision point, say “32A, 2W” and looking at the error rates along the row in Table 1, 2 and 3. ", + "bbox": [ + 174, + 208, + 825, + 347 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "One concern, we had in the early stages of our investigation, with joint training of a low-precision small network and a high precision large network was the influence of the small network’s accuracy on the accuracy of the large network. When using the joint cost function, the smaller network’s probability scores are matched with the predictions from the teacher network. The joint cost is added as a term to the total loss function (equation 1). This led us to posit that the larger network’s learning capability will be affected by the inherent impairment in the smaller low-precision network. Further, since the smaller student network learns form the larger teacher network, a vicious cycle might form where the student network’s accuracy will further drop because the teacher network’s learning capability is being impeded. However, in practice, we did not see this phenomenon occurring - in each case where the teacher network was jointly trained with a student network, the accuracy of the teacher network was always within $0 . 1 \\%$ to $0 . 2 \\%$ of the accuracy of the teacher network without it jointly supervising a student network. This could be because of our choice of $\\alpha$ , $\\beta$ and $\\gamma$ values. ", + "bbox": [ + 173, + 354, + 825, + 520 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Section 4, we mentioned about temperature, $\\tau$ , for Softmax function and hyper-parameters $\\alpha = 1$ , $\\beta = 0 . 5$ and $\\gamma = 0 . 5$ . Since, we train directly on the logits of the teacher network, we did not have to experiment with the appropriate value of $\\tau , \\tau$ is required when training on the soft targets produced by the teacher network. Although we did not do extensive studies experimenting with training on soft targets as opposed to logits, we find that $\\tau = 1$ gives us best results when training on soft targets. Hinton et al. (2015) mention that when the student network is significantly smaller than the teacher network, small values of $\\tau$ are more effective than large values. For few of the low-precision configurations, we experimented with $\\alpha = \\beta = \\gamma = 1$ , and, $\\alpha = 0 . 9$ , $\\beta = 1$ and $\\gamma = 0 . 1$ or 0.3. Each of these configurations, yielded a lower performance model compared to our original choice for these parameters. ", + "bbox": [ + 173, + 527, + 825, + 666 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For the third term in equation 1, we experimented with a mean-squared error loss function and also a loss function with logits from both the student and the teacher network (i.e. $\\mathcal { H } ( z ^ { T } , z ^ { A } ) )$ . We did not find any improvement in accuracy compared to our original choice of the cost function formulation. A thorough investigation of the behavior of the networks with other values of hyper-parameters and different loss functions is an agenda for our future work. ", + "bbox": [ + 174, + 674, + 825, + 742 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Overall, we find the distillation process to be quite effective in getting us high accuracy low-precision models. All our low-precision models surpass previously reported low-precision accuracy figures. For example, TTQ scheme achieves $3 3 . 4 \\%$ Top-1 error rate for ResNet-18 with 2-bits weight. Our best ResNet-18 model, using scheme-A, with 2-bits weight achieves ${ \\sim } 3 1 . 5 \\%$ error rate, improving the model accuracy by ${ \\sim } 2 \\%$ over TTQ. Similarly, the scheme in Mellempudi et al. (2017) achieves $2 9 . 2 \\%$ Top-1 error with 2-bits weight and 8-bits activation. The best performing $A$ pprentice network at this precision achieves $2 7 . 2 \\%$ Top-1 error. For Scheme-B and Scheme-C, which we describe next, Scheme-A serves as the new baseline. ", + "bbox": [ + 174, + 750, + 825, + 861 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 SCHEME-B: DISTILLING KNOWLEDGE FROM A TEACHER ", + "text_level": 1, + "bbox": [ + 179, + 104, + 604, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this scheme, we start with a trained teacher network. Referring back to Figure 2, the input image is passed to both the teacher and the student network, except that the learning with back-propagation happens only in the low precision student network which is trained from scratch. This is the scheme used by Bucilua et al. ˇ (2006) and Hinton et al. (2015) for training their student networks. In this scheme, the first term in equation 1 zeroes out and only the last two terms in the equation contribute toward the loss function. ", + "bbox": [ + 173, + 130, + 825, + 213 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/776aa02a37e8a7359b0d5d75f2da39efde532341c5e092bd7590bc5fbde6419b.jpg", + "image_caption": [ + "Figure 5: Top-1 error rate versus epochs of four student networks using scheme-A and scheme-B. " + ], + "image_footnote": [], + "bbox": [ + 194, + 228, + 805, + 559 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "With scheme-B, one can pre-compute and store the logit values for the input images on disk and access them during training the student network. This saves the forward pass computations in the teacher network. Scheme-B might also help the scenario where a student network attempts to learn the “dark knowledge” from a teacher network that has already been trained on some private or sensitive data (in addition to the data the student network is interested in training on). ", + "bbox": [ + 174, + 613, + 825, + 683 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "With scheme-A, we had the hypothesis that the student network would be influenced by not only the “dark knowledge” in the teacher network but also the path the teacher adopts to learn the knowledge. With scheme-B we find, that the student network gets to similar accuracy numbers as the teacher network albeit at fewer number of epochs. ", + "bbox": [ + 174, + 690, + 825, + 746 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "With this scheme, the training accuracies are similar to that reported in Table 1, 2 and 3. The low-precision student networks, however, learn in fewer number of epochs. Figure 5 plots the Top-1 error rates for few of the configurations from our experiment suite. In each of these plots, the student network in scheme-B converges around 80th-85th epoch compared to about 105 epochs in schemeA. In general, we find the student networks with scheme-B to learn in about $10 \\% { - } 2 0 \\%$ fewer epochs than the student networks trained using scheme-A. ", + "bbox": [ + 174, + 752, + 825, + 837 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.4 SCHEME-C: FINE-TUNING THE STUDENT MODEL ", + "text_level": 1, + "bbox": [ + 174, + 854, + 550, + 869 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Scheme-C is very similar to scheme-B, except that the student network is primed with full precision training weights before the start of the training process. At the beginning of the training process, the weights and activations are lowered and the student network is sort of fine-tuned on the dataset. ", + "bbox": [ + 176, + 882, + 823, + 922 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Similar to scheme-B, only the final two terms in equation 1 comprise the loss function and the lowprecision student network is trained with back-propagation algorithm. Since, the network starts from a good initial point, comparatively low learning rate is used throughout the training process. There is no clear recipe for learning rates (and change of learning rate with epochs) which works across all the configurations. In general, we find training with a learning rate of 1e-3 for 10 to 15 epochs, followed by 1e-4 for another 5 to 10 epochs, followed by 1e-5 for another 5 epochs to give us the best accuracy. Some configurations run for about 40 to 50 epochs before stabilizing. For these configurations, we find training using scheme-B with warm startup (train the student network at full-precision for about 25-30 epochs before lowering the precision) to be equally good. Wu (2016) investigate a similar scheme for binary precision on AlexNet. Our experiments show that distillation is an overkill for AlexNet and one can get comparable accuracies using techniques proposed in (Tang et al., 2017; Mishra et al., 2017). Further, Wu (2016) hypothesize that distillation scheme will work on larger networks, we show in this paper how to make it work. Tann et al. (2017) use a similar scheme for AlexNet and mention starting from a non-global optimal checkpoint gives better accuracy, though we did not find this observation to hold in our experiments. ", + "bbox": [ + 174, + 104, + 825, + 311 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We find the final accuracy of the models obtained using scheme-C to be (marginally) better than those obtained using scheme-A or scheme-B. Table 4 shows error rates of few configurations of low-precision student network obtained using scheme-A (or scheme-B) and scheme-C. For ResNet50 student network, the accuracy with ternary weights is further improved by $0 . 6 \\%$ compared to that obtained using scheme-A. Note that the performance of ternary networks obtained using schemeA are already state-of-the-art. Hence, for ResNet-50 ternary networks, $2 4 . 7 \\%$ Top-1 error rate is the new state-of-the-art. With this, ternary ResNet-50 is within $0 . 9 \\%$ of baseline accuracy $( 2 3 . 8 \\%$ vs. $2 4 . 7 \\%$ ). Similarly, with 4-bits weight and 8-bits activations, ResNet-50 model obtained using scheme-C is $0 . 4 \\%$ better than that obtained with scheme-A (closing the gap to be within $1 . 3 \\%$ of full-precision ResNet-50 accuracy). ", + "bbox": [ + 174, + 319, + 825, + 458 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/55e87daaa579c913cf7001bd757a8242642a4226d05bc53ad2a5b79129b89061.jpg", + "table_caption": [ + "Table 4: Top-1 ImageNet-1K validation set error rate $( \\% )$ with scheme-A and scheme-C for ResNet-34 and ResNet-50 student networks with ternary and 4-bits precision. " + ], + "table_footnote": [], + "table_body": "
ResNet-34 student with ResNet-50 teacher32A,2W with scheme-A or B32A,2W with scheme-C
ResNet-50 student with ResNet101 teacher27.2 25.326.9 24.7
ResNet-34 student with ResNet-50 teacher8A,4W with scheme-A or B8A,4W with scheme-C
26.926.8
ResNet-50 student with ResNet101 teacher25.525.1
", + "bbox": [ + 187, + 503, + 808, + 640 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Scheme-C is useful when one already has a trained network which can be fine-tuned using knowledge distillation schemes to produce a low-precision variant of the trained network. ", + "bbox": [ + 171, + 655, + 823, + 684 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "5.5 DISCUSSION - TERNARY PRECISION VERSUS SPARSITY ", + "text_level": 1, + "bbox": [ + 174, + 703, + 589, + 717 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "As mentioned earlier, low-precision is a form of model compression. There are many works which target network sparsification and pruning techniques to compress a model. With ternary precision models, the model size reduces by a factor of $2 / 3 2$ compared to full-precision models. With $\\mathcal { A }$ pprentice, we show how one can get a performant model with ternary precision. Many works targeting network pruning and sparsification target a full-precision model to implement their scheme. To be comparable in model size to ternary networks, a full-precision model needs to be sparsified by $9 3 . 7 5 \\%$ . Further, to be effective, a sparse model needs to store a key for every non-zero value denoting the position of the value in the weight tensor. This adds storage overhead and a sparse model needs to be about $9 5 \\%$ sparse to be at-par in memory size as a 2-bit model. Note that ternary precision also has inherent sparsity (zero is a term in the ternary symbol dictionary) – we find our ternary models to be about $50 \\%$ sparse. In work by Wen et al. (2016) and Han et al. (2015b), sparsification of full-precision networks is proposed but the sparsity achieved is less than $9 3 . 7 5 \\%$ . Further, the network accuracy using techniques in both these works lead to larger degradation in accuracy compared to our ternary models. Overall, we believe, our ternary precision models to be state-of-the-art not only in accuracy (we better the accuracy compared to prior ternary precision models) but also when one considers the size of the model at the accuracy level achieved by low-precision or sparse networks. ", + "bbox": [ + 174, + 729, + 825, + 924 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "6 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 166, + 328, + 183 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "While low-precision networks have system-level benefits, the drawback of such models is degraded accuracy when compared to full-precision models. We present three schemes based on knowledge distillation concept to improve the accuracy of low-precision networks and close the gap between the accuracy of these models and full-precision models. Each of the three schemes improve the accuracy of the low-precision network configuration compared to prior proposals. We motivate the need for a smaller model size in low batch, real-time and resource constrained inference deployment systems. We envision the low-precision models produced by our schemes to simplify the inference deployment process on resource constrained systems and on cloud-based deployment systems where low latency is a critical requirement. ", + "bbox": [ + 174, + 196, + 825, + 323 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 103, + 285, + 117 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. 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" + ], + "image_footnote": [], + "bbox": [ + 178, + 143, + 812, + 520 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In addition to ImageNet dataset, we also experiment with Apprentice scheme on CIFAR-10 dataset. CIFAR-10 dataset (Krizhevsky, 2009) consists of 50K training images and 10K testing images in 10 classes. We use various depths of ResNet topology for this study. Our implementation of ResNet for CIFAR-10 closely follows the configuration in He et al. (2015). The network inputs are $3 2 \\times 3 2$ images. The first layer is a $3 \\times 3$ convolutional layer followed by a stack of $6 n$ layers with $3 \\times 3$ convolutions on feature map sizes 32, 16 and 8; with $2 n$ layers for each feature map size. The numbers of filters are 16, 32 and 64 in each set of $2 n$ layers. This is followed by a global average pooling, a 10-way fully connected layer and a softmax layer. Thus, in total there are $6 n { + 2 }$ weight layers. ", + "bbox": [ + 173, + 575, + 825, + 700 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Figure 6a shows the impact of lowering precision as the depth of ResNet varies. As the network becomes larger in size, the impact of lowering precision is diminished (relative to the accuracy of the network at that depth when using full-precision). For example, with ResNet-110, full-precision Top-1 error rate is $6 . 1 9 \\%$ . At the same depth, ternarizing the model gives similar error rate $( 6 . 2 4 \\% )$ . Comparing this with ResNet-20, the gap between full-precision and ternary model (2-bits weight and 32-bits activations) is $0 . 8 \\%$ $7 . 9 \\%$ vs. $8 . 7 \\%$ Top-1 error). Overall, we find that ternarizing a model closely follows accuracy of baseline full-precision model. However, lowering both weights and activations almost always leads to large accuracy degradation. Accuracy of 2-bits weight and 8-bits activation network is $0 . 8 \\% 1 . 6 \\%$ worse than full-precision model. Using $A$ pprentice scheme this gap is considerably lowered. ", + "bbox": [ + 173, + 708, + 825, + 847 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Figure 6b shows the impact of lowering precision when a low-precision (student) network is paired with a full-precision (teacher) network. For this analysis we use scheme-A where we jointly train both the teacher and student network. The mix of ResNet depths we used for this study are ResNet20, 32, 44, 56, 110 and 182. ResNet-20 student network was paired with deeper ResNets from this mix, i.e. ResNet-32, 44, 56, 110 and 182 (as five separate experiments). Similarly, ResNet-44 student network was paired with deeper ResNet-56 and 110 (as two different set of experiments). ResNet-110 student network used ResNet-182 as its teacher network. For a particular ResNet depth, the figure plots the minimum error rate across each of the experiments. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We find $A$ pprentice scheme to improve the baseline full-precision accuracy. The scheme also helps close the gap between the new improved baseline accuracy and the accuracy when lowering the precision of the weights and activations. The gap between 2-bits weight and 8-bits activation network is now $0 . 4 \\% - 0 . 8 \\%$ worse than full-precision model. With ImageNet dataset, the accuracy of full-precision networks also improves but very marginally (by $0 . 3 \\%$ ). However, the impact of distillation technique on CIFAR-10 is quite pronounced - for example, top-1 error lowers by $1 . 1 \\%$ for ResNet-110. ", + "bbox": [ + 173, + 152, + 825, + 250 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/58b5bef731307b1943dcb500f0772a4d88af24ac4034b21f19ae656eba91abc5.jpg", + "image_caption": [ + "Figure 7: Distillation followed by quantization. " + ], + "image_footnote": [], + "bbox": [ + 300, + 276, + 687, + 664 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We experimented with a variation of scheme-C where a network is first compressed using distillation scheme (using a deeper ResNet as the teacher network) followed by lowering the precision and finetuning. The fine-tuning is done for 35-40 epochs with a very low learning-rate without the influence of any teacher network (no distillation). For this experiment, the student network starts from a higher accuracy compressed model compared to scheme-C (since distillation improves accuracy of student network at full-precision as well). Figure 7 shows the results with this experimental setting. For each configuration, we find the error-rate to lie in between the error-rates shown in Figure 6a and Figure 6b for the corresponding configuration, i.e. this scheme is better than low-precision training from scratch but not as good as training with methodology described in scheme-A. On an average, we find scheme-A to give $0 . 7 \\%$ better accuracy at low-precision configurations compared to the scheme mentioned here highlighting the benefits of “joint” low-precision training from scratch with distillation (Apprentice scheme). Many works proposing low-precision knobs advocate for training from scratch or training (for a significant number of epochs) with warm-startup – the conclusions from this experiment are in line with the observations in these papers. ", + "bbox": [ + 174, + 729, + 825, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "8 FUTURE RESEARCH ", + "text_level": 1, + "bbox": [ + 176, + 102, + 369, + 117 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Some works proposing low-precision networks advocate for making the layers wider (or the model larger) to recover accuracy at low-precision. These works propose making the layers wider by $2 \\mathrm { x }$ or $3 \\mathrm { x }$ . While these works show the benefits of low-precision, making the model larger increases the number of raw computations. Future work could investigate low-precision and less layer widening factor (say $1 . 1 0 \\mathrm { x }$ or $1 . 2 5 \\mathrm { x }$ ). This would help inference latency while maintaining accuracy at-par with baseline full-precision networks. ", + "bbox": [ + 174, + 133, + 825, + 217 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "As mentioned in section 5.5, sparsifying a model more than a certain percentage leads to accuracy loss. Investigating hyper-sparse network models without accuracy loss using distillation based schemes is another interesting avenue of further research. 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Our approach,", + "type": "text" + }, + { + "bbox": [ + 380, + 364, + 388, + 374 + ], + "score": 0.28, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 364, + 470, + 376 + ], + "score": 1.0, + "content": "pprentice, achieves", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 376, + 469, + 387 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 469, + 387 + ], + "score": 1.0, + "content": "state-of-the-art accuracies using ternary precision and 4-bit precision for variants", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 384, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 384, + 470, + 399 + ], + "score": 1.0, + "content": "of ResNet architecture on ImageNet dataset. We present three schemes using", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 142, + 397, + 470, + 409 + ], + "spans": [ + { + "bbox": [ + 142, + 397, + 470, + 409 + ], + "score": 1.0, + "content": "which one can apply knowledge distillation techniques to various stages of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 408, + 248, + 420 + ], + "spans": [ + { + "bbox": [ + 142, + 408, + 248, + 420 + ], + "score": 1.0, + "content": "train-and-deploy pipeline.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15, + "bbox_fs": [ + 141, + 254, + 470, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 442, + 206, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 440, + 208, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 208, + 457 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 504, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "score": 1.0, + "content": "Background: Today’s high performing deep neural networks (DNNs) for computer vision applica-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 468, + 491 + ], + "score": 1.0, + "content": "tions comprise of multiple layers and involve numerous parameters. These networks have", + "type": "text" + }, + { + "bbox": [ + 468, + 478, + 505, + 490 + ], + "score": 0.28, + "content": "{ \\mathcal { O } } ( { \\mathrm { G i g a - } }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 356, + 502 + ], + "score": 1.0, + "content": "FLOPS) compute requirements and generate models which are", + "type": "text" + }, + { + "bbox": [ + 357, + 489, + 366, + 500 + ], + "score": 0.47, + "content": "\\mathcal { O }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "(Mega-Bytes) in storage (Canziani", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "et al., 2016). Further, the memory and compute requirements during training and inference are quite", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "different (Mishra et al., 2017). Training is performed on big datasets with large batch-sizes where", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "memory footprint of activations dominates the model memory footprint. On the other hand, batch-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "size during inference is typically small and the model’s memory footprint dominates the runtime", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 546, + 199, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 199, + 556 + ], + "score": 1.0, + "content": "memory requirements.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 466, + 506, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 504, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "Because of complexity in compute, memory and storage requirements, training phase of the net-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "works is performed on CPU and/or GPU clusters in a distributed computing environment. Once", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "trained, a challenging aspect is deployment of trained models on resource constrained inference", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "systems such as portable devices or sensor networks, and for applications in which real-time predic-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "tions are required. Performing inference on edge-devices comes with severe constraints on memory,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 629 + ], + "score": 1.0, + "content": "compute and power. Additionally, ensemble based methods, which one can potentially use to get", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 627, + 444, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 444, + 640 + ], + "score": 1.0, + "content": "improved accuracy predictions, become prohibitive in resource constrained systems.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 560, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 642, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 506, + 656 + ], + "score": 1.0, + "content": "Quantization using low-precision numerics (Vanhoucke et al., 2011; Zhou et al., 2016; Lin et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "2015; Miyashita et al., 2016; Gupta et al., 2015; Zhu et al., 2016; Rastegari et al., 2016; Courbariaux", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "et al., 2015; Umuroglu et al., 2016; Mishra et al., 2017) and model compression (Bucilua et al. ˇ ,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "2006; Hinton et al., 2015; Romero et al., 2014) have emerged as popular solutions for resource", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "constrained deployment scenarios. With quantization, a low-precision version of network model is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "generated and deployed on the device. Operating in lower precision mode reduces compute as well", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "as data movement and storage requirements. However, majority of existing works in low-precision", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 732 + ], + "score": 1.0, + "content": "DNNs sacrifice accuracy over baseline full-precision networks. With model compression, a smaller", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 83, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 504, + 94 + ], + "score": 1.0, + "content": "low memory footprint network is trained to mimic the behaviour of the original complex network.", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "During this training, a process called, knowledge distillation is used to “transfer knowledge” from", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "the complex network to the smaller network. Work by Hinton et al. (2015) shows that the knowledge", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "distillation scheme can yield networks at comparable or slightly better accuracy than the original", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "complex model. However, to the best of our knowledge, all prior works using model compression", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 297, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 297, + 149 + ], + "score": 1.0, + "content": "techniques target compression at full-precision.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 642, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 504, + 94 + ], + "score": 1.0, + "content": "low memory footprint network is trained to mimic the behaviour of the original complex network.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "During this training, a process called, knowledge distillation is used to “transfer knowledge” from", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "the complex network to the smaller network. Work by Hinton et al. (2015) shows that the knowledge", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "distillation scheme can yield networks at comparable or slightly better accuracy than the original", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "complex model. However, to the best of our knowledge, all prior works using model compression", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 297, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 297, + 149 + ], + "score": 1.0, + "content": "techniques target compression at full-precision.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Our proposal: In this paper, we study the combination of network quantization with model com-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "pression and show that accuracies of low-precision networks can be significantly improved by using", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "knowledge distillation techniques. Previous studies on model compression use a large network as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "the teacher network and a small network as the student network. The small student network learns", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "from teacher network using distillation process. The network architecture of the student network", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "is typically different from that of the teacher network – for e.g. Hinton et al. (2015) investigate a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "student network that has fewer number of neurons in the hidden layers compared to the teacher net-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "work. In our work, the student network has similar topology as that of teacher network, except that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "the student network has low-precision neurons compared to the teacher network which has neurons", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 214, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 214, + 265 + ], + "score": 1.0, + "content": "operating at full-precision.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 108, + 270, + 502, + 303 + ], + "lines": [ + { + "bbox": [ + 106, + 268, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 504, + 282 + ], + "score": 1.0, + "content": "We call our approach Apprentice1 and study three schemes which produce low-precision net-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "works using knowledge distillation techniques. Each of these three schemes produce state-of-the-art", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 292, + 287, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 287, + 303 + ], + "score": 1.0, + "content": "ternary precision and 4-bit precision models.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "In the first scheme, a low-precision network and a full-precision network are jointly trained from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 332 + ], + "score": 1.0, + "content": "scratch using knowledge distillation scheme. Later in the paper we describe the rationale behind", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "this approach. Using this scheme, a new state-of-the-art accuracy is obtained for ternary and 4-bit", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "precision for ResNet-18, ResNet-34 and ResNet-50 on ImageNet dataset. In fact, using this scheme", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "the accuracy of the full-precision model also slightly improves. This scheme then serves as the new", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 308, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 308, + 375 + ], + "score": 1.0, + "content": "baseline for the other two schemes we investigate.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 504, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "In the second scheme, we start with a full-precision trained network and transfer knowledge from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 392, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 506, + 403 + ], + "score": 1.0, + "content": "this trained network continuously to train a low-precision network from scratch. We find that the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "low-precision network converges faster (albeit to similar accuracies as the first scheme) when a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 414, + 285, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 285, + 426 + ], + "score": 1.0, + "content": "trained complex network guides its training.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 504, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "In the third scheme, we start with a trained full-precision large network and an apprentice network", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "score": 1.0, + "content": "that has been initialised with full-precision weights. The apprentice network’s precision is lowered", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "and is fine-tuned using knowledge distillation techniques. We find that the low-precision network’s", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "accuracy marginally improves and surpasses the accuracy obtained via the first scheme. This scheme", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 475, + 494, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 494, + 487 + ], + "score": 1.0, + "content": "then sets the new state-of-the-art accuracies for the ResNet models at ternary and 4-bit precision.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 505 + ], + "score": 1.0, + "content": "Overall, the contributions of this paper are the techniques to obtain low-precision DNNs using", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "knowledge distillation technique. 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In our work, the apprentice is a low-precision network which is learning", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 451, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 451, + 733 + ], + "score": 1.0, + "content": "the knowledge of a high precision network (skilled employer) during a fixed number of epochs.", + "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, + 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, + 504, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 83, + 506, + 149 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "Our proposal: In this paper, we study the combination of network quantization with model com-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "pression and show that accuracies of low-precision networks can be significantly improved by using", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "knowledge distillation techniques. Previous studies on model compression use a large network as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "the teacher network and a small network as the student network. The small student network learns", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 505, + 210 + ], + "score": 1.0, + "content": "from teacher network using distillation process. The network architecture of the student network", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "is typically different from that of the teacher network – for e.g. Hinton et al. (2015) investigate a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "student network that has fewer number of neurons in the hidden layers compared to the teacher net-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "work. 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Each of these three schemes produce state-of-the-art", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 292, + 287, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 287, + 303 + ], + "score": 1.0, + "content": "ternary precision and 4-bit precision models.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 268, + 505, + 303 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 308, + 504, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "In the first scheme, a low-precision network and a full-precision network are jointly trained from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 332 + ], + "score": 1.0, + "content": "scratch using knowledge distillation scheme. Later in the paper we describe the rationale behind", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "this approach. Using this scheme, a new state-of-the-art accuracy is obtained for ternary and 4-bit", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "precision for ResNet-18, ResNet-34 and ResNet-50 on ImageNet dataset. In fact, using this scheme", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "the accuracy of the full-precision model also slightly improves. This scheme then serves as the new", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 308, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 308, + 375 + ], + "score": 1.0, + "content": "baseline for the other two schemes we investigate.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 308, + 506, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 380, + 504, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "In the second scheme, we start with a full-precision trained network and transfer knowledge from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 392, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 506, + 403 + ], + "score": 1.0, + "content": "this trained network continuously to train a low-precision network from scratch. We find that the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "low-precision network converges faster (albeit to similar accuracies as the first scheme) when a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 414, + 285, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 285, + 426 + ], + "score": 1.0, + "content": "trained complex network guides its training.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 380, + 506, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 504, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "In the third scheme, we start with a trained full-precision large network and an apprentice network", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "score": 1.0, + "content": "that has been initialised with full-precision weights. 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We discuss later in the paper", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "the network accuracies obtained using methods proposed in literature. These accuracies serve as the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 390, + 330, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 330, + 402 + ], + "score": 1.0, + "content": "starting point and baselines we compare to in our work.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "title", + "bbox": [ + 108, + 426, + 208, + 439 + ], + "lines": [ + { + "bbox": [ + 104, + 425, + 211, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 211, + 441 + ], + "score": 1.0, + "content": "3 RELATED WORK", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 456, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 504, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 504, + 468 + ], + "score": 1.0, + "content": "Low-precision networks: Low-precision DNNs are an active area of research. Most low-precision", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "networks acknowledge the over parameterization aspect of today’s DNN architectures and/or the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "aspect that lowering the precision of neurons post-training often does not impact the final perfor-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "mance. Reducing precision of weights for efficient inference pipeline has been very well studied.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 514 + ], + "score": 1.0, + "content": "Works like Binary connect (BC) (Courbariaux et al., 2015), Ternary-weight networks (TWN) (Li &", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "Liu, 2016), fine-grained ternary quantization (Mellempudi et al., 2017) and INQ (Zhou et al., 2017)", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 104, + 521, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 104, + 521, + 506, + 536 + ], + "score": 1.0, + "content": "target precision reduction of network weights. Accuracy is almost always affected when quantizing", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 533, + 504, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 463, + 546 + ], + "score": 1.0, + "content": "the weights significantly below 8-bits of precision. For AlexNet on ImageNet, TWN loses", + "type": "text" + }, + { + "bbox": [ + 463, + 533, + 478, + 544 + ], + "score": 0.83, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 533, + 504, + 546 + ], + "score": 1.0, + "content": "Top-1", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 543, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 506, + 558 + ], + "score": 1.0, + "content": "accuracy. Schemes like INQ, work in Sung et al. (2015) and Mellempudi et al. (2017) do fine-tuning", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 555, + 238, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 238, + 568 + ], + "score": 1.0, + "content": "to quantize the network weights.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 660 + ], + "lines": [ + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "Work in XNOR-NET (Rastegari et al., 2016), binary neural networks (Courbariaux & Bengio, 2016),", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 581, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 596 + ], + "score": 1.0, + "content": "DoReFa (Zhou et al., 2016) and trained ternary quantization (TTQ) (Zhu et al., 2016) target training", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 504, + 606 + ], + "score": 1.0, + "content": "pipeline. While TTQ targets weight quantization, most works targeting activation quantization show", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "that quantizing activations always hurt accuracy. XNOR-NET approach degrades Top-1 accuracy", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 120, + 628 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 616, + 140, + 627 + ], + "score": 0.86, + "content": "12 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 615, + 209, + 628 + ], + "score": 1.0, + "content": "and DoReFa by", + "type": "text" + }, + { + "bbox": [ + 209, + 616, + 225, + 627 + ], + "score": 0.84, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "when quantizing both weights and activations to 1-bit (for AlexNet", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "on ImageNet). Work by Gupta et al. (2015) advocates for low-precision fixed-point numbers for", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "training. They show 16-bits to be sufficient for training on CIFAR10 dataset. 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This", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "then facilitates more information flowing to the model parameters during back-propagation opera-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 153, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 505, + 167 + ], + "score": 1.0, + "content": "tion. FitNets (Romero et al., 2014) extend this work by using intermediate hidden layer outputs as", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 177 + ], + "score": 1.0, + "content": "target values for training a deeper, but thinner, student model. 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They achieve", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 360, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 360, + 265 + ], + "score": 1.0, + "content": "a compression factor of 10x for ResNet-34 on CIFAR datasets.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "Sparsity and hashing: Few other popular techniques for model compression are pruning (LeCun", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 504, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 504, + 293 + ], + "score": 1.0, + "content": "et al., 1990; Han et al., 2015a; Wen et al., 2016; Han et al., 2015b), hashing (Weinberger et al., 2009)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "and weight sharing (Chen et al., 2015b; Denil et al., 2013). 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They achieve", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 360, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 360, + 265 + ], + "score": 1.0, + "content": "a compression factor of 10x for ResNet-34 on CIFAR datasets.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 110, + 506, + 265 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "Sparsity and hashing: Few other popular techniques for model compression are pruning (LeCun", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 504, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 504, + 293 + ], + "score": 1.0, + "content": "et al., 1990; Han et al., 2015a; Wen et al., 2016; Han et al., 2015b), hashing (Weinberger et al., 2009)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "and weight sharing (Chen et al., 2015b; Denil et al., 2013). 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The teacher network is a high precision network and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 254, + 291, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 291, + 266 + ], + "score": 1.0, + "content": "the apprentice network is a low-precision network.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 291, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "Apprentice we combine both these techniques to improve the network accuracy as well as the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "runtime efficiency of DNNs. Using the teacher-student setup described in the last section, we inves-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 314, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 326 + ], + "score": 1.0, + "content": "tigate three schemes using which one can obtain a low-precision model for the student network. The", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "first scheme (scheme-A) jointly trains both the networks - full-precision teacher and low-precision", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "student network. The second scheme (scheme-B) trains only the low-precision student network but", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 506, + 360 + ], + "score": 1.0, + "content": "distills knowledge from a trained full-precision teacher network throughout the training process.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "The third scheme (scheme-C) starts with a trained full-precision teacher and a full-precision student", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "network but fine-tunes the student network after lowering its precision. Before we get into the details", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 393 + ], + "score": 1.0, + "content": "of each of these schemes, we discuss the accuracy numbers obtained using low-precision schemes", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 390, + 478, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 478, + 404 + ], + "score": 1.0, + "content": "described in literature. These accuracy figures serve as the baseline for comparative analysis.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 108, + 423, + 435, + 433 + ], + "lines": [ + { + "bbox": [ + 106, + 422, + 438, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 438, + 434 + ], + "score": 1.0, + "content": "5.1 TOP-1 ERROR WITH PRIOR PROPOSALS FOR LOW-PRECISION NETWORKS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "We focus on sub 8-bits precision for inference deployments, specifically ternary and 4-bits precision.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "We found TTQ (Zhu et al., 2016) scheme achieving the state-of-the-art accuracy with ternary pre-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "cision for weights and full-precision (32-bits floating-point) for activations. On Imagenet-1K (Rus-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 259, + 490 + ], + "score": 1.0, + "content": "sakovsky et al., 2015), TTQ achieves", + "type": "text" + }, + { + "bbox": [ + 259, + 479, + 287, + 489 + ], + "score": 0.85, + "content": "3 3 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "Top-1 error rate with a ResNet-18 model. We imple-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 489, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 500 + ], + "score": 1.0, + "content": "mented TTQ scheme for ResNet-34 and ResNet-50 models trained on Imagenet-1K and achieved", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 499, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 134, + 511 + ], + "score": 0.86, + "content": "2 8 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 499, + 151, + 513 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 152, + 501, + 179, + 511 + ], + "score": 0.86, + "content": "2 5 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 499, + 505, + 513 + ], + "score": 1.0, + "content": "Top-1 error rates, respectively. This scheme is our baseline for 2-bits weight and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "full-precision activations. For 2-bits weight and 8-bits activation, we find work by Mellempudi et al.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 523, + 504, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 504, + 534 + ], + "score": 1.0, + "content": "(2017) to achieve the best accuracies reported in literature. For ResNet-50, Mellempudi et al. 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Figure 2 shows the overall training framework. We use ResNet topology", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "for both the teacher and student network. When using a certain depth for the student network, we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 671, + 364, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 364, + 684 + ], + "score": 1.0, + "content": "pick the teacher network to have either the same or larger depth.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "In Bucilua et al. ˇ (2006) and Hinton et al. 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The teacher network is a high precision network and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 254, + 291, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 291, + 266 + ], + "score": 1.0, + "content": "the apprentice network is a low-precision network.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 291, + 505, + 402 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "Apprentice we combine both these techniques to improve the network accuracy as well as the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "runtime efficiency of DNNs. 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This scheme is our baseline for 2-bits weight and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "full-precision activations. For 2-bits weight and 8-bits activation, we find work by Mellempudi et al.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 523, + 504, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 504, + 534 + ], + "score": 1.0, + "content": "(2017) to achieve the best accuracies reported in literature. For ResNet-50, Mellempudi et al. (2017)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 134, + 545 + ], + "score": 1.0, + "content": "obtain", + "type": "text" + }, + { + "bbox": [ + 135, + 533, + 167, + 544 + ], + "score": 0.87, + "content": "2 9 . 2 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "Top-1 error. We consider this work to be our baseline for 2-bits weight and 8-bits", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 544, + 181, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 181, + 555 + ], + "score": 1.0, + "content": "activation models.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 445, + 505, + 555 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 561, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 104, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "For 4-bits precision, we find WRPN scheme (Mishra et al., 2017) to report the highest accuracy. We", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 572, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 583 + ], + "score": 1.0, + "content": "implemented this scheme for 4-bits weight and 8-bits activations. For ResNet-34 and ResNet-50", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 583, + 480, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 284, + 595 + ], + "score": 1.0, + "content": "models trained on Imagenet-1K, we achieve", + "type": "text" + }, + { + "bbox": [ + 284, + 583, + 312, + 594 + ], + "score": 0.86, + "content": "2 9 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 583, + 329, + 595 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 330, + 583, + 357, + 594 + ], + "score": 0.85, + "content": "2 8 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 583, + 480, + 595 + ], + "score": 1.0, + "content": "Top-1 error rates, respectively.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 560, + 505, + 595 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 615, + 406, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 614, + 407, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 407, + 627 + ], + "score": 1.0, + "content": "5.2 SCHEME-A: JOINT TRAINING OF TEACHER-STUDENT NETWORKS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 638, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 504, + 650 + ], + "score": 1.0, + "content": "In the first scheme that we investigate, a full-precision teacher network is jointly trained with a low-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "precision student network. Figure 2 shows the overall training framework. We use ResNet topology", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "for both the teacher and student network. When using a certain depth for the student network, we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 671, + 364, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 364, + 684 + ], + "score": 1.0, + "content": "pick the teacher network to have either the same or larger depth.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 638, + 505, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 700 + ], + "score": 1.0, + "content": "In Bucilua et al. ˇ (2006) and Hinton et al. (2015), only the student network trains while distilling", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "knowledge from the teacher network. In our case, we jointly train with the rationale that the teacher", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 711, + 504, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 504, + 721 + ], + "score": 1.0, + "content": "network would continuously guide the student network not only with the final trained logits, but also", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 720, + 444, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 444, + 732 + ], + "score": 1.0, + "content": "on what path the teacher takes towards generating those final higher accuracy logits.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 686, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 504, + 94 + ], + "score": 1.0, + "content": "We implement pre-activation version of ResNet (He et al., 2016) in TensorFlow (Abadi et al., 2015).", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "The training process closely follows the recipe mentioned in Torch implementation of ResNet - we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "use a batch size of 256 and no hyper-parameters are changed from what is mentioned in the recipe.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "For the teacher network, we experiment with ResNet-34, ResNet-50 and ResNet-101 as options.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 125, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 104, + 125, + 506, + 139 + ], + "score": 1.0, + "content": "For the student network, we experiment with low-precision variants of ResNet-18, ResNet-34 and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 154, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 154, + 148 + ], + "score": 1.0, + "content": "ResNet-50.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 506, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "For low-precision numerics, when using ternary precision we use the ternary weight network", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 393, + 178 + ], + "score": 1.0, + "content": "scheme (Li & Liu, 2016) where the weight tensors are quantized into", + "type": "text" + }, + { + "bbox": [ + 394, + 165, + 436, + 177 + ], + "score": 0.93, + "content": "\\{ - 1 , 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "with a per-layer", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 187 + ], + "score": 1.0, + "content": "scaling coefficient computed based on the mean of the positive terms in the weight tensor. We use", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "the WRPN scheme (Mishra et al., 2017) to quantize weights and activations to 4-bits or 8-bits. We", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 209 + ], + "score": 1.0, + "content": "do not lower the precision of the first layer and the final layer in the apprentice network. This is based", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "on the observation in almost all prior works that lowering the precision of these layers degrades the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "accuracy dramatically. 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For this figure, the difference in Top-1 error of the best low-precision student network is", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 394, + 723 + ], + "score": 1.0, + "content": "calculated from the baseline full-precision network (i.e. ResNet-18 with", + "type": "text" + }, + { + "bbox": [ + 394, + 710, + 421, + 720 + ], + "score": 0.86, + "content": "3 0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "Top-1 error), i.e. we", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "score": 1.0, + "content": "want to see how close a low-precision student network can come to a full-precision baseline model.", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 65 + } + ], + "page_idx": 5, + "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, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 504, + 94 + ], + "score": 1.0, + "content": "We implement pre-activation version of ResNet (He et al., 2016) in TensorFlow (Abadi et al., 2015).", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "The training process closely follows the recipe mentioned in Torch implementation of ResNet - we", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "use a batch size of 256 and no hyper-parameters are changed from what is mentioned in the recipe.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "For the teacher network, we experiment with ResNet-34, ResNet-50 and ResNet-101 as options.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 125, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 104, + 125, + 506, + 139 + ], + "score": 1.0, + "content": "For the student network, we experiment with low-precision variants of ResNet-18, ResNet-34 and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 154, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 154, + 148 + ], + "score": 1.0, + "content": "ResNet-50.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 104, + 82, + 506, + 148 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 506, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "For low-precision numerics, when using ternary precision we use the ternary weight network", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 393, + 178 + ], + "score": 1.0, + "content": "scheme (Li & Liu, 2016) where the weight tensors are quantized into", + "type": "text" + }, + { + "bbox": [ + 394, + 165, + 436, + 177 + ], + "score": 0.93, + "content": "\\{ - 1 , 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "with a per-layer", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 187 + ], + "score": 1.0, + "content": "scaling coefficient computed based on the mean of the positive terms in the weight tensor. 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Figure 4a and Figure 4b show the difference in Top-1 error achieved by our best", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "low-precision ResNet-34 and ResNet-50 student networks, respectively, and compares with results", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 104, + 125, + 312, + 140 + ], + "score": 1.0, + "content": "obtained using methods proposed in literature. Our", + "type": "text" + }, + { + "bbox": [ + 312, + 127, + 321, + 137 + ], + "score": 0.46, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "pprentice scheme significantly closes the gap", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "between full-precision baseline networks and low-precision variants of the same networks. In most", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 467, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 425, + 160 + ], + "score": 1.0, + "content": "cases we see our scheme to better the previously reported accuracy numbers by", + "type": "text" + }, + { + "bbox": [ + 425, + 149, + 463, + 159 + ], + "score": 0.85, + "content": "1 . 5 \\% - 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 149, + 467, + 160 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "Discussion: In scheme-A, we use a teacher network that is always as large or larger in number of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 188 + ], + "score": 1.0, + "content": "parameters than the student network. We experimented with a ternary ResNet-34 student network", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 485, + 199 + ], + "score": 1.0, + "content": "which was paired with a full-precision ResNet-18. The ternary model for ResNet-34 is about", + "type": "text" + }, + { + "bbox": [ + 485, + 187, + 504, + 198 + ], + "score": 0.74, + "content": "8 . 5 \\mathrm { x }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "smaller in size compared to the full-precision ResNet-18 model. The final trained accuracy of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 285, + 222 + ], + "score": 1.0, + "content": "ResNet-34 ternary model with this setup is", + "type": "text" + }, + { + "bbox": [ + 285, + 209, + 308, + 220 + ], + "score": 0.87, + "content": "2 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "worse than that obtained by pairing the ternary", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "ResNet-34 network with a ResNet-50 teacher network. This suggests that the distillation scheme", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "works only when the teacher network is higher in accuracy than the student network (and not neces-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "sarily bigger in capacity). Further, the benefit from using a larger teacher network saturates at some", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "point. This can be seen by picking up a precision point, say “32A, 2W” and looking at the error", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 264, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 264, + 276 + ], + "score": 1.0, + "content": "rates along the row in Table 1, 2 and 3.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "One concern, we had in the early stages of our investigation, with joint training of a low-precision", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "small network and a high precision large network was the influence of the small network’s accuracy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "on the accuracy of the large network. When using the joint cost function, the smaller network’s", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "probability scores are matched with the predictions from the teacher network. The joint cost is added", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "as a term to the total loss function (equation 1). This led us to posit that the larger network’s learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 336, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 347 + ], + "score": 1.0, + "content": "capability will be affected by the inherent impairment in the smaller low-precision network. Further,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "score": 1.0, + "content": "since the smaller student network learns form the larger teacher network, a vicious cycle might", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 356, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 371 + ], + "score": 1.0, + "content": "form where the student network’s accuracy will further drop because the teacher network’s learning", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 367, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 506, + 382 + ], + "score": 1.0, + "content": "capability is being impeded. However, in practice, we did not see this phenomenon occurring - in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "each case where the teacher network was jointly trained with a student network, the accuracy of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 249, + 402 + ], + "score": 1.0, + "content": "teacher network was always within", + "type": "text" + }, + { + "bbox": [ + 250, + 390, + 272, + 401 + ], + "score": 0.86, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 390, + 283, + 402 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 284, + 390, + 306, + 401 + ], + "score": 0.86, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 390, + 506, + 402 + ], + "score": 1.0, + "content": "of the accuracy of the teacher network without it", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 402, + 491, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 416, + 414 + ], + "score": 1.0, + "content": "jointly supervising a student network. This could be because of our choice of", + "type": "text" + }, + { + "bbox": [ + 416, + 403, + 424, + 411 + ], + "score": 0.64, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 402, + 428, + 414 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 428, + 402, + 435, + 413 + ], + "score": 0.79, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 402, + 453, + 414 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 453, + 403, + 461, + 413 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 402, + 491, + 414 + ], + "score": 1.0, + "content": "values.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 418, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 292, + 431 + ], + "score": 1.0, + "content": "In Section 4, we mentioned about temperature,", + "type": "text" + }, + { + "bbox": [ + 293, + 421, + 299, + 429 + ], + "score": 0.73, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 417, + 475, + 431 + ], + "score": 1.0, + "content": ", for Softmax function and hyper-parameters", + "type": "text" + }, + { + "bbox": [ + 475, + 419, + 501, + 429 + ], + "score": 0.89, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 417, + 505, + 431 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 140, + 441 + ], + "score": 0.91, + "content": "\\beta = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 429, + 157, + 441 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 157, + 430, + 190, + 441 + ], + "score": 0.9, + "content": "\\gamma = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 429, + 505, + 441 + ], + "score": 1.0, + "content": ". Since, we train directly on the logits of the teacher network, we did not have to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 270, + 453 + ], + "score": 1.0, + "content": "experiment with the appropriate value of", + "type": "text" + }, + { + "bbox": [ + 271, + 442, + 289, + 451 + ], + "score": 0.28, + "content": "\\tau , \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "is required when training on the soft targets produced", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "by the teacher network. Although we did not do extensive studies experimenting with training on", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 296, + 475 + ], + "score": 1.0, + "content": "soft targets as opposed to logits, we find that", + "type": "text" + }, + { + "bbox": [ + 297, + 462, + 326, + 472 + ], + "score": 0.91, + "content": "\\tau = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "gives us best results when training on soft", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "targets. Hinton et al. (2015) mention that when the student network is significantly smaller than the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 236, + 497 + ], + "score": 1.0, + "content": "teacher network, small values of", + "type": "text" + }, + { + "bbox": [ + 237, + 486, + 244, + 494 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "are more effective than large values. For few of the low-precision", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 261, + 507 + ], + "score": 1.0, + "content": "configurations, we experimented with", + "type": "text" + }, + { + "bbox": [ + 261, + 495, + 330, + 507 + ], + "score": 0.93, + "content": "\\alpha = \\beta = \\gamma = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 495, + 354, + 507 + ], + "score": 1.0, + "content": ", and,", + "type": "text" + }, + { + "bbox": [ + 354, + 495, + 389, + 506 + ], + "score": 0.83, + "content": "\\alpha = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 495, + 393, + 507 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 394, + 495, + 421, + 506 + ], + "score": 0.88, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 495, + 439, + 507 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 440, + 496, + 474, + 507 + ], + "score": 0.9, + "content": "\\gamma = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "or 0.3.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "Each of these configurations, yielded a lower performance model compared to our original choice", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 516, + 192, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 192, + 530 + ], + "score": 1.0, + "content": "for these parameters.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "For the third term in equation 1, we experimented with a mean-squared error loss function and also a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 407, + 557 + ], + "score": 1.0, + "content": "loss function with logits from both the student and the teacher network (i.e.", + "type": "text" + }, + { + "bbox": [ + 407, + 544, + 453, + 557 + ], + "score": 0.92, + "content": "\\mathcal { H } ( z ^ { T } , z ^ { A } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 543, + 505, + 557 + ], + "score": 1.0, + "content": ". We did not", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "find any improvement in accuracy compared to our original choice of the cost function formulation.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "A thorough investigation of the behavior of the networks with other values of hyper-parameters and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 578, + 333, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 333, + 590 + ], + "score": 1.0, + "content": "different loss functions is an agenda for our future work.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "score": 1.0, + "content": "Overall, we find the distillation process to be quite effective in getting us high accuracy low-precision", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "models. All our low-precision models surpass previously reported low-precision accuracy figures.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 253, + 628 + ], + "score": 1.0, + "content": "For example, TTQ scheme achieves", + "type": "text" + }, + { + "bbox": [ + 253, + 617, + 281, + 627 + ], + "score": 0.86, + "content": "3 3 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "Top-1 error rate for ResNet-18 with 2-bits weight. Our", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 382, + 641 + ], + "score": 1.0, + "content": "best ResNet-18 model, using scheme-A, with 2-bits weight achieves", + "type": "text" + }, + { + "bbox": [ + 383, + 628, + 417, + 638 + ], + "score": 0.9, + "content": "{ \\sim } 3 1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "error rate, improving", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 199, + 651 + ], + "score": 1.0, + "content": "the model accuracy by", + "type": "text" + }, + { + "bbox": [ + 199, + 639, + 221, + 649 + ], + "score": 0.89, + "content": "{ \\sim } 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "over TTQ. Similarly, the scheme in Mellempudi et al. (2017) achieves", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 134, + 660 + ], + "score": 0.85, + "content": "2 9 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 649, + 435, + 662 + ], + "score": 1.0, + "content": "Top-1 error with 2-bits weight and 8-bits activation. The best performing", + "type": "text" + }, + { + "bbox": [ + 436, + 650, + 444, + 660 + ], + "score": 0.55, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "pprentice net-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 229, + 672 + ], + "score": 1.0, + "content": "work at this precision achieves", + "type": "text" + }, + { + "bbox": [ + 229, + 660, + 257, + 671 + ], + "score": 0.85, + "content": "2 7 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "Top-1 error. For Scheme-B and Scheme-C, which we describe", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 672, + 282, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 282, + 683 + ], + "score": 1.0, + "content": "next, Scheme-A serves as the new baseline.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5 + } + ], + "page_idx": 7, + "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, + 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": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 104, + 83, + 506, + 160 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 177 + ], + "score": 1.0, + "content": "Discussion: In scheme-A, we use a teacher network that is always as large or larger in number of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 188 + ], + "score": 1.0, + "content": "parameters than the student network. We experimented with a ternary ResNet-34 student network", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 485, + 199 + ], + "score": 1.0, + "content": "which was paired with a full-precision ResNet-18. The ternary model for ResNet-34 is about", + "type": "text" + }, + { + "bbox": [ + 485, + 187, + 504, + 198 + ], + "score": 0.74, + "content": "8 . 5 \\mathrm { x }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "smaller in size compared to the full-precision ResNet-18 model. The final trained accuracy of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 285, + 222 + ], + "score": 1.0, + "content": "ResNet-34 ternary model with this setup is", + "type": "text" + }, + { + "bbox": [ + 285, + 209, + 308, + 220 + ], + "score": 0.87, + "content": "2 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "worse than that obtained by pairing the ternary", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "ResNet-34 network with a ResNet-50 teacher network. This suggests that the distillation scheme", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "works only when the teacher network is higher in accuracy than the student network (and not neces-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "sarily bigger in capacity). Further, the benefit from using a larger teacher network saturates at some", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "point. This can be seen by picking up a precision point, say “32A, 2W” and looking at the error", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 264, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 264, + 276 + ], + "score": 1.0, + "content": "rates along the row in Table 1, 2 and 3.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 164, + 506, + 276 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "One concern, we had in the early stages of our investigation, with joint training of a low-precision", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 104, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "small network and a high precision large network was the influence of the small network’s accuracy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "on the accuracy of the large network. When using the joint cost function, the smaller network’s", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "probability scores are matched with the predictions from the teacher network. The joint cost is added", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 104, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "as a term to the total loss function (equation 1). This led us to posit that the larger network’s learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 336, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 347 + ], + "score": 1.0, + "content": "capability will be affected by the inherent impairment in the smaller low-precision network. Further,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "score": 1.0, + "content": "since the smaller student network learns form the larger teacher network, a vicious cycle might", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 356, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 371 + ], + "score": 1.0, + "content": "form where the student network’s accuracy will further drop because the teacher network’s learning", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 367, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 104, + 367, + 506, + 382 + ], + "score": 1.0, + "content": "capability is being impeded. However, in practice, we did not see this phenomenon occurring - in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "each case where the teacher network was jointly trained with a student network, the accuracy of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 249, + 402 + ], + "score": 1.0, + "content": "teacher network was always within", + "type": "text" + }, + { + "bbox": [ + 250, + 390, + 272, + 401 + ], + "score": 0.86, + "content": "0 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 390, + 283, + 402 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 284, + 390, + 306, + 401 + ], + "score": 0.86, + "content": "0 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 390, + 506, + 402 + ], + "score": 1.0, + "content": "of the accuracy of the teacher network without it", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 402, + 491, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 416, + 414 + ], + "score": 1.0, + "content": "jointly supervising a student network. This could be because of our choice of", + "type": "text" + }, + { + "bbox": [ + 416, + 403, + 424, + 411 + ], + "score": 0.64, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 402, + 428, + 414 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 428, + 402, + 435, + 413 + ], + "score": 0.79, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 402, + 453, + 414 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 453, + 403, + 461, + 413 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 402, + 491, + 414 + ], + "score": 1.0, + "content": "values.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 281, + 506, + 414 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 418, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 292, + 431 + ], + "score": 1.0, + "content": "In Section 4, we mentioned about temperature,", + "type": "text" + }, + { + "bbox": [ + 293, + 421, + 299, + 429 + ], + "score": 0.73, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 417, + 475, + 431 + ], + "score": 1.0, + "content": ", for Softmax function and hyper-parameters", + "type": "text" + }, + { + "bbox": [ + 475, + 419, + 501, + 429 + ], + "score": 0.89, + "content": "\\alpha = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 417, + 505, + 431 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 140, + 441 + ], + "score": 0.91, + "content": "\\beta = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 429, + 157, + 441 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 157, + 430, + 190, + 441 + ], + "score": 0.9, + "content": "\\gamma = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 429, + 505, + 441 + ], + "score": 1.0, + "content": ". Since, we train directly on the logits of the teacher network, we did not have to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 441, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 270, + 453 + ], + "score": 1.0, + "content": "experiment with the appropriate value of", + "type": "text" + }, + { + "bbox": [ + 271, + 442, + 289, + 451 + ], + "score": 0.28, + "content": "\\tau , \\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 441, + 506, + 453 + ], + "score": 1.0, + "content": "is required when training on the soft targets produced", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "by the teacher network. Although we did not do extensive studies experimenting with training on", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 296, + 475 + ], + "score": 1.0, + "content": "soft targets as opposed to logits, we find that", + "type": "text" + }, + { + "bbox": [ + 297, + 462, + 326, + 472 + ], + "score": 0.91, + "content": "\\tau = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "gives us best results when training on soft", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "targets. Hinton et al. (2015) mention that when the student network is significantly smaller than the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 236, + 497 + ], + "score": 1.0, + "content": "teacher network, small values of", + "type": "text" + }, + { + "bbox": [ + 237, + 486, + 244, + 494 + ], + "score": 0.79, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "are more effective than large values. For few of the low-precision", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 261, + 507 + ], + "score": 1.0, + "content": "configurations, we experimented with", + "type": "text" + }, + { + "bbox": [ + 261, + 495, + 330, + 507 + ], + "score": 0.93, + "content": "\\alpha = \\beta = \\gamma = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 495, + 354, + 507 + ], + "score": 1.0, + "content": ", and,", + "type": "text" + }, + { + "bbox": [ + 354, + 495, + 389, + 506 + ], + "score": 0.83, + "content": "\\alpha = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 495, + 393, + 507 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 394, + 495, + 421, + 506 + ], + "score": 0.88, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 495, + 439, + 507 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 440, + 496, + 474, + 507 + ], + "score": 0.9, + "content": "\\gamma = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "or 0.3.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "Each of these configurations, yielded a lower performance model compared to our original choice", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 516, + 192, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 192, + 530 + ], + "score": 1.0, + "content": "for these parameters.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 417, + 506, + 530 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "For the third term in equation 1, we experimented with a mean-squared error loss function and also a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 407, + 557 + ], + "score": 1.0, + "content": "loss function with logits from both the student and the teacher network (i.e.", + "type": "text" + }, + { + "bbox": [ + 407, + 544, + 453, + 557 + ], + "score": 0.92, + "content": "\\mathcal { H } ( z ^ { T } , z ^ { A } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 543, + 505, + 557 + ], + "score": 1.0, + "content": ". We did not", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "find any improvement in accuracy compared to our original choice of the cost function formulation.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "A thorough investigation of the behavior of the networks with other values of hyper-parameters and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 578, + 333, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 333, + 590 + ], + "score": 1.0, + "content": "different loss functions is an agenda for our future work.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 533, + 506, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "score": 1.0, + "content": "Overall, we find the distillation process to be quite effective in getting us high accuracy low-precision", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "models. All our low-precision models surpass previously reported low-precision accuracy figures.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 253, + 628 + ], + "score": 1.0, + "content": "For example, TTQ scheme achieves", + "type": "text" + }, + { + "bbox": [ + 253, + 617, + 281, + 627 + ], + "score": 0.86, + "content": "3 3 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "Top-1 error rate for ResNet-18 with 2-bits weight. Our", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 382, + 641 + ], + "score": 1.0, + "content": "best ResNet-18 model, using scheme-A, with 2-bits weight achieves", + "type": "text" + }, + { + "bbox": [ + 383, + 628, + 417, + 638 + ], + "score": 0.9, + "content": "{ \\sim } 3 1 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "error rate, improving", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 637, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 199, + 651 + ], + "score": 1.0, + "content": "the model accuracy by", + "type": "text" + }, + { + "bbox": [ + 199, + 639, + 221, + 649 + ], + "score": 0.89, + "content": "{ \\sim } 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 637, + 505, + 651 + ], + "score": 1.0, + "content": "over TTQ. Similarly, the scheme in Mellempudi et al. (2017) achieves", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 134, + 660 + ], + "score": 0.85, + "content": "2 9 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 649, + 435, + 662 + ], + "score": 1.0, + "content": "Top-1 error with 2-bits weight and 8-bits activation. The best performing", + "type": "text" + }, + { + "bbox": [ + 436, + 650, + 444, + 660 + ], + "score": 0.55, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "pprentice net-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 229, + 672 + ], + "score": 1.0, + "content": "work at this precision achieves", + "type": "text" + }, + { + "bbox": [ + 229, + 660, + 257, + 671 + ], + "score": 0.85, + "content": "2 7 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "Top-1 error. For Scheme-B and Scheme-C, which we describe", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 672, + 282, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 282, + 683 + ], + "score": 1.0, + "content": "next, Scheme-A serves as the new baseline.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 594, + 506, + 683 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 110, + 83, + 370, + 93 + ], + "lines": [ + { + "bbox": [ + 108, + 82, + 372, + 95 + ], + "spans": [ + { + "bbox": [ + 108, + 82, + 372, + 95 + ], + "score": 1.0, + "content": "5.3 SCHEME-B: DISTILLING KNOWLEDGE FROM A TEACHER", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 116 + ], + "score": 1.0, + "content": "In this scheme, we start with a trained teacher network. Referring back to Figure 2, the input image", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "is passed to both the teacher and the student network, except that the learning with back-propagation", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 126, + 504, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 504, + 137 + ], + "score": 1.0, + "content": "happens only in the low precision student network which is trained from scratch. This is the scheme", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "used by Bucilua et al. ˇ (2006) and Hinton et al. (2015) for training their student networks. In this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 504, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 504, + 159 + ], + "score": 1.0, + "content": "scheme, the first term in equation 1 zeroes out and only the last two terms in the equation contribute", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 159, + 207, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 207, + 170 + ], + "score": 1.0, + "content": "toward the loss function.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "image", + "bbox": [ + 119, + 181, + 493, + 443 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 181, + 493, + 443 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 181, + 493, + 443 + ], + "spans": [ + { + "bbox": [ + 119, + 181, + 493, + 443 + ], + "score": 0.976, + "type": "image", + "image_path": "776aa02a37e8a7359b0d5d75f2da39efde532341c5e092bd7590bc5fbde6419b.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 119, + 181, + 493, + 268.3333333333333 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 119, + 268.3333333333333, + 493, + 355.66666666666663 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 119, + 355.66666666666663, + 493, + 442.99999999999994 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 128, + 460, + 482, + 471 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 459, + 484, + 472 + ], + "spans": [ + { + "bbox": [ + 126, + 459, + 484, + 472 + ], + "score": 1.0, + "content": "Figure 5: Top-1 error rate versus epochs of four student networks using scheme-A and scheme-B.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + } + ], + "index": 9.0 + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "score": 1.0, + "content": "With scheme-B, one can pre-compute and store the logit values for the input images on disk and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "access them during training the student network. This saves the forward pass computations in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "score": 1.0, + "content": "teacher network. Scheme-B might also help the scenario where a student network attempts to learn", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "score": 1.0, + "content": "the “dark knowledge” from a teacher network that has already been trained on some private or", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 529, + 448, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 448, + 542 + ], + "score": 1.0, + "content": "sensitive data (in addition to the data the student network is interested in training on).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "With scheme-A, we had the hypothesis that the student network would be influenced by not only the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "“dark knowledge” in the teacher network but also the path the teacher adopts to learn the knowledge.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "With scheme-B we find, that the student network gets to similar accuracy numbers as the teacher", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 579, + 276, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 276, + 591 + ], + "score": 1.0, + "content": "network albeit at fewer number of epochs.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "With this scheme, the training accuracies are similar to that reported in Table 1, 2 and 3. The", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "low-precision student networks, however, learn in fewer number of epochs. Figure 5 plots the Top-1", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "error rates for few of the configurations from our experiment suite. In each of these plots, the student", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 630, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 641 + ], + "score": 1.0, + "content": "network in scheme-B converges around 80th-85th epoch compared to about 105 epochs in scheme-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 641, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 408, + 652 + ], + "score": 1.0, + "content": "A. In general, we find the student networks with scheme-B to learn in about", + "type": "text" + }, + { + "bbox": [ + 408, + 641, + 449, + 651 + ], + "score": 0.67, + "content": "10 \\% { - } 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 641, + 505, + 652 + ], + "score": 1.0, + "content": "fewer epochs", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 651, + 310, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 310, + 663 + ], + "score": 1.0, + "content": "than the student networks trained using scheme-A.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 677, + 337, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 340, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 340, + 690 + ], + "score": 1.0, + "content": "5.4 SCHEME-C: FINE-TUNING THE STUDENT MODEL", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "Scheme-C is very similar to scheme-B, except that the student network is primed with full precision", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "training weights before the start of the training process. At the beginning of the training process,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "the weights and activations are lowered and the student network is sort of fine-tuned on the dataset.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + } + ], + "page_idx": 8, + "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, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 110, + 83, + 370, + 93 + ], + "lines": [ + { + "bbox": [ + 108, + 82, + 372, + 95 + ], + "spans": [ + { + "bbox": [ + 108, + 82, + 372, + 95 + ], + "score": 1.0, + "content": "5.3 SCHEME-B: DISTILLING KNOWLEDGE FROM A TEACHER", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 116 + ], + "score": 1.0, + "content": "In this scheme, we start with a trained teacher network. Referring back to Figure 2, the input image", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "is passed to both the teacher and the student network, except that the learning with back-propagation", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 126, + 504, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 504, + 137 + ], + "score": 1.0, + "content": "happens only in the low precision student network which is trained from scratch. This is the scheme", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "used by Bucilua et al. ˇ (2006) and Hinton et al. (2015) for training their student networks. In this", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 504, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 504, + 159 + ], + "score": 1.0, + "content": "scheme, the first term in equation 1 zeroes out and only the last two terms in the equation contribute", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 159, + 207, + 170 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 207, + 170 + ], + "score": 1.0, + "content": "toward the loss function.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 102, + 505, + 170 + ] + }, + { + "type": "image", + "bbox": [ + 119, + 181, + 493, + 443 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 181, + 493, + 443 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 181, + 493, + 443 + ], + "spans": [ + { + "bbox": [ + 119, + 181, + 493, + 443 + ], + "score": 0.976, + "type": "image", + "image_path": "776aa02a37e8a7359b0d5d75f2da39efde532341c5e092bd7590bc5fbde6419b.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 119, + 181, + 493, + 268.3333333333333 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 119, + 268.3333333333333, + 493, + 355.66666666666663 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 119, + 355.66666666666663, + 493, + 442.99999999999994 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 128, + 460, + 482, + 471 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 459, + 484, + 472 + ], + "spans": [ + { + "bbox": [ + 126, + 459, + 484, + 472 + ], + "score": 1.0, + "content": "Figure 5: Top-1 error rate versus epochs of four student networks using scheme-A and scheme-B.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + } + ], + "index": 9.0 + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 505, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "score": 1.0, + "content": "With scheme-B, one can pre-compute and store the logit values for the input images on disk and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "access them during training the student network. This saves the forward pass computations in the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "score": 1.0, + "content": "teacher network. Scheme-B might also help the scenario where a student network attempts to learn", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "score": 1.0, + "content": "the “dark knowledge” from a teacher network that has already been trained on some private or", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 529, + 448, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 448, + 542 + ], + "score": 1.0, + "content": "sensitive data (in addition to the data the student network is interested in training on).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 485, + 506, + 542 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 559 + ], + "score": 1.0, + "content": "With scheme-A, we had the hypothesis that the student network would be influenced by not only the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "“dark knowledge” in the teacher network but also the path the teacher adopts to learn the knowledge.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "With scheme-B we find, that the student network gets to similar accuracy numbers as the teacher", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 579, + 276, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 276, + 591 + ], + "score": 1.0, + "content": "network albeit at fewer number of epochs.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 546, + 505, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 596, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 609 + ], + "score": 1.0, + "content": "With this scheme, the training accuracies are similar to that reported in Table 1, 2 and 3. The", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "low-precision student networks, however, learn in fewer number of epochs. Figure 5 plots the Top-1", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "error rates for few of the configurations from our experiment suite. In each of these plots, the student", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 630, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 641 + ], + "score": 1.0, + "content": "network in scheme-B converges around 80th-85th epoch compared to about 105 epochs in scheme-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 641, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 408, + 652 + ], + "score": 1.0, + "content": "A. In general, we find the student networks with scheme-B to learn in about", + "type": "text" + }, + { + "bbox": [ + 408, + 641, + 449, + 651 + ], + "score": 0.67, + "content": "10 \\% { - } 2 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 641, + 505, + 652 + ], + "score": 1.0, + "content": "fewer epochs", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 651, + 310, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 310, + 663 + ], + "score": 1.0, + "content": "than the student networks trained using scheme-A.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 596, + 505, + 663 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 677, + 337, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 340, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 340, + 690 + ], + "score": 1.0, + "content": "5.4 SCHEME-C: FINE-TUNING THE STUDENT MODEL", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "Scheme-C is very similar to scheme-B, except that the student network is primed with full precision", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "training weights before the start of the training process. At the beginning of the training process,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "the weights and activations are lowered and the student network is sort of fine-tuned on the dataset.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 698, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Similar to scheme-B, only the final two terms in equation 1 comprise the loss function and the low-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "precision student network is trained with back-propagation algorithm. Since, the network starts from", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "a good initial point, comparatively low learning rate is used throughout the training process. There is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "no clear recipe for learning rates (and change of learning rate with epochs) which works across all the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "configurations. In general, we find training with a learning rate of 1e-3 for 10 to 15 epochs, followed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "by 1e-4 for another 5 to 10 epochs, followed by 1e-5 for another 5 epochs to give us the best accuracy.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "Some configurations run for about 40 to 50 epochs before stabilizing. For these configurations, we", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "find training using scheme-B with warm startup (train the student network at full-precision for about", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "25-30 epochs before lowering the precision) to be equally good. Wu (2016) investigate a similar", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "scheme for binary precision on AlexNet. Our experiments show that distillation is an overkill for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "AlexNet and one can get comparable accuracies using techniques proposed in (Tang et al., 2017;", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 202, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 506, + 216 + ], + "score": 1.0, + "content": "Mishra et al., 2017). Further, Wu (2016) hypothesize that distillation scheme will work on larger", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 226 + ], + "score": 1.0, + "content": "networks, we show in this paper how to make it work. Tann et al. (2017) use a similar scheme for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "score": 1.0, + "content": "AlexNet and mention starting from a non-global optimal checkpoint gives better accuracy, though", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 235, + 344, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 344, + 248 + ], + "score": 1.0, + "content": "we did not find this observation to hold in our experiments.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "We find the final accuracy of the models obtained using scheme-C to be (marginally) better than", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "score": 1.0, + "content": "those obtained using scheme-A or scheme-B. Table 4 shows error rates of few configurations of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "low-precision student network obtained using scheme-A (or scheme-B) and scheme-C. For ResNet-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 413, + 299 + ], + "score": 1.0, + "content": "50 student network, the accuracy with ternary weights is further improved by", + "type": "text" + }, + { + "bbox": [ + 413, + 286, + 435, + 297 + ], + "score": 0.87, + "content": "0 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "compared to that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "obtained using scheme-A. Note that the performance of ternary networks obtained using scheme-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 308, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 398, + 319 + ], + "score": 1.0, + "content": "A are already state-of-the-art. Hence, for ResNet-50 ternary networks,", + "type": "text" + }, + { + "bbox": [ + 399, + 308, + 426, + 318 + ], + "score": 0.86, + "content": "2 4 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 308, + 505, + 319 + ], + "score": 1.0, + "content": "Top-1 error rate is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 504, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 364, + 330 + ], + "score": 1.0, + "content": "the new state-of-the-art. 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Similarly, with 4-bits weight and 8-bits activations, ResNet-50 model obtained using", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 159, + 353 + ], + "score": 1.0, + "content": "scheme-C is", + "type": "text" + }, + { + "bbox": [ + 159, + 341, + 182, + 351 + ], + "score": 0.86, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 340, + 470, + 353 + ], + "score": 1.0, + "content": "better than that obtained with scheme-A (closing the gap to be within", + "type": "text" + }, + { + "bbox": [ + 470, + 341, + 492, + 351 + ], + "score": 0.85, + "content": "1 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 351, + 251, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 251, + 364 + ], + "score": 1.0, + "content": "full-precision ResNet-50 accuracy).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5 + }, + { + "type": "table", + "bbox": [ + 115, + 399, + 495, + 507 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 114, + 374, + 496, + 397 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 374, + 497, + 387 + ], + "spans": [ + { + "bbox": [ + 114, + 374, + 338, + 387 + ], + "score": 1.0, + "content": "Table 4: Top-1 ImageNet-1K validation set error rate", + "type": "text" + }, + { + "bbox": [ + 338, + 375, + 354, + 386 + ], + "score": 0.82, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 374, + 497, + 387 + ], + "score": 1.0, + "content": "with scheme-A and scheme-C for", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 114, + 385, + 426, + 398 + ], + "spans": [ + { + "bbox": [ + 114, + 385, + 426, + 398 + ], + "score": 1.0, + "content": "ResNet-34 and ResNet-50 student networks with ternary and 4-bits precision.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "table_body", + "bbox": [ + 115, + 399, + 495, + 507 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 399, + 495, + 507 + ], + "spans": [ + { + "bbox": [ + 115, + 399, + 495, + 507 + ], + "score": 0.981, + "html": "
ResNet-34 student with ResNet-50 teacher32A,2W with scheme-A or B32A,2W with scheme-C
ResNet-50 student with ResNet101 teacher27.2 25.326.9 24.7
ResNet-34 student with ResNet-50 teacher8A,4W with scheme-A or B8A,4W with scheme-C
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ResNet-50 student with ResNet101 teacher25.525.1
", + "type": "table", + "image_path": "55e87daaa579c913cf7001bd757a8242642a4226d05bc53ad2a5b79129b89061.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 115, + 399, + 495, + 435.0 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 115, + 435.0, + 495, + 471.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 115, + 471.0, + 495, + 507.0 + ], + "spans": [], + "index": 29 + } + ] + } + ], + "index": 26.75 + }, + { + "type": "text", + "bbox": [ + 105, + 519, + 504, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 504, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 504, + 531 + ], + "score": 1.0, + "content": "Scheme-C is useful when one already has a trained network which can be fine-tuned using knowl-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 531, + 441, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 441, + 542 + ], + "score": 1.0, + "content": "edge distillation schemes to produce a low-precision variant of the trained network.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 107, + 557, + 361, + 568 + ], + "lines": [ + { + "bbox": [ + 106, + 557, + 363, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 557, + 363, + 569 + ], + "score": 1.0, + "content": "5.5 DISCUSSION - TERNARY PRECISION VERSUS SPARSITY", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "As mentioned earlier, low-precision is a form of model compression. There are many works which", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "target network sparsification and pruning techniques to compress a model. With ternary preci-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 314, + 613 + ], + "score": 1.0, + "content": "sion models, the model size reduces by a factor of", + "type": "text" + }, + { + "bbox": [ + 314, + 600, + 335, + 612 + ], + "score": 0.41, + "content": "2 / 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "compared to full-precision models. With", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 107, + 611, + 115, + 622 + ], + "score": 0.29, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "pprentice, we show how one can get a performant model with ternary precision. Many works tar-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "geting network pruning and sparsification target a full-precision model to implement their scheme.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "To be comparable in model size to ternary networks, a full-precision model needs to be sparsified", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 118, + 657 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 644, + 151, + 654 + ], + "score": 0.88, + "content": "9 3 . 7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 643, + 505, + 657 + ], + "score": 1.0, + "content": ". Further, to be effective, a sparse model needs to store a key for every non-zero value de-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "noting the position of the value in the weight tensor. This adds storage overhead and a sparse model", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 178, + 678 + ], + "score": 1.0, + "content": "needs to be about", + "type": "text" + }, + { + "bbox": [ + 178, + 666, + 198, + 676 + ], + "score": 0.89, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "sparse to be at-par in memory size as a 2-bit model. Note that ternary preci-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "sion also has inherent sparsity (zero is a term in the ternary symbol dictionary) – we find our ternary", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 185, + 700 + ], + "score": 1.0, + "content": "models to be about", + "type": "text" + }, + { + "bbox": [ + 185, + 688, + 204, + 698 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "sparse. In work by Wen et al. (2016) and Han et al. (2015b), sparsification", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 416, + 711 + ], + "score": 1.0, + "content": "of full-precision networks is proposed but the sparsity achieved is less than", + "type": "text" + }, + { + "bbox": [ + 416, + 699, + 448, + 709 + ], + "score": 0.88, + "content": "9 3 . 7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". Further, the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "network accuracy using techniques in both these works lead to larger degradation in accuracy com-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "pared to our ternary models. Overall, we believe, our ternary precision models to be state-of-the-art", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 39.5 + } + ], + "page_idx": 9, + "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": [ + 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": [ + 107, + 83, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Similar to scheme-B, only the final two terms in equation 1 comprise the loss function and the low-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "precision student network is trained with back-propagation algorithm. Since, the network starts from", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "a good initial point, comparatively low learning rate is used throughout the training process. There is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "no clear recipe for learning rates (and change of learning rate with epochs) which works across all the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "configurations. In general, we find training with a learning rate of 1e-3 for 10 to 15 epochs, followed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "by 1e-4 for another 5 to 10 epochs, followed by 1e-5 for another 5 epochs to give us the best accuracy.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 161 + ], + "score": 1.0, + "content": "Some configurations run for about 40 to 50 epochs before stabilizing. For these configurations, we", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "find training using scheme-B with warm startup (train the student network at full-precision for about", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "25-30 epochs before lowering the precision) to be equally good. Wu (2016) investigate a similar", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "scheme for binary precision on AlexNet. Our experiments show that distillation is an overkill for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "AlexNet and one can get comparable accuracies using techniques proposed in (Tang et al., 2017;", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 202, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 506, + 216 + ], + "score": 1.0, + "content": "Mishra et al., 2017). Further, Wu (2016) hypothesize that distillation scheme will work on larger", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 226 + ], + "score": 1.0, + "content": "networks, we show in this paper how to make it work. Tann et al. (2017) use a similar scheme for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 505, + 239 + ], + "score": 1.0, + "content": "AlexNet and mention starting from a non-global optimal checkpoint gives better accuracy, though", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 235, + 344, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 344, + 248 + ], + "score": 1.0, + "content": "we did not find this observation to hold in our experiments.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 82, + 506, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 363 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 265 + ], + "score": 1.0, + "content": "We find the final accuracy of the models obtained using scheme-C to be (marginally) better than", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 275 + ], + "score": 1.0, + "content": "those obtained using scheme-A or scheme-B. Table 4 shows error rates of few configurations of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "low-precision student network obtained using scheme-A (or scheme-B) and scheme-C. For ResNet-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 413, + 299 + ], + "score": 1.0, + "content": "50 student network, the accuracy with ternary weights is further improved by", + "type": "text" + }, + { + "bbox": [ + 413, + 286, + 435, + 297 + ], + "score": 0.87, + "content": "0 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "compared to that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "obtained using scheme-A. Note that the performance of ternary networks obtained using scheme-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 308, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 398, + 319 + ], + "score": 1.0, + "content": "A are already state-of-the-art. Hence, for ResNet-50 ternary networks,", + "type": "text" + }, + { + "bbox": [ + 399, + 308, + 426, + 318 + ], + "score": 0.86, + "content": "2 4 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 308, + 505, + 319 + ], + "score": 1.0, + "content": "Top-1 error rate is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 504, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 364, + 330 + ], + "score": 1.0, + "content": "the new state-of-the-art. 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ResNet-34 student with ResNet-50 teacher32A,2W with scheme-A or B32A,2W with scheme-C
ResNet-50 student with ResNet101 teacher27.2 25.326.9 24.7
ResNet-34 student with ResNet-50 teacher8A,4W with scheme-A or B8A,4W with scheme-C
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ResNet-50 student with ResNet101 teacher25.525.1
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There are many works which", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "target network sparsification and pruning techniques to compress a model. With ternary preci-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 314, + 613 + ], + "score": 1.0, + "content": "sion models, the model size reduces by a factor of", + "type": "text" + }, + { + "bbox": [ + 314, + 600, + 335, + 612 + ], + "score": 0.41, + "content": "2 / 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "compared to full-precision models. With", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 107, + 611, + 115, + 622 + ], + "score": 0.29, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "pprentice, we show how one can get a performant model with ternary precision. Many works tar-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "geting network pruning and sparsification target a full-precision model to implement their scheme.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "To be comparable in model size to ternary networks, a full-precision model needs to be sparsified", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 118, + 657 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 644, + 151, + 654 + ], + "score": 0.88, + "content": "9 3 . 7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 643, + 505, + 657 + ], + "score": 1.0, + "content": ". Further, to be effective, a sparse model needs to store a key for every non-zero value de-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "noting the position of the value in the weight tensor. This adds storage overhead and a sparse model", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 178, + 678 + ], + "score": 1.0, + "content": "needs to be about", + "type": "text" + }, + { + "bbox": [ + 178, + 666, + 198, + 676 + ], + "score": 0.89, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "sparse to be at-par in memory size as a 2-bit model. Note that ternary preci-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "sion also has inherent sparsity (zero is a term in the ternary symbol dictionary) – we find our ternary", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 185, + 700 + ], + "score": 1.0, + "content": "models to be about", + "type": "text" + }, + { + "bbox": [ + 185, + 688, + 204, + 698 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "sparse. In work by Wen et al. (2016) and Han et al. (2015b), sparsification", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 416, + 711 + ], + "score": 1.0, + "content": "of full-precision networks is proposed but the sparsity achieved is less than", + "type": "text" + }, + { + "bbox": [ + 416, + 699, + 448, + 709 + ], + "score": 0.88, + "content": "9 3 . 7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". 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As the network", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "becomes larger in size, the impact of lowering precision is diminished (relative to the accuracy of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "the network at that depth when using full-precision). 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Thus, in total there are", + "type": "text" + }, + { + "bbox": [ + 451, + 534, + 474, + 544 + ], + "score": 0.82, + "content": "6 n { + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "weight", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 544, + 136, + 558 + ], + "spans": [ + { + "bbox": [ + 104, + 544, + 136, + 558 + ], + "score": 1.0, + "content": "layers.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 456, + 506, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "Figure 6a shows the impact of lowering precision as the depth of ResNet varies. As the network", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "becomes larger in size, the impact of lowering precision is diminished (relative to the accuracy of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "the network at that depth when using full-precision). For example, with ResNet-110, full-precision", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 180, + 606 + ], + "score": 1.0, + "content": "Top-1 error rate is", + "type": "text" + }, + { + "bbox": [ + 180, + 594, + 207, + 605 + ], + "score": 0.87, + "content": "6 . 1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 595, + 468, + 606 + ], + "score": 1.0, + "content": ". At the same depth, ternarizing the model gives similar error rate", + "type": "text" + }, + { + "bbox": [ + 468, + 594, + 501, + 605 + ], + "score": 0.86, + "content": "( 6 . 2 4 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 595, + 504, + 606 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "Comparing this with ResNet-20, the gap between full-precision and ternary model (2-bits weight", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 214, + 629 + ], + "score": 1.0, + "content": "and 32-bits activations) is", + "type": "text" + }, + { + "bbox": [ + 214, + 616, + 236, + 627 + ], + "score": 0.85, + "content": "0 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 615, + 241, + 629 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 242, + 616, + 264, + 627 + ], + "score": 0.87, + "content": "7 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 615, + 282, + 629 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 282, + 616, + 305, + 627 + ], + "score": 0.86, + "content": "8 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "Top-1 error). Overall, we find that ternarizing a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "model closely follows accuracy of baseline full-precision model. However, lowering both weights", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "and activations almost always leads to large accuracy degradation. Accuracy of 2-bits weight and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 217, + 661 + ], + "score": 1.0, + "content": "8-bits activation network is", + "type": "text" + }, + { + "bbox": [ + 217, + 649, + 263, + 660 + ], + "score": 0.78, + "content": "0 . 8 \\% 1 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 648, + 422, + 661 + ], + "score": 1.0, + "content": "worse than full-precision model. Using", + "type": "text" + }, + { + "bbox": [ + 423, + 649, + 431, + 659 + ], + "score": 0.45, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "pprentice scheme", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 659, + 240, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 240, + 672 + ], + "score": 1.0, + "content": "this gap is considerably lowered.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 561, + 506, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Figure 6b shows the impact of lowering precision when a low-precision (student) network is paired", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "with a full-precision (teacher) network. For this analysis we use scheme-A where we jointly train", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "both the teacher and student network. The mix of ResNet depths we used for this study are ResNet-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "20, 32, 44, 56, 110 and 182. ResNet-20 student network was paired with deeper ResNets from", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "this mix, i.e. ResNet-32, 44, 56, 110 and 182 (as five separate experiments). Similarly, ResNet-44", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "student network was paired with deeper ResNet-56 and 110 (as two different set of experiments).", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "ResNet-110 student network used ResNet-182 as its teacher network. For a particular ResNet depth,", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 391, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 391, + 117 + ], + "score": 1.0, + "content": "the figure plots the minimum error rate across each of the experiments.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 676, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "student network was paired with deeper ResNet-56 and 110 (as two different set of experiments).", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 505, + 107 + ], + "score": 1.0, + "content": "ResNet-110 student network used ResNet-182 as its teacher network. For a particular ResNet depth,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 391, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 391, + 117 + ], + "score": 1.0, + "content": "the figure plots the minimum error rate across each of the experiments.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 198 + ], + "lines": [ + { + "bbox": [ + 106, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 121, + 138, + 135 + ], + "score": 1.0, + "content": "We find", + "type": "text" + }, + { + "bbox": [ + 138, + 122, + 147, + 132 + ], + "score": 0.31, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "pprentice scheme to improve the baseline full-precision accuracy. The scheme also helps", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "close the gap between the new improved baseline accuracy and the accuracy when lowering the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "precision of the weights and activations. The gap between 2-bits weight and 8-bits activation net-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 159, + 167 + ], + "score": 1.0, + "content": "work is now", + "type": "text" + }, + { + "bbox": [ + 159, + 154, + 205, + 165 + ], + "score": 0.8, + "content": "0 . 4 \\% - 0 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "worse than full-precision model. With ImageNet dataset, the accuracy of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 359, + 177 + ], + "score": 1.0, + "content": "full-precision networks also improves but very marginally (by", + "type": "text" + }, + { + "bbox": [ + 359, + 165, + 382, + 176 + ], + "score": 0.84, + "content": "0 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 165, + 504, + 177 + ], + "score": 1.0, + "content": "). However, the impact of dis-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 175, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 104, + 175, + 468, + 190 + ], + "score": 1.0, + "content": "tillation technique on CIFAR-10 is quite pronounced - for example, top-1 error lowers by", + "type": "text" + }, + { + "bbox": [ + 468, + 176, + 489, + 187 + ], + "score": 0.85, + "content": "1 . 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 175, + 506, + 190 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 158, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 158, + 198 + ], + "score": 1.0, + "content": "ResNet-110.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "image", + "bbox": [ + 184, + 219, + 421, + 526 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 184, + 219, + 421, + 526 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 184, + 219, + 421, + 526 + ], + "spans": [ + { + "bbox": [ + 184, + 219, + 421, + 526 + ], + "score": 0.961, + "type": "image", + "image_path": "58b5bef731307b1943dcb500f0772a4d88af24ac4034b21f19ae656eba91abc5.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 184, + 219, + 421, + 232.34782608695653 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 184, + 232.34782608695653, + 421, + 245.69565217391306 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 184, + 245.69565217391306, + 421, + 259.04347826086956 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 184, + 259.04347826086956, + 421, + 272.39130434782606 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 184, + 272.39130434782606, + 421, + 285.73913043478257 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 184, + 285.73913043478257, + 421, + 299.08695652173907 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 184, + 299.08695652173907, + 421, + 312.43478260869557 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 184, + 312.43478260869557, + 421, + 325.7826086956521 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 184, + 325.7826086956521, + 421, + 339.1304347826086 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 184, + 339.1304347826086, + 421, + 352.4782608695651 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 184, + 352.4782608695651, + 421, + 365.8260869565216 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 184, + 365.8260869565216, + 421, + 379.1739130434781 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 184, + 379.1739130434781, + 421, + 392.5217391304346 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 184, + 392.5217391304346, + 421, + 405.8695652173911 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 184, + 405.8695652173911, + 421, + 419.2173913043476 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 184, + 419.2173913043476, + 421, + 432.5652173913041 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 184, + 432.5652173913041, + 421, + 445.9130434782606 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 184, + 445.9130434782606, + 421, + 459.2608695652171 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 184, + 459.2608695652171, + 421, + 472.6086956521736 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 184, + 472.6086956521736, + 421, + 485.9565217391301 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 184, + 485.9565217391301, + 421, + 499.3043478260866 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 184, + 499.3043478260866, + 421, + 512.6521739130432 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 184, + 512.6521739130432, + 421, + 525.9999999999997 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 218, + 545, + 393, + 557 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 216, + 543, + 394, + 559 + ], + "spans": [ + { + "bbox": [ + 216, + 543, + 394, + 559 + ], + "score": 1.0, + "content": "Figure 7: Distillation followed by quantization.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + } + ], + "index": 27.0 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "We experimented with a variation of scheme-C where a network is first compressed using distillation", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "scheme (using a deeper ResNet as the teacher network) followed by lowering the precision and fine-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "tuning. The fine-tuning is done for 35-40 epochs with a very low learning-rate without the influence", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "of any teacher network (no distillation). For this experiment, the student network starts from a higher", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 635 + ], + "score": 1.0, + "content": "accuracy compressed model compared to scheme-C (since distillation improves accuracy of student", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "network at full-precision as well). Figure 7 shows the results with this experimental setting. For", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "each configuration, we find the error-rate to lie in between the error-rates shown in Figure 6a and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "Figure 6b for the corresponding configuration, i.e. this scheme is better than low-precision training", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 664, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 680 + ], + "score": 1.0, + "content": "from scratch but not as good as training with methodology described in scheme-A. 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This would help inference latency while maintaining accuracy at-par", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 161, + 259, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 259, + 173 + ], + "score": 1.0, + "content": "with baseline full-precision networks.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 106, + 506, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 178, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 191 + ], + "score": 1.0, + "content": "As mentioned in section 5.5, sparsifying a model more than a certain percentage leads to accu-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 505, + 201 + ], + "score": 1.0, + "content": "racy loss. Investigating hyper-sparse network models without accuracy loss using distillation based", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 200, + 337, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 337, + 212 + ], + "score": 1.0, + "content": "schemes is another interesting avenue of further research.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 177, + 505, + 212 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/B1ae1lZRb/B1ae1lZRb_model.json b/parse/train/B1ae1lZRb/B1ae1lZRb_model.json new file mode 100644 index 0000000000000000000000000000000000000000..82577bd504c478bcf9396553ffdd9ba43c03c779 --- /dev/null +++ b/parse/train/B1ae1lZRb/B1ae1lZRb_model.json @@ -0,0 +1,21376 @@ +[ + { + "layout_dets": [ + { + "category_id": 1, + "poly": [ + 398, + 707, + 1302, + 707, + 1302, + 1165, + 398, + 1165 + ], + "score": 0.983 + }, + { + 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b/parse/train/B1eyO1BFPr/B1eyO1BFPr.md new file mode 100644 index 0000000000000000000000000000000000000000..166c23bb9a5108d3ab786547de8a4ba0be189525 --- /dev/null +++ b/parse/train/B1eyO1BFPr/B1eyO1BFPr.md @@ -0,0 +1,978 @@ +# DON’T USE LARGE MINI-BATCHES, USE LOCAL SGD + +Tao Lin EPFL, Switzerland tao.lin@epfl.ch + +Sebastian U. Stich +EPFL, Switzerland +sebastian.stich@epfl.ch +Kumar Kshitij Patel +IIT Kanpur, India +kumarkshitijpatel@gmail.com +Martin Jaggi +EPFL, Switzerland +martin.jaggi@epfl.ch + +# ABSTRACT + +Mini-batch stochastic gradient methods (SGD) are state of the art for distributed training of deep neural networks. Drastic increases in the mini-batch sizes have lead to key efficiency and scalability gains in recent years. However, progress faces a major roadblock, as models trained with large batches often do not generalize well, i.e. they do not show good accuracy on new data. + +As a remedy, we propose a post-local SGD and show that it significantly improves the generalization performance compared to large-batch training on standard benchmarks while enjoying the same efficiency (time-to-accuracy) and scalability. We further provide an extensive study of the communication efficiency vs. performance trade-offs associated with a host of local SGD variants. + +# 1 INTRODUCTION + +Fast and efficient training of large scale deep-learning models relies on distributed hardware and on distributed optimization algorithms. For efficient use of system resources, these algorithms crucially must (i) enable parallelization while being communication efficient, and (ii) exhibit good generalization behaviour, i.e. good performance on unseen data (test-set). Most machine learning applications currently depend on stochastic gradient descent (SGD) (Robbins & Monro, 1985) and in particular its mini-batch variant (Bottou, 2010; Dekel et al., 2012). However, this algorithm faces generalization difficulties in the regime of very large batch sizes as we will review now. + +Mini-batch SGD. For a sum-structured optimization problem of the form $\begin{array} { r } { \operatorname* { m i n } _ { { \pmb w } \in \mathbb { R } ^ { d } } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } f _ { i } ( { \pmb w } ) } \end{array}$ where $\pmb { w } \in \mathbb { R } ^ { d }$ denotes the parameters of the model and $f _ { i } \colon { \mathbb { R } ^ { d } } \to { \mathbb { R } }$ the loss function of the $i$ -th training example, the mini-batch SGD update for $K \geqslant 1$ workers is given as + +$$ +\begin{array} { r } { \pmb { w } _ { ( t + 1 ) } : = \pmb { w } _ { ( t ) } - \gamma _ { ( t ) } \left[ \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \frac { 1 } { B } \sum _ { i \in \mathcal { I } _ { ( t ) } ^ { k } } \nabla f _ { i } \big ( \pmb { w } _ { ( t ) } \big ) \right] , } \end{array} +$$ + +where $\gamma _ { ( t ) } > 0$ denotes the learning rate and $\smash { \mathcal { T } _ { ( t ) , } ^ { k } \subseteq [ N ] }$ the subset (mini-batch) of training datapoints selected by worker $k$ (typically selected uniformly at random from the locally available datapoints on worker $k$ ). For convenience, we will assume the same batch size $B$ per worker. + +Local SGD. Motivated to better balance the available system resources (computation vs. communication), local SGD (a.k.a. local-update SGD, parallel SGD, or federated averaging) has recently attracted increased research interest (Mcdonald et al., 2009; Zinkevich et al., 2010; McDonald et al., 2010; Zhang et al., 2014; 2016; McMahan et al., 2017). In local SGD, each worker $k \in [ K ]$ evolves a local model by performing $H$ sequential SGD updates with mini-batch size $\scriptstyle B _ { \mathrm { l o c } }$ , before communication (synchronization by averaging) among the workers. Formally, + +$$ +\begin{array} { r } { \boldsymbol { w } _ { ( t ) + h + 1 } ^ { k } : = \boldsymbol { w } _ { ( t ) + h } ^ { k } - \gamma _ { ( t ) } \left[ \frac { 1 } { B _ { \mathrm { l a c } } } \sum _ { i \in \mathcal { I } _ { ( t ) + h } ^ { k } } \nabla f _ { i } \big ( \boldsymbol { w } _ { ( t ) + h } ^ { k } \big ) \right] , \quad \boldsymbol { w } _ { ( t + 1 ) } ^ { k } : = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \boldsymbol { w } _ { ( t ) + H } ^ { k } , } \end{array} +$$ + +where $\pmb { w } _ { ( t ) \pmb { \mathrm { 1 } } } ^ { k }$ denotes the local model on machine k $k$ after $t$ global synchronization rounds and subsequent $\check { h } \in [ H ]$ local steps is defined analogously). Mini-batch SGD is a special case if $H = 1$ and $B _ { \mathrm { l o c } } = B$ . Furthermore, the communication patterns of both algorithms are identical if $B = H B _ { \mathrm { l o c } }$ . However, as the updates (eq. (2)) are different from the mini-batch updates (eq. (1)) for any $H > 1$ , the generalization behavior (test error) of both algorithms is expected to be different. + +![](images/c37533587195448eb17540ebaf8a184c24be6e289ddca82ed069b0a906e70708.jpg) + +Large batch SGD. Recent schemes for scaling training to a large number of workers rely on standard mini-batch SGD (1) with very large overall batch sizes (Shallue et al., 2018; You et al., 2018; Goyal et al., 2017), i.e. increasing the global batch size linearly with the number of workers $K$ . However, the batch size interacts differently with the overall system efficiency on one hand, and with generalization performance on the other hand. While a larger batch size in general increases throughput, it may negatively affect the final accuracy on both the train- and test-set. Two main scenarios of particular interest can be decoupled as follows: + +Scenario 1. The communication restricted setting, where the synchronization time is much higher than the gradient computation time. In this case the batch size of best efficiency is typically large, and is achieved by using partial computation (gradient accumulation) while waiting for communication. We in particular study the interesting case when the mini-batch sizes of both algorithms satisfy the relation $B = H B _ { \mathrm { l o c } }$ , as in this case both algorithms (local SGD and minibatch SGD) evaluate the same number of stochastic gradients between synchronization steps. + +Scenario 2. The regime of poor generalization of large-batch SGD, that is the use of very large overall batches (often a significant fraction of the training set size), which is known to cause drastically decreased generalization performance (Chen & Huo, 2016; Keskar et al., 2017; Hoffer et al., 2017; Shallue et al., 2018; Golmant et al., 2018). If sticking to standard mini-batch SGD and maintaining the level of parallelisation, the batch size $B$ would have to be reduced below the device (locally optimal) capacity in order to alleviate this generalization issue, which however impacts training time1. + +Main Results. Key aspects of the empirical performance of local SGD compared to mini-batch baselines are illustrated in Figure 1. In scenario 1), comparing local SGD with $H = 4$ (A4) with mini-batch SGD of same effective batch size $B = 4 B _ { \mathrm { l o c } }$ (A3) reveals a stark difference, both in terms of train and test error (local SGD achieves lower training loss and higher test accuracy). This motivates the use of local SGD as an alternative to large-batch training—a hypothesis that we confirm in our experiments. Further, in scenario 2), mini-batch SGD with smaller batch size $\scriptstyle B = B _ { \mathrm { l o c } }$ (A2) is observed to suffer from poor generalization, although the training curve matches the singlemachine baseline (A1). The generalization gap can thus not be explained as an optimization issue alone. Our proposed post-local SGD (A5) (defined by starting local SGD from the model obtained by large-batch SGD (A2) at epoch 150) closes this generalization gap with the single-machine baseline (A1) and is also more communication efficient than the mini-batch competitors. In direct comparison, post-local SGD is more communication-efficient than mini-batch SGD (while less than local SGD). It achieves better generalization performance than both these algorithms. + +Contributions. Our main contributions can thus be summarized as follows: + +Trade-offs in Local SGD: We provide the first comprehensive empirically study of the trade-offs in local SGD for deep learning—when varying the number of workers $K$ , number of local steps $H$ and mini-batch sizes—for both scenarios 1) on communication efficiency and 2) on generalization. + +Post-local SGD: We propose post-local SGD, a simple but very efficient training scheme to address the current generalization issue of large-batch training. It allows us to scale the training to much higher number of parallel devices. Large batches trained by post-local SGD enjoy improved communication efficiency, while at the same time strongly outperforming most competing small and large batch baselines in terms of accuracy. Our empirical experiments on standard benchmarks show that post-local SGD can reach flatter minima than large-batch SGD on those problems. + +# 2 RELATED WORK + +The generalization gap in large-batch training. State-of-the-art distributed deep learning frameworks (Abadi et al., 2016; Paszke et al., 2017; Seide & Agarwal, 2016) resort to synchronized large-batch SGD training, allowing scaling by adding more computational units and performing data-parallel synchronous SGD with mini-batches divided between devices. Training with large batch size (e.g. batch size $> 1 0 ^ { 3 }$ on ImageNet) typically degrades the performance both in terms of training and test error (often denoted as generalization gap) (Chen & Huo, 2016; Li, 2017; Li et al., 2014; Keskar et al., 2017; Shallue et al., 2018; McCandlish et al., 2018; Golmant et al., 2018; Masters & Luschi, 2018). Goyal et al. (2017) argue that the test error degrades because of optimization issues and propose to use a “learning rate warm-up” phase with linear scaling of the step-size. You et al. (2017a) propose Layer-wise Adaptive Rate Scaling (LARS) to scale to larger mini-batch size, but the generalization gap does not vanish. Hoffer et al. (2017) argue that the generalization gap can be closed when increasing the number of iterations along with the batch size. However, this diminishes the efficiency gains of parallel training. + +Keskar et al. (2017) empirically show that larger batch sizes correlate with sharper minima (Hochreiter & Schmidhuber, 1997) found by SGD and that flat minima are preferred for better generalization. This interpretation—despite being debated in Dinh et al. (2017)—was further developed in Hoffer et al. (2017); Yao et al. (2018); Izmailov et al. (2018). Neelakantan et al. (2015) propose to add isotropic white noise to the gradients to avoid over-fitting and better optimization. Zhu et al. (2019); Xing et al. (2018) further analyze “structured” anisotropic noise and highlight the importance of anisotropic noise (over isotropic noise) for improving generalization. The importance of the scale of the noise for non-convex optimization has also been studied in Smith & Le (2018); Chaudhari & Soatto (2018). Wen et al. (2019) propose to inject noise (sampled from the expensive empirical Fisher matrix) to large-batch SGD. However, to our best knowledge, none of the prior work (except our post-local SGD) can provide a computation efficient way to inject noise to achieve as good generalization performance as small-batch SGD, for both of CIFAR and ImageNet experiments. + +Local SGD and convergence theory. While mini-batch SGD is very well studied (Zinkevich et al., 2010; Dekel et al., 2012; Takác et al., 2013), the theoretical foundations of local SGD variants ˇ are still developing. Jain et al. (2018) study one-shot averaging on quadratic functions and Bijral et al. (2016) study local SGD in the setting of a general graph of workers. A main research question is whether local SGD provides a linear speedup with respect to the number of workers $K$ , similar to mini-batch SGD. Recent work partially confirms this, under the assumption that $H$ is not too large compared to the total iterations $T$ . Stich (2019) and Patel & Dieuleveut (2019) show convergence at rate $\mathcal { O } \big ( ( K T H B _ { \mathrm { l o c } } ) ^ { - 1 } \big )$ on strongly convex and smooth objective functions when $H = \mathcal { O } ( T ^ { 1 / 2 } )$ . For smooth non-convex objective functions, Zhou & Cong (2018) show a rate of $\mathcal { O } \big ( ( K T \dot { B } _ { \mathrm { l o c } } ) ^ { - 1 / 2 } \big )$ (for the decrement of the stochastic gradient), Yu et al. (2019) give an improved result $\mathcal { O } \big ( ( H K T B _ { \mathrm { l o c } } ) ^ { - 1 / 2 } \big )$ when $H = \mathcal { O } ( T ^ { 1 / 4 } )$ . For a discussion of more recent theoretical results we refer to (Kairouz et al., 2019). Alistarh et al. (2018) study convergence under adversarial delays. Zhang et al. (2016) empirically study the effect of the averaging frequency on the quality of the solution for some problem cases and observe that more frequent averaging at the beginning of the optimization can help. Similarly, Bijral et al. (2016) argue to average more frequently at the beginning. + +# 3 POST-LOCAL SGD AND HIERARCHICAL LOCAL SGD + +In this section we present two novel variants of local SGD. First, we propose post-local SGD to reach high generalization accuracy (cf. Section 4.2 below), and second, hierarchical SGD designed from a systems perspective aiming at optimal resource adaptivity (computation vs. communication trade-off). + +Post-local SGD: Large-batch Training Alternative for Better Generalization. We propose postlocal SGD, a variant where local SGD is only started in the second phase of training, after $t ^ { \prime }$ initial steps2 with standard mini-batch SGD. Formally, the update in (2) is performed with a iteration dependent $H _ { ( t ) }$ given as + +$$ +H _ { ( t ) } = \left\{ { \begin{array} { l l } { 1 , \quad { \mathrm { i f ~ } } t \leqslant t ^ { \prime } , } \\ { H , \quad { \mathrm { i f ~ } } t > t ^ { \prime } . } \end{array} } \right. \qquad \quad \qquad ( m i n i - b a t c h S G D ) +$$ + +(post-local SGD) + +As the proposed scheme is identical to mini-batch SGD in the first phase (with local batch size $\boldsymbol { B } = \boldsymbol { B } _ { \mathrm { l o c } } ,$ ), we can leverage previously tuned learning rate warm-up strategies and schedules for large-batch training (Goyal et al., 2017) without additional tuning. Note that we only use ‘small’ local mini-batches of size $B = B _ { \mathrm { l o c } }$ in the warm-up phase, and switch to the communication efficient larger effective batches $H B _ { \mathrm { l o c } }$ in the second phase, while also achieving better generalization in our experiments (cf. also the discussion in Section 5). We would like to point out that it is crucial to use local SGD in the second phase, as e.g. just resorting to large batch training does achieve worse performance (see e.g. 3rd row in Table 2 below). + +Hierarchical Local SGD: Optimal Use of Systems Resources in Heterogeneous Systems. Real world systems come with different communication bandwidths on several levels, e.g. with GPUs or other accelerators grouped hierarchically within a chip, machine, rack or even at the level of entire data-centers. In this scenario, we propose to employ local SGD as an inner loop on each level of the hierarchy, adapted to the corresponding computation vs communication trade-off of that particular level. The resulting scheme, hierarchical local SGD, can offer significant benefits in terms of system adaptivity and performance, as we show with experiments and a discussion in Appendix D. + +# 4 EXPERIMENTAL RESULTS + +In this section we systematically evaluate the aforementioned variants of local SGD on deep learning tasks. In summary, the main findings presented in Sections 4.1 and 4.2 below are: + +Local SGD, on the one hand, can serve as a communication-efficient alternative to mini-batch SGD for different practical purposes (communication restricted Scenario 1). For example in Figure 2(a) (with fixed $\scriptstyle B _ { \mathrm { l o c } }$ and $K$ ), in terms of $9 2 . 4 8 \%$ best test accuracy achieved by our mini-batch SGD implementation for $K = 1 6$ , practitioners can either choose to achieve reasonable good training quality $( 9 1 . 2 \%$ , matching (He et al., 2016a)) with a $2 . 5 9 \times$ speedup (time-to-accuracy) in training time $( H = 8 )$ ), or achieve slight better test performance $( 9 2 . 5 7 \% )$ ) with slightly reduced communication efficiency $1 . 7 6 \times$ speedup for $H = 2$ ). + +Post-local SGD, on the other hand, in addition to overcoming communication restrictions does provide a state-of-the-art remedy for the generalization issue of large-batch training (Scenario 2). Unlike local SGD with large $H$ and $K$ (e.g. $H = 1 6 , K = 1 6 )$ which can encounter optimization issues during initial training (which can impact later generalization performance), we found that post-local SGD elegantly enables an ideal trade-off between optimization and generalization within a fixed training budget (e.g. Table 3, Table 5, Figure 3 and many others in Appendix C.5), and can achieve the same or even better performance than small mini-batch baselines. + +Setup. We briefly outline the general experimental setup, and refer to Appendix A for full details. + +Datasets. We evaluate all methods on the following two main (standard) tasks: (1) Image classification for CIFAR-10/100 (Krizhevsky & Hinton, 2009), and (2) Image classification for ImageNet (Russakovsky et al., 2015). The detailed data augmentation scheme refers to Appendix A. + +Models. We use ResNet-20 (He et al., 2016a) with CIFAR-10 as a base configuration to understand different properties of (post-)local SGD. We then empirical evaluate the large-batch training performance of post-local SGD, for ResNet-20, DensetNet-40-12 (Huang et al., 2017b) and WideResNet28-10 (Zagoruyko & Komodakis, 2016) on CIFAR-10/100. Finally, we train ResNet-50 (He et al., 2016a) on ImageNet to investigate the accuracy and scalability of (post-)local SGD training. + +Implementation and platform. Our algorithms are implemented3 in PyTorch (Paszke et al., 2017), with a flexible configuration of the machine topology supported by Kubernetes. The cluster consists of Intel Xeon E5-2680 v3 servers and each server has 2 NVIDIA TITAN Xp GPUs. We use the notion $a \times b$ -GPU to denote the topology of the cluster, i.e., $a$ nodes and each with $b$ GPUs. + +Specific learning schemes for large-batch SGD. We rely on the recently proposed schemes for efficient large batch training (Goyal et al., 2017), which are formalized by (i) linearly scaling the learning rate w.r.t. the global mini-batch size; (ii) gradual warm-up of the learning rate from a small value. See Appendix A.3 for more details. + +Distributed training procedure on CIFAR-10/100. The experiments follow the common mini-batch SGD training scheme for CIFAR (He et al., 2016a;b; Huang et al., 2017b) and all competing methods access the same total number of data samples (i.e. gradients) regardless of the number of local steps. Training ends when the distributed algorithms have accessed the same number of samples as the single-worker baseline. The data is disjointly partitioned and reshuffled globally every epoch. The learning rate scheme follows (He et al., 2016a; Huang et al., 2017b), where we drop the initial learning rate by a factor of 10 when the model has accessed $5 0 \%$ and $7 5 \%$ of the total number of training samples. Unless mentioned specifically, the used learning rate is scaled by the global mini-batch size $B K$ for mini-batch SGD and $B _ { \mathrm { l o c } } K$ for local SGD) where the initial learning rate is fine-tuned for each model and each task for the single worker. See Appendix A.4 for more details. + +# 4.1 SUPERIOR SCALABILITY OF LOCAL SGD OVER MINI-BATCH SGD + +First, we empirically study local SGD training for the communication restricted case (i.e. Scenario 1). As local SGD (with local batch size $B _ { \mathrm { l o c . } }$ ) needs $H$ times fewer communication rounds than minibatch SGD (with the same batch size $\begin{array} { r } { B = B _ { \mathrm { l o c } } } \end{array}$ q we expect local SGD to significantly outperform mini-batch SGD in terms of time-to-accuracy and scalability and verify this experimentally. We further observe that local SGD shows better generalization performance than mini-batch SGD. + +Significantly better scalability when increasing the number of workers on CIFAR, in terms of time-to-accuracy. Figure 1 demonstrates the speedup in time-to-accuracy for training ResNet-20 for CIFAR-10, with varying number of GPUs $K$ and the number of local steps $H$ , both from 1 to 16. + +![](images/ffb0e8260d462d480f2cea8a935d85617c75c8f5acbfc16ef79217e425e4d386.jpg) +Table 1: Scaling behavior of local SGD in clock-time for increasing number of workers $K$ , for different number of local steps $H$ , for training ResNet-20 on CIFAR-10 with $B _ { \mathrm { l o c } } = 1 2 8$ . The reported speedup (averaged over three runs) is over single GPU training time for reaching the baseline top-1 test accuracy $( 9 1 . 2 \%$ as in (He et al., 2016a)). We use a $8 \times 2$ -GPU cluster with 10 Gbps network. $H = 1$ recovers mini-batch SGD. + +We demonstrate in Figure 1 that local SGD scales $2 \times$ better than its mini-batch SGD counterpart, in terms of time-to-accuracy as we increase the number of workers $K$ on a commodity cluster. The local update steps $( H )$ result in a strong advantage over the standard large-batch training. Mini-batch SGD fixes the batch size to $\boldsymbol { B } = B _ { \mathrm { l o c } }$ , and while increasing the number of workers $K$ gets impacted by the communication bottleneck (section 1), even as parallelism per device remains unchanged. In this experiment, local SGD on 8 GPUs even achieves a $2 \times$ lower time-to-accuracy than mini-batch SGD with 16 GPUs. Moreover, the (near) linear scaling performance for $H = 8$ in Figure 1, shows that the main hyper-parameter $H$ of local SGD is robust and consistently different from its mini-batch counterpart, when scaling the number of workers. + +Effectiveness and scalability of local SGD to even larger datasets (e.g., ImageNet) and larger clusters. Local SGD presents a competitive alternative to the current large-batch ImageNet training methods. Figure 8 in Appendix B.3.2 shows that we can efficiently train state-of-the-art ResNet-50 (at least $1 . 5 \times$ speedup to reach $7 5 \%$ top-1 accuracy) for ImageNet (Goyal et al., 2017; You et al., 2017a) via local SGD on a $1 6 \times 2$ -GPU cluster. + +Local SGD significantly outperforms mini-batch SGD at the same effective batch size and communication ratio. Figure 2 compares local SGD with mini-batch SGD of the same effective batch size, that is the same number of gradient computations per communication round as described in Scenario 1 above (Section 1). + +![](images/915c1d109060a8d68f5bd0bd854005ce56675f41858a11b9ab53fd3bff0bc03f.jpg) +(a) Top-1 test accuracy of local SGD. $H = 1$ is mini-batch SGD with optimal hyper-parameters. + +![](images/54d9ed9579b73f9cfd5fac7ed167e9bc812855c68ba6f9ad3ea362cb26eca9bf.jpg) +(b) Top-1 test accuracy of local SGD (circle) vs. mini-batch SGD (triangle), with same effective batch size $B = H B _ { \mathrm { l o c } }$ per worker. The points for a given $K$ have the same communication cost. + +Figure 2: Training ResNet-20 on CIFAR-10 under different $K$ and $H$ , with fixed $B _ { \mathrm { l o c } } = 1 2 8$ . All results are averaged over three runs and all settings access to the same total number of training samples. We fine-tune the learning rate of mini-batch SGD for each setting. + +# 4.2 (POST)-LOCAL SGD CLOSES THE GENERALIZATION GAP OF LARGE-BATCH TRAINING + +Even though local SGD has demonstrated effective communication efficiency with guaranteed test performance (Scenario 1), we observed in the previous section that it still encounters difficulties when scaling to very large mini-batches (Figure 2(a)), though it is much less affected than mini-batch SGD (Figure 2(b)). Post-local SGD can address the generalization issue of large batch training (Scenario 2) as we show now. First, we would like to highlight some key findings in Table 2. + +Table 2: Test performance (generalization) for local SGD variants and mini-batch SGD (highlighting selected data from Figure 2, Table 3 and Figure 3). Mini-batch SGD suffers from the generalization gap (and sometimes even from optimization issues due to insufficient training) when scaling to larger $H$ and $K$ . Same experimental setup as in Figure 2 (fine-tuned mini-batch SGD baselines). The $\hookrightarrow$ denotes the transition at the first learning rate decay. The reported results are averaged over three runs. + +
AlgorithmTop-1 acc.Effect. batch sizeAlgorithmTop-1 acc.Effect. batch size
Mini-batch SGD (K=4)92.6%KB=512Mini-batch SGD (K =16)92.5%KB=2048
Mini-batch SGD (K =4)89.5%KB=8192Mini-batch SGD (K =16)76.3%KB=16384
Mini-batch SGD (K=4)92.5%KB=512→8192Mini-batch SGD(K=16)92.0%KHBloc=2048→ 16384
Local SGD(H=16,K=4)92.5%KHBloc =8192Local SGD(H=8,K=16)92.0%KHBloc =16384
Post-local SGD(H=16,K=4)92.7%KHBloc =512→8192Post-local SGD (H=8,K=16)92.9%KHBloc =2048→ 16384
+ +We now focus on the generalization issues (Section 1) of large-batch SGD, isolating the potential negative impact from the optimization difficulty (e.g. the insufficient training epochs (Shallue et al., 2018)) from the performance on the test set. Our evaluation starts from a (standard) constant effective mini-batch size of 2048 or 4096 for mini-batch $\mathrm { S G D ^ { 4 } }$ on CIFAR dataset, as in (Keskar et al., 2017; Hoffer et al., 2017; Yao et al., 2018). + +Post-local SGD generalizes better and faster than mini-batch SGD. Table 3 summarizes the generalization performance of post-local SGD on large batch size $( K = 1 6 , B _ { \mathrm { l o c } } = 1 2 8 )$ across different architectures on CIFAR tasks for $H = 1 6$ and $H = 3 2$ . Under the same setup, Table 9 in the Appendix C.3 evaluates the speedup of training, while Figure 1 demonstrates the learning curves of mini-batch SGD and post-local SGD, highlighting the generalization difficulty of large-batch SGD. We can witness that post-local SGD achieves at least $1 . 3 \times$ speedup over the whole training procedure compared to the mini-batch SGD counterpart ( $K = 1 6$ , $B = 1 2 8$ ), while enjoying the significantly improved generalization performance. + +Table 3: Top-1 test accuracy of training different CNN models via post-local SGD on $K = 1 6$ GPUs with a large global batch size $K B _ { \mathrm { l o c } } = 2 0 4 8$ ). The reported results are the average of three runs and all settings access to the same total number of training samples. We compare to small and large mini-batch baselines. The $\star$ indicates a fine-tuned learning rate, where the tuning procedure refers to Appendix A.4. + +
CIFAR-10CIFAR-100
Small batch baseline*Large batch baseline * K=2,B=128K=16,B=128 K=16,Bloc=128K=16,Blc =128Post-local SGD (H=16)Post-local SGD (H=32)Small batch baseline *Large batch baseline *Post-local SGD (H=16)Post-local SGD (H=32) K=2,B=128K=16,B= 128K=16,Blo =128 K= 16,Bloc=128
ResNet-2092.63 ±0.2692.48 3±0.1792.80 ±0.1693.02 ±0.2468.84 ±0.0668.17 ±0.1869.24 ±0.2669.38 ±0.20
DenseNet-40-1294.41 ±0.1494.36 ±0.2094.43 ±0.1294.58 ±0.1174.85 ±0.1474.08 3±0.4674.45 ±0.3075.03 ±0.05
WideResNet-28-1095.89 ±0.1095.43 ±0.3795.94 ±0.0695.76 ±0.2579.78 3±0.1679.31 ±0.2380.28 ±0.1380.65 ±0.16
+ +We further demonstrate the generalization performance and the scalability of post-local SGD, for diverse tasks (e.g., Language Modeling), and for even larger global batch sizes $K B _ { \mathrm { l o c } } = 4 0 9 6$ for CIFAR-100 and 4096 and 8192 respectively for ImageNet), in Appendix C.5 . For example, Table 11 presents the severe generalization issue $2 \%$ drop) of the fine-tuned large-batch SGD training $\mathit { K B } = 4 0 9 6 )$ ) for above three CNNs on CIFAR-100, and cannot be addressed by increasing the training steps (Table 12). Post-local SGD $K B _ { \mathrm { l o c } } = 4 0 9 6 $ ) with default hyper-parameters can perfectly close this generalization gap or even better than the fine-tuned small mini-batch baselines5. For ImageNet training in Figure 16, the post-local SGD outperforms mini-batch SGD baseline for both of $K B = 4 0 9 6$ (76.18 and 75.87 respectively) and $K B = 8 1 9 2$ (75.65 and 75.64 respectively) with $1 . 3 5 \times$ speedup for the post-local SGD training phase. + +The effectiveness of post-local SGD training for different $H$ and $K$ . As seen in Figure 3(a), applying any number of local steps over the case of large-batch training (when $K B _ { \mathrm { l o c } } = 2 0 4 8 )$ improves the generalization performance compared to mini-batch SGD. Figure 3(b) illustrates that post-local SGD is better than mini-batch SGD in general for different number of workers $K$ (as well as different $K B _ { \mathrm { l o c . } }$ ). Thus, post-local SGD presents consistently excellent generalization performance. + +Post-local SGD can improve the training efficiency upon other compression techniques. The results in Table 4 illustrate that post-local SGD can be combined with other communication-efficient techniques (e.g. the sign-based compression schemes) for further improved communication efficiency and better test generalization performance. + +Post-local SGD can improve upon other SOTA optimizers. The benefits of post-local SGD training on ImageNet are even more pronounced for larger batches (e.g. $K B _ { \mathrm { l o c } } > 8 1 9 2$ ) (Goyal et al., + +![](images/12e9053ce26a7322e78a113c4ec510ecb64763022dfc66e574fdc1625ed4ce5b.jpg) + +(a) Performance of post-local SGD for different numbers of local steps $H$ on 16 GPUs. $H = 1$ is mini-batch SGD and $K B _ { \mathrm { l o c } } = 2 0 4 8$ . + +![](images/566dd196bc28a309c7958315b78d2971a03b5087a34eeb488fae4dcd6f363d8f.jpg) + +(b) Performance of post-local SGD for different numbers of workers $K$ for $H = 1 6$ and $H = 3 2$ . The local batch size is fixed to $B _ { \mathrm { l o c } } = B = 1 2 8$ both for local and mini-batch SGD). + +Figure 3: Top-1 test accuracy of training ResNet-20 on CIFAR-10. Box-plot figures are derived from 3 runs. Table 4: Top-1 test accuracy of training ResNet-20 on CIFAR via sign-based compression scheme and postlocal SGD $K B _ { \mathrm { l o c } } = 2 0 4 8$ and $K = 1 \bar { 6 }$ ). $H = 1$ in the table for the simplicity corresponds to the original sign-based algorithm and we fine-tune their hyper-parameters. The reported results are the average of three runs with different seeds. The learning rate schedule is fixed for different local update steps (for $H > 1$ ). For the detailed tuning procedure, training scheme, as well as the pseudo code of the used variants, we refer to Appendix C.5.5. + +
CIFAR-10 H=16CIFAR-100
H=1H =32H =64H=1H =16H=32H =64
signSGD (Bernstein etal.,018)91.61 ±0.2891.77±0.0492.22±0.1192.00 ±0.1567.21 ±0.1167.43 ±0.3668.32 ±0.1868.12 ±0.11
EF-signSGD(Karimireddy etal.,2019)92.45 ±0.2892.63±0.1692.72±0.0592.76 ±0.0668.19 ±0.1069.14 ±0.3669.00 ±0.1469.49 ±0.21
+ +2017; Shallue et al., 2018; Golmant et al., 2018). Considering other optimizers, Table 5 below shows that post-local SGD can improve upon LARS6 (You et al., 2017a). + +Table 5: The Top-1 test accuracy of training ResNet50 on ImageNet. We report the performance (w/ or w/o post-local SGD) achieved by using SGD with Nesterov Momentum, learning schemes in (Goyal et al., 2017), and LARS. We follow the general experimental setup described in Section A.4.2, and use $K = 3 2$ and $H = 4$ . + +
SGD + Momentum + LARSSGD + Momentum + LARS + Post-local SGD
KBloc=8,19275.9976.13
KBloc=16,38474.5275.23
+ +# 5 DISCUSSION AND INTERPRETATION + +Generalization issues of large-batch SGD training are not yet very well understood from a theoretical perspective. Hence, a profound theoretical study of the generalization local SGD is beyond the scope of this work but we hope the favorable experimental results will trigger future research in this direction. In this section we discuss some related work and we argue that local SGD can be seen as a way to inject and control stochastic noise to the whole training procedure. + +# Connecting Local Updates with Stochastic Noise Injection. The update eq. (1) can alternatively be written as + +$$ +{ \pmb w } _ { t + 1 } = { \pmb w } _ { t } - \gamma \hat { \pmb g } _ { t } = { \pmb w } _ { t } - \gamma { \pmb g } _ { t } + \gamma \big ( { \pmb g } _ { t } - \hat { \pmb g } _ { t } \big ) = { \pmb w } _ { t } - \gamma { \pmb g } _ { t } + \gamma \epsilon , +$$ + +where $\begin{array} { r } { \hat { \pmb g } _ { t } : = \frac { 1 } { B } \sum _ { i } \nabla f _ { i } ( { \pmb w } _ { t } ) } \end{array}$ , $\pmb { g } _ { t } : = \mathbb { E } \hat { \pmb { g } } _ { t } = \nabla f ( \pmb { w } _ { t } )$ and $\boldsymbol { \epsilon } : = \boldsymbol { g } _ { t } - \hat { \boldsymbol { g } } _ { t }$ . For most data-sets the noise $\epsilon$ can be well approximated by a zero-mean Gaussian with variance $\Sigma ( w )$ . The variance matrix $\Sigma ( w )$ , following Hoffer et al. (2017) and Hu et al. (2017), for uniform sampling of mini-batch indices can be approximated as + +$$ +\begin{array} { r } { \Sigma ( \pmb { w } ) \approx \left( \frac { 1 } { B } - \frac { 1 } { N } \right) \pmb { K } ( \pmb { w } ) = \left( \frac { 1 } { B } - \frac { 1 } { N } \right) \left( \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \nabla f _ { i } ( \pmb { w } ) \nabla f _ { i } ( \pmb { w } ) ^ { \top } \right) . } \end{array} +$$ + +Recent works (Jastrz˛ebski et al., 2018; Li et al., 2017; Mandt et al., 2017; Zhu et al., 2019) interpret eq. (3) as an Euler-Maruyama approximation to the continuous-time Stochastic Differential Equation (SDE): $d \pmb { w } _ { t } = - \pmb { g } _ { t } d t + \sqrt { \gamma \frac { N - B } { B N } } \pmb { R } ( \pmb { w } ) d \pmb { \theta } ( t )$ where $\begin{array} { r } { R ( w ) R ( w ) ^ { \top } = K ( w ) } \end{array}$ and $\pmb \theta ( t ) \sim \mathcal { N } ( 0 , \pmb I )$ . When $B \ll N$ (in eq. (4)), the learning rate and the batch size only appear in the SDE through the ratio $\rho : = \gamma B ^ { - 1 }$ . Jastrz˛ebski et al. (2018) argue that thus mainly the ratio $\rho$ controls the stochastic noise and training dynamics, and that $\rho$ positively correlates with wider minima and better generalization— matching with recent experimental successes of ImageNet training (Goyal et al., 2017). + +This explanation fails in the large batch regime. When the batch size grows too large, the relative magnitude of the stochastic noise decreases as $\gamma B ^ { - 1 } \not \approx \gamma ( N - B ) \bar { / } B N$ , i.e. $\rho$ is not anymore uniquely describing the training dynamics for large batch size and small dataset size when $B \ll N$ (Jastrz˛ebski et al., 2018). This could be a reason why the generalization difficulty of large-batch training remains, as clearly illustrated e.g. in Figure 1, Table 3, Figure $3 ( \mathbf { b } )$ , and as well as in (Shallue et al., 2018; Golmant et al., 2018). + +The local update step of local SGD is a natural and computation-free way to inject well-structured stochastic noise to the SGD training dynamics. By using the same ratio $\frac { \gamma } { B }$ as mini-batch SGD, the main difference of the local SGD training dynamics comes from the stochastic noise $\epsilon$ during the local update phase ( $K$ times smaller local mini-batch with $H$ local update steps). The $\epsilon$ will be approximately sampled with the variance matrix $K \pmb { \Sigma } ( \pmb { w } )$ instead of $\Sigma ( w )$ in mini-batch SGD, causing the stochastic noise determined by $K$ and $H$ to increase. This could be one of the reasons for post-local SGD to generalize as good as mini-batch SGD (small mini-batch size) and leading to flatter minima than large-batch SGD in our experiments. Similar positive effects of adding well-structured stochastic noise to SGD dynamics on non-convex problems have recently also been observed in (Smith & Le, 2018; Chaudhari & Soatto, 2018; Zhu et al., 2019; Xing et al., 2018; Wen et al., 2019). + +# 5.1 POST-LOCAL SGD CONVERGES TO FLATTER MINIMA + +Figure 4(a) evaluates the spectrum of the Hessian for different local minima. We observe that largebatch SGD tends to get stuck at locations with high Hessian spectrum while post-local SGD tends to prefer low curvature solutions with better generalization error. Figure 4(b) linearly interpolates two minima obtained by mini-batch SGD and post-local SGD, and Figure 13 (in the Appendix) visualizes the sharpness of the model trained by different methods. These results give evidence that post-local SGD converges to flatter minima than mini-batch SGD for training ResNet-20 on CIFAR-10. + +![](images/ca6f0fbe5c2cca0d0b89a95e97cd636645e6fb91abaa800b2732cedcc3fbe483.jpg) + +![](images/ea6cb72160f8f995ad5fc3b3bfc4a15494b305c0c53298badf63fca80a31071f.jpg) +(b) 1-d linear interpolation between models wpost-local SGD $\lambda = 0$ ) and ${ \pmb w } _ { \mathrm { m i n i } }$ -batch SGD $\lambda = 1 ,$ ), i.e., $\hat { \pmb { w } } = \lambda \pmb { w } _ { \mathrm { m i n i } }$ -batch $_ \mathrm { \{ G D + } ( 1 - \lambda ) w$ post-local SGD. The solid lines correspond to evaluate $\hat { \pmb { w } }$ on the whole training dataset while the dashed lines are on the test dataset. The model parameters only differ from the post-local SGD phase. + +(a) The dominant eigenvalue of the Hessian, for model trained from different schemes. It is evaluated on the test dataset per epoch, and only the phase related to the post-local SGD strategy is visualized. Mini-batch SGD or post-local SGD with small $H$ (e.g., $H = 2 , 4$ ) have noticeably larger dominant eigenvalue. + +Figure 4: Understanding the generalization ability of post-local SGD for large-batch training (ResNet-20 on CIFAR-10 with $B K = \bar { B } _ { \mathrm { l o c } } \bar { K } = 2 0 4 8 )$ . We use fixed $B = B _ { \mathrm { l o c } } = 1 2 8$ with $K = 1 6$ GPUs. The detailed experimental setup as well as more visualization of results are available in Appendix C.4. + +# 6 CONCLUSION + +We leverage the idea of local SGD for training in distributed and heterogeneous environments. Ours is the first work to extensively study the trade-off between communication efficiency and generalization performance of local SGD. Our local SGD variant, called post-local SGD, not only outperforms large-batch SGD’s generalization performance but also matches that of small-batch SGD. In our experiments post-local SGD converged to flatter minima compared to traditional large-batch SGD, which partially explains its improved generalization performance. We also provide extensive experiments with another variant hierarchical local SGD, showing its adaptivity to available system resources. Overall, local SGD comes off as a simpler and more efficient algorithm, replacing complex ad-hoc tricks used for current large-batch SGD training. + +# ACKNOWLEDGEMENTS + +The authors thank the anonymous reviewers and Thijs Vogels for their precious comments and feedback. We acknowledge funding from SNSF grant 200021_175796, as well as a Google Focused Research Award. + +# REFERENCES + +Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. 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NIPS - Advances in Neural Information Processing Systems 23, 2010. + +# Supplementary Material + +For easier navigation through the paper and the extensive appendix, we include a table of contents. + +# CONTENTS + +# 1 Introduction 1 + +2 Related Work 3 + +3 Post-Local SGD and Hierarchical Local SGD 4 + +4 Experimental Results 4 +4.1 Superior Scalability of Local SGD over mini-batch SGD 5 +4.2 (Post)-Local SGD closes the Generalization Gap of Large-batch Training 6 + +# 5 Discussion and Interpretation 8 + +5.1 Post-local SGD converges to flatter minima . . 9 + +6 Conclusion 10 + +# A Details on Deep Learning Experimental Setup 15 + +A.1 Dataset . . 15 +A.2 Models and Model Initialization . 15 +A.3 Large Batch Learning Schemes 16 +A.4 Hyperparameter Choices and Training Procedure, over Different Models/Datasets 16 +A.4.1 CIFAR-10/CIFAR-100 16 +A.4.2 ImageNet 17 +A.5 System Performance Evaluation . 17 + +# B Local SGD Training 18 + +B.1 Formal Definition of the Local SGD Algorithm 18 +B.2 Numerical Illustration of Local SGD on a Convex Problem . 18 +B.3 More Results on Local SGD Training . . 19 +B.3.1 Training CIFAR-10 via local SGD 19 +B.3.2 Training ImageNet via Local SGD 20 +B.3.3 Local SGD Scales to Larger Batch Sizes than Mini-batch SGD 21 +B.4 Practical Improvement Possibilities for Standard Local SGD Training 21 +B.4.1 Local SGD with Momentum 21 +B.4.2 Warm-up of the Number of Local SGD Steps . . 23 + +# C Post-local SGD Training 26 + +C.1 The Algorithm of Post-local SGD 26 +C.2 The Effectiveness of Turning on Post-local SGD after the First Learning Rate Decay 27 +C.3 The Speedup of Post-local SGD Training on CIFAR 27 +C.4 Understanding the Generalization of Post-local SGD 28 +C.5 Post-local SGD Training on Diverse Tasks 28 +C.5.1 Post-local SGD Training on CIFAR-100 for Global Mini-batch Size $K B _ { \mathrm { l o c } } = 4 0 9 6$ . . 28 +C.5.2 Post-local SGD Training on Language Modeling . 29 +C.5.3 Post-local SGD Training on ImageNet 31 +C.5.4 Post-local SGD VS. Other Noise Injection Methods 31 +C.5.5 Post-local SGD with Other Compression Schemes 32 + +# D Hierarchical Local SGD 35 + +D.1 The Illustration of Hierarchical Local SGD 35 +D.2 The Algorithm of Hierarchical Local SGD 36 +D.3 Hierarchical Local SGD Training 38 +D.3.1 The Performance of Hierarchical Local SGD. 38 + +E Communication Schemes 39 + +F Discussion and Future Work 40 + +# A DETAILS ON DEEP LEARNING EXPERIMENTAL SETUP + +# A.1 DATASET + +We use the following tasks. + +• Image classification for CIFAR-10/100 (Krizhevsky & Hinton, 2009). Each consists of a training set of 50K and a test set of 10K color images of $3 2 \times 3 2$ pixels, as well as 10 and 100 target classes respectively. We adopt the standard data augmentation scheme and preprocessing scheme (He et al., 2016a; Huang et al., 2016). For preprocessing, we normalize the data using the channel means and standard deviations. +Image classification for ImageNet (Russakovsky et al., 2015). The ILSVRC 2012 classification dataset consists of 1.28 million images for training, and 50K for validation, with 1K target classes. We use ImageNet-1k (Deng et al., 2009) and adopt the same data preprocessing and augmentation scheme as in He et al. (2016a;b); Simonyan & Zisserman (2015). The network input image is a $2 2 4 \times 2 2 4$ pixel random crop from augmented images, with per-pixel mean subtracted. Language Modeling for WikiText-2 (Merity et al., 2017). WikiText-2 is sourced from curated Wikipedia articles. It is frequently used for machine translation and language modelling, and features a vocabulary of over 30, 000 words. Compared to the preprocessed version of Penn Treebank (PTB), WikiText-2 is over 2 times larger. + +# A.2 MODELS AND MODEL INITIALIZATION + +We use ResNet-20 (He et al., 2016a) with CIFAR-10 as a base configuration to understand different properties of (post-)local SGD. We then empirically evaluate the large-batch training performance of post-local SGD, for ResNet-20, DensetNet-40-12 (Huang et al., 2017b) and WideResNet-28- 10 (Zagoruyko & Komodakis, 2016) on CIFAR-10/100, and for LSTM on WikiText-2 (Merity et al., 2017). Finally, we train ResNet-50 (He et al., 2016a) on ImageNet to investigate the accuracy and scalability of (post-)local SGD training. + +For the weight initialization we follow Goyal et al. (2017), where we adopt the initialization introduced by He et al. (2015) for convolutional layers and initialize fully-connected layers by a zero-mean Gaussian distribution with the standard deviation of 0.01. + +Table 6 demonstrates the scaling ratio of our mainly used Neural Network architectures. The scaling ratio (You et al., 2017b) identifies the ratio between computation and communication, wherein + +DNN models, the computation is proportional to the number of floating point operations required for processing an input while the communication is proportional to model size (or the number of parameters). Our local SGD training scheme will show more advantages over models with small “computation and communication scaling ratio”. + +Table 6: Scaling ratio for different models. + +
ModelCommunication# parametersComputation# flops per imageComputation/Communication scaling ratio
ResNet-20 (CIFAR-10)0.27 million0.041 billion151.85
ResNet-20 (CIFAR-100)0.27 million0.041 billion151.85
ResNet-50 (ImageNet-1k)25.00 million7.7 billion308.00
DenseNet-40-12 (CIFAR-10)1.06 million0.28 billion264.15
DenseNet-40-12 (CIFAR-100)1.10 million0.28 billion254.55
WideResNet-28-10 (CIFAR-10)36.48 million5.24 billion143.64
WideResNet-28-10 (CIFAR-100)36.54 million5.24 billion143.40
+ +# A.3 LARGE BATCH LEARNING SCHEMES + +The work of Goyal et al. (2017) proposes common configurations to tackle large-batch training for the ImageNet dataset. We specifically refer to their crucial techniques w.r.t. learning rate as “large batch learning schemes” in our main text. For a precise definition, this is formalized by the following two configurations: + +• Scaling the learning rate: When the mini-batch size is multiplied by $k$ , multiply the learning rate by $k$ . +• Learning rate gradual warm-up: We gradually ramp up the learning rate from a small to a large value. In (our) experiments, with a large mini-batch of size $k n$ , we start from a learning rate of $\eta$ and increment it by a constant amount at each iteration such that it reaches $\hat { \eta } = k \eta$ after 5 epochs. More precisely, the incremental step size for each iteration is calculated from ηˆη5N{pknq , where N is the number of total training samples, k is the number of computing units and $n$ is the local mini-batch size. + +A.4 HYPERPARAMETER CHOICES AND TRAINING PROCEDURE, OVER DIFFERENT MODELS/DATASETS + +# A.4.1 CIFAR-10/CIFAR-100 + +The experiments follow the common mini-batch SGD training scheme for CIFAR (He et al., 2016a;b; Huang et al., 2017b) and all competing methods access the same total amount of data samples regardless of the number of local steps. The training procedure is terminated when the distributed algorithms have accessed the same number of samples as a standalone worker would access. For example, ResNet-20, DensetNet-40-12 and WideResNet-28-10 would access 300, 300 and 250 epochs respectively. The data is partitioned among the GPUs and reshuffled globally every epoch. The local mini-batches are then sampled among the local data available on each GPU, and its size is fixed to $B _ { \mathrm { l o c } } = 1 2 8$ . + +The learning rate scheme follows works (He et al., 2016a; Huang et al., 2017b), where we drop the initial learning rate by 10 when the model has accessed $5 0 \%$ and $7 5 \%$ of the total number of training samples. The initial learning rates of ResNet-20, DensetNet-40-12 and WideResNet-28-10 are fine-tuned on single GPU (which are 0.2, 0.2 and 0.1 respectively), and can be scaled by the global mini-batch size when using large-batch learning schemes. + +In addition to this, we use a Nesterov momentum of 0.9 without dampening, which is applied independently to each local model. For all architectures, following He et al. (2016a), we do not apply weight decay on the learnable Batch Normalization (BN) coefficients. The weight decay of ResNet-20, DensetNet-40-12 and WideResNet-28-10 are 1e-4, 1e-4 and 5e-4 respectively. For the BN for distributed training we again follow Goyal et al. (2017) and compute the BN statistics independently for each worker. + +Unless mentioned specifically, local SGD uses the exact same optimization scheme as mini-batch SGD. + +The procedure of fine-tuning. There is no optimal learning rate scaling rule for large-batch SGD across different mini-batch sizes, tasks and architectures, as revealed in the Figure 8 of Shallue et al. (2018). Our tuning procedure is built on the insight of their Figure 8, where we grid-search the optimal learning rate for each mini-batch size, starting from the linearly scaled learning rate. For example, compared to mini-batch size 128, mini-batch size 2048 with default large-batch learning schemes need to linearly scale the learning rate by the factor of 16. In order to find out its optimal learning rate, we will evaluate a linear-spaced grid of five different factors (i.e., t15, 15.5, 16, 16.5, 17u). If the best performance was ever at one of the extremes of the grid, we would try new grid points so that the best performance was contained in the middle of the parameters. Please note that this unbounded grid search—even though initialized at the linearly scaled learning rate—does also cover other suggested scaling heuristics, for instance square root scaling, and finds the best scaling for each scenario. + +Note that in our experiments of large-batch SGD, either with the default large-batch learning schemes, or tuning/using the optimal learning rate, we always warm-up the learning rate for the first 5 epochs. + +# A.4.2 IMAGENET + +ResNet-50 training is limited to 90 passes over the data in total, and the data is disjointly partitioned and is re-shuffled globally every epoch. All competing methods access the same total number of data samples (i.e. gradients) regardless of the number of local steps. We adopt the large-batch learning schemes as in Goyal et al. (2017) below. We linearly scale the learning rate based on Number of $\mathrm { G P U s } \ \times \ \frac { 0 . 1 } { 2 5 6 } \ \times \ \mathrm { \bar { \it B } _ { g l o b } } )$ where 0.1 and 256 is the base learning rate and mini-batch size 256 respectively for standard single GPU training. The local mini-batch size is set to 128. For learning rate scaling, we perform gradual warmup for the first 5 epochs, and decay the scaled learning rate by the factor of 10 when local models have access 30, 60, 80 epochs of training samples respectively. + +# A.5 SYSTEM PERFORMANCE EVALUATION + +![](images/18500ec8c29dfa00f5f3a908bd839f0cefa6711cbcbfe78aa7ecd0442d71a0ab.jpg) +Figure 5 investigates the increased latency of transmitting data among CPU cores. +Figure 5: The data transmission cost (in seconds) of an all-reduce operation for $1 0 0 ~ \mathrm { M B }$ , over the different number of cores, using PyTorch’s built-in MPI all-reduce operation. Each evaluation is the average result of 100 data transmissions on a Kubernetes cluster. The network bandwidth is 10 Gbps, and we use 48 cores per physical machine. + +Table 7 evaluates the time of running forward and backward with different mini-batch size, for training ResNet20 on CIFAR-10. We can witness that a larger mini-batch size we use, the better parallelism can a GPU have. + +Table 7: The system performance of running forward and backward pass on a single GPU, for training ResNet20 on CIFAR-10. The “Ratio” indicates the Time of evaluating 4096 samples with specified mini-batch sizeTime of evaluating 4096 samples with mini-batch size 4096 . The “Time” is in seconds. + +
Titan XPTesla V100
Mini-Batch SizeTime per iteration (over 100 iterations)RatioTime per iteration (over 100 iterations)Ratio
320.0581.4900.0289.028
640.1001.2840.0304.836
1280.1751.1240.0342.741
2560.3231.0370.0431.733
5120.7371.1830.0731.471
10241.4691.1790.1241.249
20482.6981.0830.2121.068
40964.98310.3971
+ +# B LOCAL SGD TRAINING + +B.1 FORMAL DEFINITION OF THE LOCAL SGD ALGORITHM + +# Algorithm 1 Local SGD + +$$ +\begin{array} { r } { \pmb { g } _ { ( t ) + h - 1 } ^ { k } : = \frac { 1 } { B _ { \mathrm { l o c } } } \sum _ { i \in \mathcal { T } _ { ( t ) + h - 1 } ^ { k } } \nabla f _ { i } \left( \pmb { w } _ { ( t ) + h - 1 } ^ { k } \right) . } \end{array} +$$ + +$$ +\begin{array} { r } { \pmb { w } _ { ( t ) + h } ^ { k } : = \pmb { w } _ { ( t ) + h - 1 } - \gamma _ { ( t ) } \pmb { g } _ { ( t ) + h - 1 } ^ { k } . } \end{array} +$$ + +$$ +{ \Delta } _ { ( t ) } ^ { k } : = { w } _ { ( t ) } ^ { k } - { w } _ { ( t ) + H } ^ { k } . +$$ + +10: get new global (synchronized) model $\pmb { w } _ { ( t + 1 ) } ^ { k }$ for all $K$ nodes: + +$$ +\begin{array} { r } { \pmb { w } _ { ( t + 1 ) } ^ { k } : = \pmb { w } _ { ( t ) } ^ { k } - \gamma _ { ( t ) } \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \Delta _ { ( t ) } ^ { k } } \end{array} +$$ + +11: end for +12: end for + +# B.2 NUMERICAL ILLUSTRATION OF LOCAL SGD ON A CONVEX PROBLEM + +In addition to our deep learning experiments, we first illustrate the convergence properties of local SGD on a small scale convex problem. For this, we consider logistic regression on the $\mathtt { w } 8 \mathtt { a }$ dataset7 $( d = 3 0 0 , n = 4 9 7 4 9 )$ . We measure the number of iterations to reach the target accuracy $\epsilon = 0 . 0 0 5$ . For each combination of $H , B _ { \mathrm { l o c } }$ and $K$ we determine the best learning rate by extensive grid search (cf. paragraph below for the detailed experimental setup). In order to mitigate extraneous effects on the measured results, we here measure time in discrete units, that is we count the number of stochastic gradient computations and communication rounds, and assume that communication of the weights is $2 5 \times$ more expensive than a gradient computation, for ease of illustration. + +Figure 6(a) shows that different combinations of the parameters $( B _ { \mathrm { l o c } } , H )$ can impact the convergence time for $K = 1 6$ . Here, local SGD with p16, 16q converges more than $2 \times$ faster than for $( 6 4 , 1 )$ and $3 \times$ faster than for $( 2 5 6 , 1 )$ . + +Figure 6(b) depicts the speedup when increasing the number of workers $K$ . Local SGD shows the best speedup for $H = 1 6$ on a small number of workers, while the advantage gradually diminishes for very large $K$ . + +![](images/7ce3b2966d064642fe7f7f2bcca8d68a991d8567830e97340f3a2127c5a64c59.jpg) + +![](images/30829ff8545149a8ba10efaf58365de9e24aac28a327a4a6d2307ddcb7cd1eb5.jpg) +(b) Speedup over the number of workers $K$ to solve a regularized logistic regression problem to target accuracy $\epsilon ~ = ~ 0 . 0 0 5$ , for $B _ { \mathrm { l o c } } { } ~ = ~ 1 6$ and $H \in { 1 , 2 , 4 , 8 , 1 6 }$ . We simulate the network traffic under the assumption that communication is $2 5 \times$ slower than a stochastic gradient computation. + +(a) Time (relative to best method) to solve a regularized logistic regression problem to target accuracy $\epsilon ~ = ~ 0 . 0 0 5$ for $K \ = \ 1 6$ workers for $H \in$ $1 , 2 , 4 , 8 , 1 6$ and local mini-batch size $B _ { \mathrm { l o c } }$ . We simulate the network traffic under the assumption that communication is $2 5 \times$ slower than a stochastic gradient computation. + +Figure 6: Numerical illustration of local SGD on a convex problem. + +Experimental Setup for Convex Experiments For the illustrative experiments here we study the convergence of local SGD on the logistic regression problem, $f ( \pmb { w } ) ~ = ~ \textstyle { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \log ( 1 \ Z )$ $\begin{array} { r } { \exp ( - b _ { i } \mathbf { a } _ { i } ^ { \top } \pmb { w } ) ) + \frac { \lambda } { 2 } \| \pmb { w } \| ^ { 2 } } \end{array}$ , where $\mathbf { a } _ { i } \in \mathbb { R } ^ { d }$ and $b _ { i } \in \{ - 1 , + 1 \}$ are the data samples, and regularization parameter $\lambda = 1 / \overline { { n } }$ . For each run, we initialize ${ \pmb w } _ { 0 } = { \pmb 0 } _ { d }$ and measure the number of stochastic gradient evaluations (and communication rounds) until be best of last iterate and weighted average of the iterates reaches the target accuracy $f ( \pmb { w } _ { t } ) - f ^ { \star } \leqslant \epsilon : = 0 . 0 0 5$ , with $f ^ { \star } : = 0 . 1 2 \bar { 6 } 4 3 3 1 7 6 2 1 \bar { 6 } 5 4 5$ . For each configuration $( K , H , B _ { \mathrm { l o c } } )$ , we report the best result found with any of the following two stepsizes: $\begin{array} { r } { \gamma _ { t } : = \operatorname* { m i n } ( 3 2 , \frac { c n } { t + 1 } ) } \end{array}$ and $\gamma _ { t } = 3 2 c$ . Here $c$ is a parameter that can take the values $c = { \bar { 2 } } ^ { i }$ for $i \in \mathbb Z$ . For each stepsize we determine the best parameter $c$ by a grid search, and consider parameter $c$ optimal, if parameters $\{ 2 ^ { - 2 } c , 2 ^ { - 1 } c , 2 c , 2 ^ { 2 } c \}$ yield worse results (i.e. more iterations to reach the target accuracy). + +# B.3 MORE RESULTS ON LOCAL SGD TRAINING + +# B.3.1 TRAINING CIFAR-10 VIA LOCAL SGD + +Better communication efficiency, with guaranteed test accuracy. Figure 7 shows that local SGD is significantly more communication efficient while guaranteeing the same accuracy and enjoys faster convergence speed. In Figure 7, the local models use a fixed local mini-batch size $B _ { \mathrm { l o c } } ~ = ~ 1 2 8$ for all updates. All methods run for the same number of total gradient computations. Mini-batch + +![](images/dce2b0a9694723b07b2ac094bd4978537858d5daf08a213d5a6e434ba082e4c5.jpg) +(a) Training accuracy vs. number of (b) Training accuracy vs. training (c) Test accuracy vs. number of global synchronization rounds. time. epochs. +Figure 7: Training CIFAR-10 with ResNet-20 via local SGD $2 \times 1$ -GPU). The local batch size $B _ { \mathrm { l o c } }$ is fixed to 128, and the number of local steps $H$ is varied from 1 to 16. All the experiments are under the same training configurations. + +SGD—the baseline method for comparison—is a special case of local SGD with $H \mathbf { \Phi } = \mathbf { \Phi } 1$ , with full global model synchronization for each local update. We see that local SGD with $H > 1$ , as illustrated in Figure 7(a), by design does $H$ times less global model synchronizations, alleviating the communication bottleneck while accessing the same number of samples (see section 1). The impact of local SGD training upon the total training time is more significant for larger number of local steps $H$ (i.e., Figure 7(b)), resulting in an at least $3 \times$ speed-up when comparing mini-batch $H = 1$ to local SGD with $H = 1 6$ . The final training accuracy remains stable across different $H$ values, and there is no difference or negligible difference in test accuracy (Figure 7(c)). + +# B.3.2 TRAINING IMAGENET VIA LOCAL SGD + +Figure 8 shows the effectiveness of scaling local SGD to the challenging ImageNet dataset. We limit ResNet-50 training to 90 passes over the data in total, and use the standard training configurations as mentioned in Appendix A.4.2. + +![](images/6062e3f588aa9e0f27dde8c400d7124cc3a94c7fef501d063e5a1f3ca78a02cc.jpg) +Figure 8: The performance of local SGD trained on ImageNet-1k with ResNet-50.We evaluate the model performance on test dataset after each complete accessing of the whole training samples. We apply the largebatch learning schemes (Goyal et al., 2017) to the ImageNet for these two methods. For local SGD, the number of local steps is set to $H = 8$ . + +Moreover, in our ImageNet experiment, the initial phase of local SGD training follows the theoretical assumption mentioned in Subsection 2, and thus we gradually warm up the number of local steps from 1 to the desired value $H$ during the first few epochs of the training. We found that exponentially increasing the number of local steps from 1 by the factor of 2 (until reaching the expected number of local steps) performs well. For example, our ImageNet training uses $H = 8$ , so the number of local steps for the first three epochs is $1 , 2 , 4$ respectively. + +# B.3.3 LOCAL SGD SCALES TO LARGER BATCH SIZES THAN MINI-BATCH SGD + +The empirical studies (Shallue et al., 2018; Golmant et al., 2018) reveal the regime of maximal data parallelism across different tasks and models, where the large-batch training would reach the limit and additional parallelism provides no benefit whatsoever. + +![](images/4e126eb32eb72dc647f8c529a1e431ea1a6361df64813ce4cce6bf1d5da1ee85.jpg) +Figure 9: The relationship between steps to top-1 test accuracy and batch size, of training ResNet-20 on CIFAR-10. The “step” is equivalent to the number of applying gradients. The global mini-batch size is increased by adding more workers $K$ with fixed $B _ { \mathrm { l o c } } = 1 2 8$ . Results are averaged over three runs, each with fine-tuned learning rate. + +On contrary to standard large-batch training, local SGD scales to larger batch size and provides additional parallelism upon the limitation of current large batch training. Figure 9 shows the example of training ResNet-20 on CIFAR-10 with $H = 2$ , which trains and generalizes better in terms of update steps while with reduced communication cost. + +# B.4 PRACTICAL IMPROVEMENT POSSIBILITIES FOR STANDARD LOCAL SGD TRAINING + +We investigate different aspects of the training to address the quality when scaling local SGD to the extreme case, e.g., hybrid momentum scheme, warming up the local SGD or fine-tuning the learning rate. In this section, we briefly present how these strategies are, and how they work in practice where we train ResNet-20 on CIFAR-10 on 16 GPUs. + +# B.4.1 LOCAL SGD WITH MOMENTUM + +Momentum mini-batch SGD is widely used in place of vanilla SGD. The distributed mini-batch SGD with vanilla momentum on $K$ training nodes follows + +$$ +{ \pmb u } _ { ( t ) } = m { \pmb u } _ { ( t - 1 ) } + \frac { 1 } { K } \sum _ { k = 1 } ^ { K } { \pmb \nabla } _ { ( t ) } ^ { k } , \qquad { \pmb w } _ { ( t + 1 ) } = { \pmb w } _ { ( t ) } - \gamma { \pmb u } _ { ( t ) } +$$ + +$\begin{array} { r } { \pmb { \nabla } _ { ( t ) } ^ { k } = \frac { 1 } { | \mathcal { T } _ { ( t ) } ^ { k } | } \sum _ { i \in \mathcal { T } _ { ( t ) } ^ { k } } \nabla f _ { i } \big ( \pmb { w } _ { ( t ) } \big ) } \end{array}$ + +After $H$ updates of mini-batch SGD, we have the following updated ${ \pmb w } _ { ( t + H ) }$ + +$$ +{ \pmb w } _ { ( t + H ) } = { \pmb w } _ { ( t ) } - \gamma \left( \sum _ { \tau = 1 } ^ { H } m ^ { \tau } { \pmb u } _ { ( t - 1 ) } + \sum _ { \tau = 0 } ^ { H - 1 } \frac { m ^ { \tau } } { K } \sum _ { k = 1 } ^ { K } \pmb { \nabla } _ { ( t ) } ^ { k } + \dots + \sum _ { \tau = 0 } ^ { 0 } \frac { m ^ { \tau } } { K } \sum _ { k = 1 } ^ { K } \pmb { \nabla } _ { ( t + H - 1 ) } ^ { k } \right) +$$ + +Coming back to the setting of local SGD, we can apply momentum acceleration on each local model, or on a global level (Chen & Huo, 2016). In the remaining part of this section, we analyze the case of applying local momentum and global momentum. For ease of understanding, we assume the learning rate $\gamma$ is the same throughout the $H$ update steps. + +Local SGD with Local Momentum. When applying local momentum on the local SGD, i.e., using independent identical momentum acceleration for each local model and only globally aggregating the gradients at the time $( t ) + H$ , we have the following local update scheme + +$$ +\begin{array} { r } { \pmb { u } _ { ( t ) } ^ { k } = m \pmb { u } _ { ( t - 1 ) } ^ { k } + \pmb { \nabla } _ { ( t ) } ^ { k } , \qquad \pmb { w } _ { ( t ) + 1 } ^ { k } = \pmb { w } _ { ( t ) } ^ { k } - \gamma \pmb { u } _ { ( t ) } ^ { k } , } \end{array} +$$ + +where ∇kptq  1|Ik | $\begin{array} { r } { \pmb { \nabla } _ { ( t ) } ^ { k } = \frac { 1 } { | \mathcal { T } _ { ( t ) } ^ { k } | } \sum _ { i \in \mathcal { T } _ { ( t ) } ^ { k } } \nabla f _ { i } \big ( \pmb { w } _ { ( t ) } \big ) } \end{array}$ . Consequently, after $H$ local steps, + +$$ +{ \pmb w } _ { ( t ) + H } ^ { k } = { \pmb w } _ { ( t ) } ^ { k } - \gamma \left( \sum _ { \tau = 1 } ^ { H } m ^ { \tau } { \pmb u } _ { ( t - 1 ) } ^ { k } + \sum _ { \tau = 0 } ^ { H - 1 } m ^ { \tau } \pmb \nabla _ { ( t ) } ^ { k } + \dots + \sum _ { \tau = 0 } ^ { 0 } m ^ { \tau } \pmb \nabla _ { ( t ) + H - 1 } ^ { k } \right) . +$$ + +Substituting the above equation into eq. (2), we have the update + +$$ +\begin{array} { l } { { \displaystyle { \pmb w } _ { ( t + 1 ) } = { \pmb w } _ { ( t ) } - \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \gamma \left( \sum _ { \tau = 1 } ^ { H } m ^ { \tau } { \pmb u } _ { ( t - 1 ) } ^ { k } \sum _ { \tau = 0 } ^ { H - 1 } m ^ { \tau } \pmb \nabla _ { ( t ) } ^ { k } + \dots + \sum _ { \tau = 0 } ^ { 0 } m ^ { \tau } \pmb \nabla _ { ( t ) + H - 1 } ^ { k } \right) } } \\ { { \displaystyle ~ = ~ { \pmb w } _ { ( t ) } - \gamma \left( \sum _ { \tau = 1 } ^ { H } \frac { m ^ { \tau } } { K } \sum _ { k = 1 } ^ { K } { \pmb u } _ { ( t - 1 ) } ^ { k } + \sum _ { \tau = 0 } ^ { H - 1 } m ^ { \tau } ( \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \pmb \nabla _ { ( t ) } ^ { k } ) + \dots + \sum _ { \tau = 0 } ^ { 0 } \frac { m ^ { \tau } } { K } \sum _ { k = 1 } ^ { K } \pmb \nabla _ { ( t ) + H - 1 } ^ { k } \right) } } \end{array} +$$ + +Comparing the mini-batch SGD with local momentum local SGD after $H$ update steps $H$ global update steps v.s. $H$ local update steps and 1 global update step), we witness that the main difference of these two update schemes is the difference between $\begin{array} { r } { \sum _ { \tau = 1 } ^ { H } m ^ { \tau } { \pmb u } _ { ( t - 1 ) } } \end{array}$ and $\begin{array} { r } { \sum _ { \tau = 1 } ^ { H } \frac { m ^ { \tau } } { K } \sum _ { k = 1 } ^ { K } \pmb { u } _ { ( t - 1 ) } ^ { k } } \end{array}$ where mini-batch SGD holds a global $\pmb { u } _ { ( t - 1 ) }$ while each local model of the local SGD has their own $\pmb { u } _ { ( t - 1 ) } ^ { k }$ . We will soon see the difference between the global momentum of mini-batch SGD and the local momentum of local SGD. + +Local SGD with Global Momentum For global momentum local SGD, i.e., a more general variant of block momentum (Chen & Huo, 2016), we would like to apply the momentum factor only to the accumulated/synchronized gradients: + +$$ +\begin{array} { l } { { \displaystyle { \pmb u } _ { ( t ) } = m { \pmb u } _ { ( t - 1 ) } + \frac { 1 } { \gamma } \sum _ { k = 1 } ^ { K } \frac { 1 } { K } \big ( { \pmb w } _ { ( t ) } ^ { k } - { \pmb w } _ { ( t ) + H } ^ { k } \big ) = m { \pmb u } _ { ( t - 1 ) } + \frac { 1 } { \gamma } \sum _ { k = 1 } ^ { K } \frac { 1 } { K } \sum _ { l = 0 } ^ { H - 1 } \gamma \pmb \nabla _ { ( t ) + l } ^ { k } , } } \\ { { \displaystyle { \pmb w } _ { ( t + 1 ) } = { \pmb w } _ { ( t ) } - \gamma { \pmb u } _ { ( t ) } = { \pmb w } _ { ( t ) } - \gamma \big ( m { \pmb u } _ { ( t - 1 ) } + \sum _ { l = 0 } ^ { H - 1 } \sum _ { k = 1 } ^ { K } \frac { 1 } { K } \pmb \nabla _ { ( t ) + l } ^ { k } \big ) } } \end{array} +$$ + +where $\begin{array} { r } { \pmb { w } _ { ( t ) + H } ^ { k } = \pmb { w } _ { ( t ) } ^ { k } - \eta \sum _ { l = 0 } ^ { H - 1 } \pmb { \nabla } _ { ( t ) + l } ^ { k } = \pmb { w } _ { ( t ) } - \eta \sum _ { l = 0 } ^ { H - 1 } \pmb { \nabla } _ { ( t ) + l } ^ { k } } \end{array}$ . Note that for local SGD, we consider summing up the gradients from each local update, i.e., the model difference before and after one global synchronization, and then apply the global momentum to the gradients over workers over previous local update steps. + +Obviously, there exists a significant difference between mini-batch momentum SGD and global momentum local SGD, at least the term °Hτ0 mτ is cancelled. + +Local SGD with Hybrid Momentum. The following equation tries to combine local momentum with global momentum, showing a naive implementation. + +First of all, based on the local momentum scheme, after $H$ local update steps, + +$$ +{ \pmb w } _ { ( t ) + H } ^ { k } = { \pmb w } _ { ( t ) } ^ { k } - \gamma \big ( \sum _ { \tau = 1 } ^ { H } m ^ { \tau } { \pmb u } _ { ( t - 1 ) } ^ { k } + \sum _ { \tau = 0 } ^ { H - 1 } m ^ { \tau } \pmb \nabla _ { ( t ) } ^ { k } + . . . + \sum _ { \tau = 0 } ^ { 0 } m ^ { \tau } \pmb \nabla _ { ( t ) + H - 1 } ^ { k } \big ) +$$ + +Together with the result from local momentum with the global momentum, we have + +$$ +\begin{array} { l } { { \displaystyle { \boldsymbol u } _ { ( t ) } = m { \boldsymbol u } _ { ( t - 1 ) } + \frac { 1 } { \gamma } \sum _ { k = 1 } ^ { K } \frac { 1 } { K } \big ( { \boldsymbol w } _ { ( t ) } ^ { k } - { \boldsymbol w } _ { ( t ) + H } ^ { k } \big ) } } \\ { { \displaystyle { \boldsymbol v } _ { ( t + 1 ) } = { \boldsymbol w } _ { ( t ) } - \gamma { \boldsymbol u } _ { ( t ) } = { \boldsymbol w } _ { ( t ) } - \gamma \left[ m { \boldsymbol u } _ { ( t - 1 ) } + \frac { 1 } { \gamma } \sum _ { k = 1 } ^ { K } \frac { 1 } { K } \big ( { \boldsymbol w } _ { ( t ) } ^ { k } - { \boldsymbol w } _ { ( t ) + H } ^ { k } \big ) \right] } } \\ { { \displaystyle \qquad = { \boldsymbol w } _ { ( t ) } - \gamma \left[ m { \boldsymbol u } _ { ( t - 1 ) } + \sum _ { \tau = 1 } ^ { H } \frac { m ^ { \tau } } { K } \sum _ { k = 1 } ^ { K } { \boldsymbol u } _ { ( t - 1 ) } ^ { k } + \sum _ { \tau = 0 } ^ { H - 1 } \frac { m ^ { \tau } } { K } \sum _ { k = 1 } ^ { K } \nabla _ { ( t ) } ^ { k } + \dots + \sum _ { \tau = 0 } ^ { 0 } \frac { m ^ { \tau } } { K } \sum _ { k = 1 } ^ { K } \nabla _ { ( t ) + 1 } ^ { k } \right] } . } \end{array} +$$ + +where $\pmb { u } _ { ( t - 1 ) }$ is the global momentum memory and $\pmb { u } _ { ( t - 1 ) }$ is the local momentum memory for each node $k$ . + +Table 8: Evaluate local momentum and global momentum for ResNet-20 on CIFAR-10 data via local SGD training ( $H = 1$ case) on $5 \times 2$ -GPU Kubernetes cluster. The local mini-batch size is 128 and base batch size is 64 (used for learning rate linear scale). Each local model will access to a disjoint data partition, using the standard learning rate scheme as He et al. (2016a). + +
local momentumglobal momentumtest top-1
0.00.090.57
0.90.092.41
0.90.192.22
0.90.292.09
0.90.392.54
0.90.492.45
0.90.592.19
0.90.691.32
0.90.718.76
0.90.814.35
0.90.912.21
0.90.9510.11
+ +Local SGD with Momentum in Practice. In practice, it is possible to combine the local momentum with global momentum to further improve the model performance. For example, a toy example in Table 8 investigates the impact of different momentum schemes on CIFAR-10 trained with ResNet-20 on a $5 \times 2$ -GPU cluster, where some factors of global momentum could further slightly improve the final test accuracy. + +However, the theoretical understanding of how local momentum and global momentum contribute to the optimization still remains unclear, which further increase the difficulty of tuning local SGD over $H , K$ . An efficient way of using local and global momentum remains a future work and in this work, we only consider the local momentum. + +# B.4.2 WARM-UP OF THE NUMBER OF LOCAL SGD STEPS + +We use the term “local step warm-up strategy”, to refer to a specific variant of post-local SGD. More precisely, instead of the two-phase regime which we presented here the used number of local steps $H$ will be gradually increased from 1 to the expected number of local steps $H$ . The warm-up strategies investigated here are “linear”, “exponential” and “constant”. + +Please note that the implemented post-local SGD over the whole text only refers to the training scheme that uses frequent communication (i.e., $H = 1$ ) before the first learning rate decay and then reduces the communication frequency (i.e., $H > 1 $ ) after the decay. + +This section then investigates the trade-off between stochastic noise and the training stability. Also note that the Figure 2(a) in the main text has already presented one aspect of the trade-off. So the exploration below mainly focuses on the other aspects and tries to understand how will the scale of the added stochastic noise impact the training stability. Infact, even the model has been stabilized to a region with good quality. + +Figure 10 and Figure 11 investigate the potential improvement through using the local step warm-up strategy for the case of training ResNet-20 on CIFAR-10. Figure 10 evaluates the warm-up strategies of “linear” and “constant” for different $H$ , while the evaluation of “exponential” warm-up strategy is omitted due to its showing similar performance as “linear” warm-up. + +However, none of the investigated strategies show convincing performance. Figure 11 further studies how the period of warm-up impacts the training performance. We can witness that even if we increase the warm-up phase to 50 epochs where the training curve of mini-batch SGD becomes stabilized, the large noise introduced by the local SGD will soon degrade the status of training and lead to potential quality loss, as in Figure 11(a). + +![](images/f14f0e095bbdfba3e7dc3f2a72ef068de7d6cfe460f866a358625048e15c79bc.jpg) +Figure 10: Investigate how local step warm-up strategy impacts the performance of training CIFAR-10 with ResNet-20 via local SGD $8 \times 2$ -GPU). The local batch size $B _ { \mathrm { l o c } }$ is fixed to 128. The warmup strategies are “linear” and “constant”, and the warm-up period used here is equivalent to the number of local steps $H$ . + +![](images/fdf6a1940c536dc20ba89b5d65b424ead9a22a3001f5e768cb401a18a76bf18f.jpg) +(c) Evaluate the impact of “exponential” local step warm-up for different period of warm-up phase. +Figure 11: Investigate how different warm-up period of the local step warm-up impacts the performance of training CIFAR-10 with ResNet-20 via local SGD ( $8 \times 2$ -GPU). The local batch size $B _ { \mathrm { l o c } }$ is fixed to 128, and the strategies to warm-up the number of local steps $H$ are “linear”, “exponential” and “constant”. + +# C POST-LOCAL SGD TRAINING + +C.1 THE ALGORITHM OF POST-LOCAL SGD + +# Algorithm 2 Post-local SGD + +input: the initial model ${ \pmb w } _ { ( 0 ) }$ ; +input: training data with labels $\mathcal { T }$ ; +input: mini-batch of size $B _ { \mathrm { l o c } }$ per local model; +input: step size $\eta$ , and momentum $m$ (optional); +input: number of synchronization steps $T$ , and the first learning rate decay is performed at $T ^ { \prime }$ ; input: number of eventual local steps $H ^ { \prime }$ ; +input: number of nodes $K$ . +1: synchronize to have the same initial models $\pmb { w } _ { ( 0 ) } ^ { k } : = \pmb { w } _ { ( 0 ) }$ . +2: for all $k : = 1 , \ldots , K$ do in parallel +3: for $t : = 1 , \dots , T$ do +4: if $t < T ^ { \prime }$ then +5: $H _ { ( t ) } = 1$ +6: else +7: $H _ { ( t ) } = H ^ { \prime }$ +8: end if +9: 10: for $h : = 1 , \ldots , H _ { ( t ) }$ doch from . +$\mathcal { T } _ { ( t ) + h - 1 } ^ { k }$ +11: compute the gradient +$\begin{array} { r } { \pmb { g } _ { ( t ) + h - 1 } ^ { k } : = \frac { 1 } { B _ { \mathrm { l o c } } } \sum _ { i \in \mathcal { T } _ { ( t ) + h - 1 } ^ { k } } \nabla f _ { i } \left( \pmb { w } _ { ( t ) + h - 1 } ^ { k } \right) . } \end{array}$ +12: update the local model to +$\begin{array} { r } { \pmb { w } _ { ( t ) + h } ^ { k } : = \pmb { w } _ { ( t ) + h - 1 } - \gamma _ { ( t ) } \pmb { g } _ { ( t ) + h - 1 } ^ { k } . } \end{array}$ +13: end for +14: all-reduce aggregation of the gradients +${ \Delta } _ { ( t ) } ^ { k } : = { w } _ { ( t ) } ^ { k } - { w } _ { ( t ) + H } ^ { k } .$ +15: get new global (synchronized) model $\pmb { w } _ { ( t + 1 ) } ^ { k }$ for all $K$ nodes: +$\begin{array} { r } { \pmb { w } _ { ( t + 1 ) } ^ { k } : = \pmb { w } _ { ( t ) } ^ { k } - \gamma _ { ( t ) } \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \Delta _ { ( t ) } ^ { k } } \end{array}$ +16: end for +17: end for + +C.2 THE EFFECTIVENESS OF TURNING ON POST-LOCAL SGD AFTER THE FIRST LEARNING RATE DECAY + +In Figure 12, we study the sufficiency as well as the necessity of “injecting” more stochastic noise (i.e., using post-local SGD) into the optimization procedure after performing the first learning rate decay. Otherwise, the delayed noise injection (i.e., starting the post-local SGD only from the second learning rate decay) not only introduces more communication cost but also meets the increased risk of converging to sharper minima. + +![](images/c75fc988863a4f9d9183c639fb0867b484fbfce79b53d95586447972d401eef2.jpg) +Figure 12: The effectiveness and necessary of turning on the post-local SGD after the first learning rate decay. The example here trains ResNet-20 on CIFAR-10 on $K = 1 6$ GPUs with $B _ { \mathrm { l o c } } K = 2 0 4 8$ . + +C.3 THE SPEEDUP OF POST-LOCAL SGD TRAINING ON CIFAR + +Table 3 and Table 10 evaluate the speedup of mini-batch SGD and post-local SGD, over different CNN models, datasets, and training phases. + +Table 9: The Speedup of mini-batch SGD and post-local SGD, over different CNN models and datasets. The speedup is evaluated by $\frac { T _ { a } } { T _ { a } ^ { K } }$ , where $T _ { a }$ is the training time of the algorithm $a$ on 1 GPU and $T _ { a } ^ { K }$ corresponds to the training on $K$ GPUs. We use 16 GPUs in total (with $B _ { \mathrm { l o c } } = 1 2 8$ ) on an $8 \times 2$ -GPU cluster with 10 Gbps network bandwidth. The experimental setup is the same as Table 3. + +
CIFAR-10CIFAR-100
H=1H=16H=32H=1H=16H=32
ResNet-209.4512.2413.008.7511.0511.67
DenseNet-40-128.3110.8011.378.0410.5910.85
WideResNet-28-125.337.948.195.297.838.14
+ +Table 10: The Speedup of mini-batch SGD and post-local SGD (only consider the phase of performing post-local SGD) over different CNN models and datasets. The speedup is evaluated by $\overset { \cdot } { \frac { T _ { a } } { T _ { a } ^ { K } } }$ , where $T _ { a }$ is the training time of the algorithm $a$ (corresponding to the second phase) on 1 GPU and $T _ { a } ^ { K }$ corresponds to the training on $K$ GPUs. We use 16 GPUs in total (with $B _ { \mathrm { l o c } } = 1 2 8$ ) on an $8 \times 2$ -GPU cluster with 10 Gbps network bandwidth. The experimental setup is the same as Table 3. + +
CIFAR-10CIFAR-100
H=1H=16H=32H=1H=16H=32
ResNet-209.4517.3320.808.7515.0017.50
DenseNet-40-128.3115.4318.008.0415.5016.69
WideResNet-28-125.3315.5217.665.2915.0917.69
+ +# C.4 UNDERSTANDING THE GENERALIZATION OF POST-LOCAL SGD + +The Sharpness Visualization Figure 13 visualizes the sharpness of the minima for training ResNet20 on CIFAR-10. 10 different random direction vectors are used for the filter normalization (Li et al., 2018), to ensure the correctness and consistence of the sharp visualization. + +![](images/56f735a7f4b6c75e7ac5e89a1020bd1535a3b630fb9da8fca9a409378db6769b.jpg) +Figure 13: Sharpness visualization of the minima for ResNet-20 trained on CIFAR-10. The training is on top of $K = 1 6$ GPUs and the local batch size is fixed to $B _ { \mathrm { l o c } } = 1 2 8$ . The dashed lines are standard mini-batch SGD and the solid lines are post-local SGD with $H = 1 6$ . The sharpness visualization of minima is performed via filter normalization (Li et al., 2018). The model is perturbed as $\pmb { w } + \lambda \pmb { d }$ by a shared random direction $\textbf { \em d }$ , and is evaluated by the whole dataset (training or test respectively). The top-1 test accuracy of mini-batch SGD is 92.25, while that of post-local SGD is 92.61. The sharpness of these two minima is consistent over 10 different random directions. + +The spectrum of the Hessian for mini-batch SGD and post-local SGD Figure 14 evaluates the spectrum of the Hessian for the model trained from mini-batch SGD and post-local SGD with different $H$ , which again demonstrates the fact that large-batch SGD tends to stop at points with high Hessian spectrum while post-local SGD could easily generalize to a low curvature solution and with better generalization. + +1-d linear interpolation between models The 1-d linear interpolation was first used by Goodfellow et al. (2015) and then widely used to study the “sharpness” and “flatness” of different minima in several works (Keskar et al., 2017; Dinh et al., 2017; Li et al., 2018). In Figure 15, the model trained by post-local SGD (wpost-local SGDq can generalize to flatter minima than that of mini-batch SGD $( { \pmb w } _ { \mathrm { m i n i - b a t c h ~ S G D } } )$ , either $\pmb { w } _ { \mathrm { p o s t - l o c a l S G D } }$ is trained from scratch, or resumed from the checkpoint of ${ \pmb w } _ { \mathrm { m i n i } }$ -batch SGD (i.e., the checkpoint is one-epoch ahead of the first learning rate decay so as to share the common weight structure with ${ \pmb w } _ { \mathrm { m i n i } }$ -batch SGD). + +# C.5 POST-LOCAL SGD TRAINING ON DIVERSE TASKS + +C.5.1 POST-LOCAL SGD TRAINING ON CIFAR-100 FOR GLOBAL MINI-BATCH SIZE $K B _ { \mathrm { L O C } } = 4 0 9 6$ + +Table 11 presents the severe quality loss (at around $2 \%$ ) of the fine-tuned large-batch SGD for training three CNNs on CIFAR-100. Our post-local SGD with default hyper-parameters (i.e., the hyper-parameters from small mini-batch size and via large-batch training schemes) can perfectly close the generalization gap or even better the fine-tuned small mini-batch baselines. + +We further justify the argument of works (Hoffer et al., 2017; Shallue et al., 2018) in Table 12, where we increase the number of training epochs and train it longer (from 300 to 400 and 500) for ResNet-20 on CIFAR-100. The results below illustrate that increasing the number of training epochs alleviates the optimization difficulty of large-batch training + +![](images/899494e264f04e8125ff8d6d4ef1cf6f70ef88b3e3e9f17daeab6f415f43d1c9.jpg) + +![](images/9ce23628094beb4bdfe8284577096792ecff97ca16db53cb551b4bd0e71b97e2.jpg) + +(a) The dominant eigenvalue of the Hessian, which is evaluated on the training dataset per epoch. Only the phase related to the strategy of post-local SGD is visualized. Mini-batch SGD or post-local SGD with very much $H$ (e.g., $H =$ 2, 4) have noticeably larger dominant eigenvalue. + +(b) The dominant eigenvalue of the Hessian, which is evaluated on the test dataset per epoch. Only the phase related to the strategy of post-local SGD is visualized. Mini-batch SGD or post-local SGD with very much $H$ (e.g., $H = 2 , 4 ,$ ) have noticeably larger dominant eigenvalue. + +![](images/94eab45aaaaeb022acb1483eadea58293a151e8081b182b1876d5aebaec7762c.jpg) + +![](images/35525a6a920d3d571bdd272852ba99a1d4d6dadcc841d69c3bbf7b54cc30fc70.jpg) +(d) Top 10 eigenvalues of the Hessian, which is evaluated on the test dataset for the best model of each training scheme. The top eigenvalues of the Hessian of the mini-batch SGD clearly present significant different pattern than the post-local SGD counterpart. + +(c) Top 10 eigenvalues of the Hessian, which is evaluated on the training dataset for the best model of each training scheme. The top eigenvalues of the Hessian of the mini-batch SGD clearly present significant different pattern than the postlocal SGD counterpart. + +Figure 14: The spectrum of the Hessian for ResNet-20 trained on CIFAR-10. The training is on top of $K = 1 6$ GPUs with $K B _ { \mathrm { l o c } } = 2 0 4 8$ . The spectrum is computed using power iteration (Martens & Sutskever, 2012; Yao et al., 2018) with the relative error of 1e-4. The top-1 test accuracy of mini-batch SGD is 92.57, while that of post-local SGD ranges from 92.33 to 93.07. Current large-batch SGD tends to stop at points with considerably “larger” Hessian spectrum, while large-batch trained with post-local SGD generalizes to solution with low curvature and with better generalization. + +Table 11: Top-1 test accuracy of training different CNN models via post-local SGD on $K = 3 2$ GPUs with a large batch size $\langle B _ { \mathrm { l o c } } K = 4 0 9 6 \rangle$ . The reported results are the average of three runs. We include the small and large batch baseline, where the models are trained by mini-batch SGD with mini-batch size 256 and 4096 respectively. The $\star$ indicates the fine-tuned learning rate. + +
CIFAR-100
small batch baseline *large batch baseline *post-local SGD (H=8)post-local SGD (H=16)
ResNet-2068.84±0.0667.34±0.3468.38±0.4868.30±0.30
DenseNet-40-1274.85±0.1473.00±0.0474.50±0.3474.96±0.30
WideResNet-28-1079.78±0.1677.82±0.6579.53±0.4579.80±0.39
+ +# C.5.2 POST-LOCAL SGD TRAINING ON LANGUAGE MODELING + +We evaluate the effectiveness of post-local SGD for training the language modeling task on WikiText2 through LSTM. We borrowed and adapted the general experimental setup of Merity et al. (2018), where we use a three-layer LSTM with hidden dimension of size 650. The loss will be averaged over all examples and timesteps. The BPTT length is set to 30. We fine-tune the value of gradient clipping (0.4) and the dropout (0.4) is only applied on the output of LSTM. The local mini-batch size $B _ { \mathrm { l o c } }$ is + +![](images/5819c6ad78152034973b7636e6fa56740e9328f1c73ae25b14ab4c3b3bd166e6.jpg) +(a) ${ \pmb w } _ { \mathrm { p o s t - l o c a l \ : S G D } }$ is trained with $H = 1 6$ , where the (b) ${ \pmb w } _ { \mathrm { p o s t - l o c a l \ : S G D } }$ is trained with $H = 3 2$ , where the training resumes from the checkpoint of ${ \pmb w } _ { \mathrm { m i n i } }$ -batch SGD training resumes from the checkpoint of ${ \pmb w } _ { \mathrm { m i n i } }$ -batch SGD which is one-epoch ahead of the first learning rate decay. which is one-epoch ahead of the first learning rate decay. + +![](images/1997d1c83cef97905d107164ab8ee49c83f98a1f8e6d837343d53ac655ee51b9.jpg) +(c) wpost-local SGD is trained from scratch with $H = 1 6$ . (d) wpost-local SGD is trained from scratch with $H = 3 2$ . +Figure 15: 1-d linear interpolation between models ${ \pmb w } _ { \mathrm { p o s t } }$ -local SGD and ${ \pmb w } _ { \mathrm { m i n i } }$ -batch SGD, i.e., $\hat { \pmb { w } } = \lambda \pmb { w } _ { \mathrm { m i n i - b a t c h ~ S G D } } +$ $( 1 - \lambda ) w$ post-local SGD, for different minima of ResNet-20 trained on CIFAR-10. The training is on top of $K = 1 6$ GPUs and the local batch size is fixed to $B _ { \mathrm { l o c } } = 1 2 8$ . The solid lines correspond to evaluate $\hat { \mathbf { \Omega } } \hat { \mathbf { \Omega } } ^ { v }$ on the whole training dataset while the dashed lines are on the test dataset. The post-local SGD in Figure 15(a) and Figure 15(d) is trained from the checkpoint of ${ \pmb w } _ { \mathrm { m i n i } }$ -batch SGD before performing the first learning rate decay, while that of Figure 15(c) and Figure 15(d) is trained from scratch. The top-1 test accuracy of mini-batch SGD is 92.25, while that of post-local SGD in Figure 15(a), Figure 15(b), Figure 15(c) and Figure 15(d), are 92.61, 92.35, 93.13 and 93.04 respectively. + +Table 12: Top-1 test accuracy of large-batch SGD and post-local SGD, for training ResNet-20 on CIFAR-100 with $B _ { \mathrm { l o c } } K = 4 0 9 6$ . The reported results are the average of three runs. We include the small and large batch baseline, where the models are trained by mini-batch SGD with 256 and 4096 respectively. The $\star$ indicates the fine-tuned learning rate. The learning rate will be decayed by 10 when the distributed algorithm has accesses $5 0 \%$ and $7 5 \%$ of the total number of training samples. + +
# of epochssmall batch baseline *large batch baseline *post-local SGD (H=8)post-local SGD (H=16)
30068.84±0.0667.34±0.3468.38±0.4868.30±0.30
40069.07±0.2767.55±0.2169.06±0.1569.05±0.26
50069.03±0.1067.42±0.6369.02±0.3868.87±0.27
+ +64 and we train the model for 120 epochs. The learning rate is again decayed at the phase when the training algorithm has accessed $5 0 \%$ and $7 5 \%$ of the total training samples. + +Table 13 below demonstrates the effectiveness of post-local SGD for large-batch training on language modeling task (without extra fine-tuning except our baselines). Note that most of the existing work focuses on improving the large-batch training issue for computer vision tasks; and it is non-trivial to scale the training of LSTM for language modeling task due to the presence of different hyperparameters. Here we provide a proof of concept result in Table 13 to show that our post-local SGD is able to improve upon standard large-batch baseline. The benefits can be further pronounced if we scale to larger batches (as the case of image classification e.g. in Table 11 and Table 12). + +Table 13: The perplexity (lower is better) of language modeling task on WikiText-2. We use $K = 1 6$ and $K B = K B _ { \mathrm { l o c } } = \bar { 1 } 0 \bar { 2 } 4$ . The reported results are evaluated on the validation dataset (average of three runs). We fine-tune the learning rate for mini-batch SGD baselines. + +
small batchbaseline*largebatch baseline*large-batch (H=8)large-batch (H=16)
86.50 ±0.3586.90±0.4986.61 ±0.3086.85 ±0.13
+ +# C.5.3 POST-LOCAL SGD TRAINING ON IMAGENET + +We evaluate the performance of post-local SGD on the challenging ImageNet training. Again we limit ResNet-50 training to 90 passes over the data in total, and use the standard training configurations as mentioned in Appendix A.4.2. The post-local SGD begins when performing the first learning rate decay. + +We can witness that post-local SGD outperforms mini-batch SGD baseline for both of mini-batch size 4096 (76.18 and 75.87 respectively) and 8192 (75.65 and 75.64 respectively). + +![](images/206a4f8eea096e43299613979904d8e667b72510f9e90e43b2306a6518aec158.jpg) +(a) The performance of post-local SGD training for ImageNet-1k with $\bar { K } B _ { \mathrm { l o c } } = 4 0 9 6$ , in terms of epoch-to-accuracy. + +![](images/02c4fc777dc6d6d3646192bbed8abe095bb1199efe20586b322313c7696bf6c5.jpg) +(b) The performance of post-local SGD training for ImageNet-1k with $\bar { K } B _ { \mathrm { l o c } } = 8 1 9 2$ , in terms of epoch-to-accuracy. + +![](images/99ddb42556cf29104400f4991e0b8cae12e7ddcbeeaa9535e7b0066b625c8677.jpg) + +![](images/ebf9a94487094d0256fe1bd57a81bc3539b2d27a32b6cc54b0214d15ec7cfc4b.jpg) +(d) The performance of post-local SGD training for ImageNet-1k with $K B _ { \mathrm { l o c } } = 8 1 9 2$ , in terms of time-to-accuracy. + +(c) The performance of post-local SGD training for ImageNet-1k with $K B _ { \mathrm { l o c } } ~ = ~ 4 0 9 6$ , in terms of time-to-accuracy. + +Figure 16: The performance of post-local SGD training for ImageNet-1k. We evaluate the model performance on test dataset after each complete accessing of the whole training samples. Note that due to the resource limitation of the main experimental platform in the paper, these experiments are on top of a $8 \times 4$ -GPU (V100) Kubernetes cluster with 10 Gbps network bandwidth. + +# C.5.4 POST-LOCAL SGD VS. OTHER NOISE INJECTION METHODS + +The role of “noise” has been actively studied in SGD for non-convex deep learning, from optimization and generalization aspects. Neelakantan et al. (2015) propose to inject isotropic white noise for better optimization. Zhu et al. (2019); Xing et al. (2018) study the “structured” anisotropic noise and find the importance of anisotropic noise in SGD for escaping from minima (over isotropic noise) in terms of generalization. Wen et al. (2019) on top of these work and try to inject noise (sampled from the expensive empirical Fisher matrix) to large-batch SGD. + +However, to our best knowledge, none of the prior work can provide a computation efficient way to inject noise to achieve as good generalization performance as small-batch SGD. Either it is practically unknown if injecting the isotropic noise (Neelakantan et al., 2015) can alleviate the issue of large-batch training, or it has been empirically justified (Wen et al., 2019) that injecting anisotropic noise from expensive (diagonal) empirical Fisher matrix fails to recover the same performance as the small mini-batch baselines. In this section, we only evaluate the impact of injecting isotropic noise (Neelakantan et al., 2015) for large-batch training, and omit the comparison to Wen et al. $( 2 0 1 9 ) ^ { 8 }$ . + +The idea of Neelakantan et al. (2015) considers to add time-dependent Gaussian noise to the gradient at every training step $t$ : $\nabla f _ { i } \big ( \mathbf { \dot { w } } _ { ( t ) } \big ) ^ { \cdot } \gets \nabla f _ { i } \big ( \mathbf { w } _ { ( t ) } \big ) + \mathcal { N } ( 0 , \dot { \pmb { \sigma } } _ { t } ^ { 2 } )$ , where $\pmb { \sigma } _ { t } ^ { 2 }$ follows $\begin{array} { r } { \sigma _ { t } ^ { 2 } : = \frac { \eta } { ( 1 + t ) ^ { \gamma ^ { \prime } } } } \end{array}$ We follow the general hyper-parameter search scheme (as mentioned in Section A.4) and fine-tune the $\eta$ and $\gamma ^ { \prime }$ in the range of $\{ 1 e ^ { - 6 } , 5 e ^ { - 6 } , 1 e ^ { - 5 } , 5 e ^ { - 5 } \}$ and $\{ 0 . 5 , 1 , 1 . 5 , 2 , 2 . 5 , 3 . 0 \}$ respectively. + +Table 14 below compares the post-local SGD to Neelakantan et al. (2015), where the noise injection scheme in Neelakantan et al. (2015) cannot address the large-batch training issue and could even deteriorate the generalization performance. Note that we also tried to search the hyper-parameters at around the values reported in Neelakantan et al. (2015) for our state-of-the-art CNNs, but it will directly result in the severe quality loss or even divergence. + +Table 14: Top-1 test accuracy of adding isotropic noise for large-batch training. We revisit the case of ResNet-20 in Table 3 (2K for CIFAR-10) and Table 11 (4K for CIFAR-100), where adding isotropic noise as in Neelakantan et al. (2015) cannot address the large-batch training difficulty. The $\star$ indicates a fine-tuned learning rate. + +
CIFAR-10,KB= 2KCIFAR-100,KB= 4K
Neelakantan et al. (2015)*92.46 ±0.2767.15 ±0.26
mini-batch SGD *92.48 ±0.1767.38 ±0.34
Post-local SGD93.02 ±0.2468.30 ±0.48
+ +Compared to another local SGD variant. Recently, Wang & Joshi (2019) proposed to decrease the number of local update steps $H$ during training. However, their scheme is inherited from the convergence analysis for optimization (not the generalization for deep learning), and in principle opposite to our interpolation and proposed strategy (i.e. increasing the local update steps during the training). + +Their evaluation (ResNet-50 on CIFAR-10 for $K = 4$ with $B _ { \mathrm { l o c } } = 1 2 8 )$ also does not cover the difficult large batch training scenario (Scenario 2), e.g., $H = 1 6$ , $K = 1 6$ , $B _ { \mathrm { l o c } } = 1 2 8$ . For the same CIFAR-10 task and $K { = } 4$ as in Wang & Joshi (2019), our smaller ResNet-20 with local SGD can simply reach a better accuracy with less communication9 (Figure 2). + +# C.5.5 POST-LOCAL SGD WITH OTHER COMPRESSION SCHEMES + +In this subsection, we demonstrate that (post-)local SGD can be integrated with other compression techniques for better training efficiency (further reduced communication cost) and improved generalization performance (w.r.t. the gradient compression methods). + +We use sign-based compression scheme (i.e. signSGD (Bernstein et al., 2018) and EFsignSGD (Karimireddy et al., 2019)) for the demonstration. The pseudo code can be found in Algorithm 3 and Algorithm 4. + +Post-local SGD with signSGD. We slightly adapt the original signSGD (Bernstein et al., 2018) to fit in the local SGD framework. The local model will be firstly updated by the sign of the local update directions (e.g. gradients, or gradients with weight decay and momentum acceleration), and then be synchronized to reach the global consensus model for the next local updates. + +Note that we can almost recover the signSGD in Bernstein et al. (2018) from Algorithm 3 when $H ^ { \prime } = 1$ , except that we will average over the sign instead of using the majority vote in Bernstein et al. (2018). Table 15 illustrates the trivial generalization performance difference of these two schemes, where the post-local SGD in Algorithm 3 is able to significantly improve the generalization performance (as in Table 4) with further improved communication efficiency. + +input: the initial model ${ \pmb w } _ { ( 0 ) } \in \mathbb { R } ^ { d }$ ; training data with labels $\mathcal { T }$ ; mini-batch of size $B _ { \mathrm { l o c } }$ per local model; step +size $\eta$ , and momentum $_ { \pmb { m } }$ (optional); number of synchronization steps $T$ , and the first learning rate decay is +performed at $T ^ { \prime }$ ; number of eventual local steps $\dot { H } ^ { \prime }$ ; number of nodes $K$ . +1: synchronize to have the same initial models $\hat { w } _ { ( 0 ) } ^ { k } : = w _ { ( 0 ) }$ . +2: for all $k : = 1 , \ldots , K$ do in parallel +3: for $t : = 1 , \dots , T$ do +4: if $t < T ^ { \prime }$ then +5: $H _ { ( t ) } = 1$ +6: else +7: $H _ { ( t ) } = H ^ { \prime }$ +8: end if +9: for $h : = 1 , \ldots , H _ { ( t ) }$ do +10: sample a mini-batch from $\mathcal { T } _ { ( t ) + h - 1 } ^ { k }$ +11: compute the gradients $\begin{array} { r } { \pmb { g } _ { ( t ) + h - 1 } ^ { k } : = \frac { 1 } { B _ { \mathrm { l o c } } } \sum _ { i \in \mathbb { Z } _ { ( t ) + h - 1 } ^ { k } } \nabla f _ { i } \big ( \pmb { w } _ { ( t ) + h - 1 } ^ { k } \big ) } \end{array}$ . ™ can involve weight +decay and momentum. +12: update the local model $\begin{array} { r } { \pmb { w } _ { ( t ) + h } ^ { k } : = \pmb { w } _ { ( t ) + h - 1 } - \gamma _ { ( t ) } \mathrm { s i g n } ( \pmb { g } _ { ( t ) + h - 1 } ^ { k } ) . } \end{array}$ . +13: end for +14: get model difference $\Delta _ { ( t ) } ^ { k } : = \boldsymbol { w } _ { ( t ) } ^ { k } - \boldsymbol { w } _ { ( t ) + H } ^ { k }$ . +15: compress the model difference: $\begin{array} { r } { \pmb { s } _ { ( t ) } ^ { k } = \mathrm { s i g n } ( \Delta _ { ( t ) } ^ { k } ) } \end{array}$ and $\begin{array} { r } { p _ { ( t ) } ^ { k } = \frac { \left\| \Delta _ { ( t ) } ^ { k } \right\| _ { 1 } } { d } } \end{array}$ +16: ptq get new global (synchronized) model $\pmb { w } _ { ( t + 1 ) } ^ { k }$ ptq for all ptq nodes: $\begin{array} { r } { \pmb { w } _ { ( t + 1 ) } ^ { k } : = \pmb { w } _ { ( t ) } ^ { k } - \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \pmb { s } _ { ( t ) } ^ { k } \pmb { p } _ { ( t ) } ^ { k } , } \end{array}$ . +17: end for +18: end for + +Algorithm 3 (Post-)Local SGD with the compression scheme in signSGD +Table 15: Top-1 test accuracy of training ResNet-20 on CIFAR $( K B _ { \mathrm { l o c } } = 2 0 4 8$ and $K = 1 6$ ). + +
CIFAR-10CIFAR-100
signSGD (average over signs)91.61 ±0.2867.15 ±0.10
signSGD (majority vote over signs)91.61 ±0.2667.21 ±0.11
+ +Post-local SGD with EF-signSGD. We extend the single-worker EF-signSGD algorithm of Karimireddy et al. (2019) (i.e. their Algorithm 1) to multiple-workers case by empirically investigating different algorithmic design choices. Our presented Algorithm 4 of EF-signSGD for multiple-workers can reach a similar performance as the mini-batch SGD counterpart (both are after the proper hyper-parameters tuning and under the same experimental setup). + +Training schemes, hyper-parameter tuning procedure for signSGD, signSGD variant in Algorithm 3, and EF-signSGD in Algorithm 4. The training schemes of the signSGD (Bernstein et al., 2018) (i.e. the one using majority vote) and the signSGD variant in Algorithm 3 are slightly different from the the experimental setup mentioned in A.4, where we only fine-tune the initial learning rate in signSGD and Algorithm 3 (when $H = 1$ ). The optimal learning rate is searched from the grid $\{ 0 . 0 \bar { 0 } 5 , 0 . 1 0 , 0 . 1 5 , \bar { 0 . 2 0 } \}$ with the general fine-tuning principle mentioned in A.4. No learning rate + +input: the initial model ${ \pmb w } _ { ( 0 ) } \in \mathbb { R } ^ { d }$ ; training data with labels $\mathcal { T }$ ; mini-batch of size $B _ { \mathrm { l o c } }$ per local model; step size $\eta$ , and momentum $_ { \mathbf { \nabla } } \mathbf { m } _ { \mathbf { \nabla } }$ (optional); number of synchronization steps $T$ , and the first learning rate decay is performed at $T ^ { \prime }$ ; number of eventual local steps $H ^ { \prime }$ ; number of nodes $K$ . error-compensated memory ep0q  0 P Rd. +1: synchronize to have the same initial models $\pmb { w } _ { ( 0 ) } ^ { k } : = \pmb { w } _ { ( 0 ) }$ . +2: initialize the local error memory $e _ { ( 0 ) } ^ { k } : = e _ { ( 0 ) }$ . +3: for all $k : = 1 , \ldots , K$ do in parallel +4: for $t : = 1 , \dots , T$ do +5: if $t < T ^ { \prime }$ then +6: $H _ { ( t ) } = 1$ +7: else +8: $H _ { ( t ) } = H ^ { \prime }$ +9: end if +10: 11: for s $h : = 1 , \ldots , H _ { ( t ) }$ doch from $\mathcal { T } _ { ( t ) + h - 1 } ^ { k }$ +12: compute the gradient $\begin{array} { r } { \pmb { g } _ { ( t ) + h - 1 } ^ { k } : = \frac { 1 } { B _ { \mathrm { l o c } } } \sum _ { i \in \mathcal { T } _ { ( t ) + h - 1 } ^ { k } } \nabla f _ { i } \left( \pmb { w } _ { ( t ) + h - 1 } ^ { k } \right) . } \end{array}$ . ™ can involve weight decay and momentum. +13: update the local model wkptqh : wptqh1  γptqgkptqh1 . +14: end for +15: get model difference $\Delta _ { ( t ) } ^ { k } : = { \pmb w } _ { ( t ) } ^ { k } - { \pmb w } _ { ( t ) + H } ^ { k } + { \pmb e } _ { ( t ) }$ . +16: compress the model difference: $\begin{array} { r } { \pmb { s } _ { ( t ) } ^ { k } = \mathrm { s i g n } ( \Delta _ { ( t ) } ^ { k } ) } \end{array}$ and $\begin{array} { r } { p _ { ( t ) } ^ { k } = \frac { \left\| \Delta _ { ( t ) } ^ { k } \right\| _ { 1 } } { d } } \end{array}$ +17: update local memory: $\pmb { e } _ { ( t + 1 ) } ^ { k } = \pmb { e } _ { ( t ) } ^ { k } - \pmb { s } _ { ( t ) } ^ { k } \pmb { p } _ { ( t ) } ^ { k }$ . +18: get new global (synchronized) model $\pmb { w } _ { ( t + 1 ) } ^ { k }$ for all $K$ nodes: $\begin{array} { r } { \pmb { w } _ { ( t + 1 ) } ^ { k } : = \pmb { w } _ { ( t ) } ^ { k } - \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \pmb { s } _ { ( t ) } ^ { k } \pmb { p } _ { ( t ) } ^ { k } } \end{array}$ . +19: end for +20: end for + +scaling up or warming up is required during the training, and the learning rate will be decayed by 10 when accessing $5 0 \%$ and $7 5 \%$ of the total training samples. + +The training schemes of the distributed EF-signSGD in Algorithm 4 in general follow the experimental setup mentioned in A.4. We grid search the optimal initial learning rate for Algorithm 4 (for the case of $H = 1$ ), and gradually warm up the learning rate from a relatively small value (0.1) to this found initial learning rate during the first 5 epochs of the training. + +Note that in our experiments, weight decay and Nesterov momentum are used for the local model updates, for both of Algorithm 3 and Algorithm 4. We empirically found these two techniques can significantly improve the training/test performance under a fixed training epoch budget. + +# D HIERARCHICAL LOCAL SGD + +The idea of local SGD can be leveraged to the more general setting of training on decentralized and heterogeneous systems, which is an increasingly important application area. Such systems have become common in the industry, e.g. with GPUs or other accelerators grouped hierarchically within machines, racks or even at the level of several data-centers. Hierarchical system architectures such as in Figure 17 motivate our hierarchical extension of local SGD. Moreover, end-user devices such as mobile phones form huge heterogeneous networks, where the benefits of efficient distributed and data-local training of machine learning models promises strong benefits in terms of data privacy. + +![](images/3538fafbeb5d128b9d18ce9102e2d7892ea69231d67457ad406a943dcc6f4125.jpg) +Figure 17: Illustration of a hierarchical network architecture of a cluster in the data center. While GPUs within each node are linked with fast connections (e.g. NVLink), connections between the servers within and between different racks have much lower bandwidth and latency (via top-of-the-rack switches and cluster switches). The hierarchy can be extended several layers further and further. Finally, edge switches face the external network at even lower bandwidth. + +# D.1 THE ILLUSTRATION OF HIERARCHICAL LOCAL SGD + +Real world systems come with different communication bandwidths on several levels. In this scenario, we propose to employ local SGD on each level of the hierarchy, adapted to each corresponding computation vs communication trade-off. The resulting scheme, hierarchical local SGD, can offer significant benefits in system adaptivity and performance. + +![](images/d7126f45b6310422bb1ec03c9d49c6a9d13b2daf71a9244f3efb43ed60e776b7.jpg) +Figure 18: An illustration of hierarchical local SGD, for $B _ { \mathrm { l o c } } = 2$ , using $H = 3$ inner local steps and $H ^ { b } = 2$ outer ‘block’ steps. Local parameter updates are depicted in red, whereas block and global synchronization is depicted in purple and black respectively. + +As the guiding example, we consider compute clusters which typically allocate a large number of GPUs grouped over several machines, and refer to each group as a GPU-block. Hierarchical local SGD continuously updates the local models on each GPU for a number of $H$ local update steps before a (fast) synchronization within a GPU-block. On the outer level, after $H ^ { b }$ such block update steps, a (slower) global synchronization over all GPU-blocks is performed. Figure 18 and Algorithm 5 depict how the hierarchical local SGD works, and the complete procedure is formalized below: + +$$ +\begin{array} { r l } & { w _ { [ ( t ) + l ] + H } ^ { k } : = w _ { [ ( t ) + l ] } ^ { k } - \displaystyle \sum _ { h = 1 } ^ { H } \frac { \gamma _ { [ ( t ) ] } } { B _ { \operatorname* { l o c } } } \cdot \sum _ { \nabla _ { [ ( t ) + l ] + h - 1 } } \nabla f _ { i } \big ( w _ { [ ( t ) + l ] + h - 1 } ^ { k } } \\ & { } \\ & { w _ { [ ( t ) + l + 1 ] } ^ { k } : = w _ { [ ( t ) + l ] } ^ { k } - \displaystyle \frac { K _ { i } } { K _ { i } } \sum _ { k = 1 } ^ { K _ { i } } \big ( w _ { [ ( t ) + l ] } ^ { k } - w _ { [ ( t ) + l ] + H } ^ { k } \big ) } \\ & { \qquad w _ { [ ( t + 1 ) ] } ^ { k } : = w _ { [ ( t ) ] } ^ { k } - \displaystyle \frac { K } { K } \sum _ { k = 1 } ^ { K } \big ( w _ { [ ( t ) ] } ^ { k } - w _ { [ ( t ) + H ^ { k } ] } ^ { k } \big ) } \end{array} +$$ + +where ${ \pmb w } _ { [ ( t ) + l ] + H } ^ { k }$ indicates the model after $l$ k1 block update steps and $H$ local update steps, and $K _ { i }$ is rptqlsHthe number of GPUs on the GPU-block $i$ . The definition of $\gamma _ { [ ( t ) ] }$ and $\mathcal { T } _ { [ ( t ) + l ] + h - 1 } ^ { k }$ follows a similar scheme. + +As the number of devices grows to the thousands (Goyal et al., 2017; You et al., 2017b), the difference between ‘within’ and ‘between’ block communication efficiency becomes more drastic. Thus, the performance benefits of our adaptive scheme compared to flat & large mini-batch SGD will be even more pronounced. + +D.2 THE ALGORITHM OF HIERARCHICAL LOCAL SGD + +# Algorithm 5 Hierarchical Local SGD + +input: the initial model wrp0qs; +input: training data with labels $\mathcal { T }$ ; +input: mini-batch of size $B _ { \mathrm { l o c } }$ per local model; +input: step size $\eta$ , and momentum $m$ (optional); +input: number of synchronization steps $T$ over nodes; +input: number of local update steps $H$ , and block update steps $H ^ { b }$ ; +input: number of nodes $K$ in total; and nodes $K ^ { \prime }$ per GPU-block. +1: synchronize to have the same initial models wkrp0qs : wrp0qs. +2: for all $k : = 1 , \ldots , K$ do in parallel +3: for $t : = 1 , \dots , T$ do +4: for $l : = 1 , \ldots , H ^ { b } \ : .$ do +5: 6: for s $h : = 1 , \ldots , H$ doatch from . $\mathscr { T } _ { [ ( t ) + l ] + h - 1 } ^ { k }$ +7: compute the gradient $\pmb { g } _ { [ ( t ) + l ] + h - 1 } ^ { k } : = \frac { 1 } { B _ { \mathrm { l o c } } } \sum _ { i \in \mathcal { T } _ { [ ( t ) + l ] + h - 1 } ^ { k } } \nabla f _ { i } \left( \pmb { w } _ { [ ( t ) + l ] + h - 1 } ^ { k } \right) .$ +8: update the local model $\pmb { w } _ { [ ( t ) + l ] + h } ^ { k } : = \pmb { w } _ { [ ( t ) + l ] + h - 1 } ^ { k } - \gamma _ { [ ( t ) ] } \pmb { g } _ { [ ( t ) + l ] + h - 1 } ^ { k } .$ +9: end for +10: inner all-reduce aggregation o ${ \Delta } _ { [ ( t ) + l ] } ^ { k } : = { \pmb w } _ { [ ( t ) + l ] } ^ { k } - { \pmb w } _ { [ ( t ) + l ] + H } ^ { k } .$ +11: get new block (synchronized) model $\boldsymbol { w } _ { [ ( t ) + l + 1 ] } ^ { k }$ for $K ^ { \prime }$ block nodes: $\begin{array} { r } { \pmb { w } _ { [ ( t ) + l + 1 ] } ^ { k } : = \pmb { w } _ { [ ( t ) + l ] } ^ { k } - \gamma _ { [ ( t ) ] } \frac { 1 } { K ^ { \prime } } \sum _ { k = 1 } ^ { K ^ { \prime } } \Delta _ { [ ( t ) + l ] } ^ { k } , } \end{array}$ +12: end for +13: outer all-reduce aggregation of the gradients $\Delta _ { [ ( t ) ] } ^ { k } : = w _ { [ ( t ) ] } ^ { k } - w _ { [ ( t ) + H ^ { b } ] } ^ { k } .$ +14: get new global (synchronized) model $\pmb { w } _ { [ ( t + 1 ) ] } ^ { k }$ for all $K$ nodes: $\begin{array} { r } { \pmb { w } _ { [ ( t + 1 ) ] } ^ { k } : = \pmb { w } _ { [ ( t ) ] } ^ { k } - \gamma _ { [ ( t ) ] } \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \Delta _ { [ ( t ) ] } ^ { k } . } \end{array}$ +15: end for +16: end for + +# D.3 HIERARCHICAL LOCAL SGD TRAINING + +Now we move to our proposed training scheme for distributed heterogeneous systems. In our experimental setup, we try to mimic the real world setting where several compute devices such as GPUs are grouped over different servers, and where network bandwidth (e.g. Ethernet) limits the communication of updates of large models. The investigation of hierarchical local SGD again trains ResNet-20 on CIFAR-10 and follows the same training procedure as local SGD where we re-formulate below. + +The experiments follow the common mini-batch SGD training scheme for CIFAR (He et al., 2016a;b) and all competing methods access the same total amount of data samples regardless of the number of local steps or block steps. More precisely, the training procedure is terminated when the distributed algorithms have accessed the same number of samples as a standalone worker would access in 300 epochs. The data is partitioned among the GPUs and reshuffled globally every epoch. The local mini-batches are then sampled among the local data available on each GPU. The learning rate scheme is the same as in He et al. (2016a), where the initial learning rate starts from 0.1 and is divided by 10 when the model has accessed $5 0 \%$ and $7 5 \%$ of the total number of training samples. In addition to this, the momentum parameter is set to 0.9 without dampening and applied independently to each local model. + +# D.3.1 THE PERFORMANCE OF HIERARCHICAL LOCAL SGD. + +Table 16: Training CIFAR-10 with ResNet-20 via local SGD on a $8 \times 2$ -GPU cluster. The local batch size $\scriptstyle B _ { \mathrm { l o c } }$ is fixed to 128 with $H ^ { b } = 1$ , and we scale the number of local steps $H$ from 1 to 1024. The reported training times are the average of three runs and all the experiments are under the same training configurations for the equivalent of 300 epochs, without specific tuning. + +
H=12481632641282565121024
Training Time (minutes)20.0713.9510.489.208.578.329.229.239.5010.3010.65
+ +Training time vs. local number of steps. Table 16 shows the performance of local SGD in terms of training time. The communication traffic comes from the global synchronization over 8 nodes, each having 2 GPUs. We can witness that increasing the number of local steps over the “datacenter” scenario cannot infinitely improve the communication performance, or would even reduce the communication benefits brought by a large number of local steps. Hierarchical local SGD with inner node synchronization reduces the difficulty of synchronizing over the complex heterogeneous environment, and hence enhances the overall system performance of the synchronization. The benefits are further pronounced when scaling up the cluster size. + +![](images/682e3dc133ddf37adafdd03e87df61f021ce1ea4af19f7b4ece9007081d58d7e.jpg) + +![](images/11ecc8f309954027e21251edaf60f4f87a0ca10f76b8767525ecd8a9460cea30.jpg) + +![](images/dbb1f6c8723df4d8bdca876d2a4d5f406d6a2712b24ae3753143a7258703c482.jpg) +e (b) Training accuracy vs. time. The (c) Training accuracy vs. time. The number of local steps is $H = 2$ , with number of local steps is $H = 2$ , with 1 second delay for each global syn- 50 seconds delay for each global synchronization. chronization. + +(a) Training accuracy vs. time. T number of local steps is $H = 2$ . + +Figure 19: The performance of hierarchical local SGD trained on CIFAR-10 with ResNet-20 ( $2 \times 2$ -GPU). Each GPU block of the hierarchical local SGD has 2 GPUs, and we have 2 blocks in total. Each figure fixes the number of local steps but varies the number of block steps from 1 to 32. All the experiments are under the same training configurations without specific tuning. + +Hierarchical local SGD shows high tolerance to network delays. Even in our small-scale experiment of two servers and each with two GPUs, hierarchical local SGD shows its ability to significantly reduce the communication cost by increasing the number of block step $H ^ { b }$ (for a fixed $H$ ), with trivial performance degradation. Moreover, hierarchical local SGD with a sufficient number of block steps offers strong robustness to network delays. For example, for fixed $H = 2$ , by increasing the number of $H ^ { b }$ , i.e. reducing the number of global synchronizations over all models, we obtain a significant gain in training time as in Figure 19(a). The impact of a network of slower communication is further studied in Figure 19(b), where the training is simulated in a realistic scenario and each global communication round comes with an additional delay of 1 second. Surprisingly, even for the global synchronization with straggling workers and severe 50 seconds delay per global communication round, Figure 19(c) demonstrates that a large number of block steps (e.g. $H ^ { b } = 1 6$ ) still manages to fully overcome the communication bottleneck with no/trivial performance damage. + +Table 17: The performance of training CIFAR-10 with ResNet-20 via hierarchical local SGD on a 16-GPU Kubernetes cluster. We simulate three different types of cluster topology, namely 8 nodes with 2 GPUs/node, 4 nodes with 4 GPUs/node, and 2 nodes with 8 GPUs/node. The configuration of hierarchical local SGD satisfies $H \cdot H ^ { b } = 1 6$ . All variants either synchronize within each node or over all GPUs, and the communication cost is estimated by only considering $\dot { H ^ { \cdot } } H ^ { b } = 1 6$ model updates during the training (the update could come from a different level of the synchronizations). The reported results are the average of three runs and all the experiments are under the same training configurations, training for the equivalent of 300 epochs, without specific tuning. + +
H=1, H=16H=2, H=8H=4, H=4H=8, H=2H= 16, H=1
# of sync.over nodes11111
# of sync.within node157310
Test acc.on 8 × 2-GPU90.02 ±0.2890.2589.9591.4191.18±0.02
Test acc.on 4 × 4-GPU91.65±0.08 91.26±0.19 91.46±0.23 91.91
±0.06±0.17±0.24
±0.16
Test acc.on 2 × 8-GPU92.1492.0591.9491.56
±0.10±0.14±0.09±0.18
+ +Hierarchical local SGD offers improved scaling and better test accuracy. Table 17 compares the mini-batch SGD with hierarchical local SGD for fixed product $H \cdot H ^ { b } = 1 6$ under different network topologies, with the same training procedure. We can observe that for a heterogeneous system with a sufficient block size, hierarchical local SGD with a sufficient number of block update steps can further improve the generalization performance of local SGD training. More precisely, when ${ \dot { H } } { \cdot } H ^ { b }$ is fixed, hierarchical local SGD with more frequent inner-node synchronizations $\cdot H ^ { b } > 1 \rangle$ ) outperforms local SGD ( $\begin{array} { r } { H ^ { b } = 1 } \end{array}$ ), while still maintaining the benefits of significantly reduced communication by the inner synchronizations within each node. In summary, as witnessed by Table 17, hierarchical local SGD outperforms both local SGD and mini-batch SGD in terms of training speed as well as model performance, especially for the training across nodes where inter-node connection is slow but intra-node communication is more efficient. + +# E COMMUNICATION SCHEMES + +This section evaluates the communication cost in terms of the number of local steps and block steps, and formalizes the whole communication problem below. + +Assume $K$ computing devices uniformly distributed over $K ^ { \prime }$ servers, where each server has $\frac { K } { K ^ { \prime } }$ devices. The hierarchical local SGD training procedure will access $N$ total samples with local mini-batch size $B$ , with $H$ local steps and $H ^ { b }$ block steps. + +The MPI communication scheme (Gropp et al., 1999) is introduced for communication cost evaluation. More precisely, we use general all-reduce, e.g., recursive halving and doubling algorithm (Thakur et al., 2005; Rabenseifner, 2004), for gradient aggregation among $K$ computation devices. For each all-reduce communication, it introduces $C \cdot \log _ { 2 } K$ communication cost, where $C$ is the message transmission time plus network latency. + +The communication cost under our hierarchical local SGD setting is mainly determined by the number of local steps and block steps. The $\textstyle { \frac { K } { T } }$ models within each server synchronize the gradients for every $H$ local mini-batch, and it only performs global gradients aggregation of $K$ local models after $H ^ { b }$ block updates. Thus, the total number of synchronizations among compute devices is reduced to $\Big \lceil \frac { N } { K B \cdot H H ^ { b } } \Big \rceil$ , and we can formulate the overall communication cost $\tilde { C }$ as: + +$$ +\begin{array} { r } { \tilde { C } \approx \left( \left\lceil \frac { N } { K B \cdot H } \right\rceil - \left\lceil \frac { N } { K B \cdot H H ^ { b } } \right\rceil \right) \cdot C _ { 1 } \cdot K ^ { \prime } \log _ { 2 } \frac { K } { K ^ { \prime } } + \left\lceil \frac { N } { K B \cdot H H ^ { b } } \right\rceil \cdot C _ { 2 } \log _ { 2 } K } \end{array} +$$ + +where $C _ { 1 }$ is the single message passing cost for compute devices within the same server, $C _ { 2 }$ is the cost of that across servers, and obviously $C _ { 1 } \ll C _ { 2 }$ . We can easily witness that the number of block steps $H ^ { b }$ is more deterministic in terms of communication reduction than local step $H$ . Empirical evaluations can be found in Section D.3. + +Also, note that our hierarchical local SGD is orthogonal to the implementation of gradient aggregation (Goyal et al., 2017) optimized for the hardware, but focusing on overcoming the aggregation cost of more general distributed scenarios, and can easily be integrated with any optimized all-reduce implementation. + +# F DISCUSSION AND FUTURE WORK + +Data distribution patterns. In our experiments, the dataset is globally shuffled once per epoch and each local worker only accesses a disjoint part of the training data. Removing shuffling altogether, and instead keeping the disjoint data parts completely local during training might be envisioned for extremely large datasets which can not be shared, or also in a federated scenario where data locality is a must for privacy reasons. This scenario is not covered by the current theoretical understanding of local SGD, but will be interesting to investigate theoretically and practically. + +Better learning rate scheduler for local SGD. We have shown in our experiments that local SGD delivers consistent and significant improvements over the state-of-the-art performance of mini-batch SGD. For ImageNet, we simply applied the same configuration of “large-batch learning schemes” by Goyal et al. (2017). However, this set of schemes was specifically developed and tuned for mini-batch SGD only, not for local SGD. For example, scaling the learning rate w.r.t. the global mini-batch size ignores the frequent local updates where each local model only accesses local minibatches for most of the time. Therefore, it is expected that specifically deriving and tuning a learning rate scheduler for local SGD would lead to even more drastic improvements over mini-batch SGD, especially on larger tasks such as ImageNet. + +Adaptive local SGD. As local SGD achieves better generalization than current mini-batch SGD approaches, an interesting question is if the number of local steps $H$ could be chosen adaptively, i.e. change during the training phase. This could potentially eliminate or at least simplify complex learning rate schedules. Furthermore, recent works (Loshchilov & Hutter, 2017; Huang et al., 2017a) leverage cyclic learning rate schedules either improving the general performance of deep neural network training, or using an ensemble multiple neural networks at no additional training cost. Adaptive local SGD could potentially achieve similar goals with reduced training cost. + +Hierarchical local SGD design. Hierarchical local SGD provides a simple but efficient training solution for devices over the complex heterogeneous system. However, its performance might be impacted by the cluster topology. For example, the topology of $8 \times 2$ -GPU in Table 17 fails to further improve the performance of local SGD by using more frequent inner node synchronizations. On contrary, sufficiently large size of the GPU block could easily benefit from the block update of hierarchical local SGD, for both communication efficiency and training quality. The design space of hierarchical local SGD for different cluster topologies should be further investigated, e.g., to investigate the two levels of model averaging frequency (within and between blocks) in terms of convergence, and the interplay of different local minima in the case of very large number of local steps. Another interesting line of work, explores heterogenous systems by allowing for different number of local steps on different clusters, thus making up for slower machines. \ No newline at end of file diff --git a/parse/train/B1eyO1BFPr/B1eyO1BFPr_content_list.json b/parse/train/B1eyO1BFPr/B1eyO1BFPr_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..6e47a00ddba4b5cf31d0a1c662a3dafab13d3298 --- /dev/null +++ b/parse/train/B1eyO1BFPr/B1eyO1BFPr_content_list.json @@ -0,0 +1,4580 @@ +[ + { + "type": "text", + "text": "DON’T USE LARGE MINI-BATCHES, USE LOCAL SGD ", + "text_level": 1, + "bbox": [ + 176, + 98, + 820, + 122 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Tao Lin EPFL, Switzerland tao.lin@epfl.ch ", + "bbox": [ + 184, + 148, + 333, + 189 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sebastian U. Stich \nEPFL, Switzerland \nsebastian.stich@epfl.ch \nKumar Kshitij Patel \nIIT Kanpur, India \nkumarkshitijpatel@gmail.com \nMartin Jaggi \nEPFL, Switzerland \nmartin.jaggi@epfl.ch ", + "bbox": [ + 586, + 147, + 813, + 189 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 184, + 210, + 450, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 583, + 210, + 781, + 253 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 290, + 544, + 304 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Mini-batch stochastic gradient methods (SGD) are state of the art for distributed training of deep neural networks. Drastic increases in the mini-batch sizes have lead to key efficiency and scalability gains in recent years. However, progress faces a major roadblock, as models trained with large batches often do not generalize well, i.e. they do not show good accuracy on new data. ", + "bbox": [ + 232, + 320, + 766, + 391 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As a remedy, we propose a post-local SGD and show that it significantly improves the generalization performance compared to large-batch training on standard benchmarks while enjoying the same efficiency (time-to-accuracy) and scalability. We further provide an extensive study of the communication efficiency vs. performance trade-offs associated with a host of local SGD variants. ", + "bbox": [ + 232, + 391, + 766, + 460 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 488, + 336, + 503 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Fast and efficient training of large scale deep-learning models relies on distributed hardware and on distributed optimization algorithms. For efficient use of system resources, these algorithms crucially must (i) enable parallelization while being communication efficient, and (ii) exhibit good generalization behaviour, i.e. good performance on unseen data (test-set). Most machine learning applications currently depend on stochastic gradient descent (SGD) (Robbins & Monro, 1985) and in particular its mini-batch variant (Bottou, 2010; Dekel et al., 2012). However, this algorithm faces generalization difficulties in the regime of very large batch sizes as we will review now. ", + "bbox": [ + 173, + 518, + 825, + 617 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Mini-batch SGD. For a sum-structured optimization problem of the form $\\begin{array} { r } { \\operatorname* { m i n } _ { { \\pmb w } \\in \\mathbb { R } ^ { d } } \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } f _ { i } ( { \\pmb w } ) } \\end{array}$ where $\\pmb { w } \\in \\mathbb { R } ^ { d }$ denotes the parameters of the model and $f _ { i } \\colon { \\mathbb { R } ^ { d } } \\to { \\mathbb { R } }$ the loss function of the $i$ -th training example, the mini-batch SGD update for $K \\geqslant 1$ workers is given as ", + "bbox": [ + 174, + 631, + 825, + 675 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/d0d18e43e04ed2ceb83b33ce21c596932b82319bf504637f57fe3650f2bba13c.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { w } _ { ( t + 1 ) } : = \\pmb { w } _ { ( t ) } - \\gamma _ { ( t ) } \\left[ \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\frac { 1 } { B } \\sum _ { i \\in \\mathcal { I } _ { ( t ) } ^ { k } } \\nabla f _ { i } \\big ( \\pmb { w } _ { ( t ) } \\big ) \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 312, + 680, + 684, + 708 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "where $\\gamma _ { ( t ) } > 0$ denotes the learning rate and $\\smash { \\mathcal { T } _ { ( t ) , } ^ { k } \\subseteq [ N ] }$ the subset (mini-batch) of training datapoints selected by worker $k$ (typically selected uniformly at random from the locally available datapoints on worker $k$ ). For convenience, we will assume the same batch size $B$ per worker. ", + "bbox": [ + 174, + 715, + 825, + 758 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Local SGD. Motivated to better balance the available system resources (computation vs. communication), local SGD (a.k.a. local-update SGD, parallel SGD, or federated averaging) has recently attracted increased research interest (Mcdonald et al., 2009; Zinkevich et al., 2010; McDonald et al., 2010; Zhang et al., 2014; 2016; McMahan et al., 2017). In local SGD, each worker $k \\in [ K ]$ evolves a local model by performing $H$ sequential SGD updates with mini-batch size $\\scriptstyle B _ { \\mathrm { l o c } }$ , before communication (synchronization by averaging) among the workers. Formally, ", + "bbox": [ + 173, + 773, + 825, + 859 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/b18feeef614e92478cb2db2c358d96812faaad28ec9de7b74b6cb4443e3744e4.jpg", + "text": "$$\n\\begin{array} { r } { \\boldsymbol { w } _ { ( t ) + h + 1 } ^ { k } : = \\boldsymbol { w } _ { ( t ) + h } ^ { k } - \\gamma _ { ( t ) } \\left[ \\frac { 1 } { B _ { \\mathrm { l a c } } } \\sum _ { i \\in \\mathcal { I } _ { ( t ) + h } ^ { k } } \\nabla f _ { i } \\big ( \\boldsymbol { w } _ { ( t ) + h } ^ { k } \\big ) \\right] , \\quad \\boldsymbol { w } _ { ( t + 1 ) } ^ { k } : = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\boldsymbol { w } _ { ( t ) + H } ^ { k } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 861, + 785, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "where $\\pmb { w } _ { ( t ) \\pmb { \\mathrm { 1 } } } ^ { k }$ denotes the local model on machine k $k$ after $t$ global synchronization rounds and subsequent $\\check { h } \\in [ H ]$ local steps is defined analogously). Mini-batch SGD is a special case if $H = 1$ and $B _ { \\mathrm { l o c } } = B$ . Furthermore, the communication patterns of both algorithms are identical if $B = H B _ { \\mathrm { l o c } }$ . However, as the updates (eq. (2)) are different from the mini-batch updates (eq. (1)) for any $H > 1$ , the generalization behavior (test error) of both algorithms is expected to be different. ", + "bbox": [ + 176, + 895, + 825, + 926 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/c37533587195448eb17540ebaf8a184c24be6e289ddca82ed069b0a906e70708.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 174, + 102, + 823, + 313 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 422, + 825, + 465 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Large batch SGD. Recent schemes for scaling training to a large number of workers rely on standard mini-batch SGD (1) with very large overall batch sizes (Shallue et al., 2018; You et al., 2018; Goyal et al., 2017), i.e. increasing the global batch size linearly with the number of workers $K$ . However, the batch size interacts differently with the overall system efficiency on one hand, and with generalization performance on the other hand. While a larger batch size in general increases throughput, it may negatively affect the final accuracy on both the train- and test-set. Two main scenarios of particular interest can be decoupled as follows: ", + "bbox": [ + 173, + 479, + 825, + 578 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Scenario 1. The communication restricted setting, where the synchronization time is much higher than the gradient computation time. In this case the batch size of best efficiency is typically large, and is achieved by using partial computation (gradient accumulation) while waiting for communication. We in particular study the interesting case when the mini-batch sizes of both algorithms satisfy the relation $B = H B _ { \\mathrm { l o c } }$ , as in this case both algorithms (local SGD and minibatch SGD) evaluate the same number of stochastic gradients between synchronization steps. ", + "bbox": [ + 174, + 583, + 825, + 667 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Scenario 2. The regime of poor generalization of large-batch SGD, that is the use of very large overall batches (often a significant fraction of the training set size), which is known to cause drastically decreased generalization performance (Chen & Huo, 2016; Keskar et al., 2017; Hoffer et al., 2017; Shallue et al., 2018; Golmant et al., 2018). If sticking to standard mini-batch SGD and maintaining the level of parallelisation, the batch size $B$ would have to be reduced below the device (locally optimal) capacity in order to alleviate this generalization issue, which however impacts training time1. ", + "bbox": [ + 174, + 671, + 825, + 768 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Main Results. Key aspects of the empirical performance of local SGD compared to mini-batch baselines are illustrated in Figure 1. In scenario 1), comparing local SGD with $H = 4$ (A4) with mini-batch SGD of same effective batch size $B = 4 B _ { \\mathrm { l o c } }$ (A3) reveals a stark difference, both in terms of train and test error (local SGD achieves lower training loss and higher test accuracy). This motivates the use of local SGD as an alternative to large-batch training—a hypothesis that we confirm in our experiments. Further, in scenario 2), mini-batch SGD with smaller batch size $\\scriptstyle B = B _ { \\mathrm { l o c } }$ (A2) is observed to suffer from poor generalization, although the training curve matches the singlemachine baseline (A1). The generalization gap can thus not be explained as an optimization issue alone. Our proposed post-local SGD (A5) (defined by starting local SGD from the model obtained by large-batch SGD (A2) at epoch 150) closes this generalization gap with the single-machine baseline (A1) and is also more communication efficient than the mini-batch competitors. In direct comparison, post-local SGD is more communication-efficient than mini-batch SGD (while less than local SGD). It achieves better generalization performance than both these algorithms. ", + "bbox": [ + 173, + 789, + 825, + 887 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Contributions. Our main contributions can thus be summarized as follows: ", + "bbox": [ + 176, + 200, + 681, + 215 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Trade-offs in Local SGD: We provide the first comprehensive empirically study of the trade-offs in local SGD for deep learning—when varying the number of workers $K$ , number of local steps $H$ and mini-batch sizes—for both scenarios 1) on communication efficiency and 2) on generalization. ", + "bbox": [ + 178, + 220, + 826, + 262 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Post-local SGD: We propose post-local SGD, a simple but very efficient training scheme to address the current generalization issue of large-batch training. It allows us to scale the training to much higher number of parallel devices. Large batches trained by post-local SGD enjoy improved communication efficiency, while at the same time strongly outperforming most competing small and large batch baselines in terms of accuracy. Our empirical experiments on standard benchmarks show that post-local SGD can reach flatter minima than large-batch SGD on those problems. ", + "bbox": [ + 178, + 266, + 825, + 349 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 378, + 344, + 395 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The generalization gap in large-batch training. State-of-the-art distributed deep learning frameworks (Abadi et al., 2016; Paszke et al., 2017; Seide & Agarwal, 2016) resort to synchronized large-batch SGD training, allowing scaling by adding more computational units and performing data-parallel synchronous SGD with mini-batches divided between devices. Training with large batch size (e.g. batch size $> 1 0 ^ { 3 }$ on ImageNet) typically degrades the performance both in terms of training and test error (often denoted as generalization gap) (Chen & Huo, 2016; Li, 2017; Li et al., 2014; Keskar et al., 2017; Shallue et al., 2018; McCandlish et al., 2018; Golmant et al., 2018; Masters & Luschi, 2018). Goyal et al. (2017) argue that the test error degrades because of optimization issues and propose to use a “learning rate warm-up” phase with linear scaling of the step-size. You et al. (2017a) propose Layer-wise Adaptive Rate Scaling (LARS) to scale to larger mini-batch size, but the generalization gap does not vanish. Hoffer et al. (2017) argue that the generalization gap can be closed when increasing the number of iterations along with the batch size. However, this diminishes the efficiency gains of parallel training. ", + "bbox": [ + 173, + 412, + 825, + 592 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Keskar et al. (2017) empirically show that larger batch sizes correlate with sharper minima (Hochreiter & Schmidhuber, 1997) found by SGD and that flat minima are preferred for better generalization. This interpretation—despite being debated in Dinh et al. (2017)—was further developed in Hoffer et al. (2017); Yao et al. (2018); Izmailov et al. (2018). Neelakantan et al. (2015) propose to add isotropic white noise to the gradients to avoid over-fitting and better optimization. Zhu et al. (2019); Xing et al. (2018) further analyze “structured” anisotropic noise and highlight the importance of anisotropic noise (over isotropic noise) for improving generalization. The importance of the scale of the noise for non-convex optimization has also been studied in Smith & Le (2018); Chaudhari & Soatto (2018). Wen et al. (2019) propose to inject noise (sampled from the expensive empirical Fisher matrix) to large-batch SGD. However, to our best knowledge, none of the prior work (except our post-local SGD) can provide a computation efficient way to inject noise to achieve as good generalization performance as small-batch SGD, for both of CIFAR and ImageNet experiments. ", + "bbox": [ + 173, + 599, + 825, + 766 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Local SGD and convergence theory. While mini-batch SGD is very well studied (Zinkevich et al., 2010; Dekel et al., 2012; Takác et al., 2013), the theoretical foundations of local SGD variants ˇ are still developing. Jain et al. (2018) study one-shot averaging on quadratic functions and Bijral et al. (2016) study local SGD in the setting of a general graph of workers. A main research question is whether local SGD provides a linear speedup with respect to the number of workers $K$ , similar to mini-batch SGD. Recent work partially confirms this, under the assumption that $H$ is not too large compared to the total iterations $T$ . Stich (2019) and Patel & Dieuleveut (2019) show convergence at rate $\\mathcal { O } \\big ( ( K T H B _ { \\mathrm { l o c } } ) ^ { - 1 } \\big )$ on strongly convex and smooth objective functions when $H = \\mathcal { O } ( T ^ { 1 / 2 } )$ . For smooth non-convex objective functions, Zhou & Cong (2018) show a rate of $\\mathcal { O } \\big ( ( K T \\dot { B } _ { \\mathrm { l o c } } ) ^ { - 1 / 2 } \\big )$ (for the decrement of the stochastic gradient), Yu et al. (2019) give an improved result $\\mathcal { O } \\big ( ( H K T B _ { \\mathrm { l o c } } ) ^ { - 1 / 2 } \\big )$ when $H = \\mathcal { O } ( T ^ { 1 / 4 } )$ . For a discussion of more recent theoretical results we refer to (Kairouz et al., 2019). Alistarh et al. (2018) study convergence under adversarial delays. Zhang et al. (2016) empirically study the effect of the averaging frequency on the quality of the solution for some problem cases and observe that more frequent averaging at the beginning of the optimization can help. Similarly, Bijral et al. (2016) argue to average more frequently at the beginning. ", + "bbox": [ + 174, + 782, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 102, + 826, + 174 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 POST-LOCAL SGD AND HIERARCHICAL LOCAL SGD ", + "text_level": 1, + "bbox": [ + 174, + 193, + 650, + 210 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section we present two novel variants of local SGD. First, we propose post-local SGD to reach high generalization accuracy (cf. Section 4.2 below), and second, hierarchical SGD designed from a systems perspective aiming at optimal resource adaptivity (computation vs. communication trade-off). ", + "bbox": [ + 174, + 224, + 826, + 267 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Post-local SGD: Large-batch Training Alternative for Better Generalization. We propose postlocal SGD, a variant where local SGD is only started in the second phase of training, after $t ^ { \\prime }$ initial steps2 with standard mini-batch SGD. Formally, the update in (2) is performed with a iteration dependent $H _ { ( t ) }$ given as ", + "bbox": [ + 173, + 273, + 825, + 330 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/285e933457897d30275fe847e5eab32fdb90ea8c4740141721908a4bdebc4a2b.jpg", + "text": "$$\nH _ { ( t ) } = \\left\\{ { \\begin{array} { l l } { 1 , \\quad { \\mathrm { i f ~ } } t \\leqslant t ^ { \\prime } , } \\\\ { H , \\quad { \\mathrm { i f ~ } } t > t ^ { \\prime } . } \\end{array} } \\right. \\qquad \\quad \\qquad ( m i n i - b a t c h S G D )\n$$", + "text_format": "latex", + "bbox": [ + 259, + 332, + 629, + 367 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(post-local SGD) ", + "bbox": [ + 709, + 342, + 821, + 356 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As the proposed scheme is identical to mini-batch SGD in the first phase (with local batch size $\\boldsymbol { B } = \\boldsymbol { B } _ { \\mathrm { l o c } } ,$ ), we can leverage previously tuned learning rate warm-up strategies and schedules for large-batch training (Goyal et al., 2017) without additional tuning. Note that we only use ‘small’ local mini-batches of size $B = B _ { \\mathrm { l o c } }$ in the warm-up phase, and switch to the communication efficient larger effective batches $H B _ { \\mathrm { l o c } }$ in the second phase, while also achieving better generalization in our experiments (cf. also the discussion in Section 5). We would like to point out that it is crucial to use local SGD in the second phase, as e.g. just resorting to large batch training does achieve worse performance (see e.g. 3rd row in Table 2 below). ", + "bbox": [ + 173, + 372, + 825, + 484 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Hierarchical Local SGD: Optimal Use of Systems Resources in Heterogeneous Systems. Real world systems come with different communication bandwidths on several levels, e.g. with GPUs or other accelerators grouped hierarchically within a chip, machine, rack or even at the level of entire data-centers. In this scenario, we propose to employ local SGD as an inner loop on each level of the hierarchy, adapted to the corresponding computation vs communication trade-off of that particular level. The resulting scheme, hierarchical local SGD, can offer significant benefits in terms of system adaptivity and performance, as we show with experiments and a discussion in Appendix D. ", + "bbox": [ + 174, + 491, + 825, + 589 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 EXPERIMENTAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 608, + 418, + 625 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section we systematically evaluate the aforementioned variants of local SGD on deep learning tasks. In summary, the main findings presented in Sections 4.1 and 4.2 below are: ", + "bbox": [ + 171, + 640, + 823, + 669 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Local SGD, on the one hand, can serve as a communication-efficient alternative to mini-batch SGD for different practical purposes (communication restricted Scenario 1). For example in Figure 2(a) (with fixed $\\scriptstyle B _ { \\mathrm { l o c } }$ and $K$ ), in terms of $9 2 . 4 8 \\%$ best test accuracy achieved by our mini-batch SGD implementation for $K = 1 6$ , practitioners can either choose to achieve reasonable good training quality $( 9 1 . 2 \\%$ , matching (He et al., 2016a)) with a $2 . 5 9 \\times$ speedup (time-to-accuracy) in training time $( H = 8 )$ ), or achieve slight better test performance $( 9 2 . 5 7 \\% )$ ) with slightly reduced communication efficiency $1 . 7 6 \\times$ speedup for $H = 2$ ). ", + "bbox": [ + 173, + 675, + 825, + 773 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Post-local SGD, on the other hand, in addition to overcoming communication restrictions does provide a state-of-the-art remedy for the generalization issue of large-batch training (Scenario 2). Unlike local SGD with large $H$ and $K$ (e.g. $H = 1 6 , K = 1 6 )$ which can encounter optimization issues during initial training (which can impact later generalization performance), we found that post-local SGD elegantly enables an ideal trade-off between optimization and generalization within a fixed training budget (e.g. Table 3, Table 5, Figure 3 and many others in Appendix C.5), and can achieve the same or even better performance than small mini-batch baselines. ", + "bbox": [ + 173, + 780, + 825, + 851 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Setup. We briefly outline the general experimental setup, and refer to Appendix A for full details. ", + "bbox": [ + 174, + 151, + 821, + 165 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Datasets. We evaluate all methods on the following two main (standard) tasks: (1) Image classification for CIFAR-10/100 (Krizhevsky & Hinton, 2009), and (2) Image classification for ImageNet (Russakovsky et al., 2015). The detailed data augmentation scheme refers to Appendix A. ", + "bbox": [ + 174, + 171, + 825, + 214 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Models. We use ResNet-20 (He et al., 2016a) with CIFAR-10 as a base configuration to understand different properties of (post-)local SGD. We then empirical evaluate the large-batch training performance of post-local SGD, for ResNet-20, DensetNet-40-12 (Huang et al., 2017b) and WideResNet28-10 (Zagoruyko & Komodakis, 2016) on CIFAR-10/100. Finally, we train ResNet-50 (He et al., 2016a) on ImageNet to investigate the accuracy and scalability of (post-)local SGD training. ", + "bbox": [ + 174, + 220, + 825, + 291 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Implementation and platform. Our algorithms are implemented3 in PyTorch (Paszke et al., 2017), with a flexible configuration of the machine topology supported by Kubernetes. The cluster consists of Intel Xeon E5-2680 v3 servers and each server has 2 NVIDIA TITAN Xp GPUs. We use the notion $a \\times b$ -GPU to denote the topology of the cluster, i.e., $a$ nodes and each with $b$ GPUs. ", + "bbox": [ + 174, + 297, + 825, + 354 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Specific learning schemes for large-batch SGD. We rely on the recently proposed schemes for efficient large batch training (Goyal et al., 2017), which are formalized by (i) linearly scaling the learning rate w.r.t. the global mini-batch size; (ii) gradual warm-up of the learning rate from a small value. See Appendix A.3 for more details. ", + "bbox": [ + 174, + 361, + 825, + 416 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Distributed training procedure on CIFAR-10/100. The experiments follow the common mini-batch SGD training scheme for CIFAR (He et al., 2016a;b; Huang et al., 2017b) and all competing methods access the same total number of data samples (i.e. gradients) regardless of the number of local steps. Training ends when the distributed algorithms have accessed the same number of samples as the single-worker baseline. The data is disjointly partitioned and reshuffled globally every epoch. The learning rate scheme follows (He et al., 2016a; Huang et al., 2017b), where we drop the initial learning rate by a factor of 10 when the model has accessed $5 0 \\%$ and $7 5 \\%$ of the total number of training samples. Unless mentioned specifically, the used learning rate is scaled by the global mini-batch size $B K$ for mini-batch SGD and $B _ { \\mathrm { l o c } } K$ for local SGD) where the initial learning rate is fine-tuned for each model and each task for the single worker. See Appendix A.4 for more details. ", + "bbox": [ + 174, + 424, + 825, + 563 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 SUPERIOR SCALABILITY OF LOCAL SGD OVER MINI-BATCH SGD ", + "text_level": 1, + "bbox": [ + 176, + 583, + 671, + 598 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "First, we empirically study local SGD training for the communication restricted case (i.e. Scenario 1). As local SGD (with local batch size $B _ { \\mathrm { l o c . } }$ ) needs $H$ times fewer communication rounds than minibatch SGD (with the same batch size $\\begin{array} { r } { B = B _ { \\mathrm { l o c } } } \\end{array}$ q we expect local SGD to significantly outperform mini-batch SGD in terms of time-to-accuracy and scalability and verify this experimentally. We further observe that local SGD shows better generalization performance than mini-batch SGD. ", + "bbox": [ + 174, + 611, + 825, + 681 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Significantly better scalability when increasing the number of workers on CIFAR, in terms of time-to-accuracy. Figure 1 demonstrates the speedup in time-to-accuracy for training ResNet-20 for CIFAR-10, with varying number of GPUs $K$ and the number of local steps $H$ , both from 1 to 16. ", + "bbox": [ + 173, + 699, + 826, + 742 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/ffb0e8260d462d480f2cea8a935d85617c75c8f5acbfc16ef79217e425e4d386.jpg", + "image_caption": [ + "Table 1: Scaling behavior of local SGD in clock-time for increasing number of workers $K$ , for different number of local steps $H$ , for training ResNet-20 on CIFAR-10 with $B _ { \\mathrm { l o c } } = 1 2 8$ . The reported speedup (averaged over three runs) is over single GPU training time for reaching the baseline top-1 test accuracy $( 9 1 . 2 \\%$ as in (He et al., 2016a)). We use a $8 \\times 2$ -GPU cluster with 10 Gbps network. $H = 1$ recovers mini-batch SGD. " + ], + "image_footnote": [], + "bbox": [ + 232, + 762, + 486, + 891 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We demonstrate in Figure 1 that local SGD scales $2 \\times$ better than its mini-batch SGD counterpart, in terms of time-to-accuracy as we increase the number of workers $K$ on a commodity cluster. The local update steps $( H )$ result in a strong advantage over the standard large-batch training. Mini-batch SGD fixes the batch size to $\\boldsymbol { B } = B _ { \\mathrm { l o c } }$ , and while increasing the number of workers $K$ gets impacted by the communication bottleneck (section 1), even as parallelism per device remains unchanged. In this experiment, local SGD on 8 GPUs even achieves a $2 \\times$ lower time-to-accuracy than mini-batch SGD with 16 GPUs. Moreover, the (near) linear scaling performance for $H = 8$ in Figure 1, shows that the main hyper-parameter $H$ of local SGD is robust and consistently different from its mini-batch counterpart, when scaling the number of workers. ", + "bbox": [ + 173, + 103, + 825, + 229 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Effectiveness and scalability of local SGD to even larger datasets (e.g., ImageNet) and larger clusters. Local SGD presents a competitive alternative to the current large-batch ImageNet training methods. Figure 8 in Appendix B.3.2 shows that we can efficiently train state-of-the-art ResNet-50 (at least $1 . 5 \\times$ speedup to reach $7 5 \\%$ top-1 accuracy) for ImageNet (Goyal et al., 2017; You et al., 2017a) via local SGD on a $1 6 \\times 2$ -GPU cluster. ", + "bbox": [ + 174, + 247, + 825, + 316 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Local SGD significantly outperforms mini-batch SGD at the same effective batch size and communication ratio. Figure 2 compares local SGD with mini-batch SGD of the same effective batch size, that is the same number of gradient computations per communication round as described in Scenario 1 above (Section 1). ", + "bbox": [ + 173, + 335, + 825, + 391 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/915c1d109060a8d68f5bd0bd854005ce56675f41858a11b9ab53fd3bff0bc03f.jpg", + "image_caption": [ + "(a) Top-1 test accuracy of local SGD. $H = 1$ is mini-batch SGD with optimal hyper-parameters. " + ], + "image_footnote": [], + "bbox": [ + 176, + 410, + 372, + 508 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/54d9ed9579b73f9cfd5fac7ed167e9bc812855c68ba6f9ad3ea362cb26eca9bf.jpg", + "image_caption": [ + "(b) Top-1 test accuracy of local SGD (circle) vs. mini-batch SGD (triangle), with same effective batch size $B = H B _ { \\mathrm { l o c } }$ per worker. The points for a given $K$ have the same communication cost. " + ], + "image_footnote": [], + "bbox": [ + 398, + 414, + 813, + 508 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 2: Training ResNet-20 on CIFAR-10 under different $K$ and $H$ , with fixed $B _ { \\mathrm { l o c } } = 1 2 8$ . All results are averaged over three runs and all settings access to the same total number of training samples. We fine-tune the learning rate of mini-batch SGD for each setting. ", + "bbox": [ + 173, + 569, + 823, + 608 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 (POST)-LOCAL SGD CLOSES THE GENERALIZATION GAP OF LARGE-BATCH TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 638, + 816, + 654 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Even though local SGD has demonstrated effective communication efficiency with guaranteed test performance (Scenario 1), we observed in the previous section that it still encounters difficulties when scaling to very large mini-batches (Figure 2(a)), though it is much less affected than mini-batch SGD (Figure 2(b)). Post-local SGD can address the generalization issue of large batch training (Scenario 2) as we show now. First, we would like to highlight some key findings in Table 2. ", + "bbox": [ + 174, + 666, + 825, + 736 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/9961f5efe990d6c424f3e6d6060d473c77b3238dfbe0d5b626f102a85adc1f92.jpg", + "table_caption": [ + "Table 2: Test performance (generalization) for local SGD variants and mini-batch SGD (highlighting selected data from Figure 2, Table 3 and Figure 3). Mini-batch SGD suffers from the generalization gap (and sometimes even from optimization issues due to insufficient training) when scaling to larger $H$ and $K$ . Same experimental setup as in Figure 2 (fine-tuned mini-batch SGD baselines). The $\\hookrightarrow$ denotes the transition at the first learning rate decay. The reported results are averaged over three runs. " + ], + "table_footnote": [], + "table_body": "
AlgorithmTop-1 acc.Effect. batch sizeAlgorithmTop-1 acc.Effect. batch size
Mini-batch SGD (K=4)92.6%KB=512Mini-batch SGD (K =16)92.5%KB=2048
Mini-batch SGD (K =4)89.5%KB=8192Mini-batch SGD (K =16)76.3%KB=16384
Mini-batch SGD (K=4)92.5%KB=512→8192Mini-batch SGD(K=16)92.0%KHBloc=2048→ 16384
Local SGD(H=16,K=4)92.5%KHBloc =8192Local SGD(H=8,K=16)92.0%KHBloc =16384
Post-local SGD(H=16,K=4)92.7%KHBloc =512→8192Post-local SGD (H=8,K=16)92.9%KHBloc =2048→ 16384
", + "bbox": [ + 197, + 828, + 808, + 921 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We now focus on the generalization issues (Section 1) of large-batch SGD, isolating the potential negative impact from the optimization difficulty (e.g. the insufficient training epochs (Shallue et al., 2018)) from the performance on the test set. Our evaluation starts from a (standard) constant effective mini-batch size of 2048 or 4096 for mini-batch $\\mathrm { S G D ^ { 4 } }$ on CIFAR dataset, as in (Keskar et al., 2017; Hoffer et al., 2017; Yao et al., 2018). ", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Post-local SGD generalizes better and faster than mini-batch SGD. Table 3 summarizes the generalization performance of post-local SGD on large batch size $( K = 1 6 , B _ { \\mathrm { l o c } } = 1 2 8 )$ across different architectures on CIFAR tasks for $H = 1 6$ and $H = 3 2$ . Under the same setup, Table 9 in the Appendix C.3 evaluates the speedup of training, while Figure 1 demonstrates the learning curves of mini-batch SGD and post-local SGD, highlighting the generalization difficulty of large-batch SGD. We can witness that post-local SGD achieves at least $1 . 3 \\times$ speedup over the whole training procedure compared to the mini-batch SGD counterpart ( $K = 1 6$ , $B = 1 2 8$ ), while enjoying the significantly improved generalization performance. ", + "bbox": [ + 173, + 190, + 825, + 303 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/b657be97f18a24df66634a05d7f9bb9d40ddb71d106b465dca9b08565a2f2a1a.jpg", + "table_caption": [ + "Table 3: Top-1 test accuracy of training different CNN models via post-local SGD on $K = 1 6$ GPUs with a large global batch size $K B _ { \\mathrm { l o c } } = 2 0 4 8$ ). The reported results are the average of three runs and all settings access to the same total number of training samples. We compare to small and large mini-batch baselines. The $\\star$ indicates a fine-tuned learning rate, where the tuning procedure refers to Appendix A.4. " + ], + "table_footnote": [], + "table_body": "
CIFAR-10CIFAR-100
Small batch baseline*Large batch baseline * K=2,B=128K=16,B=128 K=16,Bloc=128K=16,Blc =128Post-local SGD (H=16)Post-local SGD (H=32)Small batch baseline *Large batch baseline *Post-local SGD (H=16)Post-local SGD (H=32) K=2,B=128K=16,B= 128K=16,Blo =128 K= 16,Bloc=128
ResNet-2092.63 ±0.2692.48 3±0.1792.80 ±0.1693.02 ±0.2468.84 ±0.0668.17 ±0.1869.24 ±0.2669.38 ±0.20
DenseNet-40-1294.41 ±0.1494.36 ±0.2094.43 ±0.1294.58 ±0.1174.85 ±0.1474.08 3±0.4674.45 ±0.3075.03 ±0.05
WideResNet-28-1095.89 ±0.1095.43 ±0.3795.94 ±0.0695.76 ±0.2579.78 3±0.1679.31 ±0.2380.28 ±0.1380.65 ±0.16
", + "bbox": [ + 176, + 381, + 825, + 474 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We further demonstrate the generalization performance and the scalability of post-local SGD, for diverse tasks (e.g., Language Modeling), and for even larger global batch sizes $K B _ { \\mathrm { l o c } } = 4 0 9 6$ for CIFAR-100 and 4096 and 8192 respectively for ImageNet), in Appendix C.5 . For example, Table 11 presents the severe generalization issue $2 \\%$ drop) of the fine-tuned large-batch SGD training $\\mathit { K B } = 4 0 9 6 )$ ) for above three CNNs on CIFAR-100, and cannot be addressed by increasing the training steps (Table 12). Post-local SGD $K B _ { \\mathrm { l o c } } = 4 0 9 6 $ ) with default hyper-parameters can perfectly close this generalization gap or even better than the fine-tuned small mini-batch baselines5. For ImageNet training in Figure 16, the post-local SGD outperforms mini-batch SGD baseline for both of $K B = 4 0 9 6$ (76.18 and 75.87 respectively) and $K B = 8 1 9 2$ (75.65 and 75.64 respectively) with $1 . 3 5 \\times$ speedup for the post-local SGD training phase. ", + "bbox": [ + 173, + 486, + 825, + 625 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The effectiveness of post-local SGD training for different $H$ and $K$ . As seen in Figure 3(a), applying any number of local steps over the case of large-batch training (when $K B _ { \\mathrm { l o c } } = 2 0 4 8 )$ improves the generalization performance compared to mini-batch SGD. Figure 3(b) illustrates that post-local SGD is better than mini-batch SGD in general for different number of workers $K$ (as well as different $K B _ { \\mathrm { l o c . } }$ ). Thus, post-local SGD presents consistently excellent generalization performance. ", + "bbox": [ + 174, + 642, + 825, + 712 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Post-local SGD can improve the training efficiency upon other compression techniques. The results in Table 4 illustrate that post-local SGD can be combined with other communication-efficient techniques (e.g. the sign-based compression schemes) for further improved communication efficiency and better test generalization performance. ", + "bbox": [ + 174, + 729, + 825, + 785 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Post-local SGD can improve upon other SOTA optimizers. The benefits of post-local SGD training on ImageNet are even more pronounced for larger batches (e.g. $K B _ { \\mathrm { l o c } } > 8 1 9 2$ ) (Goyal et al., ", + "bbox": [ + 173, + 803, + 823, + 832 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/12e9053ce26a7322e78a113c4ec510ecb64763022dfc66e574fdc1625ed4ce5b.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 178, + 98, + 450, + 236 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(a) Performance of post-local SGD for different numbers of local steps $H$ on 16 GPUs. $H = 1$ is mini-batch SGD and $K B _ { \\mathrm { l o c } } = 2 0 4 8$ . ", + "bbox": [ + 173, + 241, + 460, + 280 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/566dd196bc28a309c7958315b78d2971a03b5087a34eeb488fae4dcd6f363d8f.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 539, + 98, + 810, + 236 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "(b) Performance of post-local SGD for different numbers of workers $K$ for $H = 1 6$ and $H = 3 2$ . The local batch size is fixed to $B _ { \\mathrm { l o c } } = B = 1 2 8$ both for local and mini-batch SGD). ", + "bbox": [ + 532, + 241, + 825, + 292 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/a791272d088ca150f63351c5c95d647c8185395a02e3235ae2509f801d37f987.jpg", + "table_caption": [ + "Figure 3: Top-1 test accuracy of training ResNet-20 on CIFAR-10. Box-plot figures are derived from 3 runs. Table 4: Top-1 test accuracy of training ResNet-20 on CIFAR via sign-based compression scheme and postlocal SGD $K B _ { \\mathrm { l o c } } = 2 0 4 8$ and $K = 1 \\bar { 6 }$ ). $H = 1$ in the table for the simplicity corresponds to the original sign-based algorithm and we fine-tune their hyper-parameters. The reported results are the average of three runs with different seeds. The learning rate schedule is fixed for different local update steps (for $H > 1$ ). For the detailed tuning procedure, training scheme, as well as the pseudo code of the used variants, we refer to Appendix C.5.5. " + ], + "table_footnote": [], + "table_body": "
CIFAR-10 H=16CIFAR-100
H=1H =32H =64H=1H =16H=32H =64
signSGD (Bernstein etal.,018)91.61 ±0.2891.77±0.0492.22±0.1192.00 ±0.1567.21 ±0.1167.43 ±0.3668.32 ±0.1868.12 ±0.11
EF-signSGD(Karimireddy etal.,2019)92.45 ±0.2892.63±0.1692.72±0.0592.76 ±0.0668.19 ±0.1069.14 ±0.3669.00 ±0.1469.49 ±0.21
", + "bbox": [ + 173, + 390, + 825, + 431 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "2017; Shallue et al., 2018; Golmant et al., 2018). Considering other optimizers, Table 5 below shows that post-local SGD can improve upon LARS6 (You et al., 2017a). ", + "bbox": [ + 173, + 457, + 820, + 486 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/58c6581e6790d4d30c422d04bfb00c99a1eccb26b47a045836149a3b9fc90d96.jpg", + "table_caption": [ + "Table 5: The Top-1 test accuracy of training ResNet50 on ImageNet. We report the performance (w/ or w/o post-local SGD) achieved by using SGD with Nesterov Momentum, learning schemes in (Goyal et al., 2017), and LARS. We follow the general experimental setup described in Section A.4.2, and use $K = 3 2$ and $H = 4$ . " + ], + "table_footnote": [], + "table_body": "
SGD + Momentum + LARSSGD + Momentum + LARS + Post-local SGD
KBloc=8,19275.9976.13
KBloc=16,38474.5275.23
", + "bbox": [ + 232, + 539, + 759, + 583 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 DISCUSSION AND INTERPRETATION ", + "text_level": 1, + "bbox": [ + 173, + 612, + 501, + 628 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Generalization issues of large-batch SGD training are not yet very well understood from a theoretical perspective. Hence, a profound theoretical study of the generalization local SGD is beyond the scope of this work but we hope the favorable experimental results will trigger future research in this direction. In this section we discuss some related work and we argue that local SGD can be seen as a way to inject and control stochastic noise to the whole training procedure. ", + "bbox": [ + 173, + 640, + 825, + 710 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Connecting Local Updates with Stochastic Noise Injection. The update eq. (1) can alternatively be written as ", + "text_level": 1, + "bbox": [ + 174, + 727, + 823, + 753 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/8a92b403bef073c2be26fc7858cc3b1015d0eaef3f238fceefc63691d7bc22c3.jpg", + "text": "$$\n{ \\pmb w } _ { t + 1 } = { \\pmb w } _ { t } - \\gamma \\hat { \\pmb g } _ { t } = { \\pmb w } _ { t } - \\gamma { \\pmb g } _ { t } + \\gamma \\big ( { \\pmb g } _ { t } - \\hat { \\pmb g } _ { t } \\big ) = { \\pmb w } _ { t } - \\gamma { \\pmb g } _ { t } + \\gamma \\epsilon ,\n$$", + "text_format": "latex", + "bbox": [ + 279, + 753, + 717, + 775 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\hat { \\pmb g } _ { t } : = \\frac { 1 } { B } \\sum _ { i } \\nabla f _ { i } ( { \\pmb w } _ { t } ) } \\end{array}$ , $\\pmb { g } _ { t } : = \\mathbb { E } \\hat { \\pmb { g } } _ { t } = \\nabla f ( \\pmb { w } _ { t } )$ and $\\boldsymbol { \\epsilon } : = \\boldsymbol { g } _ { t } - \\hat { \\boldsymbol { g } } _ { t }$ . For most data-sets the noise $\\epsilon$ can be well approximated by a zero-mean Gaussian with variance $\\Sigma ( w )$ . The variance matrix $\\Sigma ( w )$ , following Hoffer et al. (2017) and Hu et al. (2017), for uniform sampling of mini-batch indices can be approximated as ", + "bbox": [ + 174, + 781, + 825, + 839 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/018f2efc0f9f511c15dc06d162e104905600abbcb317df0e833725d695d5ddd1.jpg", + "text": "$$\n\\begin{array} { r } { \\Sigma ( \\pmb { w } ) \\approx \\left( \\frac { 1 } { B } - \\frac { 1 } { N } \\right) \\pmb { K } ( \\pmb { w } ) = \\left( \\frac { 1 } { B } - \\frac { 1 } { N } \\right) \\left( \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\nabla f _ { i } ( \\pmb { w } ) \\nabla f _ { i } ( \\pmb { w } ) ^ { \\top } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 263, + 847, + 735, + 873 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Recent works (Jastrz˛ebski et al., 2018; Li et al., 2017; Mandt et al., 2017; Zhu et al., 2019) interpret eq. (3) as an Euler-Maruyama approximation to the continuous-time Stochastic Differential Equation (SDE): $d \\pmb { w } _ { t } = - \\pmb { g } _ { t } d t + \\sqrt { \\gamma \\frac { N - B } { B N } } \\pmb { R } ( \\pmb { w } ) d \\pmb { \\theta } ( t )$ where $\\begin{array} { r } { R ( w ) R ( w ) ^ { \\top } = K ( w ) } \\end{array}$ and $\\pmb \\theta ( t ) \\sim \\mathcal { N } ( 0 , \\pmb I )$ . When $B \\ll N$ (in eq. (4)), the learning rate and the batch size only appear in the SDE through the ratio $\\rho : = \\gamma B ^ { - 1 }$ . Jastrz˛ebski et al. (2018) argue that thus mainly the ratio $\\rho$ controls the stochastic noise and training dynamics, and that $\\rho$ positively correlates with wider minima and better generalization— matching with recent experimental successes of ImageNet training (Goyal et al., 2017). ", + "bbox": [ + 173, + 103, + 826, + 210 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This explanation fails in the large batch regime. When the batch size grows too large, the relative magnitude of the stochastic noise decreases as $\\gamma B ^ { - 1 } \\not \\approx \\gamma ( N - B ) \\bar { / } B N$ , i.e. $\\rho$ is not anymore uniquely describing the training dynamics for large batch size and small dataset size when $B \\ll N$ (Jastrz˛ebski et al., 2018). This could be a reason why the generalization difficulty of large-batch training remains, as clearly illustrated e.g. in Figure 1, Table 3, Figure $3 ( \\mathbf { b } )$ , and as well as in (Shallue et al., 2018; Golmant et al., 2018). ", + "bbox": [ + 173, + 217, + 825, + 301 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The local update step of local SGD is a natural and computation-free way to inject well-structured stochastic noise to the SGD training dynamics. By using the same ratio $\\frac { \\gamma } { B }$ as mini-batch SGD, the main difference of the local SGD training dynamics comes from the stochastic noise $\\epsilon$ during the local update phase ( $K$ times smaller local mini-batch with $H$ local update steps). The $\\epsilon$ will be approximately sampled with the variance matrix $K \\pmb { \\Sigma } ( \\pmb { w } )$ instead of $\\Sigma ( w )$ in mini-batch SGD, causing the stochastic noise determined by $K$ and $H$ to increase. This could be one of the reasons for post-local SGD to generalize as good as mini-batch SGD (small mini-batch size) and leading to flatter minima than large-batch SGD in our experiments. Similar positive effects of adding well-structured stochastic noise to SGD dynamics on non-convex problems have recently also been observed in (Smith & Le, 2018; Chaudhari & Soatto, 2018; Zhu et al., 2019; Xing et al., 2018; Wen et al., 2019). ", + "bbox": [ + 173, + 308, + 825, + 446 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.1 POST-LOCAL SGD CONVERGES TO FLATTER MINIMA ", + "text_level": 1, + "bbox": [ + 173, + 468, + 580, + 482 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Figure 4(a) evaluates the spectrum of the Hessian for different local minima. We observe that largebatch SGD tends to get stuck at locations with high Hessian spectrum while post-local SGD tends to prefer low curvature solutions with better generalization error. Figure 4(b) linearly interpolates two minima obtained by mini-batch SGD and post-local SGD, and Figure 13 (in the Appendix) visualizes the sharpness of the model trained by different methods. These results give evidence that post-local SGD converges to flatter minima than mini-batch SGD for training ResNet-20 on CIFAR-10. ", + "bbox": [ + 173, + 492, + 826, + 575 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/ca6f0fbe5c2cca0d0b89a95e97cd636645e6fb91abaa800b2732cedcc3fbe483.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 181, + 592, + 473, + 739 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/ea6cb72160f8f995ad5fc3b3bfc4a15494b305c0c53298badf63fca80a31071f.jpg", + "image_caption": [ + "(b) 1-d linear interpolation between models wpost-local SGD $\\lambda = 0$ ) and ${ \\pmb w } _ { \\mathrm { m i n i } }$ -batch SGD $\\lambda = 1 ,$ ), i.e., $\\hat { \\pmb { w } } = \\lambda \\pmb { w } _ { \\mathrm { m i n i } }$ -batch $_ \\mathrm { \\{ G D + } ( 1 - \\lambda ) w$ post-local SGD. The solid lines correspond to evaluate $\\hat { \\pmb { w } }$ on the whole training dataset while the dashed lines are on the test dataset. The model parameters only differ from the post-local SGD phase. " + ], + "image_footnote": [], + "bbox": [ + 519, + 589, + 813, + 738 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "(a) The dominant eigenvalue of the Hessian, for model trained from different schemes. It is evaluated on the test dataset per epoch, and only the phase related to the post-local SGD strategy is visualized. Mini-batch SGD or post-local SGD with small $H$ (e.g., $H = 2 , 4$ ) have noticeably larger dominant eigenvalue. ", + "bbox": [ + 174, + 746, + 486, + 821 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Figure 4: Understanding the generalization ability of post-local SGD for large-batch training (ResNet-20 on CIFAR-10 with $B K = \\bar { B } _ { \\mathrm { l o c } } \\bar { K } = 2 0 4 8 )$ . We use fixed $B = B _ { \\mathrm { l o c } } = 1 2 8$ with $K = 1 6$ GPUs. The detailed experimental setup as well as more visualization of results are available in Appendix C.4. ", + "bbox": [ + 173, + 843, + 821, + 882 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 318, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We leverage the idea of local SGD for training in distributed and heterogeneous environments. Ours is the first work to extensively study the trade-off between communication efficiency and generalization performance of local SGD. Our local SGD variant, called post-local SGD, not only outperforms large-batch SGD’s generalization performance but also matches that of small-batch SGD. In our experiments post-local SGD converged to flatter minima compared to traditional large-batch SGD, which partially explains its improved generalization performance. We also provide extensive experiments with another variant hierarchical local SGD, showing its adaptivity to available system resources. Overall, local SGD comes off as a simpler and more efficient algorithm, replacing complex ad-hoc tricks used for current large-batch SGD training. ", + "bbox": [ + 174, + 133, + 825, + 258 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 280, + 367, + 294 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The authors thank the anonymous reviewers and Thijs Vogels for their precious comments and feedback. We acknowledge funding from SNSF grant 200021_175796, as well as a Google Focused Research Award. ", + "bbox": [ + 176, + 310, + 825, + 352 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 372, + 285, + 387 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. 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NIPS - Advances in Neural Information Processing Systems 23, 2010. ", + "bbox": [ + 173, + 731, + 823, + 758 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Supplementary Material ", + "text_level": 1, + "bbox": [ + 173, + 98, + 532, + 127 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For easier navigation through the paper and the extensive appendix, we include a table of contents. ", + "bbox": [ + 173, + 147, + 818, + 162 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "CONTENTS ", + "text_level": 1, + "bbox": [ + 174, + 180, + 269, + 198 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "1 Introduction 1 ", + "text_level": 1, + "bbox": [ + 173, + 213, + 823, + 232 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "2 Related Work 3 ", + "bbox": [ + 173, + 246, + 821, + 261 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "3 Post-Local SGD and Hierarchical Local SGD 4 ", + "bbox": [ + 171, + 276, + 823, + 291 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "4 Experimental Results 4 \n4.1 Superior Scalability of Local SGD over mini-batch SGD 5 \n4.2 (Post)-Local SGD closes the Generalization Gap of Large-batch Training 6 ", + "bbox": [ + 173, + 309, + 826, + 364 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "5 Discussion and Interpretation 8 ", + "text_level": 1, + "bbox": [ + 173, + 381, + 826, + 395 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "5.1 Post-local SGD converges to flatter minima . . 9 ", + "bbox": [ + 184, + 401, + 826, + 416 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "6 Conclusion 10 ", + "bbox": [ + 173, + 431, + 823, + 446 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A Details on Deep Learning Experimental Setup 15 ", + "text_level": 1, + "bbox": [ + 176, + 463, + 823, + 478 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 Dataset . . 15 \nA.2 Models and Model Initialization . 15 \nA.3 Large Batch Learning Schemes 16 \nA.4 Hyperparameter Choices and Training Procedure, over Different Models/Datasets 16 \nA.4.1 CIFAR-10/CIFAR-100 16 \nA.4.2 ImageNet 17 \nA.5 System Performance Evaluation . 17 ", + "bbox": [ + 194, + 483, + 826, + 621 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B Local SGD Training 18 ", + "text_level": 1, + "bbox": [ + 173, + 637, + 823, + 651 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B.1 Formal Definition of the Local SGD Algorithm 18 \nB.2 Numerical Illustration of Local SGD on a Convex Problem . 18 \nB.3 More Results on Local SGD Training . . 19 \nB.3.1 Training CIFAR-10 via local SGD 19 \nB.3.2 Training ImageNet via Local SGD 20 \nB.3.3 Local SGD Scales to Larger Batch Sizes than Mini-batch SGD 21 \nB.4 Practical Improvement Possibilities for Standard Local SGD Training 21 \nB.4.1 Local SGD with Momentum 21 \nB.4.2 Warm-up of the Number of Local SGD Steps . . 23 ", + "bbox": [ + 194, + 656, + 826, + 833 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C Post-local SGD Training 26 ", + "text_level": 1, + "bbox": [ + 173, + 848, + 825, + 863 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C.1 The Algorithm of Post-local SGD 26 \nC.2 The Effectiveness of Turning on Post-local SGD after the First Learning Rate Decay 27 \nC.3 The Speedup of Post-local SGD Training on CIFAR 27 \nC.4 Understanding the Generalization of Post-local SGD 28 \nC.5 Post-local SGD Training on Diverse Tasks 28 \nC.5.1 Post-local SGD Training on CIFAR-100 for Global Mini-batch Size $K B _ { \\mathrm { l o c } } = 4 0 9 6$ . . 28 \nC.5.2 Post-local SGD Training on Language Modeling . 29 \nC.5.3 Post-local SGD Training on ImageNet 31 \nC.5.4 Post-local SGD VS. Other Noise Injection Methods 31 \nC.5.5 Post-local SGD with Other Compression Schemes 32 ", + "bbox": [ + 196, + 869, + 826, + 925 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 199, + 102, + 825, + 239 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D Hierarchical Local SGD 35 ", + "text_level": 1, + "bbox": [ + 174, + 255, + 825, + 270 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D.1 The Illustration of Hierarchical Local SGD 35 \nD.2 The Algorithm of Hierarchical Local SGD 36 \nD.3 Hierarchical Local SGD Training 38 \nD.3.1 The Performance of Hierarchical Local SGD. 38 ", + "bbox": [ + 194, + 276, + 826, + 352 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "E Communication Schemes 39", + "bbox": [ + 174, + 367, + 826, + 382 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "F Discussion and Future Work 40 ", + "bbox": [ + 174, + 400, + 823, + 414 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A DETAILS ON DEEP LEARNING EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 434, + 656, + 450 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1 DATASET ", + "text_level": 1, + "bbox": [ + 176, + 464, + 282, + 479 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We use the following tasks. ", + "bbox": [ + 176, + 491, + 352, + 506 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• Image classification for CIFAR-10/100 (Krizhevsky & Hinton, 2009). Each consists of a training set of 50K and a test set of 10K color images of $3 2 \\times 3 2$ pixels, as well as 10 and 100 target classes respectively. We adopt the standard data augmentation scheme and preprocessing scheme (He et al., 2016a; Huang et al., 2016). For preprocessing, we normalize the data using the channel means and standard deviations. \nImage classification for ImageNet (Russakovsky et al., 2015). The ILSVRC 2012 classification dataset consists of 1.28 million images for training, and 50K for validation, with 1K target classes. We use ImageNet-1k (Deng et al., 2009) and adopt the same data preprocessing and augmentation scheme as in He et al. (2016a;b); Simonyan & Zisserman (2015). The network input image is a $2 2 4 \\times 2 2 4$ pixel random crop from augmented images, with per-pixel mean subtracted. Language Modeling for WikiText-2 (Merity et al., 2017). WikiText-2 is sourced from curated Wikipedia articles. It is frequently used for machine translation and language modelling, and features a vocabulary of over 30, 000 words. Compared to the preprocessed version of Penn Treebank (PTB), WikiText-2 is over 2 times larger. ", + "bbox": [ + 176, + 516, + 826, + 713 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2 MODELS AND MODEL INITIALIZATION ", + "text_level": 1, + "bbox": [ + 176, + 729, + 486, + 744 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We use ResNet-20 (He et al., 2016a) with CIFAR-10 as a base configuration to understand different properties of (post-)local SGD. We then empirically evaluate the large-batch training performance of post-local SGD, for ResNet-20, DensetNet-40-12 (Huang et al., 2017b) and WideResNet-28- 10 (Zagoruyko & Komodakis, 2016) on CIFAR-10/100, and for LSTM on WikiText-2 (Merity et al., 2017). Finally, we train ResNet-50 (He et al., 2016a) on ImageNet to investigate the accuracy and scalability of (post-)local SGD training. ", + "bbox": [ + 174, + 756, + 825, + 840 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "For the weight initialization we follow Goyal et al. (2017), where we adopt the initialization introduced by He et al. (2015) for convolutional layers and initialize fully-connected layers by a zero-mean Gaussian distribution with the standard deviation of 0.01. ", + "bbox": [ + 174, + 847, + 825, + 888 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Table 6 demonstrates the scaling ratio of our mainly used Neural Network architectures. The scaling ratio (You et al., 2017b) identifies the ratio between computation and communication, wherein ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "DNN models, the computation is proportional to the number of floating point operations required for processing an input while the communication is proportional to model size (or the number of parameters). Our local SGD training scheme will show more advantages over models with small “computation and communication scaling ratio”. ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/58085895160205104296dfe6e60caee78025de1d6a9faf234dd1baf53b968594.jpg", + "table_caption": [ + "Table 6: Scaling ratio for different models. " + ], + "table_footnote": [], + "table_body": "
ModelCommunication# parametersComputation# flops per imageComputation/Communication scaling ratio
ResNet-20 (CIFAR-10)0.27 million0.041 billion151.85
ResNet-20 (CIFAR-100)0.27 million0.041 billion151.85
ResNet-50 (ImageNet-1k)25.00 million7.7 billion308.00
DenseNet-40-12 (CIFAR-10)1.06 million0.28 billion264.15
DenseNet-40-12 (CIFAR-100)1.10 million0.28 billion254.55
WideResNet-28-10 (CIFAR-10)36.48 million5.24 billion143.64
WideResNet-28-10 (CIFAR-100)36.54 million5.24 billion143.40
", + "bbox": [ + 176, + 198, + 820, + 381 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.3 LARGE BATCH LEARNING SCHEMES ", + "text_level": 1, + "bbox": [ + 176, + 405, + 470, + 420 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The work of Goyal et al. (2017) proposes common configurations to tackle large-batch training for the ImageNet dataset. We specifically refer to their crucial techniques w.r.t. learning rate as “large batch learning schemes” in our main text. For a precise definition, this is formalized by the following two configurations: ", + "bbox": [ + 176, + 430, + 825, + 487 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "• Scaling the learning rate: When the mini-batch size is multiplied by $k$ , multiply the learning rate by $k$ . \n• Learning rate gradual warm-up: We gradually ramp up the learning rate from a small to a large value. In (our) experiments, with a large mini-batch of size $k n$ , we start from a learning rate of $\\eta$ and increment it by a constant amount at each iteration such that it reaches $\\hat { \\eta } = k \\eta$ after 5 epochs. More precisely, the incremental step size for each iteration is calculated from ηˆ\u0001η5N{pknq , where N is the number of total training samples, k is the number of computing units and $n$ is the local mini-batch size. ", + "bbox": [ + 215, + 497, + 825, + 619 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.4 HYPERPARAMETER CHOICES AND TRAINING PROCEDURE, OVER DIFFERENT MODELS/DATASETS ", + "bbox": [ + 176, + 636, + 750, + 664 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.4.1 CIFAR-10/CIFAR-100 ", + "text_level": 1, + "bbox": [ + 176, + 675, + 398, + 690 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The experiments follow the common mini-batch SGD training scheme for CIFAR (He et al., 2016a;b; Huang et al., 2017b) and all competing methods access the same total amount of data samples regardless of the number of local steps. The training procedure is terminated when the distributed algorithms have accessed the same number of samples as a standalone worker would access. For example, ResNet-20, DensetNet-40-12 and WideResNet-28-10 would access 300, 300 and 250 epochs respectively. The data is partitioned among the GPUs and reshuffled globally every epoch. The local mini-batches are then sampled among the local data available on each GPU, and its size is fixed to $B _ { \\mathrm { l o c } } = 1 2 8$ . ", + "bbox": [ + 173, + 700, + 825, + 811 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The learning rate scheme follows works (He et al., 2016a; Huang et al., 2017b), where we drop the initial learning rate by 10 when the model has accessed $5 0 \\%$ and $7 5 \\%$ of the total number of training samples. The initial learning rates of ResNet-20, DensetNet-40-12 and WideResNet-28-10 are fine-tuned on single GPU (which are 0.2, 0.2 and 0.1 respectively), and can be scaled by the global mini-batch size when using large-batch learning schemes. ", + "bbox": [ + 174, + 818, + 825, + 888 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In addition to this, we use a Nesterov momentum of 0.9 without dampening, which is applied independently to each local model. For all architectures, following He et al. (2016a), we do not apply weight decay on the learnable Batch Normalization (BN) coefficients. The weight decay of ResNet-20, DensetNet-40-12 and WideResNet-28-10 are 1e-4, 1e-4 and 5e-4 respectively. For the BN for distributed training we again follow Goyal et al. (2017) and compute the BN statistics independently for each worker. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Unless mentioned specifically, local SGD uses the exact same optimization scheme as mini-batch SGD. ", + "bbox": [ + 176, + 166, + 823, + 194 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "The procedure of fine-tuning. There is no optimal learning rate scaling rule for large-batch SGD across different mini-batch sizes, tasks and architectures, as revealed in the Figure 8 of Shallue et al. (2018). Our tuning procedure is built on the insight of their Figure 8, where we grid-search the optimal learning rate for each mini-batch size, starting from the linearly scaled learning rate. For example, compared to mini-batch size 128, mini-batch size 2048 with default large-batch learning schemes need to linearly scale the learning rate by the factor of 16. In order to find out its optimal learning rate, we will evaluate a linear-spaced grid of five different factors (i.e., t15, 15.5, 16, 16.5, 17u). If the best performance was ever at one of the extremes of the grid, we would try new grid points so that the best performance was contained in the middle of the parameters. Please note that this unbounded grid search—even though initialized at the linearly scaled learning rate—does also cover other suggested scaling heuristics, for instance square root scaling, and finds the best scaling for each scenario. ", + "bbox": [ + 174, + 212, + 825, + 364 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Note that in our experiments of large-batch SGD, either with the default large-batch learning schemes, or tuning/using the optimal learning rate, we always warm-up the learning rate for the first 5 epochs. ", + "bbox": [ + 176, + 371, + 825, + 400 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.4.2 IMAGENET ", + "text_level": 1, + "bbox": [ + 176, + 416, + 310, + 431 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "ResNet-50 training is limited to 90 passes over the data in total, and the data is disjointly partitioned and is re-shuffled globally every epoch. All competing methods access the same total number of data samples (i.e. gradients) regardless of the number of local steps. We adopt the large-batch learning schemes as in Goyal et al. (2017) below. We linearly scale the learning rate based on \u0000Number of $\\mathrm { G P U s } \\ \\times \\ \\frac { 0 . 1 } { 2 5 6 } \\ \\times \\ \\mathrm { \\bar { \\it B } _ { g l o b } } )$ where 0.1 and 256 is the base learning rate and mini-batch size 256 respectively for standard single GPU training. The local mini-batch size is set to 128. For learning rate scaling, we perform gradual warmup for the first 5 epochs, and decay the scaled learning rate by the factor of 10 when local models have access 30, 60, 80 epochs of training samples respectively. ", + "bbox": [ + 174, + 440, + 825, + 553 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.5 SYSTEM PERFORMANCE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 571, + 482, + 585 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/18500ec8c29dfa00f5f3a908bd839f0cefa6711cbcbfe78aa7ecd0442d71a0ab.jpg", + "image_caption": [ + "Figure 5 investigates the increased latency of transmitting data among CPU cores. ", + "Figure 5: The data transmission cost (in seconds) of an all-reduce operation for $1 0 0 ~ \\mathrm { M B }$ , over the different number of cores, using PyTorch’s built-in MPI all-reduce operation. Each evaluation is the average result of 100 data transmissions on a Kubernetes cluster. The network bandwidth is 10 Gbps, and we use 48 cores per physical machine. " + ], + "image_footnote": [], + "bbox": [ + 338, + 631, + 656, + 792 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Table 7 evaluates the time of running forward and backward with different mini-batch size, for training ResNet20 on CIFAR-10. We can witness that a larger mini-batch size we use, the better parallelism can a GPU have. ", + "bbox": [ + 174, + 882, + 825, + 922 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/7eadbf67e8f7e0033c7725867d64b2638d1d07f2a9f55107f710c1313525b834.jpg", + "table_caption": [ + "Table 7: The system performance of running forward and backward pass on a single GPU, for training ResNet20 on CIFAR-10. The “Ratio” indicates the Time of evaluating 4096 samples with specified mini-batch sizeTime of evaluating 4096 samples with mini-batch size 4096 . The “Time” is in seconds. " + ], + "table_footnote": [], + "table_body": "
Titan XPTesla V100
Mini-Batch SizeTime per iteration (over 100 iterations)RatioTime per iteration (over 100 iterations)Ratio
320.0581.4900.0289.028
640.1001.2840.0304.836
1280.1751.1240.0342.741
2560.3231.0370.0431.733
5120.7371.1830.0731.471
10241.4691.1790.1241.249
20482.6981.0830.2121.068
40964.98310.3971
", + "bbox": [ + 176, + 142, + 825, + 318 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B LOCAL SGD TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 343, + 406, + 359 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.1 FORMAL DEFINITION OF THE LOCAL SGD ALGORITHM ", + "bbox": [ + 174, + 375, + 606, + 390 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Algorithm 1 Local SGD ", + "text_level": 1, + "bbox": [ + 174, + 407, + 331, + 421 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/8837a2d6ddde9bc82ef8a5e4757da8dc9a06a255fbd71339945b7ec6a40d97b9.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { g } _ { ( t ) + h - 1 } ^ { k } : = \\frac { 1 } { B _ { \\mathrm { l o c } } } \\sum _ { i \\in \\mathcal { T } _ { ( t ) + h - 1 } ^ { k } } \\nabla f _ { i } \\left( \\pmb { w } _ { ( t ) + h - 1 } ^ { k } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 390, + 595, + 679, + 618 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/33aa9d83c7a614cb045608f16ec41954cb2d875f05e3318d5cc1634c7a944135.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { w } _ { ( t ) + h } ^ { k } : = \\pmb { w } _ { ( t ) + h - 1 } - \\gamma _ { ( t ) } \\pmb { g } _ { ( t ) + h - 1 } ^ { k } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 418, + 651, + 651, + 670 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/6692b99af156469a491c8efff94ca47d976f236ed419dd6c87a8a6b69312b357.jpg", + "text": "$$\n{ \\Delta } _ { ( t ) } ^ { k } : = { w } _ { ( t ) } ^ { k } - { w } _ { ( t ) + H } ^ { k } .\n$$", + "text_format": "latex", + "bbox": [ + 449, + 707, + 606, + 727 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "10: get new global (synchronized) model $\\pmb { w } _ { ( t + 1 ) } ^ { k }$ for all $K$ nodes: ", + "bbox": [ + 173, + 743, + 599, + 760 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/8792f3a734fbfa6e21133d4b94d2e797f158428e87a367293dd658f55046425b.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { w } _ { ( t + 1 ) } ^ { k } : = \\pmb { w } _ { ( t ) } ^ { k } - \\gamma _ { ( t ) } \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\Delta _ { ( t ) } ^ { k } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 416, + 766, + 637, + 786 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "11: end for \n12: end for ", + "bbox": [ + 174, + 791, + 259, + 816 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.2 NUMERICAL ILLUSTRATION OF LOCAL SGD ON A CONVEX PROBLEM ", + "text_level": 1, + "bbox": [ + 174, + 845, + 705, + 861 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In addition to our deep learning experiments, we first illustrate the convergence properties of local SGD on a small scale convex problem. For this, we consider logistic regression on the $\\mathtt { w } 8 \\mathtt { a }$ dataset7 $( d = 3 0 0 , n = 4 9 7 4 9 )$ . We measure the number of iterations to reach the target accuracy $\\epsilon = 0 . 0 0 5$ . For each combination of $H , B _ { \\mathrm { l o c } }$ and $K$ we determine the best learning rate by extensive grid search (cf. paragraph below for the detailed experimental setup). In order to mitigate extraneous effects on the measured results, we here measure time in discrete units, that is we count the number of stochastic gradient computations and communication rounds, and assume that communication of the weights is $2 5 \\times$ more expensive than a gradient computation, for ease of illustration. ", + "bbox": [ + 174, + 872, + 826, + 901 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Figure 6(a) shows that different combinations of the parameters $( B _ { \\mathrm { l o c } } , H )$ can impact the convergence time for $K = 1 6$ . Here, local SGD with p16, 16q converges more than $2 \\times$ faster than for $( 6 4 , 1 )$ and $3 \\times$ faster than for $( 2 5 6 , 1 )$ . ", + "bbox": [ + 174, + 194, + 825, + 237 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Figure 6(b) depicts the speedup when increasing the number of workers $K$ . Local SGD shows the best speedup for $H = 1 6$ on a small number of workers, while the advantage gradually diminishes for very large $K$ . ", + "bbox": [ + 174, + 243, + 825, + 285 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/7ce3b2966d064642fe7f7f2bcca8d68a991d8567830e97340f3a2127c5a64c59.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 194, + 316, + 444, + 484 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/30829ff8545149a8ba10efaf58365de9e24aac28a327a4a6d2307ddcb7cd1eb5.jpg", + "image_caption": [ + "(b) Speedup over the number of workers $K$ to solve a regularized logistic regression problem to target accuracy $\\epsilon ~ = ~ 0 . 0 0 5$ , for $B _ { \\mathrm { l o c } } { } ~ = ~ 1 6$ and $H \\in { 1 , 2 , 4 , 8 , 1 6 }$ . We simulate the network traffic under the assumption that communication is $2 5 \\times$ slower than a stochastic gradient computation. " + ], + "image_footnote": [], + "bbox": [ + 534, + 315, + 789, + 484 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(a) Time (relative to best method) to solve a regularized logistic regression problem to target accuracy $\\epsilon ~ = ~ 0 . 0 0 5$ for $K \\ = \\ 1 6$ workers for $H \\in$ $1 , 2 , 4 , 8 , 1 6$ and local mini-batch size $B _ { \\mathrm { l o c } }$ . We simulate the network traffic under the assumption that communication is $2 5 \\times$ slower than a stochastic gradient computation. ", + "bbox": [ + 173, + 489, + 480, + 578 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Figure 6: Numerical illustration of local SGD on a convex problem. ", + "bbox": [ + 294, + 594, + 697, + 608 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Experimental Setup for Convex Experiments For the illustrative experiments here we study the convergence of local SGD on the logistic regression problem, $f ( \\pmb { w } ) ~ = ~ \\textstyle { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } \\log ( 1 \\ Z )$ $\\begin{array} { r } { \\exp ( - b _ { i } \\mathbf { a } _ { i } ^ { \\top } \\pmb { w } ) ) + \\frac { \\lambda } { 2 } \\| \\pmb { w } \\| ^ { 2 } } \\end{array}$ , where $\\mathbf { a } _ { i } \\in \\mathbb { R } ^ { d }$ and $b _ { i } \\in \\{ - 1 , + 1 \\}$ are the data samples, and regularization parameter $\\lambda = 1 / \\overline { { n } }$ . For each run, we initialize ${ \\pmb w } _ { 0 } = { \\pmb 0 } _ { d }$ and measure the number of stochastic gradient evaluations (and communication rounds) until be best of last iterate and weighted average of the iterates reaches the target accuracy $f ( \\pmb { w } _ { t } ) - f ^ { \\star } \\leqslant \\epsilon : = 0 . 0 0 5$ , with $f ^ { \\star } : = 0 . 1 2 \\bar { 6 } 4 3 3 1 7 6 2 1 \\bar { 6 } 5 4 5$ . For each configuration $( K , H , B _ { \\mathrm { l o c } } )$ , we report the best result found with any of the following two stepsizes: $\\begin{array} { r } { \\gamma _ { t } : = \\operatorname* { m i n } ( 3 2 , \\frac { c n } { t + 1 } ) } \\end{array}$ and $\\gamma _ { t } = 3 2 c$ . Here $c$ is a parameter that can take the values $c = { \\bar { 2 } } ^ { i }$ for $i \\in \\mathbb Z$ . For each stepsize we determine the best parameter $c$ by a grid search, and consider parameter $c$ optimal, if parameters $\\{ 2 ^ { - 2 } c , 2 ^ { - 1 } c , 2 c , 2 ^ { 2 } c \\}$ yield worse results (i.e. more iterations to reach the target accuracy). ", + "bbox": [ + 173, + 643, + 826, + 801 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.3 MORE RESULTS ON LOCAL SGD TRAINING ", + "text_level": 1, + "bbox": [ + 178, + 818, + 519, + 832 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.3.1 TRAINING CIFAR-10 VIA LOCAL SGD ", + "text_level": 1, + "bbox": [ + 173, + 843, + 503, + 858 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Better communication efficiency, with guaranteed test accuracy. Figure 7 shows that local SGD is significantly more communication efficient while guaranteeing the same accuracy and enjoys faster convergence speed. In Figure 7, the local models use a fixed local mini-batch size $B _ { \\mathrm { l o c } } ~ = ~ 1 2 8$ for all updates. All methods run for the same number of total gradient computations. Mini-batch ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/dce2b0a9694723b07b2ac094bd4978537858d5daf08a213d5a6e434ba082e4c5.jpg", + "image_caption": [ + "(a) Training accuracy vs. number of (b) Training accuracy vs. training (c) Test accuracy vs. number of global synchronization rounds. time. epochs. ", + "Figure 7: Training CIFAR-10 with ResNet-20 via local SGD $2 \\times 1$ -GPU). The local batch size $B _ { \\mathrm { l o c } }$ is fixed to 128, and the number of local steps $H$ is varied from 1 to 16. All the experiments are under the same training configurations. " + ], + "image_footnote": [], + "bbox": [ + 179, + 106, + 813, + 237 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "SGD—the baseline method for comparison—is a special case of local SGD with $H \\mathbf { \\Phi } = \\mathbf { \\Phi } 1$ , with full global model synchronization for each local update. We see that local SGD with $H > 1$ , as illustrated in Figure 7(a), by design does $H$ times less global model synchronizations, alleviating the communication bottleneck while accessing the same number of samples (see section 1). The impact of local SGD training upon the total training time is more significant for larger number of local steps $H$ (i.e., Figure 7(b)), resulting in an at least $3 \\times$ speed-up when comparing mini-batch $H = 1$ to local SGD with $H = 1 6$ . The final training accuracy remains stable across different $H$ values, and there is no difference or negligible difference in test accuracy (Figure 7(c)). ", + "bbox": [ + 174, + 364, + 825, + 477 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "B.3.2 TRAINING IMAGENET VIA LOCAL SGD ", + "text_level": 1, + "bbox": [ + 174, + 496, + 508, + 511 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Figure 8 shows the effectiveness of scaling local SGD to the challenging ImageNet dataset. We limit ResNet-50 training to 90 passes over the data in total, and use the standard training configurations as mentioned in Appendix A.4.2. ", + "bbox": [ + 173, + 522, + 825, + 564 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/6062e3f588aa9e0f27dde8c400d7124cc3a94c7fef501d063e5a1f3ca78a02cc.jpg", + "image_caption": [ + "Figure 8: The performance of local SGD trained on ImageNet-1k with ResNet-50.We evaluate the model performance on test dataset after each complete accessing of the whole training samples. We apply the largebatch learning schemes (Goyal et al., 2017) to the ImageNet for these two methods. For local SGD, the number of local steps is set to $H = 8$ . " + ], + "image_footnote": [], + "bbox": [ + 362, + 585, + 630, + 757 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Moreover, in our ImageNet experiment, the initial phase of local SGD training follows the theoretical assumption mentioned in Subsection 2, and thus we gradually warm up the number of local steps from 1 to the desired value $H$ during the first few epochs of the training. We found that exponentially increasing the number of local steps from 1 by the factor of 2 (until reaching the expected number of local steps) performs well. For example, our ImageNet training uses $H = 8$ , so the number of local steps for the first three epochs is $1 , 2 , 4$ respectively. ", + "bbox": [ + 174, + 839, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "B.3.3 LOCAL SGD SCALES TO LARGER BATCH SIZES THAN MINI-BATCH SGD ", + "text_level": 1, + "bbox": [ + 173, + 103, + 736, + 118 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "The empirical studies (Shallue et al., 2018; Golmant et al., 2018) reveal the regime of maximal data parallelism across different tasks and models, where the large-batch training would reach the limit and additional parallelism provides no benefit whatsoever. ", + "bbox": [ + 173, + 127, + 823, + 170 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/4e126eb32eb72dc647f8c529a1e431ea1a6361df64813ce4cce6bf1d5da1ee85.jpg", + "image_caption": [ + "Figure 9: The relationship between steps to top-1 test accuracy and batch size, of training ResNet-20 on CIFAR-10. The “step” is equivalent to the number of applying gradients. The global mini-batch size is increased by adding more workers $K$ with fixed $B _ { \\mathrm { l o c } } = 1 2 8$ . Results are averaged over three runs, each with fine-tuned learning rate. " + ], + "image_footnote": [], + "bbox": [ + 364, + 184, + 632, + 357 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "On contrary to standard large-batch training, local SGD scales to larger batch size and provides additional parallelism upon the limitation of current large batch training. Figure 9 shows the example of training ResNet-20 on CIFAR-10 with $H = 2$ , which trains and generalizes better in terms of update steps while with reduced communication cost. ", + "bbox": [ + 174, + 436, + 825, + 493 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "B.4 PRACTICAL IMPROVEMENT POSSIBILITIES FOR STANDARD LOCAL SGD TRAINING ", + "text_level": 1, + "bbox": [ + 173, + 510, + 794, + 525 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We investigate different aspects of the training to address the quality when scaling local SGD to the extreme case, e.g., hybrid momentum scheme, warming up the local SGD or fine-tuning the learning rate. In this section, we briefly present how these strategies are, and how they work in practice where we train ResNet-20 on CIFAR-10 on 16 GPUs. ", + "bbox": [ + 174, + 535, + 825, + 592 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "B.4.1 LOCAL SGD WITH MOMENTUM ", + "text_level": 1, + "bbox": [ + 174, + 607, + 454, + 622 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Momentum mini-batch SGD is widely used in place of vanilla SGD. The distributed mini-batch SGD with vanilla momentum on $K$ training nodes follows ", + "bbox": [ + 173, + 631, + 823, + 660 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/6511c42a6e15a05d3b688ccfb41f4c319df000a55f4da5af8a6ee5224d0a4db8.jpg", + "text": "$$\n{ \\pmb u } _ { ( t ) } = m { \\pmb u } _ { ( t - 1 ) } + \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } { \\pmb \\nabla } _ { ( t ) } ^ { k } , \\qquad { \\pmb w } _ { ( t + 1 ) } = { \\pmb w } _ { ( t ) } - \\gamma { \\pmb u } _ { ( t ) }\n$$", + "text_format": "latex", + "bbox": [ + 295, + 666, + 702, + 709 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "$\\begin{array} { r } { \\pmb { \\nabla } _ { ( t ) } ^ { k } = \\frac { 1 } { | \\mathcal { T } _ { ( t ) } ^ { k } | } \\sum _ { i \\in \\mathcal { T } _ { ( t ) } ^ { k } } \\nabla f _ { i } \\big ( \\pmb { w } _ { ( t ) } \\big ) } \\end{array}$ ", + "bbox": [ + 174, + 715, + 433, + 739 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "After $H$ updates of mini-batch SGD, we have the following updated ${ \\pmb w } _ { ( t + H ) }$ ", + "bbox": [ + 173, + 744, + 679, + 761 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/194bb0cc6dad1e7c12752e00993d0f87c1bfbdc9a9061b38d40169ebe959bea3.jpg", + "text": "$$\n{ \\pmb w } _ { ( t + H ) } = { \\pmb w } _ { ( t ) } - \\gamma \\left( \\sum _ { \\tau = 1 } ^ { H } m ^ { \\tau } { \\pmb u } _ { ( t - 1 ) } + \\sum _ { \\tau = 0 } ^ { H - 1 } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\pmb { \\nabla } _ { ( t ) } ^ { k } + \\dots + \\sum _ { \\tau = 0 } ^ { 0 } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\pmb { \\nabla } _ { ( t + H - 1 ) } ^ { k } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 194, + 767, + 802, + 811 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Coming back to the setting of local SGD, we can apply momentum acceleration on each local model, or on a global level (Chen & Huo, 2016). In the remaining part of this section, we analyze the case of applying local momentum and global momentum. For ease of understanding, we assume the learning rate $\\gamma$ is the same throughout the $H$ update steps. ", + "bbox": [ + 173, + 824, + 825, + 881 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Local SGD with Local Momentum. When applying local momentum on the local SGD, i.e., using independent identical momentum acceleration for each local model and only globally aggregating the gradients at the time $( t ) + H$ , we have the following local update scheme ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 656, + 119 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/6120084ac338ea02919d5d53752e1af956753b262a65d56e1154eb05efcdd30e.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { u } _ { ( t ) } ^ { k } = m \\pmb { u } _ { ( t - 1 ) } ^ { k } + \\pmb { \\nabla } _ { ( t ) } ^ { k } , \\qquad \\pmb { w } _ { ( t ) + 1 } ^ { k } = \\pmb { w } _ { ( t ) } ^ { k } - \\gamma \\pmb { u } _ { ( t ) } ^ { k } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 121, + 676, + 142 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where ∇kptq \u0010 1|Ik | $\\begin{array} { r } { \\pmb { \\nabla } _ { ( t ) } ^ { k } = \\frac { 1 } { | \\mathcal { T } _ { ( t ) } ^ { k } | } \\sum _ { i \\in \\mathcal { T } _ { ( t ) } ^ { k } } \\nabla f _ { i } \\big ( \\pmb { w } _ { ( t ) } \\big ) } \\end{array}$ . Consequently, after $H$ local steps, ", + "bbox": [ + 174, + 143, + 661, + 166 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/d6840f4af716bd93dc935d18c1c7aff8ae3c56c5b48c070fca9e2c00088c9c67.jpg", + "text": "$$\n{ \\pmb w } _ { ( t ) + H } ^ { k } = { \\pmb w } _ { ( t ) } ^ { k } - \\gamma \\left( \\sum _ { \\tau = 1 } ^ { H } m ^ { \\tau } { \\pmb u } _ { ( t - 1 ) } ^ { k } + \\sum _ { \\tau = 0 } ^ { H - 1 } m ^ { \\tau } \\pmb \\nabla _ { ( t ) } ^ { k } + \\dots + \\sum _ { \\tau = 0 } ^ { 0 } m ^ { \\tau } \\pmb \\nabla _ { ( t ) + H - 1 } ^ { k } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 223, + 167, + 772, + 212 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Substituting the above equation into eq. (2), we have the update ", + "bbox": [ + 173, + 219, + 591, + 234 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/208655ec0e132a311e7579c52c7c2768cecdfd60ca81b45a8938af03bb5a228d.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\pmb w } _ { ( t + 1 ) } = { \\pmb w } _ { ( t ) } - \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\gamma \\left( \\sum _ { \\tau = 1 } ^ { H } m ^ { \\tau } { \\pmb u } _ { ( t - 1 ) } ^ { k } \\sum _ { \\tau = 0 } ^ { H - 1 } m ^ { \\tau } \\pmb \\nabla _ { ( t ) } ^ { k } + \\dots + \\sum _ { \\tau = 0 } ^ { 0 } m ^ { \\tau } \\pmb \\nabla _ { ( t ) + H - 1 } ^ { k } \\right) } } \\\\ { { \\displaystyle ~ = ~ { \\pmb w } _ { ( t ) } - \\gamma \\left( \\sum _ { \\tau = 1 } ^ { H } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } { \\pmb u } _ { ( t - 1 ) } ^ { k } + \\sum _ { \\tau = 0 } ^ { H - 1 } m ^ { \\tau } ( \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\pmb \\nabla _ { ( t ) } ^ { k } ) + \\dots + \\sum _ { \\tau = 0 } ^ { 0 } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\pmb \\nabla _ { ( t ) + H - 1 } ^ { k } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 234, + 807, + 323 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Comparing the mini-batch SGD with local momentum local SGD after $H$ update steps $H$ global update steps v.s. $H$ local update steps and 1 global update step), we witness that the main difference of these two update schemes is the difference between $\\begin{array} { r } { \\sum _ { \\tau = 1 } ^ { H } m ^ { \\tau } { \\pmb u } _ { ( t - 1 ) } } \\end{array}$ and $\\begin{array} { r } { \\sum _ { \\tau = 1 } ^ { H } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\pmb { u } _ { ( t - 1 ) } ^ { k } } \\end{array}$ where mini-batch SGD holds a global $\\pmb { u } _ { ( t - 1 ) }$ while each local model of the local SGD has their own $\\pmb { u } _ { ( t - 1 ) } ^ { k }$ . We will soon see the difference between the global momentum of mini-batch SGD and the local momentum of local SGD. ", + "bbox": [ + 173, + 328, + 826, + 422 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Local SGD with Global Momentum For global momentum local SGD, i.e., a more general variant of block momentum (Chen & Huo, 2016), we would like to apply the momentum factor only to the accumulated/synchronized gradients: ", + "bbox": [ + 173, + 435, + 825, + 478 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/798821934f84793a1c22e251a3bd1664bf67e962d05363b720743226b43d48e1.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\pmb u } _ { ( t ) } = m { \\pmb u } _ { ( t - 1 ) } + \\frac { 1 } { \\gamma } \\sum _ { k = 1 } ^ { K } \\frac { 1 } { K } \\big ( { \\pmb w } _ { ( t ) } ^ { k } - { \\pmb w } _ { ( t ) + H } ^ { k } \\big ) = m { \\pmb u } _ { ( t - 1 ) } + \\frac { 1 } { \\gamma } \\sum _ { k = 1 } ^ { K } \\frac { 1 } { K } \\sum _ { l = 0 } ^ { H - 1 } \\gamma \\pmb \\nabla _ { ( t ) + l } ^ { k } , } } \\\\ { { \\displaystyle { \\pmb w } _ { ( t + 1 ) } = { \\pmb w } _ { ( t ) } - \\gamma { \\pmb u } _ { ( t ) } = { \\pmb w } _ { ( t ) } - \\gamma \\big ( m { \\pmb u } _ { ( t - 1 ) } + \\sum _ { l = 0 } ^ { H - 1 } \\sum _ { k = 1 } ^ { K } \\frac { 1 } { K } \\pmb \\nabla _ { ( t ) + l } ^ { k } \\big ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 479, + 779, + 564 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\pmb { w } _ { ( t ) + H } ^ { k } = \\pmb { w } _ { ( t ) } ^ { k } - \\eta \\sum _ { l = 0 } ^ { H - 1 } \\pmb { \\nabla } _ { ( t ) + l } ^ { k } = \\pmb { w } _ { ( t ) } - \\eta \\sum _ { l = 0 } ^ { H - 1 } \\pmb { \\nabla } _ { ( t ) + l } ^ { k } } \\end{array}$ . Note that for local SGD, we consider summing up the gradients from each local update, i.e., the model difference before and after one global synchronization, and then apply the global momentum to the gradients over workers over previous local update steps. ", + "bbox": [ + 173, + 568, + 826, + 628 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Obviously, there exists a significant difference between mini-batch momentum SGD and global momentum local SGD, at least the term °Hτ\u00100 mτ is cancelled. ", + "bbox": [ + 174, + 635, + 821, + 667 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Local SGD with Hybrid Momentum. The following equation tries to combine local momentum with global momentum, showing a naive implementation. ", + "bbox": [ + 173, + 679, + 825, + 709 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "First of all, based on the local momentum scheme, after $H$ local update steps, ", + "bbox": [ + 174, + 715, + 684, + 731 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/84743c0a6d60b96b903b92e9acc95d6fef1bfbdb9906ee29c14820e4d22d9786.jpg", + "text": "$$\n{ \\pmb w } _ { ( t ) + H } ^ { k } = { \\pmb w } _ { ( t ) } ^ { k } - \\gamma \\big ( \\sum _ { \\tau = 1 } ^ { H } m ^ { \\tau } { \\pmb u } _ { ( t - 1 ) } ^ { k } + \\sum _ { \\tau = 0 } ^ { H - 1 } m ^ { \\tau } \\pmb \\nabla _ { ( t ) } ^ { k } + . . . + \\sum _ { \\tau = 0 } ^ { 0 } m ^ { \\tau } \\pmb \\nabla _ { ( t ) + H - 1 } ^ { k } \\big )\n$$", + "text_format": "latex", + "bbox": [ + 233, + 731, + 763, + 773 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Together with the result from local momentum with the global momentum, we have ", + "bbox": [ + 173, + 781, + 723, + 796 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/b06a2825e59084a4f1e872111c862efebf1bc5f6e513186406faaab114bff0ae.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\boldsymbol u } _ { ( t ) } = m { \\boldsymbol u } _ { ( t - 1 ) } + \\frac { 1 } { \\gamma } \\sum _ { k = 1 } ^ { K } \\frac { 1 } { K } \\big ( { \\boldsymbol w } _ { ( t ) } ^ { k } - { \\boldsymbol w } _ { ( t ) + H } ^ { k } \\big ) } } \\\\ { { \\displaystyle { \\boldsymbol v } _ { ( t + 1 ) } = { \\boldsymbol w } _ { ( t ) } - \\gamma { \\boldsymbol u } _ { ( t ) } = { \\boldsymbol w } _ { ( t ) } - \\gamma \\left[ m { \\boldsymbol u } _ { ( t - 1 ) } + \\frac { 1 } { \\gamma } \\sum _ { k = 1 } ^ { K } \\frac { 1 } { K } \\big ( { \\boldsymbol w } _ { ( t ) } ^ { k } - { \\boldsymbol w } _ { ( t ) + H } ^ { k } \\big ) \\right] } } \\\\ { { \\displaystyle \\qquad = { \\boldsymbol w } _ { ( t ) } - \\gamma \\left[ m { \\boldsymbol u } _ { ( t - 1 ) } + \\sum _ { \\tau = 1 } ^ { H } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } { \\boldsymbol u } _ { ( t - 1 ) } ^ { k } + \\sum _ { \\tau = 0 } ^ { H - 1 } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\nabla _ { ( t ) } ^ { k } + \\dots + \\sum _ { \\tau = 0 } ^ { 0 } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\nabla _ { ( t ) + 1 } ^ { k } \\right] } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 797, + 839, + 930 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "where $\\pmb { u } _ { ( t - 1 ) }$ is the global momentum memory and $\\pmb { u } _ { ( t - 1 ) }$ is the local momentum memory for each node $k$ . ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Table 8: Evaluate local momentum and global momentum for ResNet-20 on CIFAR-10 data via local SGD training ( $H = 1$ case) on $5 \\times 2$ -GPU Kubernetes cluster. The local mini-batch size is 128 and base batch size is 64 (used for learning rate linear scale). Each local model will access to a disjoint data partition, using the standard learning rate scheme as He et al. (2016a). ", + "bbox": [ + 173, + 142, + 825, + 194 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/0f0f17f22e89bd0535184ca6946c90ba3b343b4af987f8586a7c2807921f3344.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
local momentumglobal momentumtest top-1
0.00.090.57
0.90.092.41
0.90.192.22
0.90.292.09
0.90.392.54
0.90.492.45
0.90.592.19
0.90.691.32
0.90.718.76
0.90.814.35
0.90.912.21
0.90.9510.11
", + "bbox": [ + 359, + 207, + 642, + 429 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Local SGD with Momentum in Practice. In practice, it is possible to combine the local momentum with global momentum to further improve the model performance. For example, a toy example in Table 8 investigates the impact of different momentum schemes on CIFAR-10 trained with ResNet-20 on a $5 \\times 2$ -GPU cluster, where some factors of global momentum could further slightly improve the final test accuracy. ", + "bbox": [ + 173, + 457, + 825, + 526 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "However, the theoretical understanding of how local momentum and global momentum contribute to the optimization still remains unclear, which further increase the difficulty of tuning local SGD over $H , K$ . An efficient way of using local and global momentum remains a future work and in this work, we only consider the local momentum. ", + "bbox": [ + 174, + 534, + 825, + 589 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "B.4.2 WARM-UP OF THE NUMBER OF LOCAL SGD STEPS ", + "text_level": 1, + "bbox": [ + 174, + 604, + 586, + 619 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We use the term “local step warm-up strategy”, to refer to a specific variant of post-local SGD. More precisely, instead of the two-phase regime which we presented here the used number of local steps $H$ will be gradually increased from 1 to the expected number of local steps $H$ . The warm-up strategies investigated here are “linear”, “exponential” and “constant”. ", + "bbox": [ + 174, + 630, + 825, + 685 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Please note that the implemented post-local SGD over the whole text only refers to the training scheme that uses frequent communication (i.e., $H = 1$ ) before the first learning rate decay and then reduces the communication frequency (i.e., $H > 1 $ ) after the decay. ", + "bbox": [ + 174, + 693, + 825, + 736 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "This section then investigates the trade-off between stochastic noise and the training stability. Also note that the Figure 2(a) in the main text has already presented one aspect of the trade-off. So the exploration below mainly focuses on the other aspects and tries to understand how will the scale of the added stochastic noise impact the training stability. Infact, even the model has been stabilized to a region with good quality. ", + "bbox": [ + 174, + 741, + 825, + 813 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Figure 10 and Figure 11 investigate the potential improvement through using the local step warm-up strategy for the case of training ResNet-20 on CIFAR-10. Figure 10 evaluates the warm-up strategies of “linear” and “constant” for different $H$ , while the evaluation of “exponential” warm-up strategy is omitted due to its showing similar performance as “linear” warm-up. ", + "bbox": [ + 174, + 819, + 825, + 875 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "However, none of the investigated strategies show convincing performance. Figure 11 further studies how the period of warm-up impacts the training performance. We can witness that even if we increase the warm-up phase to 50 epochs where the training curve of mini-batch SGD becomes stabilized, the large noise introduced by the local SGD will soon degrade the status of training and lead to potential quality loss, as in Figure 11(a). ", + "bbox": [ + 174, + 882, + 823, + 924 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 132 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/f14f0e095bbdfba3e7dc3f2a72ef068de7d6cfe460f866a358625048e15c79bc.jpg", + "image_caption": [ + "Figure 10: Investigate how local step warm-up strategy impacts the performance of training CIFAR-10 with ResNet-20 via local SGD $8 \\times 2$ -GPU). The local batch size $B _ { \\mathrm { l o c } }$ is fixed to 128. The warmup strategies are “linear” and “constant”, and the warm-up period used here is equivalent to the number of local steps $H$ . " + ], + "image_footnote": [], + "bbox": [ + 184, + 147, + 826, + 481 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/fdf6a1940c536dc20ba89b5d65b424ead9a22a3001f5e768cb401a18a76bf18f.jpg", + "image_caption": [ + "(c) Evaluate the impact of “exponential” local step warm-up for different period of warm-up phase. ", + "Figure 11: Investigate how different warm-up period of the local step warm-up impacts the performance of training CIFAR-10 with ResNet-20 via local SGD ( $8 \\times 2$ -GPU). The local batch size $B _ { \\mathrm { l o c } }$ is fixed to 128, and the strategies to warm-up the number of local steps $H$ are “linear”, “exponential” and “constant”. " + ], + "image_footnote": [], + "bbox": [ + 199, + 148, + 789, + 739 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "C POST-LOCAL SGD TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 102, + 454, + 118 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "C.1 THE ALGORITHM OF POST-LOCAL SGD ", + "bbox": [ + 173, + 133, + 495, + 148 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Algorithm 2 Post-local SGD ", + "text_level": 1, + "bbox": [ + 174, + 166, + 356, + 180 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "input: the initial model ${ \\pmb w } _ { ( 0 ) }$ ; \ninput: training data with labels $\\mathcal { T }$ ; \ninput: mini-batch of size $B _ { \\mathrm { l o c } }$ per local model; \ninput: step size $\\eta$ , and momentum $m$ (optional); \ninput: number of synchronization steps $T$ , and the first learning rate decay is performed at $T ^ { \\prime }$ ; input: number of eventual local steps $H ^ { \\prime }$ ; \ninput: number of nodes $K$ . \n1: synchronize to have the same initial models $\\pmb { w } _ { ( 0 ) } ^ { k } : = \\pmb { w } _ { ( 0 ) }$ . \n2: for all $k : = 1 , \\ldots , K$ do in parallel \n3: for $t : = 1 , \\dots , T$ do \n4: if $t < T ^ { \\prime }$ then \n5: $H _ { ( t ) } = 1$ \n6: else \n7: $H _ { ( t ) } = H ^ { \\prime }$ \n8: end if \n9: 10: for $h : = 1 , \\ldots , H _ { ( t ) }$ doch from . \n$\\mathcal { T } _ { ( t ) + h - 1 } ^ { k }$ \n11: compute the gradient \n$\\begin{array} { r } { \\pmb { g } _ { ( t ) + h - 1 } ^ { k } : = \\frac { 1 } { B _ { \\mathrm { l o c } } } \\sum _ { i \\in \\mathcal { T } _ { ( t ) + h - 1 } ^ { k } } \\nabla f _ { i } \\left( \\pmb { w } _ { ( t ) + h - 1 } ^ { k } \\right) . } \\end{array}$ \n12: update the local model to \n$\\begin{array} { r } { \\pmb { w } _ { ( t ) + h } ^ { k } : = \\pmb { w } _ { ( t ) + h - 1 } - \\gamma _ { ( t ) } \\pmb { g } _ { ( t ) + h - 1 } ^ { k } . } \\end{array}$ \n13: end for \n14: all-reduce aggregation of the gradients \n${ \\Delta } _ { ( t ) } ^ { k } : = { w } _ { ( t ) } ^ { k } - { w } _ { ( t ) + H } ^ { k } .$ \n15: get new global (synchronized) model $\\pmb { w } _ { ( t + 1 ) } ^ { k }$ for all $K$ nodes: \n$\\begin{array} { r } { \\pmb { w } _ { ( t + 1 ) } ^ { k } : = \\pmb { w } _ { ( t ) } ^ { k } - \\gamma _ { ( t ) } \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\Delta _ { ( t ) } ^ { k } } \\end{array}$ \n16: end for \n17: end for ", + "bbox": [ + 171, + 188, + 735, + 642 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "C.2 THE EFFECTIVENESS OF TURNING ON POST-LOCAL SGD AFTER THE FIRST LEARNING RATE DECAY ", + "bbox": [ + 173, + 103, + 820, + 132 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "In Figure 12, we study the sufficiency as well as the necessity of “injecting” more stochastic noise (i.e., using post-local SGD) into the optimization procedure after performing the first learning rate decay. Otherwise, the delayed noise injection (i.e., starting the post-local SGD only from the second learning rate decay) not only introduces more communication cost but also meets the increased risk of converging to sharper minima. ", + "bbox": [ + 173, + 145, + 825, + 215 + ], + "page_idx": 26 + }, + { + "type": "image", + "img_path": "images/c75fc988863a4f9d9183c639fb0867b484fbfce79b53d95586447972d401eef2.jpg", + "image_caption": [ + "Figure 12: The effectiveness and necessary of turning on the post-local SGD after the first learning rate decay. The example here trains ResNet-20 on CIFAR-10 on $K = 1 6$ GPUs with $B _ { \\mathrm { l o c } } K = 2 0 4 8$ . " + ], + "image_footnote": [], + "bbox": [ + 362, + 237, + 630, + 373 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "C.3 THE SPEEDUP OF POST-LOCAL SGD TRAINING ON CIFAR ", + "bbox": [ + 174, + 443, + 625, + 457 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Table 3 and Table 10 evaluate the speedup of mini-batch SGD and post-local SGD, over different CNN models, datasets, and training phases. ", + "bbox": [ + 174, + 470, + 820, + 500 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Table 9: The Speedup of mini-batch SGD and post-local SGD, over different CNN models and datasets. The speedup is evaluated by $\\frac { T _ { a } } { T _ { a } ^ { K } }$ , where $T _ { a }$ is the training time of the algorithm $a$ on 1 GPU and $T _ { a } ^ { K }$ corresponds to the training on $K$ GPUs. We use 16 GPUs in total (with $B _ { \\mathrm { l o c } } = 1 2 8$ ) on an $8 \\times 2$ -GPU cluster with 10 Gbps network bandwidth. The experimental setup is the same as Table 3. ", + "bbox": [ + 174, + 517, + 825, + 573 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/fa78ab2303df8a144f795cd6f0c38452086a56774773e06f01c26657f09fe52b.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
CIFAR-10CIFAR-100
H=1H=16H=32H=1H=16H=32
ResNet-209.4512.2413.008.7511.0511.67
DenseNet-40-128.3110.8011.378.0410.5910.85
WideResNet-28-125.337.948.195.297.838.14
", + "bbox": [ + 259, + 584, + 738, + 694 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Table 10: The Speedup of mini-batch SGD and post-local SGD (only consider the phase of performing post-local SGD) over different CNN models and datasets. The speedup is evaluated by $\\overset { \\cdot } { \\frac { T _ { a } } { T _ { a } ^ { K } } }$ , where $T _ { a }$ is the training time of the algorithm $a$ (corresponding to the second phase) on 1 GPU and $T _ { a } ^ { K }$ corresponds to the training on $K$ GPUs. We use 16 GPUs in total (with $B _ { \\mathrm { l o c } } = 1 2 8$ ) on an $8 \\times 2$ -GPU cluster with 10 Gbps network bandwidth. The experimental setup is the same as Table 3. ", + "bbox": [ + 173, + 731, + 825, + 758 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 762, + 823, + 801 + ], + "page_idx": 26 + }, + { + "type": "table", + "img_path": "images/326c940ea119517170f414ab7d0675b0ea453695506ba5da46e573794f8fdbe2.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
CIFAR-10CIFAR-100
H=1H=16H=32H=1H=16H=32
ResNet-209.4517.3320.808.7515.0017.50
DenseNet-40-128.3115.4318.008.0415.5016.69
WideResNet-28-125.3315.5217.665.2915.0917.69
", + "bbox": [ + 259, + 813, + 738, + 921 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "C.4 UNDERSTANDING THE GENERALIZATION OF POST-LOCAL SGD ", + "text_level": 1, + "bbox": [ + 176, + 103, + 658, + 117 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "The Sharpness Visualization Figure 13 visualizes the sharpness of the minima for training ResNet20 on CIFAR-10. 10 different random direction vectors are used for the filter normalization (Li et al., 2018), to ensure the correctness and consistence of the sharp visualization. ", + "bbox": [ + 174, + 132, + 826, + 175 + ], + "page_idx": 27 + }, + { + "type": "image", + "img_path": "images/56f735a7f4b6c75e7ac5e89a1020bd1535a3b630fb9da8fca9a409378db6769b.jpg", + "image_caption": [ + "Figure 13: Sharpness visualization of the minima for ResNet-20 trained on CIFAR-10. The training is on top of $K = 1 6$ GPUs and the local batch size is fixed to $B _ { \\mathrm { l o c } } = 1 2 8$ . The dashed lines are standard mini-batch SGD and the solid lines are post-local SGD with $H = 1 6$ . The sharpness visualization of minima is performed via filter normalization (Li et al., 2018). The model is perturbed as $\\pmb { w } + \\lambda \\pmb { d }$ by a shared random direction $\\textbf { \\em d }$ , and is evaluated by the whole dataset (training or test respectively). The top-1 test accuracy of mini-batch SGD is 92.25, while that of post-local SGD is 92.61. The sharpness of these two minima is consistent over 10 different random directions. " + ], + "image_footnote": [], + "bbox": [ + 181, + 207, + 815, + 380 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "The spectrum of the Hessian for mini-batch SGD and post-local SGD Figure 14 evaluates the spectrum of the Hessian for the model trained from mini-batch SGD and post-local SGD with different $H$ , which again demonstrates the fact that large-batch SGD tends to stop at points with high Hessian spectrum while post-local SGD could easily generalize to a low curvature solution and with better generalization. ", + "bbox": [ + 174, + 518, + 825, + 588 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "1-d linear interpolation between models The 1-d linear interpolation was first used by Goodfellow et al. (2015) and then widely used to study the “sharpness” and “flatness” of different minima in several works (Keskar et al., 2017; Dinh et al., 2017; Li et al., 2018). In Figure 15, the model trained by post-local SGD (wpost-local SGDq can generalize to flatter minima than that of mini-batch SGD $( { \\pmb w } _ { \\mathrm { m i n i - b a t c h ~ S G D } } )$ , either $\\pmb { w } _ { \\mathrm { p o s t - l o c a l S G D } }$ is trained from scratch, or resumed from the checkpoint of ${ \\pmb w } _ { \\mathrm { m i n i } }$ -batch SGD (i.e., the checkpoint is one-epoch ahead of the first learning rate decay so as to share the common weight structure with ${ \\pmb w } _ { \\mathrm { m i n i } }$ -batch SGD). ", + "bbox": [ + 174, + 612, + 825, + 710 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "C.5 POST-LOCAL SGD TRAINING ON DIVERSE TASKS ", + "text_level": 1, + "bbox": [ + 173, + 733, + 563, + 748 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "C.5.1 POST-LOCAL SGD TRAINING ON CIFAR-100 FOR GLOBAL MINI-BATCH SIZE $K B _ { \\mathrm { L O C } } = 4 0 9 6$ ", + "bbox": [ + 176, + 762, + 772, + 792 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Table 11 presents the severe quality loss (at around $2 \\%$ ) of the fine-tuned large-batch SGD for training three CNNs on CIFAR-100. Our post-local SGD with default hyper-parameters (i.e., the hyper-parameters from small mini-batch size and via large-batch training schemes) can perfectly close the generalization gap or even better the fine-tuned small mini-batch baselines. ", + "bbox": [ + 174, + 804, + 823, + 861 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "We further justify the argument of works (Hoffer et al., 2017; Shallue et al., 2018) in Table 12, where we increase the number of training epochs and train it longer (from 300 to 400 and 500) for ResNet-20 on CIFAR-100. The results below illustrate that increasing the number of training epochs alleviates the optimization difficulty of large-batch training ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 27 + }, + { + "type": "image", + "img_path": "images/899494e264f04e8125ff8d6d4ef1cf6f70ef88b3e3e9f17daeab6f415f43d1c9.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 179, + 112, + 450, + 248 + ], + "page_idx": 28 + }, + { + "type": "image", + "img_path": "images/9ce23628094beb4bdfe8284577096792ecff97ca16db53cb551b4bd0e71b97e2.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 540, + 112, + 810, + 248 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "(a) The dominant eigenvalue of the Hessian, which is evaluated on the training dataset per epoch. Only the phase related to the strategy of post-local SGD is visualized. Mini-batch SGD or post-local SGD with very much $H$ (e.g., $H =$ 2, 4) have noticeably larger dominant eigenvalue. ", + "bbox": [ + 173, + 253, + 465, + 330 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "(b) The dominant eigenvalue of the Hessian, which is evaluated on the test dataset per epoch. Only the phase related to the strategy of post-local SGD is visualized. Mini-batch SGD or post-local SGD with very much $H$ (e.g., $H = 2 , 4 ,$ ) have noticeably larger dominant eigenvalue. ", + "bbox": [ + 532, + 253, + 826, + 330 + ], + "page_idx": 28 + }, + { + "type": "image", + "img_path": "images/94eab45aaaaeb022acb1483eadea58293a151e8081b182b1876d5aebaec7762c.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 179, + 338, + 450, + 476 + ], + "page_idx": 28 + }, + { + "type": "image", + "img_path": "images/35525a6a920d3d571bdd272852ba99a1d4d6dadcc841d69c3bbf7b54cc30fc70.jpg", + "image_caption": [ + "(d) Top 10 eigenvalues of the Hessian, which is evaluated on the test dataset for the best model of each training scheme. The top eigenvalues of the Hessian of the mini-batch SGD clearly present significant different pattern than the post-local SGD counterpart. " + ], + "image_footnote": [], + "bbox": [ + 539, + 338, + 810, + 476 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "(c) Top 10 eigenvalues of the Hessian, which is evaluated on the training dataset for the best model of each training scheme. The top eigenvalues of the Hessian of the mini-batch SGD clearly present significant different pattern than the postlocal SGD counterpart. ", + "bbox": [ + 173, + 482, + 465, + 558 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Figure 14: The spectrum of the Hessian for ResNet-20 trained on CIFAR-10. The training is on top of $K = 1 6$ GPUs with $K B _ { \\mathrm { l o c } } = 2 0 4 8$ . The spectrum is computed using power iteration (Martens & Sutskever, 2012; Yao et al., 2018) with the relative error of 1e-4. The top-1 test accuracy of mini-batch SGD is 92.57, while that of post-local SGD ranges from 92.33 to 93.07. Current large-batch SGD tends to stop at points with considerably “larger” Hessian spectrum, while large-batch trained with post-local SGD generalizes to solution with low curvature and with better generalization. ", + "bbox": [ + 173, + 568, + 825, + 643 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Table 11: Top-1 test accuracy of training different CNN models via post-local SGD on $K = 3 2$ GPUs with a large batch size $\\langle B _ { \\mathrm { l o c } } K = 4 0 9 6 \\rangle$ . The reported results are the average of three runs. We include the small and large batch baseline, where the models are trained by mini-batch SGD with mini-batch size 256 and 4096 respectively. The $\\star$ indicates the fine-tuned learning rate. ", + "bbox": [ + 173, + 651, + 825, + 704 + ], + "page_idx": 28 + }, + { + "type": "table", + "img_path": "images/a7004ed4cd5162e5cf3397966133ac3a3b87b848f3de88bc5a5fd8fe4c720f99.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
CIFAR-100
small batch baseline *large batch baseline *post-local SGD (H=8)post-local SGD (H=16)
ResNet-2068.84±0.0667.34±0.3468.38±0.4868.30±0.30
DenseNet-40-1274.85±0.1473.00±0.0474.50±0.3474.96±0.30
WideResNet-28-1079.78±0.1677.82±0.6579.53±0.4579.80±0.39
", + "bbox": [ + 176, + 715, + 825, + 799 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "C.5.2 POST-LOCAL SGD TRAINING ON LANGUAGE MODELING ", + "text_level": 1, + "bbox": [ + 174, + 827, + 627, + 842 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "We evaluate the effectiveness of post-local SGD for training the language modeling task on WikiText2 through LSTM. We borrowed and adapted the general experimental setup of Merity et al. (2018), where we use a three-layer LSTM with hidden dimension of size 650. The loss will be averaged over all examples and timesteps. The BPTT length is set to 30. We fine-tune the value of gradient clipping (0.4) and the dropout (0.4) is only applied on the output of LSTM. The local mini-batch size $B _ { \\mathrm { l o c } }$ is ", + "bbox": [ + 174, + 853, + 826, + 924 + ], + "page_idx": 28 + }, + { + "type": "image", + "img_path": "images/5819c6ad78152034973b7636e6fa56740e9328f1c73ae25b14ab4c3b3bd166e6.jpg", + "image_caption": [ + "(a) ${ \\pmb w } _ { \\mathrm { p o s t - l o c a l \\ : S G D } }$ is trained with $H = 1 6$ , where the (b) ${ \\pmb w } _ { \\mathrm { p o s t - l o c a l \\ : S G D } }$ is trained with $H = 3 2$ , where the training resumes from the checkpoint of ${ \\pmb w } _ { \\mathrm { m i n i } }$ -batch SGD training resumes from the checkpoint of ${ \\pmb w } _ { \\mathrm { m i n i } }$ -batch SGD which is one-epoch ahead of the first learning rate decay. which is one-epoch ahead of the first learning rate decay. " + ], + "image_footnote": [], + "bbox": [ + 181, + 102, + 816, + 258 + ], + "page_idx": 29 + }, + { + "type": "image", + "img_path": "images/1997d1c83cef97905d107164ab8ee49c83f98a1f8e6d837343d53ac655ee51b9.jpg", + "image_caption": [ + "(c) wpost-local SGD is trained from scratch with $H = 1 6$ . (d) wpost-local SGD is trained from scratch with $H = 3 2$ . ", + "Figure 15: 1-d linear interpolation between models ${ \\pmb w } _ { \\mathrm { p o s t } }$ -local SGD and ${ \\pmb w } _ { \\mathrm { m i n i } }$ -batch SGD, i.e., $\\hat { \\pmb { w } } = \\lambda \\pmb { w } _ { \\mathrm { m i n i - b a t c h ~ S G D } } +$ $( 1 - \\lambda ) w$ post-local SGD, for different minima of ResNet-20 trained on CIFAR-10. The training is on top of $K = 1 6$ GPUs and the local batch size is fixed to $B _ { \\mathrm { l o c } } = 1 2 8$ . The solid lines correspond to evaluate $\\hat { \\mathbf { \\Omega } } \\hat { \\mathbf { \\Omega } } ^ { v }$ on the whole training dataset while the dashed lines are on the test dataset. The post-local SGD in Figure 15(a) and Figure 15(d) is trained from the checkpoint of ${ \\pmb w } _ { \\mathrm { m i n i } }$ -batch SGD before performing the first learning rate decay, while that of Figure 15(c) and Figure 15(d) is trained from scratch. The top-1 test accuracy of mini-batch SGD is 92.25, while that of post-local SGD in Figure 15(a), Figure 15(b), Figure 15(c) and Figure 15(d), are 92.61, 92.35, 93.13 and 93.04 respectively. " + ], + "image_footnote": [], + "bbox": [ + 181, + 320, + 815, + 435 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Table 12: Top-1 test accuracy of large-batch SGD and post-local SGD, for training ResNet-20 on CIFAR-100 with $B _ { \\mathrm { l o c } } K = 4 0 9 6$ . The reported results are the average of three runs. We include the small and large batch baseline, where the models are trained by mini-batch SGD with 256 and 4096 respectively. The $\\star$ indicates the fine-tuned learning rate. The learning rate will be decayed by 10 when the distributed algorithm has accesses $5 0 \\%$ and $7 5 \\%$ of the total number of training samples. ", + "bbox": [ + 173, + 601, + 825, + 665 + ], + "page_idx": 29 + }, + { + "type": "table", + "img_path": "images/aeea1c99752e4dada03728c05424353c0a7572e35185ed74b902f4eb13294b53.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
# of epochssmall batch baseline *large batch baseline *post-local SGD (H=8)post-local SGD (H=16)
30068.84±0.0667.34±0.3468.38±0.4868.30±0.30
40069.07±0.2767.55±0.2169.06±0.1569.05±0.26
50069.03±0.1067.42±0.6369.02±0.3868.87±0.27
", + "bbox": [ + 178, + 676, + 823, + 750 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "64 and we train the model for 120 epochs. The learning rate is again decayed at the phase when the training algorithm has accessed $5 0 \\%$ and $7 5 \\%$ of the total training samples. ", + "bbox": [ + 173, + 790, + 823, + 820 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Table 13 below demonstrates the effectiveness of post-local SGD for large-batch training on language modeling task (without extra fine-tuning except our baselines). Note that most of the existing work focuses on improving the large-batch training issue for computer vision tasks; and it is non-trivial to scale the training of LSTM for language modeling task due to the presence of different hyperparameters. Here we provide a proof of concept result in Table 13 to show that our post-local SGD is able to improve upon standard large-batch baseline. The benefits can be further pronounced if we scale to larger batches (as the case of image classification e.g. in Table 11 and Table 12). ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 29 + }, + { + "type": "table", + "img_path": "images/f56780634bf8c5f41bdad9bb4d3f4e7586a67a1ff0f1c5a8a410209915043f23.jpg", + "table_caption": [ + "Table 13: The perplexity (lower is better) of language modeling task on WikiText-2. We use $K = 1 6$ and $K B = K B _ { \\mathrm { l o c } } = \\bar { 1 } 0 \\bar { 2 } 4$ . The reported results are evaluated on the validation dataset (average of three runs). We fine-tune the learning rate for mini-batch SGD baselines. " + ], + "table_footnote": [], + "table_body": "
small batchbaseline*largebatch baseline*large-batch (H=8)large-batch (H=16)
86.50 ±0.3586.90±0.4986.61 ±0.3086.85 ±0.13
", + "bbox": [ + 207, + 151, + 790, + 180 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "C.5.3 POST-LOCAL SGD TRAINING ON IMAGENET ", + "text_level": 1, + "bbox": [ + 173, + 204, + 542, + 220 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "We evaluate the performance of post-local SGD on the challenging ImageNet training. Again we limit ResNet-50 training to 90 passes over the data in total, and use the standard training configurations as mentioned in Appendix A.4.2. The post-local SGD begins when performing the first learning rate decay. ", + "bbox": [ + 173, + 229, + 825, + 286 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "We can witness that post-local SGD outperforms mini-batch SGD baseline for both of mini-batch size 4096 (76.18 and 75.87 respectively) and 8192 (75.65 and 75.64 respectively). ", + "bbox": [ + 173, + 292, + 823, + 321 + ], + "page_idx": 30 + }, + { + "type": "image", + "img_path": "images/206a4f8eea096e43299613979904d8e667b72510f9e90e43b2306a6518aec158.jpg", + "image_caption": [ + "(a) The performance of post-local SGD training for ImageNet-1k with $\\bar { K } B _ { \\mathrm { l o c } } = 4 0 9 6$ , in terms of epoch-to-accuracy. " + ], + "image_footnote": [], + "bbox": [ + 178, + 344, + 450, + 517 + ], + "page_idx": 30 + }, + { + "type": "image", + "img_path": "images/02c4fc777dc6d6d3646192bbed8abe095bb1199efe20586b322313c7696bf6c5.jpg", + "image_caption": [ + "(b) The performance of post-local SGD training for ImageNet-1k with $\\bar { K } B _ { \\mathrm { l o c } } = 8 1 9 2$ , in terms of epoch-to-accuracy. " + ], + "image_footnote": [], + "bbox": [ + 537, + 344, + 810, + 517 + ], + "page_idx": 30 + }, + { + "type": "image", + "img_path": "images/99ddb42556cf29104400f4991e0b8cae12e7ddcbeeaa9535e7b0066b625c8677.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 178, + 570, + 450, + 742 + ], + "page_idx": 30 + }, + { + "type": "image", + "img_path": "images/ebf9a94487094d0256fe1bd57a81bc3539b2d27a32b6cc54b0214d15ec7cfc4b.jpg", + "image_caption": [ + "(d) The performance of post-local SGD training for ImageNet-1k with $K B _ { \\mathrm { l o c } } = 8 1 9 2$ , in terms of time-to-accuracy. " + ], + "image_footnote": [], + "bbox": [ + 537, + 570, + 810, + 742 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "(c) The performance of post-local SGD training for ImageNet-1k with $K B _ { \\mathrm { l o c } } ~ = ~ 4 0 9 6$ , in terms of time-to-accuracy. ", + "bbox": [ + 173, + 748, + 464, + 786 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Figure 16: The performance of post-local SGD training for ImageNet-1k. We evaluate the model performance on test dataset after each complete accessing of the whole training samples. Note that due to the resource limitation of the main experimental platform in the paper, these experiments are on top of a $8 \\times 4$ -GPU (V100) Kubernetes cluster with 10 Gbps network bandwidth. ", + "bbox": [ + 173, + 795, + 825, + 847 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "C.5.4 POST-LOCAL SGD VS. OTHER NOISE INJECTION METHODS ", + "text_level": 1, + "bbox": [ + 174, + 871, + 648, + 886 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "The role of “noise” has been actively studied in SGD for non-convex deep learning, from optimization and generalization aspects. Neelakantan et al. (2015) propose to inject isotropic white noise for better optimization. Zhu et al. (2019); Xing et al. (2018) study the “structured” anisotropic noise and find the importance of anisotropic noise in SGD for escaping from minima (over isotropic noise) in terms of generalization. Wen et al. (2019) on top of these work and try to inject noise (sampled from the expensive empirical Fisher matrix) to large-batch SGD. ", + "bbox": [ + 173, + 895, + 820, + 924 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "However, to our best knowledge, none of the prior work can provide a computation efficient way to inject noise to achieve as good generalization performance as small-batch SGD. Either it is practically unknown if injecting the isotropic noise (Neelakantan et al., 2015) can alleviate the issue of large-batch training, or it has been empirically justified (Wen et al., 2019) that injecting anisotropic noise from expensive (diagonal) empirical Fisher matrix fails to recover the same performance as the small mini-batch baselines. In this section, we only evaluate the impact of injecting isotropic noise (Neelakantan et al., 2015) for large-batch training, and omit the comparison to Wen et al. $( 2 0 1 9 ) ^ { 8 }$ . ", + "bbox": [ + 173, + 166, + 825, + 277 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "The idea of Neelakantan et al. (2015) considers to add time-dependent Gaussian noise to the gradient at every training step $t$ : $\\nabla f _ { i } \\big ( \\mathbf { \\dot { w } } _ { ( t ) } \\big ) ^ { \\cdot } \\gets \\nabla f _ { i } \\big ( \\mathbf { w } _ { ( t ) } \\big ) + \\mathcal { N } ( 0 , \\dot { \\pmb { \\sigma } } _ { t } ^ { 2 } )$ , where $\\pmb { \\sigma } _ { t } ^ { 2 }$ follows $\\begin{array} { r } { \\sigma _ { t } ^ { 2 } : = \\frac { \\eta } { ( 1 + t ) ^ { \\gamma ^ { \\prime } } } } \\end{array}$ We follow the general hyper-parameter search scheme (as mentioned in Section A.4) and fine-tune the $\\eta$ and $\\gamma ^ { \\prime }$ in the range of $\\{ 1 e ^ { - 6 } , 5 e ^ { - 6 } , 1 e ^ { - 5 } , 5 e ^ { - 5 } \\}$ and $\\{ 0 . 5 , 1 , 1 . 5 , 2 , 2 . 5 , 3 . 0 \\}$ respectively. ", + "bbox": [ + 174, + 285, + 825, + 344 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Table 14 below compares the post-local SGD to Neelakantan et al. (2015), where the noise injection scheme in Neelakantan et al. (2015) cannot address the large-batch training issue and could even deteriorate the generalization performance. Note that we also tried to search the hyper-parameters at around the values reported in Neelakantan et al. (2015) for our state-of-the-art CNNs, but it will directly result in the severe quality loss or even divergence. ", + "bbox": [ + 174, + 351, + 825, + 421 + ], + "page_idx": 31 + }, + { + "type": "table", + "img_path": "images/4d590cb5a81f53bedc2244e03d4c40f4dcbb13fe029f352de0c4e225d2fbf5d5.jpg", + "table_caption": [ + "Table 14: Top-1 test accuracy of adding isotropic noise for large-batch training. We revisit the case of ResNet-20 in Table 3 (2K for CIFAR-10) and Table 11 (4K for CIFAR-100), where adding isotropic noise as in Neelakantan et al. (2015) cannot address the large-batch training difficulty. The $\\star$ indicates a fine-tuned learning rate. " + ], + "table_footnote": [], + "table_body": "
CIFAR-10,KB= 2KCIFAR-100,KB= 4K
Neelakantan et al. (2015)*92.46 ±0.2767.15 ±0.26
mini-batch SGD *92.48 ±0.1767.38 ±0.34
Post-local SGD93.02 ±0.2468.30 ±0.48
", + "bbox": [ + 236, + 502, + 759, + 570 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Compared to another local SGD variant. Recently, Wang & Joshi (2019) proposed to decrease the number of local update steps $H$ during training. However, their scheme is inherited from the convergence analysis for optimization (not the generalization for deep learning), and in principle opposite to our interpolation and proposed strategy (i.e. increasing the local update steps during the training). ", + "bbox": [ + 174, + 603, + 825, + 674 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Their evaluation (ResNet-50 on CIFAR-10 for $K = 4$ with $B _ { \\mathrm { l o c } } = 1 2 8 )$ also does not cover the difficult large batch training scenario (Scenario 2), e.g., $H = 1 6$ , $K = 1 6$ , $B _ { \\mathrm { l o c } } = 1 2 8$ . For the same CIFAR-10 task and $K { = } 4$ as in Wang & Joshi (2019), our smaller ResNet-20 with local SGD can simply reach a better accuracy with less communication9 (Figure 2). ", + "bbox": [ + 174, + 680, + 825, + 736 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "C.5.5 POST-LOCAL SGD WITH OTHER COMPRESSION SCHEMES ", + "text_level": 1, + "bbox": [ + 176, + 757, + 632, + 771 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "In this subsection, we demonstrate that (post-)local SGD can be integrated with other compression techniques for better training efficiency (further reduced communication cost) and improved generalization performance (w.r.t. the gradient compression methods). ", + "bbox": [ + 174, + 784, + 825, + 825 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "We use sign-based compression scheme (i.e. signSGD (Bernstein et al., 2018) and EFsignSGD (Karimireddy et al., 2019)) for the demonstration. The pseudo code can be found in Algorithm 3 and Algorithm 4. ", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Post-local SGD with signSGD. We slightly adapt the original signSGD (Bernstein et al., 2018) to fit in the local SGD framework. The local model will be firstly updated by the sign of the local update directions (e.g. gradients, or gradients with weight decay and momentum acceleration), and then be synchronized to reach the global consensus model for the next local updates. ", + "bbox": [ + 174, + 162, + 825, + 218 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Note that we can almost recover the signSGD in Bernstein et al. (2018) from Algorithm 3 when $H ^ { \\prime } = 1$ , except that we will average over the sign instead of using the majority vote in Bernstein et al. (2018). Table 15 illustrates the trivial generalization performance difference of these two schemes, where the post-local SGD in Algorithm 3 is able to significantly improve the generalization performance (as in Table 4) with further improved communication efficiency. ", + "bbox": [ + 173, + 226, + 825, + 296 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "input: the initial model ${ \\pmb w } _ { ( 0 ) } \\in \\mathbb { R } ^ { d }$ ; training data with labels $\\mathcal { T }$ ; mini-batch of size $B _ { \\mathrm { l o c } }$ per local model; step \nsize $\\eta$ , and momentum $_ { \\pmb { m } }$ (optional); number of synchronization steps $T$ , and the first learning rate decay is \nperformed at $T ^ { \\prime }$ ; number of eventual local steps $\\dot { H } ^ { \\prime }$ ; number of nodes $K$ . \n1: synchronize to have the same initial models $\\hat { w } _ { ( 0 ) } ^ { k } : = w _ { ( 0 ) }$ . \n2: for all $k : = 1 , \\ldots , K$ do in parallel \n3: for $t : = 1 , \\dots , T$ do \n4: if $t < T ^ { \\prime }$ then \n5: $H _ { ( t ) } = 1$ \n6: else \n7: $H _ { ( t ) } = H ^ { \\prime }$ \n8: end if \n9: for $h : = 1 , \\ldots , H _ { ( t ) }$ do \n10: sample a mini-batch from $\\mathcal { T } _ { ( t ) + h - 1 } ^ { k }$ \n11: compute the gradients $\\begin{array} { r } { \\pmb { g } _ { ( t ) + h - 1 } ^ { k } : = \\frac { 1 } { B _ { \\mathrm { l o c } } } \\sum _ { i \\in \\mathbb { Z } _ { ( t ) + h - 1 } ^ { k } } \\nabla f _ { i } \\big ( \\pmb { w } _ { ( t ) + h - 1 } ^ { k } \\big ) } \\end{array}$ . ™ can involve weight \ndecay and momentum. \n12: update the local model $\\begin{array} { r } { \\pmb { w } _ { ( t ) + h } ^ { k } : = \\pmb { w } _ { ( t ) + h - 1 } - \\gamma _ { ( t ) } \\mathrm { s i g n } ( \\pmb { g } _ { ( t ) + h - 1 } ^ { k } ) . } \\end{array}$ . \n13: end for \n14: get model difference $\\Delta _ { ( t ) } ^ { k } : = \\boldsymbol { w } _ { ( t ) } ^ { k } - \\boldsymbol { w } _ { ( t ) + H } ^ { k }$ . \n15: compress the model difference: $\\begin{array} { r } { \\pmb { s } _ { ( t ) } ^ { k } = \\mathrm { s i g n } ( \\Delta _ { ( t ) } ^ { k } ) } \\end{array}$ and $\\begin{array} { r } { p _ { ( t ) } ^ { k } = \\frac { \\left\\| \\Delta _ { ( t ) } ^ { k } \\right\\| _ { 1 } } { d } } \\end{array}$ \n16: ptq get new global (synchronized) model $\\pmb { w } _ { ( t + 1 ) } ^ { k }$ ptq for all ptq nodes: $\\begin{array} { r } { \\pmb { w } _ { ( t + 1 ) } ^ { k } : = \\pmb { w } _ { ( t ) } ^ { k } - \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\pmb { s } _ { ( t ) } ^ { k } \\pmb { p } _ { ( t ) } ^ { k } , } \\end{array}$ . \n17: end for \n18: end for ", + "bbox": [ + 173, + 330, + 826, + 638 + ], + "page_idx": 32 + }, + { + "type": "table", + "img_path": "images/a7440cc6f7266dbed27a191bed1740ba313e1af216bacf15b5b04935bb4d7d4c.jpg", + "table_caption": [ + "Algorithm 3 (Post-)Local SGD with the compression scheme in signSGD ", + "Table 15: Top-1 test accuracy of training ResNet-20 on CIFAR $( K B _ { \\mathrm { l o c } } = 2 0 4 8$ and $K = 1 6$ ). " + ], + "table_footnote": [], + "table_body": "
CIFAR-10CIFAR-100
signSGD (average over signs)91.61 ±0.2867.15 ±0.10
signSGD (majority vote over signs)91.61 ±0.2667.21 ±0.11
", + "bbox": [ + 299, + 679, + 694, + 728 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Post-local SGD with EF-signSGD. We extend the single-worker EF-signSGD algorithm of Karimireddy et al. (2019) (i.e. their Algorithm 1) to multiple-workers case by empirically investigating different algorithmic design choices. Our presented Algorithm 4 of EF-signSGD for multiple-workers can reach a similar performance as the mini-batch SGD counterpart (both are after the proper hyper-parameters tuning and under the same experimental setup). ", + "bbox": [ + 173, + 752, + 825, + 823 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Training schemes, hyper-parameter tuning procedure for signSGD, signSGD variant in Algorithm 3, and EF-signSGD in Algorithm 4. The training schemes of the signSGD (Bernstein et al., 2018) (i.e. the one using majority vote) and the signSGD variant in Algorithm 3 are slightly different from the the experimental setup mentioned in A.4, where we only fine-tune the initial learning rate in signSGD and Algorithm 3 (when $H = 1$ ). The optimal learning rate is searched from the grid $\\{ 0 . 0 \\bar { 0 } 5 , 0 . 1 0 , 0 . 1 5 , \\bar { 0 . 2 0 } \\}$ with the general fine-tuning principle mentioned in A.4. No learning rate ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "input: the initial model ${ \\pmb w } _ { ( 0 ) } \\in \\mathbb { R } ^ { d }$ ; training data with labels $\\mathcal { T }$ ; mini-batch of size $B _ { \\mathrm { l o c } }$ per local model; step size $\\eta$ , and momentum $_ { \\mathbf { \\nabla } } \\mathbf { m } _ { \\mathbf { \\nabla } }$ (optional); number of synchronization steps $T$ , and the first learning rate decay is performed at $T ^ { \\prime }$ ; number of eventual local steps $H ^ { \\prime }$ ; number of nodes $K$ . error-compensated memory ep0q \u0010 0 P Rd. \n1: synchronize to have the same initial models $\\pmb { w } _ { ( 0 ) } ^ { k } : = \\pmb { w } _ { ( 0 ) }$ . \n2: initialize the local error memory $e _ { ( 0 ) } ^ { k } : = e _ { ( 0 ) }$ . \n3: for all $k : = 1 , \\ldots , K$ do in parallel \n4: for $t : = 1 , \\dots , T$ do \n5: if $t < T ^ { \\prime }$ then \n6: $H _ { ( t ) } = 1$ \n7: else \n8: $H _ { ( t ) } = H ^ { \\prime }$ \n9: end if \n10: 11: for s $h : = 1 , \\ldots , H _ { ( t ) }$ doch from $\\mathcal { T } _ { ( t ) + h - 1 } ^ { k }$ \n12: compute the gradient $\\begin{array} { r } { \\pmb { g } _ { ( t ) + h - 1 } ^ { k } : = \\frac { 1 } { B _ { \\mathrm { l o c } } } \\sum _ { i \\in \\mathcal { T } _ { ( t ) + h - 1 } ^ { k } } \\nabla f _ { i } \\left( \\pmb { w } _ { ( t ) + h - 1 } ^ { k } \\right) . } \\end{array}$ . ™ can involve weight decay and momentum. \n13: update the local model wkptq\u0000h :\u0010 wptq\u0000h\u00011 \u0001 γptqgkptq\u0000h\u00011 . \n14: end for \n15: get model difference $\\Delta _ { ( t ) } ^ { k } : = { \\pmb w } _ { ( t ) } ^ { k } - { \\pmb w } _ { ( t ) + H } ^ { k } + { \\pmb e } _ { ( t ) }$ . \n16: compress the model difference: $\\begin{array} { r } { \\pmb { s } _ { ( t ) } ^ { k } = \\mathrm { s i g n } ( \\Delta _ { ( t ) } ^ { k } ) } \\end{array}$ and $\\begin{array} { r } { p _ { ( t ) } ^ { k } = \\frac { \\left\\| \\Delta _ { ( t ) } ^ { k } \\right\\| _ { 1 } } { d } } \\end{array}$ \n17: update local memory: $\\pmb { e } _ { ( t + 1 ) } ^ { k } = \\pmb { e } _ { ( t ) } ^ { k } - \\pmb { s } _ { ( t ) } ^ { k } \\pmb { p } _ { ( t ) } ^ { k }$ . \n18: get new global (synchronized) model $\\pmb { w } _ { ( t + 1 ) } ^ { k }$ for all $K$ nodes: $\\begin{array} { r } { \\pmb { w } _ { ( t + 1 ) } ^ { k } : = \\pmb { w } _ { ( t ) } ^ { k } - \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\pmb { s } _ { ( t ) } ^ { k } \\pmb { p } _ { ( t ) } ^ { k } } \\end{array}$ . \n19: end for \n20: end for ", + "bbox": [ + 174, + 123, + 826, + 477 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "scaling up or warming up is required during the training, and the learning rate will be decayed by 10 when accessing $5 0 \\%$ and $7 5 \\%$ of the total training samples. ", + "bbox": [ + 173, + 502, + 823, + 531 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "The training schemes of the distributed EF-signSGD in Algorithm 4 in general follow the experimental setup mentioned in A.4. We grid search the optimal initial learning rate for Algorithm 4 (for the case of $H = 1$ ), and gradually warm up the learning rate from a relatively small value (0.1) to this found initial learning rate during the first 5 epochs of the training. ", + "bbox": [ + 174, + 537, + 825, + 594 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Note that in our experiments, weight decay and Nesterov momentum are used for the local model updates, for both of Algorithm 3 and Algorithm 4. We empirically found these two techniques can significantly improve the training/test performance under a fixed training epoch budget. ", + "bbox": [ + 174, + 601, + 825, + 643 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "D HIERARCHICAL LOCAL SGD ", + "text_level": 1, + "bbox": [ + 176, + 102, + 450, + 118 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "The idea of local SGD can be leveraged to the more general setting of training on decentralized and heterogeneous systems, which is an increasingly important application area. Such systems have become common in the industry, e.g. with GPUs or other accelerators grouped hierarchically within machines, racks or even at the level of several data-centers. Hierarchical system architectures such as in Figure 17 motivate our hierarchical extension of local SGD. Moreover, end-user devices such as mobile phones form huge heterogeneous networks, where the benefits of efficient distributed and data-local training of machine learning models promises strong benefits in terms of data privacy. ", + "bbox": [ + 173, + 135, + 825, + 232 + ], + "page_idx": 34 + }, + { + "type": "image", + "img_path": "images/3538fafbeb5d128b9d18ce9102e2d7892ea69231d67457ad406a943dcc6f4125.jpg", + "image_caption": [ + "Figure 17: Illustration of a hierarchical network architecture of a cluster in the data center. While GPUs within each node are linked with fast connections (e.g. NVLink), connections between the servers within and between different racks have much lower bandwidth and latency (via top-of-the-rack switches and cluster switches). The hierarchy can be extended several layers further and further. Finally, edge switches face the external network at even lower bandwidth. " + ], + "image_footnote": [], + "bbox": [ + 303, + 251, + 687, + 388 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "D.1 THE ILLUSTRATION OF HIERARCHICAL LOCAL SGD ", + "text_level": 1, + "bbox": [ + 173, + 501, + 583, + 515 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Real world systems come with different communication bandwidths on several levels. In this scenario, we propose to employ local SGD on each level of the hierarchy, adapted to each corresponding computation vs communication trade-off. The resulting scheme, hierarchical local SGD, can offer significant benefits in system adaptivity and performance. ", + "bbox": [ + 174, + 527, + 826, + 583 + ], + "page_idx": 34 + }, + { + "type": "image", + "img_path": "images/d7126f45b6310422bb1ec03c9d49c6a9d13b2daf71a9244f3efb43ed60e776b7.jpg", + "image_caption": [ + "Figure 18: An illustration of hierarchical local SGD, for $B _ { \\mathrm { l o c } } = 2$ , using $H = 3$ inner local steps and $H ^ { b } = 2$ outer ‘block’ steps. Local parameter updates are depicted in red, whereas block and global synchronization is depicted in purple and black respectively. " + ], + "image_footnote": [], + "bbox": [ + 385, + 601, + 609, + 800 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "As the guiding example, we consider compute clusters which typically allocate a large number of GPUs grouped over several machines, and refer to each group as a GPU-block. Hierarchical local SGD continuously updates the local models on each GPU for a number of $H$ local update steps before a (fast) synchronization within a GPU-block. On the outer level, after $H ^ { b }$ such block update steps, a (slower) global synchronization over all GPU-blocks is performed. Figure 18 and Algorithm 5 depict how the hierarchical local SGD works, and the complete procedure is formalized below: ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/be83b0b4004d77ab70091e380d424ed7c4347ae1689b9a88ca4efd951d12aa62.jpg", + "text": "$$\n\\begin{array} { r l } & { w _ { [ ( t ) + l ] + H } ^ { k } : = w _ { [ ( t ) + l ] } ^ { k } - \\displaystyle \\sum _ { h = 1 } ^ { H } \\frac { \\gamma _ { [ ( t ) ] } } { B _ { \\operatorname* { l o c } } } \\cdot \\sum _ { \\nabla _ { [ ( t ) + l ] + h - 1 } } \\nabla f _ { i } \\big ( w _ { [ ( t ) + l ] + h - 1 } ^ { k } } \\\\ & { } \\\\ & { w _ { [ ( t ) + l + 1 ] } ^ { k } : = w _ { [ ( t ) + l ] } ^ { k } - \\displaystyle \\frac { K _ { i } } { K _ { i } } \\sum _ { k = 1 } ^ { K _ { i } } \\big ( w _ { [ ( t ) + l ] } ^ { k } - w _ { [ ( t ) + l ] + H } ^ { k } \\big ) } \\\\ & { \\qquad w _ { [ ( t + 1 ) ] } ^ { k } : = w _ { [ ( t ) ] } ^ { k } - \\displaystyle \\frac { K } { K } \\sum _ { k = 1 } ^ { K } \\big ( w _ { [ ( t ) ] } ^ { k } - w _ { [ ( t ) + H ^ { k } ] } ^ { k } \\big ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 130, + 691, + 252 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "where ${ \\pmb w } _ { [ ( t ) + l ] + H } ^ { k }$ indicates the model after $l$ k\u00101 block update steps and $H$ local update steps, and $K _ { i }$ is rptq\u0000ls\u0000Hthe number of GPUs on the GPU-block $i$ . The definition of $\\gamma _ { [ ( t ) ] }$ and $\\mathcal { T } _ { [ ( t ) + l ] + h - 1 } ^ { k }$ follows a similar scheme. ", + "bbox": [ + 173, + 250, + 826, + 299 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "As the number of devices grows to the thousands (Goyal et al., 2017; You et al., 2017b), the difference between ‘within’ and ‘between’ block communication efficiency becomes more drastic. Thus, the performance benefits of our adaptive scheme compared to flat & large mini-batch SGD will be even more pronounced. ", + "bbox": [ + 173, + 304, + 825, + 361 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "D.2 THE ALGORITHM OF HIERARCHICAL LOCAL SGD ", + "bbox": [ + 173, + 377, + 570, + 392 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Algorithm 5 Hierarchical Local SGD ", + "text_level": 1, + "bbox": [ + 174, + 256, + 408, + 268 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "input: the initial model wrp0qs; \ninput: training data with labels $\\mathcal { T }$ ; \ninput: mini-batch of size $B _ { \\mathrm { l o c } }$ per local model; \ninput: step size $\\eta$ , and momentum $m$ (optional); \ninput: number of synchronization steps $T$ over nodes; \ninput: number of local update steps $H$ , and block update steps $H ^ { b }$ ; \ninput: number of nodes $K$ in total; and nodes $K ^ { \\prime }$ per GPU-block. \n1: synchronize to have the same initial models wkrp0qs :\u0010 wrp0qs. \n2: for all $k : = 1 , \\ldots , K$ do in parallel \n3: for $t : = 1 , \\dots , T$ do \n4: for $l : = 1 , \\ldots , H ^ { b } \\ : .$ do \n5: 6: for s $h : = 1 , \\ldots , H$ doatch from . $\\mathscr { T } _ { [ ( t ) + l ] + h - 1 } ^ { k }$ \n7: compute the gradient $\\pmb { g } _ { [ ( t ) + l ] + h - 1 } ^ { k } : = \\frac { 1 } { B _ { \\mathrm { l o c } } } \\sum _ { i \\in \\mathcal { T } _ { [ ( t ) + l ] + h - 1 } ^ { k } } \\nabla f _ { i } \\left( \\pmb { w } _ { [ ( t ) + l ] + h - 1 } ^ { k } \\right) .$ \n8: update the local model $\\pmb { w } _ { [ ( t ) + l ] + h } ^ { k } : = \\pmb { w } _ { [ ( t ) + l ] + h - 1 } ^ { k } - \\gamma _ { [ ( t ) ] } \\pmb { g } _ { [ ( t ) + l ] + h - 1 } ^ { k } .$ \n9: end for \n10: inner all-reduce aggregation o ${ \\Delta } _ { [ ( t ) + l ] } ^ { k } : = { \\pmb w } _ { [ ( t ) + l ] } ^ { k } - { \\pmb w } _ { [ ( t ) + l ] + H } ^ { k } .$ \n11: get new block (synchronized) model $\\boldsymbol { w } _ { [ ( t ) + l + 1 ] } ^ { k }$ for $K ^ { \\prime }$ block nodes: $\\begin{array} { r } { \\pmb { w } _ { [ ( t ) + l + 1 ] } ^ { k } : = \\pmb { w } _ { [ ( t ) + l ] } ^ { k } - \\gamma _ { [ ( t ) ] } \\frac { 1 } { K ^ { \\prime } } \\sum _ { k = 1 } ^ { K ^ { \\prime } } \\Delta _ { [ ( t ) + l ] } ^ { k } , } \\end{array}$ \n12: end for \n13: outer all-reduce aggregation of the gradients $\\Delta _ { [ ( t ) ] } ^ { k } : = w _ { [ ( t ) ] } ^ { k } - w _ { [ ( t ) + H ^ { b } ] } ^ { k } .$ \n14: get new global (synchronized) model $\\pmb { w } _ { [ ( t + 1 ) ] } ^ { k }$ for all $K$ nodes: $\\begin{array} { r } { \\pmb { w } _ { [ ( t + 1 ) ] } ^ { k } : = \\pmb { w } _ { [ ( t ) ] } ^ { k } - \\gamma _ { [ ( t ) ] } \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\Delta _ { [ ( t ) ] } ^ { k } . } \\end{array}$ \n15: end for \n16: end for ", + "bbox": [ + 173, + 275, + 720, + 772 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "D.3 HIERARCHICAL LOCAL SGD TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 103, + 495, + 117 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Now we move to our proposed training scheme for distributed heterogeneous systems. In our experimental setup, we try to mimic the real world setting where several compute devices such as GPUs are grouped over different servers, and where network bandwidth (e.g. Ethernet) limits the communication of updates of large models. The investigation of hierarchical local SGD again trains ResNet-20 on CIFAR-10 and follows the same training procedure as local SGD where we re-formulate below. ", + "bbox": [ + 174, + 130, + 825, + 213 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "The experiments follow the common mini-batch SGD training scheme for CIFAR (He et al., 2016a;b) and all competing methods access the same total amount of data samples regardless of the number of local steps or block steps. More precisely, the training procedure is terminated when the distributed algorithms have accessed the same number of samples as a standalone worker would access in 300 epochs. The data is partitioned among the GPUs and reshuffled globally every epoch. The local mini-batches are then sampled among the local data available on each GPU. The learning rate scheme is the same as in He et al. (2016a), where the initial learning rate starts from 0.1 and is divided by 10 when the model has accessed $5 0 \\%$ and $7 5 \\%$ of the total number of training samples. In addition to this, the momentum parameter is set to 0.9 without dampening and applied independently to each local model. ", + "bbox": [ + 174, + 219, + 825, + 359 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "D.3.1 THE PERFORMANCE OF HIERARCHICAL LOCAL SGD. ", + "text_level": 1, + "bbox": [ + 174, + 376, + 604, + 390 + ], + "page_idx": 37 + }, + { + "type": "table", + "img_path": "images/9ac7d93e96bfff28941dbf476fd3f937de0780bb304f8aec034a0d3b388b1459.jpg", + "table_caption": [ + "Table 16: Training CIFAR-10 with ResNet-20 via local SGD on a $8 \\times 2$ -GPU cluster. The local batch size $\\scriptstyle B _ { \\mathrm { l o c } }$ is fixed to 128 with $H ^ { b } = 1$ , and we scale the number of local steps $H$ from 1 to 1024. The reported training times are the average of three runs and all the experiments are under the same training configurations for the equivalent of 300 epochs, without specific tuning. " + ], + "table_footnote": [], + "table_body": "
H=12481632641282565121024
Training Time (minutes)20.0713.9510.489.208.578.329.229.239.5010.3010.65
", + "bbox": [ + 178, + 468, + 825, + 503 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Training time vs. local number of steps. Table 16 shows the performance of local SGD in terms of training time. The communication traffic comes from the global synchronization over 8 nodes, each having 2 GPUs. We can witness that increasing the number of local steps over the “datacenter” scenario cannot infinitely improve the communication performance, or would even reduce the communication benefits brought by a large number of local steps. Hierarchical local SGD with inner node synchronization reduces the difficulty of synchronizing over the complex heterogeneous environment, and hence enhances the overall system performance of the synchronization. The benefits are further pronounced when scaling up the cluster size. ", + "bbox": [ + 173, + 529, + 825, + 642 + ], + "page_idx": 37 + }, + { + "type": "image", + "img_path": "images/682e3dc133ddf37adafdd03e87df61f021ce1ea4af19f7b4ece9007081d58d7e.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 395, + 661, + 596, + 790 + ], + "page_idx": 37 + }, + { + "type": "image", + "img_path": "images/11ecc8f309954027e21251edaf60f4f87a0ca10f76b8767525ecd8a9460cea30.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 611, + 661, + 812, + 790 + ], + "page_idx": 37 + }, + { + "type": "image", + "img_path": "images/dbb1f6c8723df4d8bdca876d2a4d5f406d6a2712b24ae3753143a7258703c482.jpg", + "image_caption": [ + "e (b) Training accuracy vs. time. The (c) Training accuracy vs. time. The number of local steps is $H = 2$ , with number of local steps is $H = 2$ , with 1 second delay for each global syn- 50 seconds delay for each global synchronization. chronization. " + ], + "image_footnote": [], + "bbox": [ + 179, + 661, + 379, + 790 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "(a) Training accuracy vs. time. T number of local steps is $H = 2$ . ", + "bbox": [ + 173, + 795, + 367, + 820 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Figure 19: The performance of hierarchical local SGD trained on CIFAR-10 with ResNet-20 ( $2 \\times 2$ -GPU). Each GPU block of the hierarchical local SGD has 2 GPUs, and we have 2 blocks in total. Each figure fixes the number of local steps but varies the number of block steps from 1 to 32. All the experiments are under the same training configurations without specific tuning. ", + "bbox": [ + 173, + 861, + 825, + 911 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Hierarchical local SGD shows high tolerance to network delays. Even in our small-scale experiment of two servers and each with two GPUs, hierarchical local SGD shows its ability to significantly reduce the communication cost by increasing the number of block step $H ^ { b }$ (for a fixed $H$ ), with trivial performance degradation. Moreover, hierarchical local SGD with a sufficient number of block steps offers strong robustness to network delays. For example, for fixed $H = 2$ , by increasing the number of $H ^ { b }$ , i.e. reducing the number of global synchronizations over all models, we obtain a significant gain in training time as in Figure 19(a). The impact of a network of slower communication is further studied in Figure 19(b), where the training is simulated in a realistic scenario and each global communication round comes with an additional delay of 1 second. Surprisingly, even for the global synchronization with straggling workers and severe 50 seconds delay per global communication round, Figure 19(c) demonstrates that a large number of block steps (e.g. $H ^ { b } = 1 6$ ) still manages to fully overcome the communication bottleneck with no/trivial performance damage. ", + "bbox": [ + 174, + 103, + 825, + 270 + ], + "page_idx": 38 + }, + { + "type": "table", + "img_path": "images/bf4eae3cb1893aed237e10b9b0284ae3ee3b31167a999f035b9664a2d8fa56ae.jpg", + "table_caption": [ + "Table 17: The performance of training CIFAR-10 with ResNet-20 via hierarchical local SGD on a 16-GPU Kubernetes cluster. We simulate three different types of cluster topology, namely 8 nodes with 2 GPUs/node, 4 nodes with 4 GPUs/node, and 2 nodes with 8 GPUs/node. The configuration of hierarchical local SGD satisfies $H \\cdot H ^ { b } = 1 6$ . All variants either synchronize within each node or over all GPUs, and the communication cost is estimated by only considering $\\dot { H ^ { \\cdot } } H ^ { b } = 1 6$ model updates during the training (the update could come from a different level of the synchronizations). The reported results are the average of three runs and all the experiments are under the same training configurations, training for the equivalent of 300 epochs, without specific tuning. " + ], + "table_footnote": [], + "table_body": "
H=1, H=16H=2, H=8H=4, H=4H=8, H=2H= 16, H=1
# of sync.over nodes11111
# of sync.within node157310
Test acc.on 8 × 2-GPU90.02 ±0.2890.2589.9591.4191.18±0.02
Test acc.on 4 × 4-GPU91.65±0.08 91.26±0.19 91.46±0.23 91.91
±0.06±0.17±0.24
±0.16
Test acc.on 2 × 8-GPU92.1492.0591.9491.56
±0.10±0.14±0.09±0.18
", + "bbox": [ + 173, + 387, + 825, + 535 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Hierarchical local SGD offers improved scaling and better test accuracy. Table 17 compares the mini-batch SGD with hierarchical local SGD for fixed product $H \\cdot H ^ { b } = 1 6$ under different network topologies, with the same training procedure. We can observe that for a heterogeneous system with a sufficient block size, hierarchical local SGD with a sufficient number of block update steps can further improve the generalization performance of local SGD training. More precisely, when ${ \\dot { H } } { \\cdot } H ^ { b }$ is fixed, hierarchical local SGD with more frequent inner-node synchronizations $\\cdot H ^ { b } > 1 \\rangle$ ) outperforms local SGD ( $\\begin{array} { r } { H ^ { b } = 1 } \\end{array}$ ), while still maintaining the benefits of significantly reduced communication by the inner synchronizations within each node. In summary, as witnessed by Table 17, hierarchical local SGD outperforms both local SGD and mini-batch SGD in terms of training speed as well as model performance, especially for the training across nodes where inter-node connection is slow but intra-node communication is more efficient. ", + "bbox": [ + 173, + 561, + 825, + 714 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "E COMMUNICATION SCHEMES ", + "text_level": 1, + "bbox": [ + 176, + 737, + 442, + 752 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "This section evaluates the communication cost in terms of the number of local steps and block steps, and formalizes the whole communication problem below. ", + "bbox": [ + 173, + 770, + 823, + 797 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Assume $K$ computing devices uniformly distributed over $K ^ { \\prime }$ servers, where each server has $\\frac { K } { K ^ { \\prime } }$ devices. The hierarchical local SGD training procedure will access $N$ total samples with local mini-batch size $B$ , with $H$ local steps and $H ^ { b }$ block steps. ", + "bbox": [ + 174, + 804, + 821, + 847 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "The MPI communication scheme (Gropp et al., 1999) is introduced for communication cost evaluation. More precisely, we use general all-reduce, e.g., recursive halving and doubling algorithm (Thakur et al., 2005; Rabenseifner, 2004), for gradient aggregation among $K$ computation devices. For each all-reduce communication, it introduces $C \\cdot \\log _ { 2 } K$ communication cost, where $C$ is the message transmission time plus network latency. ", + "bbox": [ + 176, + 853, + 825, + 924 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "The communication cost under our hierarchical local SGD setting is mainly determined by the number of local steps and block steps. The $\\textstyle { \\frac { K } { T } }$ models within each server synchronize the gradients for every $H$ local mini-batch, and it only performs global gradients aggregation of $K$ local models after $H ^ { b }$ block updates. Thus, the total number of synchronizations among compute devices is reduced to $\\Big \\lceil \\frac { N } { K B \\cdot H H ^ { b } } \\Big \\rceil$ , and we can formulate the overall communication cost $\\tilde { C }$ as: ", + "bbox": [ + 173, + 103, + 825, + 179 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/722a911482635b9df77720159abae16b40ad3e9440685998d9cda53a0283811d.jpg", + "text": "$$\n\\begin{array} { r } { \\tilde { C } \\approx \\left( \\left\\lceil \\frac { N } { K B \\cdot H } \\right\\rceil - \\left\\lceil \\frac { N } { K B \\cdot H H ^ { b } } \\right\\rceil \\right) \\cdot C _ { 1 } \\cdot K ^ { \\prime } \\log _ { 2 } \\frac { K } { K ^ { \\prime } } + \\left\\lceil \\frac { N } { K B \\cdot H H ^ { b } } \\right\\rceil \\cdot C _ { 2 } \\log _ { 2 } K } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 258, + 186, + 740, + 207 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "where $C _ { 1 }$ is the single message passing cost for compute devices within the same server, $C _ { 2 }$ is the cost of that across servers, and obviously $C _ { 1 } \\ll C _ { 2 }$ . We can easily witness that the number of block steps $H ^ { b }$ is more deterministic in terms of communication reduction than local step $H$ . Empirical evaluations can be found in Section D.3. ", + "bbox": [ + 174, + 212, + 825, + 267 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Also, note that our hierarchical local SGD is orthogonal to the implementation of gradient aggregation (Goyal et al., 2017) optimized for the hardware, but focusing on overcoming the aggregation cost of more general distributed scenarios, and can easily be integrated with any optimized all-reduce implementation. ", + "bbox": [ + 174, + 273, + 825, + 330 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "F DISCUSSION AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 174, + 351, + 486, + 367 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Data distribution patterns. In our experiments, the dataset is globally shuffled once per epoch and each local worker only accesses a disjoint part of the training data. Removing shuffling altogether, and instead keeping the disjoint data parts completely local during training might be envisioned for extremely large datasets which can not be shared, or also in a federated scenario where data locality is a must for privacy reasons. This scenario is not covered by the current theoretical understanding of local SGD, but will be interesting to investigate theoretically and practically. ", + "bbox": [ + 174, + 382, + 825, + 465 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Better learning rate scheduler for local SGD. We have shown in our experiments that local SGD delivers consistent and significant improvements over the state-of-the-art performance of mini-batch SGD. For ImageNet, we simply applied the same configuration of “large-batch learning schemes” by Goyal et al. (2017). However, this set of schemes was specifically developed and tuned for mini-batch SGD only, not for local SGD. For example, scaling the learning rate w.r.t. the global mini-batch size ignores the frequent local updates where each local model only accesses local minibatches for most of the time. Therefore, it is expected that specifically deriving and tuning a learning rate scheduler for local SGD would lead to even more drastic improvements over mini-batch SGD, especially on larger tasks such as ImageNet. ", + "bbox": [ + 174, + 481, + 826, + 607 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Adaptive local SGD. As local SGD achieves better generalization than current mini-batch SGD approaches, an interesting question is if the number of local steps $H$ could be chosen adaptively, i.e. change during the training phase. This could potentially eliminate or at least simplify complex learning rate schedules. Furthermore, recent works (Loshchilov & Hutter, 2017; Huang et al., 2017a) leverage cyclic learning rate schedules either improving the general performance of deep neural network training, or using an ensemble multiple neural networks at no additional training cost. Adaptive local SGD could potentially achieve similar goals with reduced training cost. ", + "bbox": [ + 174, + 621, + 825, + 719 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Hierarchical local SGD design. Hierarchical local SGD provides a simple but efficient training solution for devices over the complex heterogeneous system. However, its performance might be impacted by the cluster topology. For example, the topology of $8 \\times 2$ -GPU in Table 17 fails to further improve the performance of local SGD by using more frequent inner node synchronizations. On contrary, sufficiently large size of the GPU block could easily benefit from the block update of hierarchical local SGD, for both communication efficiency and training quality. The design space of hierarchical local SGD for different cluster topologies should be further investigated, e.g., to investigate the two levels of model averaging frequency (within and between blocks) in terms of convergence, and the interplay of different local minima in the case of very large number of local steps. Another interesting line of work, explores heterogenous systems by allowing for different number of local steps on different clusters, thus making up for slower machines. 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Most machine learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "applications currently depend on stochastic gradient descent (SGD) (Robbins & Monro, 1985) and in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "particular its mini-batch variant (Bottou, 2010; Dekel et al., 2012). 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Motivated to better balance the available system resources (computation vs. commu-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "nication), local SGD (a.k.a. local-update SGD, parallel SGD, or federated averaging) has recently", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 648 + ], + "score": 1.0, + "content": "attracted increased research interest (Mcdonald et al., 2009; Zinkevich et al., 2010; McDonald", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 646, + 504, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 470, + 659 + ], + "score": 1.0, + "content": "et al., 2010; Zhang et al., 2014; 2016; McMahan et al., 2017). In local SGD, each worker", + "type": "text" + }, + { + "bbox": [ + 470, + 646, + 504, + 658 + ], + "score": 0.92, + "content": "k \\in [ K ]", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 257, + 670 + ], + "score": 1.0, + "content": "evolves a local model by performing", + "type": "text" + }, + { + "bbox": [ + 258, + 658, + 268, + 668 + ], + "score": 0.78, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 657, + 455, + 670 + ], + "score": 1.0, + "content": "sequential SGD updates with mini-batch size", + "type": "text" + }, + { + "bbox": [ + 455, + 658, + 472, + 668 + ], + "score": 0.91, + "content": "\\scriptstyle B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 657, + 505, + 670 + ], + "score": 1.0, + "content": ", before", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 669, + 420, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 420, + 681 + ], + "score": 1.0, + "content": "communication (synchronization by averaging) among the workers. Formally,", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 612, + 506, + 681 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 682, + 481, + 704 + ], + "lines": [ + { + "bbox": [ + 117, + 682, + 481, + 704 + ], + "spans": [ + { + "bbox": [ + 117, + 682, + 481, + 704 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\boldsymbol { w } _ { ( t ) + h + 1 } ^ { k } : = \\boldsymbol { w } _ { ( t ) + h } ^ { k } - \\gamma _ { ( t ) } \\left[ \\frac { 1 } { B _ { \\mathrm { l a c } } } \\sum _ { i \\in \\mathcal { I } _ { ( t ) + h } ^ { k } } \\nabla f _ { i } \\big ( \\boldsymbol { w } _ { ( t ) + h } ^ { k } \\big ) \\right] , \\quad \\boldsymbol { w } _ { ( t + 1 ) } ^ { k } : = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\boldsymbol { w } _ { ( t ) + H } ^ { k } , } \\end{array}", + "type": "interline_equation", + "image_path": "b18feeef614e92478cb2db2c358d96812faaad28ec9de7b74b6cb4443e3744e4.jpg" + } + ] + } + ], + "index": 45, + "virtual_lines": [ + { + "bbox": [ + 117, + 682, + 481, + 704 + ], + "spans": [], + "index": 45 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 505, + 734 + ], + "lines": [ + { + "bbox": [ + 102, + 703, + 505, + 730 + ], + "spans": [ + { + "bbox": [ + 102, + 703, + 134, + 730 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 709, + 165, + 723 + ], + "score": 0.9, + "content": "\\pmb { w } _ { ( t ) \\pmb { \\mathrm { 1 } } } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 703, + 321, + 730 + ], + "score": 1.0, + "content": "denotes the local model on machine k", + "type": "text" + }, + { + "bbox": [ + 321, + 710, + 328, + 720 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 703, + 352, + 730 + ], + "score": 1.0, + "content": "after", + "type": "text" + }, + { + "bbox": [ + 352, + 711, + 357, + 720 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 703, + 505, + 730 + ], + "score": 1.0, + "content": "global synchronization rounds and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 719, + 507, + 736 + ], + "spans": [ + { + "bbox": [ + 104, + 719, + 154, + 736 + ], + "score": 1.0, + "content": "subsequent", + "type": "text" + }, + { + "bbox": [ + 154, + 721, + 188, + 732 + ], + "score": 0.91, + "content": "\\check { h } \\in [ H ]", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 719, + 235, + 736 + ], + "score": 1.0, + "content": "local steps", + "type": "text" + }, + { + "bbox": [ + 266, + 719, + 507, + 736 + ], + "score": 1.0, + "content": "is defined analogously). 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Furthermore, the communication patterns of both algorithms are identical", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 115, + 359 + ], + "score": 1.0, + "content": "if", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 115, + 347, + 163, + 357 + ], + "score": 0.91, + "content": "B = H B _ { \\mathrm { l o c } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 164, + 346, + 506, + 359 + ], + "score": 1.0, + "content": ". However, as the updates (eq. (2)) are different from the mini-batch updates (eq. (1))", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 137, + 370 + ], + "score": 1.0, + "content": "for any", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 137, + 358, + 165, + 367 + ], + "score": 0.89, + "content": "H > 1", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 166, + 357, + 506, + 370 + ], + "score": 1.0, + "content": ", the generalization behavior (test error) of both algorithms is expected to be different.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 46.5, + "bbox_fs": [ + 102, + 703, + 507, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 81, + 504, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 81, + 504, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 81, + 504, + 248 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 504, + 248 + ], + "score": 0.809, + "type": "image", + "image_path": "c37533587195448eb17540ebaf8a184c24be6e289ddca82ed069b0a906e70708.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 81, + 504, + 136.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 136.66666666666666, + 504, + 192.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 192.33333333333331, + 504, + 247.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 335, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 115, + 348 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 115, + 336, + 144, + 345 + ], + "score": 0.91, + "content": "H = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 335, + 162, + 348 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 162, + 336, + 201, + 346 + ], + "score": 0.92, + "content": "B _ { \\mathrm { l o c } } = B", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 335, + 506, + 348 + ], + "score": 1.0, + "content": ". Furthermore, the communication patterns of both algorithms are identical", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 346, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 115, + 359 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 115, + 347, + 163, + 357 + ], + "score": 0.91, + "content": "B = H B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 346, + 506, + 359 + ], + "score": 1.0, + "content": ". However, as the updates (eq. (2)) are different from the mini-batch updates (eq. (1))", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 137, + 370 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 137, + 358, + 165, + 367 + ], + "score": 0.89, + "content": "H > 1", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 357, + 506, + 370 + ], + "score": 1.0, + "content": ", the generalization behavior (test error) of both algorithms is expected to be different.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "Large batch SGD. Recent schemes for scaling training to a large number of workers rely on", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "score": 1.0, + "content": "standard mini-batch SGD (1) with very large overall batch sizes (Shallue et al., 2018; You et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 403, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 492, + 415 + ], + "score": 1.0, + "content": "2018; Goyal et al., 2017), i.e. increasing the global batch size linearly with the number of workers", + "type": "text" + }, + { + "bbox": [ + 493, + 403, + 503, + 413 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 403, + 506, + 415 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "However, the batch size interacts differently with the overall system efficiency on one hand, and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "with generalization performance on the other hand. While a larger batch size in general increases", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "throughput, it may negatively affect the final accuracy on both the train- and test-set. Two main", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 447, + 347, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 347, + 459 + ], + "score": 1.0, + "content": "scenarios of particular interest can be decoupled as follows:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "Scenario 1. The communication restricted setting, where the synchronization time is much higher", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 117, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 117, + 473, + 505, + 487 + ], + "score": 1.0, + "content": "than the gradient computation time. In this case the batch size of best efficiency is typically", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 117, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 117, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "large, and is achieved by using partial computation (gradient accumulation) while waiting for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 117, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 117, + 495, + 505, + 508 + ], + "score": 1.0, + "content": "communication. We in particular study the interesting case when the mini-batch sizes of both", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 118, + 507, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 118, + 507, + 242, + 519 + ], + "score": 1.0, + "content": "algorithms satisfy the relation", + "type": "text" + }, + { + "bbox": [ + 242, + 507, + 287, + 518 + ], + "score": 0.92, + "content": "B = H B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 507, + 506, + 519 + ], + "score": 1.0, + "content": ", as in this case both algorithms (local SGD and mini-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 117, + 517, + 492, + 531 + ], + "spans": [ + { + "bbox": [ + 117, + 517, + 492, + 531 + ], + "score": 1.0, + "content": "batch SGD) evaluate the same number of stochastic gradients between synchronization steps.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "score": 1.0, + "content": "Scenario 2. The regime of poor generalization of large-batch SGD, that is the use of very large", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 118, + 543, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 118, + 543, + 506, + 555 + ], + "score": 1.0, + "content": "overall batches (often a significant fraction of the training set size), which is known to cause", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 117, + 553, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 117, + 553, + 506, + 565 + ], + "score": 1.0, + "content": "drastically decreased generalization performance (Chen & Huo, 2016; Keskar et al., 2017; Hoffer", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 117, + 565, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 117, + 565, + 505, + 576 + ], + "score": 1.0, + "content": "et al., 2017; Shallue et al., 2018; Golmant et al., 2018). 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Key aspects of the empirical performance of local SGD compared to mini-batch", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 433, + 649 + ], + "score": 1.0, + "content": "baselines are illustrated in Figure 1. 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This", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "motivates the use of local SGD as an alternative to large-batch training—a hypothesis that we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 679, + 504, + 694 + ], + "spans": [ + { + "bbox": [ + 104, + 679, + 468, + 694 + ], + "score": 1.0, + "content": "confirm in our experiments. Further, in scenario 2), mini-batch SGD with smaller batch size", + "type": "text" + }, + { + "bbox": [ + 468, + 680, + 504, + 691 + ], + "score": 0.91, + "content": "\\scriptstyle B = B _ { \\mathrm { l o c } }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "score": 1.0, + "content": "(A2) is observed to suffer from poor generalization, although the training curve matches the single-", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 119, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "1Note that in terms of efficiency on current GPUs, the computation time on device for small batch sizes is", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 397, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 254, + 732 + ], + "score": 1.0, + "content": "not constant but scales non-linearly with", + "type": "text" + }, + { + "bbox": [ + 254, + 722, + 262, + 730 + ], + "score": 0.78, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 721, + 397, + 732 + ], + "score": 1.0, + "content": ", as shown in Table 7 in Appendix A.", + "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, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 81, + 504, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 81, + 504, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 81, + 504, + 248 + ], + "spans": [ + { + "bbox": [ + 107, + 81, + 504, + 248 + ], + "score": 0.809, + "type": "image", + "image_path": "c37533587195448eb17540ebaf8a184c24be6e289ddca82ed069b0a906e70708.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 81, + 504, + 136.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 136.66666666666666, + 504, + 192.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 192.33333333333331, + 504, + 247.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 335, + 505, + 369 + ], + "lines": [], + "index": 4, + "bbox_fs": [ + 105, + 335, + 506, + 370 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "Large batch SGD. Recent schemes for scaling training to a large number of workers rely on", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 405 + ], + "score": 1.0, + "content": "standard mini-batch SGD (1) with very large overall batch sizes (Shallue et al., 2018; You et al.,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 403, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 492, + 415 + ], + "score": 1.0, + "content": "2018; Goyal et al., 2017), i.e. increasing the global batch size linearly with the number of workers", + "type": "text" + }, + { + "bbox": [ + 493, + 403, + 503, + 413 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 403, + 506, + 415 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "However, the batch size interacts differently with the overall system efficiency on one hand, and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "with generalization performance on the other hand. While a larger batch size in general increases", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 448 + ], + "score": 1.0, + "content": "throughput, it may negatively affect the final accuracy on both the train- and test-set. Two main", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 447, + 347, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 347, + 459 + ], + "score": 1.0, + "content": "scenarios of particular interest can be decoupled as follows:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 381, + 506, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 506, + 475 + ], + "score": 1.0, + "content": "Scenario 1. 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We in particular study the interesting case when the mini-batch sizes of both", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 118, + 507, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 118, + 507, + 242, + 519 + ], + "score": 1.0, + "content": "algorithms satisfy the relation", + "type": "text" + }, + { + "bbox": [ + 242, + 507, + 287, + 518 + ], + "score": 0.92, + "content": "B = H B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 507, + 506, + 519 + ], + "score": 1.0, + "content": ", as in this case both algorithms (local SGD and mini-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 117, + 517, + 492, + 531 + ], + "spans": [ + { + "bbox": [ + 117, + 517, + 492, + 531 + ], + "score": 1.0, + "content": "batch SGD) evaluate the same number of stochastic gradients between synchronization steps.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 106, + 462, + 506, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 609 + ], + "lines": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "score": 1.0, + "content": "Scenario 2. The regime of poor generalization of large-batch SGD, that is the use of very large", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 118, + 543, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 118, + 543, + 506, + 555 + ], + "score": 1.0, + "content": "overall batches (often a significant fraction of the training set size), which is known to cause", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 117, + 553, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 117, + 553, + 506, + 565 + ], + "score": 1.0, + "content": "drastically decreased generalization performance (Chen & Huo, 2016; Keskar et al., 2017; Hoffer", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 117, + 565, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 117, + 565, + 505, + 576 + ], + "score": 1.0, + "content": "et al., 2017; Shallue et al., 2018; Golmant et al., 2018). 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This", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "motivates the use of local SGD as an alternative to large-batch training—a hypothesis that we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 679, + 504, + 694 + ], + "spans": [ + { + "bbox": [ + 104, + 679, + 468, + 694 + ], + "score": 1.0, + "content": "confirm in our experiments. Further, in scenario 2), mini-batch SGD with smaller batch size", + "type": "text" + }, + { + "bbox": [ + 468, + 680, + 504, + 691 + ], + "score": 0.91, + "content": "\\scriptstyle B = B _ { \\mathrm { l o c } }", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "score": 1.0, + "content": "(A2) is observed to suffer from poor generalization, although the training curve matches the single-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "machine baseline (A1). The generalization gap can thus not be explained as an optimization issue", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "alone. Our proposed post-local SGD (A5) (defined by starting local SGD from the model obtained by", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "large-batch SGD (A2) at epoch 150) closes this generalization gap with the single-machine baseline", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "(A1) and is also more communication efficient than the mini-batch competitors. In direct comparison,", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "score": 1.0, + "content": "post-local SGD is more communication-efficient than mini-batch SGD (while less than local SGD).", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 398, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 398, + 149 + ], + "score": 1.0, + "content": "It achieves better generalization performance than both these algorithms.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 625, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "machine baseline (A1). The generalization gap can thus not be explained as an optimization issue", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "alone. Our proposed post-local SGD (A5) (defined by starting local SGD from the model obtained by", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "large-batch SGD (A2) at epoch 150) closes this generalization gap with the single-machine baseline", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "(A1) and is also more communication efficient than the mini-batch competitors. In direct comparison,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 140 + ], + "score": 1.0, + "content": "post-local SGD is more communication-efficient than mini-batch SGD (while less than local SGD).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 398, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 398, + 149 + ], + "score": 1.0, + "content": "It achieves better generalization performance than both these algorithms.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 108, + 159, + 417, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 159, + 418, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 418, + 173 + ], + "score": 1.0, + "content": "Contributions. 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It allows us to scale the training to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 118, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 118, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "much higher number of parallel devices. Large batches trained by post-local SGD enjoy improved", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 118, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 118, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "communication efficiency, while at the same time strongly outperforming most competing small", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 118, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 118, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "and large batch baselines in terms of accuracy. Our empirical experiments on standard benchmarks", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 118, + 266, + 489, + 278 + ], + "spans": [ + { + "bbox": [ + 118, + 266, + 489, + 278 + ], + "score": 1.0, + "content": "show that post-local SGD can reach flatter minima than large-batch SGD on those problems.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 300, + 211, + 313 + ], + "lines": [ + { + "bbox": [ + 104, + 299, + 213, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 299, + 213, + 316 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 341 + ], + "score": 1.0, + "content": "The generalization gap in large-batch training. State-of-the-art distributed deep learning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "frameworks (Abadi et al., 2016; Paszke et al., 2017; Seide & Agarwal, 2016) resort to synchronized", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 346, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 104, + 346, + 506, + 364 + ], + "score": 1.0, + "content": "large-batch SGD training, allowing scaling by adding more computational units and performing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 358, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 374 + ], + "score": 1.0, + "content": "data-parallel synchronous SGD with mini-batches divided between devices. Training with large", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 211, + 384 + ], + "score": 1.0, + "content": "batch size (e.g. batch size", + "type": "text" + }, + { + "bbox": [ + 211, + 370, + 236, + 381 + ], + "score": 0.87, + "content": "> 1 0 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "on ImageNet) typically degrades the performance both in terms of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "training and test error (often denoted as generalization gap) (Chen & Huo, 2016; Li, 2017; Li et al.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "2014; Keskar et al., 2017; Shallue et al., 2018; McCandlish et al., 2018; Golmant et al., 2018; Masters", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 402, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 417 + ], + "score": 1.0, + "content": "& Luschi, 2018). Goyal et al. (2017) argue that the test error degrades because of optimization issues", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 414, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 506, + 427 + ], + "score": 1.0, + "content": "and propose to use a “learning rate warm-up” phase with linear scaling of the step-size. You et al.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "(2017a) propose Layer-wise Adaptive Rate Scaling (LARS) to scale to larger mini-batch size, but", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "the generalization gap does not vanish. Hoffer et al. (2017) argue that the generalization gap can be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "closed when increasing the number of iterations along with the batch size. However, this diminishes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 265, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 265, + 472 + ], + "score": 1.0, + "content": "the efficiency gains of parallel training.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "Keskar et al. (2017) empirically show that larger batch sizes correlate with sharper minima (Hochreiter", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "& Schmidhuber, 1997) found by SGD and that flat minima are preferred for better generalization.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "This interpretation—despite being debated in Dinh et al. (2017)—was further developed in Hoffer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 508, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 520 + ], + "score": 1.0, + "content": "et al. (2017); Yao et al. (2018); Izmailov et al. (2018). Neelakantan et al. (2015) propose to add", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 519, + 507, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 507, + 532 + ], + "score": 1.0, + "content": "isotropic white noise to the gradients to avoid over-fitting and better optimization. Zhu et al. (2019);", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "Xing et al. (2018) further analyze “structured” anisotropic noise and highlight the importance of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "anisotropic noise (over isotropic noise) for improving generalization. The importance of the scale", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "of the noise for non-convex optimization has also been studied in Smith & Le (2018); Chaudhari", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "& Soatto (2018). Wen et al. (2019) propose to inject noise (sampled from the expensive empirical", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 505, + 587 + ], + "score": 1.0, + "content": "Fisher matrix) to large-batch SGD. 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While mini-batch SGD is very well studied (Zinkevich et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "2010; Dekel et al., 2012; Takác et al., 2013), the theoretical foundations of local SGD variants ˇ", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 643, + 504, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 504, + 655 + ], + "score": 1.0, + "content": "are still developing. Jain et al. (2018) study one-shot averaging on quadratic functions and Bijral", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "et al. (2016) study local SGD in the setting of a general graph of workers. A main research", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "score": 1.0, + "content": "question is whether local SGD provides a linear speedup with respect to the number of workers", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 676, + 117, + 686 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 676, + 484, + 689 + ], + "score": 1.0, + "content": ", similar to mini-batch SGD. 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Stich (2019) and Patel & Dieuleveut (2019) show", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 695, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 188, + 712 + ], + "score": 1.0, + "content": "convergence at rate", + "type": "text" + }, + { + "bbox": [ + 189, + 697, + 267, + 710 + ], + "score": 0.93, + "content": "\\mathcal { O } \\big ( ( K T H B _ { \\mathrm { l o c } } ) ^ { - 1 } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 695, + 506, + 712 + ], + "score": 1.0, + "content": "on strongly convex and smooth objective functions when", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 707, + 507, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 166, + 721 + ], + "score": 0.88, + "content": "H = \\mathcal { O } ( T ^ { 1 / 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 707, + 507, + 724 + ], + "score": 1.0, + "content": ". 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Our main contributions can thus be summarized as follows:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 159, + 418, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 175, + 506, + 208 + ], + "lines": [ + { + "bbox": [ + 109, + 175, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 109, + 175, + 505, + 187 + ], + "score": 1.0, + "content": "Trade-offs in Local SGD: We provide the first comprehensive empirically study of the trade-offs", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 118, + 186, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 118, + 186, + 393, + 199 + ], + "score": 1.0, + "content": "in local SGD for deep learning—when varying the number of workers", + "type": "text" + }, + { + "bbox": [ + 394, + 187, + 404, + 196 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 186, + 493, + 199 + ], + "score": 1.0, + "content": ", number of local steps", + "type": "text" + }, + { + "bbox": [ + 494, + 187, + 504, + 196 + ], + "score": 0.71, + "content": "H", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 118, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 118, + 197, + 506, + 210 + ], + "score": 1.0, + "content": "and mini-batch sizes—for both scenarios 1) on communication efficiency and 2) on generalization.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 109, + 175, + 506, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 211, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 108, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 108, + 210, + 506, + 223 + ], + "score": 1.0, + "content": "Post-local SGD: We propose post-local SGD, a simple but very efficient training scheme to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 117, + 222, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 117, + 222, + 506, + 235 + ], + "score": 1.0, + "content": "address the current generalization issue of large-batch training. It allows us to scale the training to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 118, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 118, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "much higher number of parallel devices. Large batches trained by post-local SGD enjoy improved", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 118, + 244, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 118, + 244, + 505, + 257 + ], + "score": 1.0, + "content": "communication efficiency, while at the same time strongly outperforming most competing small", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 118, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 118, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "and large batch baselines in terms of accuracy. Our empirical experiments on standard benchmarks", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 118, + 266, + 489, + 278 + ], + "spans": [ + { + "bbox": [ + 118, + 266, + 489, + 278 + ], + "score": 1.0, + "content": "show that post-local SGD can reach flatter minima than large-batch SGD on those problems.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 108, + 210, + 506, + 278 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 300, + 211, + 313 + ], + "lines": [ + { + "bbox": [ + 104, + 299, + 213, + 316 + ], + "spans": [ + { + "bbox": [ + 104, + 299, + 213, + 316 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 341 + ], + "score": 1.0, + "content": "The generalization gap in large-batch training. State-of-the-art distributed deep learning", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 350 + ], + "score": 1.0, + "content": "frameworks (Abadi et al., 2016; Paszke et al., 2017; Seide & Agarwal, 2016) resort to synchronized", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 346, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 104, + 346, + 506, + 364 + ], + "score": 1.0, + "content": "large-batch SGD training, allowing scaling by adding more computational units and performing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 358, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 374 + ], + "score": 1.0, + "content": "data-parallel synchronous SGD with mini-batches divided between devices. Training with large", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 370, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 211, + 384 + ], + "score": 1.0, + "content": "batch size (e.g. batch size", + "type": "text" + }, + { + "bbox": [ + 211, + 370, + 236, + 381 + ], + "score": 0.87, + "content": "> 1 0 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "on ImageNet) typically degrades the performance both in terms of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "training and test error (often denoted as generalization gap) (Chen & Huo, 2016; Li, 2017; Li et al.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "2014; Keskar et al., 2017; Shallue et al., 2018; McCandlish et al., 2018; Golmant et al., 2018; Masters", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 402, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 417 + ], + "score": 1.0, + "content": "& Luschi, 2018). Goyal et al. (2017) argue that the test error degrades because of optimization issues", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 414, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 506, + 427 + ], + "score": 1.0, + "content": "and propose to use a “learning rate warm-up” phase with linear scaling of the step-size. You et al.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "(2017a) propose Layer-wise Adaptive Rate Scaling (LARS) to scale to larger mini-batch size, but", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "the generalization gap does not vanish. Hoffer et al. (2017) argue that the generalization gap can be", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "closed when increasing the number of iterations along with the batch size. However, this diminishes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 457, + 265, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 265, + 472 + ], + "score": 1.0, + "content": "the efficiency gains of parallel training.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 325, + 506, + 472 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 475, + 505, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 488 + ], + "score": 1.0, + "content": "Keskar et al. (2017) empirically show that larger batch sizes correlate with sharper minima (Hochreiter", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 498 + ], + "score": 1.0, + "content": "& Schmidhuber, 1997) found by SGD and that flat minima are preferred for better generalization.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "This interpretation—despite being debated in Dinh et al. (2017)—was further developed in Hoffer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 508, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 520 + ], + "score": 1.0, + "content": "et al. (2017); Yao et al. (2018); Izmailov et al. (2018). Neelakantan et al. (2015) propose to add", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 519, + 507, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 507, + 532 + ], + "score": 1.0, + "content": "isotropic white noise to the gradients to avoid over-fitting and better optimization. Zhu et al. (2019);", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "Xing et al. (2018) further analyze “structured” anisotropic noise and highlight the importance of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "anisotropic noise (over isotropic noise) for improving generalization. The importance of the scale", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "of the noise for non-convex optimization has also been studied in Smith & Le (2018); Chaudhari", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 505, + 576 + ], + "score": 1.0, + "content": "& Soatto (2018). Wen et al. (2019) propose to inject noise (sampled from the expensive empirical", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 505, + 587 + ], + "score": 1.0, + "content": "Fisher matrix) to large-batch SGD. However, to our best knowledge, none of the prior work (except", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 598 + ], + "score": 1.0, + "content": "our post-local SGD) can provide a computation efficient way to inject noise to achieve as good", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 596, + 492, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 492, + 609 + ], + "score": 1.0, + "content": "generalization performance as small-batch SGD, for both of CIFAR and ImageNet experiments.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 475, + 507, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 633 + ], + "score": 1.0, + "content": "Local SGD and convergence theory. While mini-batch SGD is very well studied (Zinkevich et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "2010; Dekel et al., 2012; Takác et al., 2013), the theoretical foundations of local SGD variants ˇ", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 643, + 504, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 643, + 504, + 655 + ], + "score": 1.0, + "content": "are still developing. Jain et al. (2018) study one-shot averaging on quadratic functions and Bijral", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "et al. (2016) study local SGD in the setting of a general graph of workers. A main research", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "score": 1.0, + "content": "question is whether local SGD provides a linear speedup with respect to the number of workers", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 676, + 117, + 686 + ], + "score": 0.78, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 676, + 484, + 689 + ], + "score": 1.0, + "content": ", similar to mini-batch SGD. Recent work partially confirms this, under the assumption that", + "type": "text" + }, + { + "bbox": [ + 484, + 676, + 495, + 686 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 686, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 289, + 699 + ], + "score": 1.0, + "content": "not too large compared to the total iterations", + "type": "text" + }, + { + "bbox": [ + 289, + 687, + 298, + 696 + ], + "score": 0.75, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 686, + 506, + 699 + ], + "score": 1.0, + "content": ". Stich (2019) and Patel & Dieuleveut (2019) show", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 695, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 188, + 712 + ], + "score": 1.0, + "content": "convergence at rate", + "type": "text" + }, + { + "bbox": [ + 189, + 697, + 267, + 710 + ], + "score": 0.93, + "content": "\\mathcal { O } \\big ( ( K T H B _ { \\mathrm { l o c } } ) ^ { - 1 } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 695, + 506, + 712 + ], + "score": 1.0, + "content": "on strongly convex and smooth objective functions when", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 707, + 507, + 724 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 166, + 721 + ], + "score": 0.88, + "content": "H = \\mathcal { O } ( T ^ { 1 / 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 707, + 507, + 724 + ], + "score": 1.0, + "content": ". For smooth non-convex objective functions, Zhou & Cong (2018) show a rate of", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 107, + 718, + 507, + 735 + ], + "spans": [ + { + "bbox": [ + 107, + 720, + 182, + 733 + ], + "score": 0.88, + "content": "\\mathcal { O } \\big ( ( K T \\dot { B } _ { \\mathrm { l o c } } ) ^ { - 1 / 2 } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 718, + 507, + 735 + ], + "score": 1.0, + "content": "(for the decrement of the stochastic gradient), Yu et al. (2019) give an improved", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 80, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 131, + 96 + ], + "score": 1.0, + "content": "result", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 132, + 81, + 216, + 95 + ], + "score": 0.91, + "content": "\\mathcal { O } \\big ( ( H K T B _ { \\mathrm { l o c } } ) ^ { - 1 / 2 } \\big )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 217, + 80, + 242, + 96 + ], + "score": 1.0, + "content": "when", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 243, + 81, + 300, + 95 + ], + "score": 0.94, + "content": "H = \\mathcal { O } ( T ^ { 1 / 4 } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 300, + 80, + 506, + 96 + ], + "score": 1.0, + "content": ". For a discussion of more recent theoretical results", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "we refer to (Kairouz et al., 2019). Alistarh et al. (2018) study convergence under adversarial delays.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "Zhang et al. (2016) empirically study the effect of the averaging frequency on the quality of the", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "solution for some problem cases and observe that more frequent averaging at the beginning of the", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 507, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 507, + 141 + ], + "score": 1.0, + "content": "optimization can help. Similarly, Bijral et al. (2016) argue to average more frequently at the beginning.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 620, + 507, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 506, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 131, + 96 + ], + "score": 1.0, + "content": "result", + "type": "text" + }, + { + "bbox": [ + 132, + 81, + 216, + 95 + ], + "score": 0.91, + "content": "\\mathcal { O } \\big ( ( H K T B _ { \\mathrm { l o c } } ) ^ { - 1 / 2 } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 80, + 242, + 96 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 243, + 81, + 300, + 95 + ], + "score": 0.94, + "content": "H = \\mathcal { O } ( T ^ { 1 / 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 80, + 506, + 96 + ], + "score": 1.0, + "content": ". For a discussion of more recent theoretical results", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "we refer to (Kairouz et al., 2019). Alistarh et al. (2018) study convergence under adversarial delays.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "Zhang et al. (2016) empirically study the effect of the averaging frequency on the quality of the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "solution for some problem cases and observe that more frequent averaging at the beginning of the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 507, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 507, + 141 + ], + "score": 1.0, + "content": "optimization can help. Similarly, Bijral et al. (2016) argue to average more frequently at the beginning.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 107, + 153, + 398, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 399, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 399, + 168 + ], + "score": 1.0, + "content": "3 POST-LOCAL SGD AND HIERARCHICAL LOCAL SGD", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 178, + 506, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "score": 1.0, + "content": "In this section we present two novel variants of local SGD. First, we propose post-local SGD to reach", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 506, + 202 + ], + "score": 1.0, + "content": "high generalization accuracy (cf. Section 4.2 below), and second, hierarchical SGD designed from a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 213 + ], + "score": 1.0, + "content": "systems perspective aiming at optimal resource adaptivity (computation vs. communication trade-off).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 217, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 507, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 507, + 230 + ], + "score": 1.0, + "content": "Post-local SGD: Large-batch Training Alternative for Better Generalization. We propose post-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 470, + 240 + ], + "score": 1.0, + "content": "local SGD, a variant where local SGD is only started in the second phase of training, after", + "type": "text" + }, + { + "bbox": [ + 470, + 228, + 478, + 238 + ], + "score": 0.83, + "content": "t ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 228, + 505, + 240 + ], + "score": 1.0, + "content": "initial", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "steps2 with standard mini-batch SGD. Formally, the update in (2) is performed with a iteration", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 249, + 206, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 150, + 264 + ], + "score": 1.0, + "content": "dependent", + "type": "text" + }, + { + "bbox": [ + 150, + 250, + 169, + 263 + ], + "score": 0.91, + "content": "H _ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 249, + 206, + 264 + ], + "score": 1.0, + "content": "given as", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 263, + 385, + 291 + ], + "lines": [ + { + "bbox": [ + 159, + 263, + 385, + 291 + ], + "spans": [ + { + "bbox": [ + 159, + 263, + 385, + 291 + ], + "score": 0.87, + "content": "H _ { ( t ) } = \\left\\{ { \\begin{array} { l l } { 1 , \\quad { \\mathrm { i f ~ } } t \\leqslant t ^ { \\prime } , } \\\\ { H , \\quad { \\mathrm { i f ~ } } t > t ^ { \\prime } . } \\end{array} } \\right. \\qquad \\quad \\qquad ( m i n i - b a t c h S G D )", + "type": "interline_equation", + "image_path": "285e933457897d30275fe847e5eab32fdb90ea8c4740141721908a4bdebc4a2b.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 159, + 263, + 385, + 277.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 159, + 277.0, + 385, + 291.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 434, + 271, + 503, + 282 + ], + "lines": [ + { + "bbox": [ + 433, + 270, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 433, + 270, + 505, + 285 + ], + "score": 1.0, + "content": "(post-local SGD)", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "As the proposed scheme is identical to mini-batch SGD in the first phase (with local batch size", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 107, + 307, + 149, + 318 + ], + "score": 0.9, + "content": "\\boldsymbol { B } = \\boldsymbol { B } _ { \\mathrm { l o c } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 306, + 506, + 320 + ], + "score": 1.0, + "content": "), we can leverage previously tuned learning rate warm-up strategies and schedules for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "large-batch training (Goyal et al., 2017) without additional tuning. Note that we only use ‘small’", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 328, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 209, + 341 + ], + "score": 1.0, + "content": "local mini-batches of size", + "type": "text" + }, + { + "bbox": [ + 209, + 329, + 248, + 340 + ], + "score": 0.92, + "content": "B = B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 328, + 505, + 341 + ], + "score": 1.0, + "content": "in the warm-up phase, and switch to the communication efficient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 201, + 353 + ], + "score": 1.0, + "content": "larger effective batches", + "type": "text" + }, + { + "bbox": [ + 201, + 340, + 228, + 351 + ], + "score": 0.91, + "content": "H B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "in the second phase, while also achieving better generalization in our", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "experiments (cf. also the discussion in Section 5). We would like to point out that it is crucial to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "use local SGD in the second phase, as e.g. just resorting to large batch training does achieve worse", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 373, + 303, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 303, + 385 + ], + "score": 1.0, + "content": "performance (see e.g. 3rd row in Table 2 below).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "Hierarchical Local SGD: Optimal Use of Systems Resources in Heterogeneous Systems. Real", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 401, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 506, + 413 + ], + "score": 1.0, + "content": "world systems come with different communication bandwidths on several levels, e.g. with GPUs or", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "other accelerators grouped hierarchically within a chip, machine, rack or even at the level of entire", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "score": 1.0, + "content": "data-centers. In this scenario, we propose to employ local SGD as an inner loop on each level of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "hierarchy, adapted to the corresponding computation vs communication trade-off of that particular", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "level. The resulting scheme, hierarchical local SGD, can offer significant benefits in terms of system", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 472, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 472, + 468 + ], + "score": 1.0, + "content": "adaptivity and performance, as we show with experiments and a discussion in Appendix D.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 482, + 256, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 481, + 258, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 258, + 497 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 105, + 507, + 504, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 521 + ], + "score": 1.0, + "content": "In this section we systematically evaluate the aforementioned variants of local SGD on deep learning", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 518, + 434, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 434, + 531 + ], + "score": 1.0, + "content": "tasks. In summary, the main findings presented in Sections 4.1 and 4.2 below are:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 535, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "Local SGD, on the one hand, can serve as a communication-efficient alternative to mini-batch SGD", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "for different practical purposes (communication restricted Scenario 1). For example in Figure 2(a)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 557, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 154, + 570 + ], + "score": 1.0, + "content": "(with fixed", + "type": "text" + }, + { + "bbox": [ + 154, + 558, + 172, + 568 + ], + "score": 0.9, + "content": "\\scriptstyle B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 558, + 191, + 570 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 558, + 202, + 568 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 558, + 257, + 570 + ], + "score": 1.0, + "content": "), in terms of", + "type": "text" + }, + { + "bbox": [ + 257, + 557, + 290, + 568 + ], + "score": 0.89, + "content": "9 2 . 4 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 558, + 504, + 570 + ], + "score": 1.0, + "content": "best test accuracy achieved by our mini-batch SGD", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 567, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 189, + 583 + ], + "score": 1.0, + "content": "implementation for", + "type": "text" + }, + { + "bbox": [ + 189, + 569, + 222, + 579 + ], + "score": 0.89, + "content": "K = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 567, + 505, + 583 + ], + "score": 1.0, + "content": ", practitioners can either choose to achieve reasonable good training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 136, + 593 + ], + "score": 1.0, + "content": "quality", + "type": "text" + }, + { + "bbox": [ + 136, + 579, + 166, + 590 + ], + "score": 0.85, + "content": "( 9 1 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 579, + 306, + 593 + ], + "score": 1.0, + "content": ", matching (He et al., 2016a)) with a", + "type": "text" + }, + { + "bbox": [ + 307, + 579, + 333, + 590 + ], + "score": 0.88, + "content": "2 . 5 9 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "speedup (time-to-accuracy) in training time", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 109, + 603 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 109, + 591, + 137, + 601 + ], + "score": 0.85, + "content": "( H = 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 590, + 311, + 603 + ], + "score": 1.0, + "content": "), or achieve slight better test performance", + "type": "text" + }, + { + "bbox": [ + 311, + 591, + 347, + 601 + ], + "score": 0.82, + "content": "( 9 2 . 5 7 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 590, + 505, + 603 + ], + "score": 1.0, + "content": ") with slightly reduced communication", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 601, + 261, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 151, + 614 + ], + "score": 1.0, + "content": "efficiency", + "type": "text" + }, + { + "bbox": [ + 151, + 602, + 178, + 612 + ], + "score": 0.84, + "content": "1 . 7 6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 601, + 228, + 614 + ], + "score": 1.0, + "content": "speedup for", + "type": "text" + }, + { + "bbox": [ + 228, + 601, + 254, + 612 + ], + "score": 0.88, + "content": "H = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 601, + 261, + 614 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 618, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "Post-local SGD, on the other hand, in addition to overcoming communication restrictions does provide", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 630, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 641 + ], + "score": 1.0, + "content": "a state-of-the-art remedy for the generalization issue of large-batch training (Scenario 2). 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However it could be tuned more generally aiming at capturing the time when trajectory starts to get into", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 699, + 507, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 290, + 714 + ], + "score": 1.0, + "content": "the influence basin of a local minimum (Robbins", + "type": "text" + }, + { + "bbox": [ + 291, + 702, + 299, + 711 + ], + "score": 0.43, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 699, + 507, + 714 + ], + "score": 1.0, + "content": "Monro, 1985; Smith et al., 2018; Loshchilov & Hutter,", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 485, + 723 + ], + "score": 1.0, + "content": "2017; Huang et al., 2017a). Results in Appendix B.4.2 and C.2 empirically evaluate the impact of different", + "type": "text" + }, + { + "bbox": [ + 486, + 711, + 492, + 721 + ], + "score": 0.82, + "content": "t ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "on", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 225, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 225, + 732 + ], + "score": 1.0, + "content": "optimization and generalization.", + "type": "text" + } + ] + } + ] + }, + { + "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 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 506, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 80, + 507, + 141 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 107, + 153, + 398, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 399, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 399, + 168 + ], + "score": 1.0, + "content": "3 POST-LOCAL SGD AND HIERARCHICAL LOCAL SGD", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 178, + 506, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 506, + 191 + ], + "score": 1.0, + "content": "In this section we present two novel variants of local SGD. First, we propose post-local SGD to reach", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 506, + 202 + ], + "score": 1.0, + "content": "high generalization accuracy (cf. Section 4.2 below), and second, hierarchical SGD designed from a", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 201, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 213 + ], + "score": 1.0, + "content": "systems perspective aiming at optimal resource adaptivity (computation vs. communication trade-off).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 178, + 506, + 213 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 217, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 507, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 507, + 230 + ], + "score": 1.0, + "content": "Post-local SGD: Large-batch Training Alternative for Better Generalization. We propose post-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 470, + 240 + ], + "score": 1.0, + "content": "local SGD, a variant where local SGD is only started in the second phase of training, after", + "type": "text" + }, + { + "bbox": [ + 470, + 228, + 478, + 238 + ], + "score": 0.83, + "content": "t ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 228, + 505, + 240 + ], + "score": 1.0, + "content": "initial", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "steps2 with standard mini-batch SGD. Formally, the update in (2) is performed with a iteration", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 249, + 206, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 150, + 264 + ], + "score": 1.0, + "content": "dependent", + "type": "text" + }, + { + "bbox": [ + 150, + 250, + 169, + 263 + ], + "score": 0.91, + "content": "H _ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 249, + 206, + 264 + ], + "score": 1.0, + "content": "given as", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 216, + 507, + 264 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 263, + 385, + 291 + ], + "lines": [ + { + "bbox": [ + 159, + 263, + 385, + 291 + ], + "spans": [ + { + "bbox": [ + 159, + 263, + 385, + 291 + ], + "score": 0.87, + "content": "H _ { ( t ) } = \\left\\{ { \\begin{array} { l l } { 1 , \\quad { \\mathrm { i f ~ } } t \\leqslant t ^ { \\prime } , } \\\\ { H , \\quad { \\mathrm { i f ~ } } t > t ^ { \\prime } . } \\end{array} } \\right. \\qquad \\quad \\qquad ( m i n i - b a t c h S G D )", + "type": "interline_equation", + "image_path": "285e933457897d30275fe847e5eab32fdb90ea8c4740141721908a4bdebc4a2b.jpg" + } + ] + } + ], + "index": 13.5, + "virtual_lines": [ + { + "bbox": [ + 159, + 263, + 385, + 277.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 159, + 277.0, + 385, + 291.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 434, + 271, + 503, + 282 + ], + "lines": [ + { + "bbox": [ + 433, + 270, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 433, + 270, + 505, + 285 + ], + "score": 1.0, + "content": "(post-local SGD)", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 433, + 270, + 505, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 295, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "As the proposed scheme is identical to mini-batch SGD in the first phase (with local batch size", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 306, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 107, + 307, + 149, + 318 + ], + "score": 0.9, + "content": "\\boldsymbol { B } = \\boldsymbol { B } _ { \\mathrm { l o c } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 306, + 506, + 320 + ], + "score": 1.0, + "content": "), we can leverage previously tuned learning rate warm-up strategies and schedules for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "large-batch training (Goyal et al., 2017) without additional tuning. Note that we only use ‘small’", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 328, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 209, + 341 + ], + "score": 1.0, + "content": "local mini-batches of size", + "type": "text" + }, + { + "bbox": [ + 209, + 329, + 248, + 340 + ], + "score": 0.92, + "content": "B = B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 328, + 505, + 341 + ], + "score": 1.0, + "content": "in the warm-up phase, and switch to the communication efficient", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 201, + 353 + ], + "score": 1.0, + "content": "larger effective batches", + "type": "text" + }, + { + "bbox": [ + 201, + 340, + 228, + 351 + ], + "score": 0.91, + "content": "H B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "in the second phase, while also achieving better generalization in our", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "experiments (cf. also the discussion in Section 5). We would like to point out that it is crucial to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "use local SGD in the second phase, as e.g. just resorting to large batch training does achieve worse", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 373, + 303, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 303, + 385 + ], + "score": 1.0, + "content": "performance (see e.g. 3rd row in Table 2 below).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 296, + 506, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 467 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "Hierarchical Local SGD: Optimal Use of Systems Resources in Heterogeneous Systems. Real", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 401, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 506, + 413 + ], + "score": 1.0, + "content": "world systems come with different communication bandwidths on several levels, e.g. with GPUs or", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "other accelerators grouped hierarchically within a chip, machine, rack or even at the level of entire", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 436 + ], + "score": 1.0, + "content": "data-centers. In this scenario, we propose to employ local SGD as an inner loop on each level of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "hierarchy, adapted to the corresponding computation vs communication trade-off of that particular", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "level. The resulting scheme, hierarchical local SGD, can offer significant benefits in terms of system", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 472, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 472, + 468 + ], + "score": 1.0, + "content": "adaptivity and performance, as we show with experiments and a discussion in Appendix D.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 390, + 506, + 468 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 482, + 256, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 481, + 258, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 258, + 497 + ], + "score": 1.0, + "content": "4 EXPERIMENTAL RESULTS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 105, + 507, + 504, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 521 + ], + "score": 1.0, + "content": "In this section we systematically evaluate the aforementioned variants of local SGD on deep learning", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 518, + 434, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 434, + 531 + ], + "score": 1.0, + "content": "tasks. In summary, the main findings presented in Sections 4.1 and 4.2 below are:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 506, + 505, + 531 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 535, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "Local SGD, on the one hand, can serve as a communication-efficient alternative to mini-batch SGD", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "for different practical purposes (communication restricted Scenario 1). For example in Figure 2(a)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 557, + 504, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 154, + 570 + ], + "score": 1.0, + "content": "(with fixed", + "type": "text" + }, + { + "bbox": [ + 154, + 558, + 172, + 568 + ], + "score": 0.9, + "content": "\\scriptstyle B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 558, + 191, + 570 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 191, + 558, + 202, + 568 + ], + "score": 0.79, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 558, + 257, + 570 + ], + "score": 1.0, + "content": "), in terms of", + "type": "text" + }, + { + "bbox": [ + 257, + 557, + 290, + 568 + ], + "score": 0.89, + "content": "9 2 . 4 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 558, + 504, + 570 + ], + "score": 1.0, + "content": "best test accuracy achieved by our mini-batch SGD", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 567, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 189, + 583 + ], + "score": 1.0, + "content": "implementation for", + "type": "text" + }, + { + "bbox": [ + 189, + 569, + 222, + 579 + ], + "score": 0.89, + "content": "K = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 567, + 505, + 583 + ], + "score": 1.0, + "content": ", practitioners can either choose to achieve reasonable good training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 136, + 593 + ], + "score": 1.0, + "content": "quality", + "type": "text" + }, + { + "bbox": [ + 136, + 579, + 166, + 590 + ], + "score": 0.85, + "content": "( 9 1 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 579, + 306, + 593 + ], + "score": 1.0, + "content": ", matching (He et al., 2016a)) with a", + "type": "text" + }, + { + "bbox": [ + 307, + 579, + 333, + 590 + ], + "score": 0.88, + "content": "2 . 5 9 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "speedup (time-to-accuracy) in training time", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 109, + 603 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 109, + 591, + 137, + 601 + ], + "score": 0.85, + "content": "( H = 8 )", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 590, + 311, + 603 + ], + "score": 1.0, + "content": "), or achieve slight better test performance", + "type": "text" + }, + { + "bbox": [ + 311, + 591, + 347, + 601 + ], + "score": 0.82, + "content": "( 9 2 . 5 7 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 590, + 505, + 603 + ], + "score": 1.0, + "content": ") with slightly reduced communication", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 601, + 261, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 151, + 614 + ], + "score": 1.0, + "content": "efficiency", + "type": "text" + }, + { + "bbox": [ + 151, + 602, + 178, + 612 + ], + "score": 0.84, + "content": "1 . 7 6 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 601, + 228, + 614 + ], + "score": 1.0, + "content": "speedup for", + "type": "text" + }, + { + "bbox": [ + 228, + 601, + 254, + 612 + ], + "score": 0.88, + "content": "H = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 601, + 261, + 614 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 535, + 506, + 614 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 618, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "Post-local SGD, on the other hand, in addition to overcoming communication restrictions does provide", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 630, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 641 + ], + "score": 1.0, + "content": "a state-of-the-art remedy for the generalization issue of large-batch training (Scenario 2). Unlike", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 197, + 653 + ], + "score": 1.0, + "content": "local SGD with large", + "type": "text" + }, + { + "bbox": [ + 197, + 641, + 208, + 650 + ], + "score": 0.78, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 640, + 227, + 653 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 227, + 641, + 238, + 650 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 640, + 262, + 653 + ], + "score": 1.0, + "content": "(e.g.", + "type": "text" + }, + { + "bbox": [ + 262, + 640, + 332, + 651 + ], + "score": 0.41, + "content": "H = 1 6 , K = 1 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "which can encounter optimization issues", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 650, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 505, + 664 + ], + "score": 1.0, + "content": "during initial training (which can impact later generalization performance), we found that post-local", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 676 + ], + "score": 1.0, + "content": "SGD elegantly enables an ideal trade-off between optimization and generalization within a fixed", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "training budget (e.g. Table 3, Table 5, Figure 3 and many others in Appendix C.5), and can achieve", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 384, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 384, + 105 + ], + "score": 1.0, + "content": "the same or even better performance than small mini-batch baselines.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 618, + 506, + 676 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "training budget (e.g. Table 3, Table 5, Figure 3 and many others in Appendix C.5), and can achieve", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 384, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 384, + 105 + ], + "score": 1.0, + "content": "the same or even better performance than small mini-batch baselines.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 120, + 503, + 131 + ], + "lines": [ + { + "bbox": [ + 106, + 119, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 106, + 119, + 505, + 133 + ], + "score": 1.0, + "content": "Setup. We briefly outline the general experimental setup, and refer to Appendix A for full details.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 136, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "Datasets. We evaluate all methods on the following two main (standard) tasks: (1) Image classification", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 148, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 506, + 160 + ], + "score": 1.0, + "content": "for CIFAR-10/100 (Krizhevsky & Hinton, 2009), and (2) Image classification for ImageNet (Rus-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 159, + 448, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 448, + 171 + ], + "score": 1.0, + "content": "sakovsky et al., 2015). The detailed data augmentation scheme refers to Appendix A.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 175, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 175, + 504, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 504, + 187 + ], + "score": 1.0, + "content": "Models. We use ResNet-20 (He et al., 2016a) with CIFAR-10 as a base configuration to understand", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "different properties of (post-)local SGD. We then empirical evaluate the large-batch training perfor-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "mance of post-local SGD, for ResNet-20, DensetNet-40-12 (Huang et al., 2017b) and WideResNet-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 507, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 507, + 221 + ], + "score": 1.0, + "content": "28-10 (Zagoruyko & Komodakis, 2016) on CIFAR-10/100. Finally, we train ResNet-50 (He et al.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 476, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 476, + 234 + ], + "score": 1.0, + "content": "2016a) on ImageNet to investigate the accuracy and scalability of (post-)local SGD training.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 507, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 507, + 250 + ], + "score": 1.0, + "content": "Implementation and platform. Our algorithms are implemented3 in PyTorch (Paszke et al., 2017),", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "with a flexible configuration of the machine topology supported by Kubernetes. The cluster consists", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "of Intel Xeon E5-2680 v3 servers and each server has 2 NVIDIA TITAN Xp GPUs. 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We rely on the recently proposed schemes for efficient", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "large batch training (Goyal et al., 2017), which are formalized by (i) linearly scaling the learning rate", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "w.r.t. the global mini-batch size; (ii) gradual warm-up of the learning rate from a small value. See", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 320, + 233, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 233, + 331 + ], + "score": 1.0, + "content": "Appendix A.3 for more details.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "Distributed training procedure on CIFAR-10/100. The experiments follow the common mini-batch", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "SGD training scheme for CIFAR (He et al., 2016a;b; Huang et al., 2017b) and all competing methods", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "access the same total number of data samples (i.e. gradients) regardless of the number of local steps.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "Training ends when the distributed algorithms have accessed the same number of samples as the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "single-worker baseline. The data is disjointly partitioned and reshuffled globally every epoch. 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See Appendix A.4 for more details.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 108, + 462, + 411, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 412, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 412, + 475 + ], + "score": 1.0, + "content": "4.1 SUPERIOR SCALABILITY OF LOCAL SGD OVER MINI-BATCH SGD", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "score": 1.0, + "content": "First, we empirically study local SGD training for the communication restricted case (i.e. Scenario 1).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 494, + 507, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 255, + 509 + ], + "score": 1.0, + "content": "As local SGD (with local batch size", + "type": "text" + }, + { + "bbox": [ + 255, + 496, + 274, + 506 + ], + "score": 0.87, + "content": "B _ { \\mathrm { l o c . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 494, + 303, + 509 + ], + "score": 1.0, + "content": ") needs", + "type": "text" + }, + { + "bbox": [ + 304, + 496, + 314, + 505 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 494, + 507, + 509 + ], + "score": 1.0, + "content": "times fewer communication rounds than mini-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 259, + 519 + ], + "score": 1.0, + "content": "batch SGD (with the same batch size", + "type": "text" + }, + { + "bbox": [ + 260, + 506, + 300, + 518 + ], + "score": 0.91, + "content": "\\begin{array} { r } { B = B _ { \\mathrm { l o c } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "q we expect local SGD to significantly outperform", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "score": 1.0, + "content": "mini-batch SGD in terms of time-to-accuracy and scalability and verify this experimentally. 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We briefly outline the general experimental setup, and refer to Appendix A for full details.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 119, + 505, + 133 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 136, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "Datasets. We evaluate all methods on the following two main (standard) tasks: (1) Image classification", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 148, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 506, + 160 + ], + "score": 1.0, + "content": "for CIFAR-10/100 (Krizhevsky & Hinton, 2009), and (2) Image classification for ImageNet (Rus-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 159, + 448, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 448, + 171 + ], + "score": 1.0, + "content": "sakovsky et al., 2015). The detailed data augmentation scheme refers to Appendix A.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 137, + 506, + 171 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 175, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 175, + 504, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 504, + 187 + ], + "score": 1.0, + "content": "Models. We use ResNet-20 (He et al., 2016a) with CIFAR-10 as a base configuration to understand", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "different properties of (post-)local SGD. We then empirical evaluate the large-batch training perfor-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "mance of post-local SGD, for ResNet-20, DensetNet-40-12 (Huang et al., 2017b) and WideResNet-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 507, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 507, + 221 + ], + "score": 1.0, + "content": "28-10 (Zagoruyko & Komodakis, 2016) on CIFAR-10/100. Finally, we train ResNet-50 (He et al.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 476, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 476, + 234 + ], + "score": 1.0, + "content": "2016a) on ImageNet to investigate the accuracy and scalability of (post-)local SGD training.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 175, + 507, + 234 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 507, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 507, + 250 + ], + "score": 1.0, + "content": "Implementation and platform. Our algorithms are implemented3 in PyTorch (Paszke et al., 2017),", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "with a flexible configuration of the machine topology supported by Kubernetes. The cluster consists", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "of Intel Xeon E5-2680 v3 servers and each server has 2 NVIDIA TITAN Xp GPUs. We use the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 475, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 134, + 281 + ], + "score": 1.0, + "content": "notion", + "type": "text" + }, + { + "bbox": [ + 135, + 270, + 157, + 280 + ], + "score": 0.9, + "content": "a \\times b", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 270, + 347, + 281 + ], + "score": 1.0, + "content": "-GPU to denote the topology of the cluster, i.e.,", + "type": "text" + }, + { + "bbox": [ + 348, + 272, + 354, + 279 + ], + "score": 0.46, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 270, + 438, + 281 + ], + "score": 1.0, + "content": "nodes and each with", + "type": "text" + }, + { + "bbox": [ + 439, + 270, + 445, + 279 + ], + "score": 0.53, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 270, + 475, + 281 + ], + "score": 1.0, + "content": "GPUs.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 235, + 507, + 281 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 505, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "Specific learning schemes for large-batch SGD. We rely on the recently proposed schemes for efficient", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 310 + ], + "score": 1.0, + "content": "large batch training (Goyal et al., 2017), which are formalized by (i) linearly scaling the learning rate", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "w.r.t. the global mini-batch size; (ii) gradual warm-up of the learning rate from a small value. See", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 320, + 233, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 233, + 331 + ], + "score": 1.0, + "content": "Appendix A.3 for more details.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 286, + 506, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "Distributed training procedure on CIFAR-10/100. The experiments follow the common mini-batch", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 360 + ], + "score": 1.0, + "content": "SGD training scheme for CIFAR (He et al., 2016a;b; Huang et al., 2017b) and all competing methods", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "access the same total number of data samples (i.e. gradients) regardless of the number of local steps.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 505, + 381 + ], + "score": 1.0, + "content": "Training ends when the distributed algorithms have accessed the same number of samples as the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "single-worker baseline. The data is disjointly partitioned and reshuffled globally every epoch. The", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "score": 1.0, + "content": "learning rate scheme follows (He et al., 2016a; Huang et al., 2017b), where we drop the initial learning", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 399, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 318, + 417 + ], + "score": 1.0, + "content": "rate by a factor of 10 when the model has accessed", + "type": "text" + }, + { + "bbox": [ + 318, + 402, + 338, + 412 + ], + "score": 0.89, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 399, + 357, + 417 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 402, + 377, + 412 + ], + "score": 0.89, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 399, + 506, + 417 + ], + "score": 1.0, + "content": "of the total number of training", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "samples. Unless mentioned specifically, the used learning rate is scaled by the global mini-batch size", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 110, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 110, + 424, + 128, + 434 + ], + "score": 0.82, + "content": "B K", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 423, + 228, + 436 + ], + "score": 1.0, + "content": "for mini-batch SGD and", + "type": "text" + }, + { + "bbox": [ + 229, + 424, + 255, + 435 + ], + "score": 0.92, + "content": "B _ { \\mathrm { l o c } } K", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "for local SGD) where the initial learning rate is fine-tuned for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 435, + 442, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 442, + 447 + ], + "score": 1.0, + "content": "each model and each task for the single worker. See Appendix A.4 for more details.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 336, + 506, + 447 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 462, + 411, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 412, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 412, + 475 + ], + "score": 1.0, + "content": "4.1 SUPERIOR SCALABILITY OF LOCAL SGD OVER MINI-BATCH SGD", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 484, + 505, + 540 + ], + "lines": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "score": 1.0, + "content": "First, we empirically study local SGD training for the communication restricted case (i.e. Scenario 1).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 494, + 507, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 255, + 509 + ], + "score": 1.0, + "content": "As local SGD (with local batch size", + "type": "text" + }, + { + "bbox": [ + 255, + 496, + 274, + 506 + ], + "score": 0.87, + "content": "B _ { \\mathrm { l o c . } }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 494, + 303, + 509 + ], + "score": 1.0, + "content": ") needs", + "type": "text" + }, + { + "bbox": [ + 304, + 496, + 314, + 505 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 494, + 507, + 509 + ], + "score": 1.0, + "content": "times fewer communication rounds than mini-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 259, + 519 + ], + "score": 1.0, + "content": "batch SGD (with the same batch size", + "type": "text" + }, + { + "bbox": [ + 260, + 506, + 300, + 518 + ], + "score": 0.91, + "content": "\\begin{array} { r } { B = B _ { \\mathrm { l o c } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "q we expect local SGD to significantly outperform", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 530 + ], + "score": 1.0, + "content": "mini-batch SGD in terms of time-to-accuracy and scalability and verify this experimentally. 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Figure 8 in Appendix B.3.2 shows that we can efficiently train state-of-the-art ResNet-50", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 229, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 141, + 242 + ], + "score": 1.0, + "content": "(at least", + "type": "text" + }, + { + "bbox": [ + 141, + 230, + 163, + 240 + ], + "score": 0.88, + "content": "1 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 229, + 235, + 242 + ], + "score": 1.0, + "content": "speedup to reach", + "type": "text" + }, + { + "bbox": [ + 236, + 229, + 255, + 240 + ], + "score": 0.87, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 229, + 506, + 242 + ], + "score": 1.0, + "content": "top-1 accuracy) for ImageNet (Goyal et al., 2017; You et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 240, + 299, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 215, + 253 + ], + "score": 1.0, + "content": "2017a) via local SGD on a", + "type": "text" + }, + { + "bbox": [ + 215, + 240, + 243, + 251 + ], + "score": 0.88, + "content": "1 6 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 240, + 299, + 253 + ], + "score": 1.0, + "content": "-GPU cluster.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 266, + 505, + 310 + ], + "lines": [ + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "Local SGD significantly outperforms mini-batch SGD at the same effective batch size and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "communication ratio. Figure 2 compares local SGD with mini-batch SGD of the same effective", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "batch size, that is the same number of gradient computations per communication round as described", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 299, + 236, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 236, + 311 + ], + "score": 1.0, + "content": "in Scenario 1 above (Section 1).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "image", + "bbox": [ + 108, + 325, + 228, + 403 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 325, + 228, + 403 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 325, + 228, + 403 + ], + "spans": [ + { + "bbox": [ + 108, + 325, + 228, + 403 + ], + "score": 0.692, + "type": "image", + "image_path": "915c1d109060a8d68f5bd0bd854005ce56675f41858a11b9ab53fd3bff0bc03f.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 108, + 325, + 228, + 364.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 108, + 364.0, + 228, + 403.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 407, + 235, + 438 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 407, + 235, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 235, + 418 + ], + "score": 1.0, + "content": "(a) Top-1 test accuracy of local", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 417, + 235, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 131, + 428 + ], + "score": 1.0, + "content": "SGD.", + "type": "text" + }, + { + "bbox": [ + 131, + 417, + 159, + 426 + ], + "score": 0.89, + "content": "H = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 417, + 235, + 428 + ], + "score": 1.0, + "content": "is mini-batch SGD", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 427, + 221, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 221, + 439 + ], + "score": 1.0, + "content": "with optimal hyper-parameters.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + } + ], + "index": 22.25 + }, + { + "type": "image", + "bbox": [ + 244, + 328, + 498, + 403 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 244, + 328, + 498, + 403 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 244, + 328, + 498, + 403 + ], + "spans": [ + { + "bbox": [ + 244, + 328, + 498, + 403 + ], + "score": 0.744, + "type": "image", + "image_path": "54d9ed9579b73f9cfd5fac7ed167e9bc812855c68ba6f9ad3ea362cb26eca9bf.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 244, + 328, + 498, + 353.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 244, + 353.0, + 498, + 378.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 244, + 378.0, + 498, + 403.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 240, + 407, + 504, + 438 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 240, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 240, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "(b) Top-1 test accuracy of local SGD (circle) vs. mini-batch SGD", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 241, + 417, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 241, + 417, + 394, + 428 + ], + "score": 1.0, + "content": "(triangle), with same effective batch size", + "type": "text" + }, + { + "bbox": [ + 395, + 417, + 441, + 427 + ], + "score": 0.91, + "content": "B = H B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 417, + 505, + 428 + ], + "score": 1.0, + "content": "per worker. 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All results are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 461, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 472 + ], + "score": 1.0, + "content": "averaged over three runs and all settings access to the same total number of training samples. We fine-tune the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 471, + 284, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 284, + 483 + ], + "score": 1.0, + "content": "learning rate of mini-batch SGD for each setting.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 506, + 500, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 502, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 502, + 520 + ], + "score": 1.0, + "content": "4.2 (POST)-LOCAL SGD CLOSES THE GENERALIZATION GAP OF LARGE-BATCH TRAINING", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Even though local SGD has demonstrated effective communication efficiency with guaranteed test", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "performance (Scenario 1), we observed in the previous section that it still encounters difficulties when", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "scaling to very large mini-batches (Figure 2(a)), though it is much less affected than mini-batch SGD", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "(Figure 2(b)). Post-local SGD can address the generalization issue of large batch training (Scenario 2)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 426, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 426, + 584 + ], + "score": 1.0, + "content": "as we show now. First, we would like to highlight some key findings in Table 2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "table", + "bbox": [ + 121, + 656, + 495, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 596, + 505, + 646 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "Table 2: Test performance (generalization) for local SGD variants and mini-batch SGD (highlighting selected", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 606, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 505, + 617 + ], + "score": 1.0, + "content": "data from Figure 2, Table 3 and Figure 3). Mini-batch SGD suffers from the generalization gap (and sometimes", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 616, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 395, + 627 + ], + "score": 1.0, + "content": "even from optimization issues due to insufficient training) when scaling to larger", + "type": "text" + }, + { + "bbox": [ + 396, + 617, + 405, + 625 + ], + "score": 0.73, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 616, + 421, + 627 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 421, + 617, + 430, + 625 + ], + "score": 0.71, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 616, + 505, + 627 + ], + "score": 1.0, + "content": ". Same experimental", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 339, + 639 + ], + "score": 1.0, + "content": "setup as in Figure 2 (fine-tuned mini-batch SGD baselines). The", + "type": "text" + }, + { + "bbox": [ + 339, + 627, + 354, + 635 + ], + "score": 0.8, + "content": "\\hookrightarrow", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "denotes the transition at the first learning", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 635, + 327, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 327, + 648 + ], + "score": 1.0, + "content": "rate decay. The reported results are averaged over three runs.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + }, + { + "type": "table_body", + "bbox": [ + 121, + 656, + 495, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 121, + 656, + 495, + 730 + ], + "spans": [ + { + "bbox": [ + 121, + 656, + 495, + 730 + ], + "score": 0.982, + "html": "
AlgorithmTop-1 acc.Effect. batch sizeAlgorithmTop-1 acc.Effect. batch size
Mini-batch SGD (K=4)92.6%KB=512Mini-batch SGD (K =16)92.5%KB=2048
Mini-batch SGD (K =4)89.5%KB=8192Mini-batch SGD (K =16)76.3%KB=16384
Mini-batch SGD (K=4)92.5%KB=512→8192Mini-batch SGD(K=16)92.0%KHBloc=2048→ 16384
Local SGD(H=16,K=4)92.5%KHBloc =8192Local SGD(H=8,K=16)92.0%KHBloc =16384
Post-local SGD(H=16,K=4)92.7%KHBloc =512→8192Post-local SGD (H=8,K=16)92.9%KHBloc =2048→ 16384
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Figure 8 in Appendix B.3.2 shows that we can efficiently train state-of-the-art ResNet-50", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 229, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 229, + 141, + 242 + ], + "score": 1.0, + "content": "(at least", + "type": "text" + }, + { + "bbox": [ + 141, + 230, + 163, + 240 + ], + "score": 0.88, + "content": "1 . 5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 229, + 235, + 242 + ], + "score": 1.0, + "content": "speedup to reach", + "type": "text" + }, + { + "bbox": [ + 236, + 229, + 255, + 240 + ], + "score": 0.87, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 229, + 506, + 242 + ], + "score": 1.0, + "content": "top-1 accuracy) for ImageNet (Goyal et al., 2017; You et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 240, + 299, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 215, + 253 + ], + "score": 1.0, + "content": "2017a) via local SGD on a", + "type": "text" + }, + { + "bbox": [ + 215, + 240, + 243, + 251 + ], + "score": 0.88, + "content": "1 6 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 240, + 299, + 253 + ], + "score": 1.0, + "content": "-GPU cluster.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 195, + 506, + 253 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 266, + 505, + 310 + ], + "lines": [ + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 505, + 278 + ], + "score": 1.0, + "content": "Local SGD significantly outperforms mini-batch SGD at the same effective batch size and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 289 + ], + "score": 1.0, + "content": "communication ratio. Figure 2 compares local SGD with mini-batch SGD of the same effective", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 506, + 300 + ], + "score": 1.0, + "content": "batch size, that is the same number of gradient computations per communication round as described", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 299, + 236, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 236, + 311 + ], + "score": 1.0, + "content": "in Scenario 1 above (Section 1).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 266, + 506, + 311 + ] + }, + { + "type": "image", + "bbox": [ + 108, + 325, + 228, + 403 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 325, + 228, + 403 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 325, + 228, + 403 + ], + "spans": [ + { + "bbox": [ + 108, + 325, + 228, + 403 + ], + "score": 0.692, + "type": "image", + "image_path": "915c1d109060a8d68f5bd0bd854005ce56675f41858a11b9ab53fd3bff0bc03f.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 108, + 325, + 228, + 364.0 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 108, + 364.0, + 228, + 403.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 407, + 235, + 438 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 407, + 235, + 418 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 235, + 418 + ], + "score": 1.0, + "content": "(a) Top-1 test accuracy of local", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 417, + 235, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 131, + 428 + ], + "score": 1.0, + "content": "SGD.", + "type": "text" + }, + { + "bbox": [ + 131, + 417, + 159, + 426 + ], + "score": 0.89, + "content": "H = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 417, + 235, + 428 + ], + "score": 1.0, + "content": "is mini-batch SGD", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 427, + 221, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 221, + 439 + ], + "score": 1.0, + "content": "with optimal hyper-parameters.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + } + ], + "index": 22.25 + }, + { + "type": "image", + "bbox": [ + 244, + 328, + 498, + 403 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 244, + 328, + 498, + 403 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 244, + 328, + 498, + 403 + ], + "spans": [ + { + "bbox": [ + 244, + 328, + 498, + 403 + ], + "score": 0.744, + "type": "image", + "image_path": "54d9ed9579b73f9cfd5fac7ed167e9bc812855c68ba6f9ad3ea362cb26eca9bf.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 244, + 328, + 498, + 353.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 244, + 353.0, + 498, + 378.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 244, + 378.0, + 498, + 403.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 240, + 407, + 504, + 438 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 240, + 406, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 240, + 406, + 505, + 418 + ], + "score": 1.0, + "content": "(b) Top-1 test accuracy of local SGD (circle) vs. mini-batch SGD", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 241, + 417, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 241, + 417, + 394, + 428 + ], + "score": 1.0, + "content": "(triangle), with same effective batch size", + "type": "text" + }, + { + "bbox": [ + 395, + 417, + 441, + 427 + ], + "score": 0.91, + "content": "B = H B _ { \\mathrm { l o c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 417, + 505, + 428 + ], + "score": 1.0, + "content": "per worker. 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All results are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 461, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 472 + ], + "score": 1.0, + "content": "averaged over three runs and all settings access to the same total number of training samples. We fine-tune the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 471, + 284, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 284, + 483 + ], + "score": 1.0, + "content": "learning rate of mini-batch SGD for each setting.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 450, + 506, + 483 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 506, + 500, + 518 + ], + "lines": [ + { + "bbox": [ + 105, + 506, + 502, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 502, + 520 + ], + "score": 1.0, + "content": "4.2 (POST)-LOCAL SGD CLOSES THE GENERALIZATION GAP OF LARGE-BATCH TRAINING", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Even though local SGD has demonstrated effective communication efficiency with guaranteed test", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "performance (Scenario 1), we observed in the previous section that it still encounters difficulties when", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "scaling to very large mini-batches (Figure 2(a)), though it is much less affected than mini-batch SGD", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "(Figure 2(b)). Post-local SGD can address the generalization issue of large batch training (Scenario 2)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 426, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 426, + 584 + ], + "score": 1.0, + "content": "as we show now. First, we would like to highlight some key findings in Table 2.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 529, + 506, + 584 + ] + }, + { + "type": "table", + "bbox": [ + 121, + 656, + 495, + 730 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 596, + 505, + 646 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "Table 2: Test performance (generalization) for local SGD variants and mini-batch SGD (highlighting selected", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 606, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 505, + 617 + ], + "score": 1.0, + "content": "data from Figure 2, Table 3 and Figure 3). Mini-batch SGD suffers from the generalization gap (and sometimes", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 616, + 505, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 395, + 627 + ], + "score": 1.0, + "content": "even from optimization issues due to insufficient training) when scaling to larger", + "type": "text" + }, + { + "bbox": [ + 396, + 617, + 405, + 625 + ], + "score": 0.73, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 616, + 421, + 627 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 421, + 617, + 430, + 625 + ], + "score": 0.71, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 616, + 505, + 627 + ], + "score": 1.0, + "content": ". Same experimental", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 625, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 339, + 639 + ], + "score": 1.0, + "content": "setup as in Figure 2 (fine-tuned mini-batch SGD baselines). The", + "type": "text" + }, + { + "bbox": [ + 339, + 627, + 354, + 635 + ], + "score": 0.8, + "content": "\\hookrightarrow", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 625, + 506, + 639 + ], + "score": 1.0, + "content": "denotes the transition at the first learning", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 635, + 327, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 327, + 648 + ], + "score": 1.0, + "content": "rate decay. The reported results are averaged over three runs.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + }, + { + "type": "table_body", + "bbox": [ + 121, + 656, + 495, + 730 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 121, + 656, + 495, + 730 + ], + "spans": [ + { + "bbox": [ + 121, + 656, + 495, + 730 + ], + "score": 0.982, + "html": "
AlgorithmTop-1 acc.Effect. batch sizeAlgorithmTop-1 acc.Effect. batch size
Mini-batch SGD (K=4)92.6%KB=512Mini-batch SGD (K =16)92.5%KB=2048
Mini-batch SGD (K =4)89.5%KB=8192Mini-batch SGD (K =16)76.3%KB=16384
Mini-batch SGD (K=4)92.5%KB=512→8192Mini-batch SGD(K=16)92.0%KHBloc=2048→ 16384
Local SGD(H=16,K=4)92.5%KHBloc =8192Local SGD(H=8,K=16)92.0%KHBloc =16384
Post-local SGD(H=16,K=4)92.7%KHBloc =512→8192Post-local SGD (H=8,K=16)92.9%KHBloc =2048→ 16384
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CIFAR-10CIFAR-100
Small batch baseline*Large batch baseline * K=2,B=128K=16,B=128 K=16,Bloc=128K=16,Blc =128Post-local SGD (H=16)Post-local SGD (H=32)Small batch baseline *Large batch baseline *Post-local SGD (H=16)Post-local SGD (H=32) K=2,B=128K=16,B= 128K=16,Blo =128 K= 16,Bloc=128
ResNet-2092.63 ±0.2692.48 3±0.1792.80 ±0.1693.02 ±0.2468.84 ±0.0668.17 ±0.1869.24 ±0.2669.38 ±0.20
DenseNet-40-1294.41 ±0.1494.36 ±0.2094.43 ±0.1294.58 ±0.1174.85 ±0.1474.08 3±0.4674.45 ±0.3075.03 ±0.05
WideResNet-28-1095.89 ±0.1095.43 ±0.3795.94 ±0.0695.76 ±0.2579.78 3±0.1679.31 ±0.2380.28 ±0.1380.65 ±0.16
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CIFAR-10CIFAR-100
Small batch baseline*Large batch baseline * K=2,B=128K=16,B=128 K=16,Bloc=128K=16,Blc =128Post-local SGD (H=16)Post-local SGD (H=32)Small batch baseline *Large batch baseline *Post-local SGD (H=16)Post-local SGD (H=32) K=2,B=128K=16,B= 128K=16,Blo =128 K= 16,Bloc=128
ResNet-2092.63 ±0.2692.48 3±0.1792.80 ±0.1693.02 ±0.2468.84 ±0.0668.17 ±0.1869.24 ±0.2669.38 ±0.20
DenseNet-40-1294.41 ±0.1494.36 ±0.2094.43 ±0.1294.58 ±0.1174.85 ±0.1474.08 3±0.4674.45 ±0.3075.03 ±0.05
WideResNet-28-1095.89 ±0.1095.43 ±0.3795.94 ±0.0695.76 ±0.2579.78 3±0.1679.31 ±0.2380.28 ±0.1380.65 ±0.16
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CIFAR-10 H=16CIFAR-100
H=1H =32H =64H=1H =16H=32H =64
signSGD (Bernstein etal.,018)91.61 ±0.2891.77±0.0492.22±0.1192.00 ±0.1567.21 ±0.1167.43 ±0.3668.32 ±0.1868.12 ±0.11
EF-signSGD(Karimireddy etal.,2019)92.45 ±0.2892.63±0.1692.72±0.0592.76 ±0.0668.19 ±0.1069.14 ±0.3669.00 ±0.1469.49 ±0.21
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SGD + Momentum + LARSSGD + Momentum + LARS + Post-local SGD
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CIFAR-10 H=16CIFAR-100
H=1H =32H =64H=1H =16H=32H =64
signSGD (Bernstein etal.,018)91.61 ±0.2891.77±0.0492.22±0.1192.00 ±0.1567.21 ±0.1167.43 ±0.3668.32 ±0.1868.12 ±0.11
EF-signSGD(Karimireddy etal.,2019)92.45 ±0.2892.63±0.1692.72±0.0592.76 ±0.0668.19 ±0.1069.14 ±0.3669.00 ±0.1469.49 ±0.21
", + "type": "table", + "image_path": "a791272d088ca150f63351c5c95d647c8185395a02e3235ae2509f801d37f987.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 106, + 309, + 505, + 320.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 106, + 320.0, + 505, + 331.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 106, + 331.0, + 505, + 342.0 + ], + "spans": [], + "index": 34 + } + ] + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 502, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 504, + 375 + ], + "score": 1.0, + "content": "2017; Shallue et al., 2018; Golmant et al., 2018). Considering other optimizers, Table 5 below shows", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 373, + 373, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 373, + 386 + ], + "score": 1.0, + "content": "that post-local SGD can improve upon LARS6 (You et al., 2017a).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 361, + 504, + 386 + ] + }, + { + "type": "table", + "bbox": [ + 142, + 427, + 465, + 462 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 396, + 506, + 426 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 396, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 407 + ], + "score": 1.0, + "content": "Table 5: The Top-1 test accuracy of training ResNet50 on ImageNet. We report the performance (w/ or w/o", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 406, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 417 + ], + "score": 1.0, + "content": "post-local SGD) achieved by using SGD with Nesterov Momentum, learning schemes in (Goyal et al., 2017),", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 416, + 502, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 429, + 427 + ], + "score": 1.0, + "content": "and LARS. We follow the general experimental setup described in Section A.4.2, and use", + "type": "text" + }, + { + "bbox": [ + 429, + 416, + 457, + 425 + ], + "score": 0.89, + "content": "K = 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 416, + 473, + 427 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 474, + 416, + 497, + 425 + ], + "score": 0.89, + "content": "H = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 416, + 502, + 427 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + }, + { + "type": "table_body", + "bbox": [ + 142, + 427, + 465, + 462 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 142, + 427, + 465, + 462 + ], + "spans": [ + { + "bbox": [ + 142, + 427, + 465, + 462 + ], + "score": 0.96, + "html": "
SGD + Momentum + LARSSGD + Momentum + LARS + Post-local SGD
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Hence, a profound theoretical study of the generalization local SGD is beyond the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 505, + 542 + ], + "score": 1.0, + "content": "scope of this work but we hope the favorable experimental results will trigger future research in this", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "direction. In this section we discuss some related work and we argue that local SGD can be seen as a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 551, + 402, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 402, + 564 + ], + "score": 1.0, + "content": "way to inject and control stochastic noise to the whole training procedure.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 507, + 506, + 564 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 576, + 504, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 574, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 589 + ], + "score": 1.0, + "content": "Connecting Local Updates with Stochastic Noise Injection. The update eq. (1) can alternatively", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 586, + 161, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 161, + 599 + ], + "score": 1.0, + "content": "be written as", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 597, + 439, + 614 + ], + "lines": [ + { + "bbox": [ + 171, + 597, + 439, + 614 + ], + "spans": [ + { + "bbox": [ + 171, + 597, + 439, + 614 + ], + "score": 0.89, + "content": "{ \\pmb w } _ { t + 1 } = { \\pmb w } _ { t } - \\gamma \\hat { \\pmb g } _ { t } = { \\pmb w } _ { t } - \\gamma { \\pmb g } _ { t } + \\gamma \\big ( { \\pmb g } _ { t } - \\hat { \\pmb g } _ { t } \\big ) = { \\pmb w } _ { t } - \\gamma { \\pmb g } _ { t } + \\gamma \\epsilon ,", + "type": "interline_equation", + "image_path": "8a92b403bef073c2be26fc7858cc3b1015d0eaef3f238fceefc63691d7bc22c3.jpg" + } + ] + } + ], + "index": 51, + "virtual_lines": [ + { + "bbox": [ + 171, + 597, + 439, + 614 + ], + "spans": [], + "index": 51 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 619, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 104, + 619, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 619, + 133, + 635 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 619, + 220, + 633 + ], + "score": 0.73, + "content": "\\begin{array} { r } { \\hat { \\pmb g } _ { t } : = \\frac { 1 } { B } \\sum _ { i } \\nabla f _ { i } ( { \\pmb w } _ { t } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 619, + 223, + 635 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 224, + 620, + 315, + 633 + ], + "score": 0.78, + "content": "\\pmb { g } _ { t } : = \\mathbb { E } \\hat { \\pmb { g } } _ { t } = \\nabla f ( \\pmb { w } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 619, + 334, + 635 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 334, + 621, + 387, + 633 + ], + "score": 0.91, + "content": "\\boldsymbol { \\epsilon } : = \\boldsymbol { g } _ { t } - \\hat { \\boldsymbol { g } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 619, + 506, + 635 + ], + "score": 1.0, + "content": ". For most data-sets the noise", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 107, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 107, + 634, + 113, + 641 + ], + "score": 0.59, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 632, + 388, + 644 + ], + "score": 1.0, + "content": "can be well approximated by a zero-mean Gaussian with variance", + "type": "text" + }, + { + "bbox": [ + 389, + 631, + 414, + 644 + ], + "score": 0.92, + "content": "\\Sigma ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 632, + 505, + 644 + ], + "score": 1.0, + "content": ". The variance matrix", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 131, + 654 + ], + "score": 0.91, + "content": "\\Sigma ( w )", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 642, + 506, + 655 + ], + "score": 1.0, + "content": ", following Hoffer et al. (2017) and Hu et al. 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When the batch size grows too large, the relative", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 182, + 507, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 304, + 197 + ], + "score": 1.0, + "content": "magnitude of the stochastic noise decreases as", + "type": "text" + }, + { + "bbox": [ + 304, + 183, + 411, + 195 + ], + "score": 0.91, + "content": "\\gamma B ^ { - 1 } \\not \\approx \\gamma ( N - B ) \\bar { / } B N", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 182, + 431, + 197 + ], + "score": 1.0, + "content": ", i.e.", + "type": "text" + }, + { + "bbox": [ + 432, + 185, + 438, + 195 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 182, + 507, + 197 + ], + "score": 1.0, + "content": "is not anymore", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 504, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 472, + 207 + ], + "score": 1.0, + "content": "uniquely describing the training dynamics for large batch size and small dataset size when", + "type": "text" + }, + { + "bbox": [ + 472, + 195, + 504, + 205 + ], + "score": 0.87, + "content": "B \\ll N", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "(Jastrz˛ebski et al., 2018). 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Similar positive effects of adding well-structured", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "stochastic noise to SGD dynamics on non-convex problems have recently also been observed in (Smith", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 479, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 479, + 355 + ], + "score": 1.0, + "content": "& Le, 2018; Chaudhari & Soatto, 2018; Zhu et al., 2019; Xing et al., 2018; Wen et al., 2019).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 106, + 371, + 355, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 356, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 356, + 384 + ], + "score": 1.0, + "content": "5.1 POST-LOCAL SGD CONVERGES TO FLATTER MINIMA", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 506, + 456 + ], + "lines": [ + { + "bbox": [ + 104, + 389, + 507, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 507, + 404 + ], + "score": 1.0, + "content": "Figure 4(a) evaluates the spectrum of the Hessian for different local minima. 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Similar positive effects of adding well-structured", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "stochastic noise to SGD dynamics on non-convex problems have recently also been observed in (Smith", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 342, + 479, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 479, + 355 + ], + "score": 1.0, + "content": "& Le, 2018; Chaudhari & Soatto, 2018; Zhu et al., 2019; Xing et al., 2018; Wen et al., 2019).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 244, + 507, + 355 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 371, + 355, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 356, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 356, + 384 + ], + "score": 1.0, + "content": "5.1 POST-LOCAL SGD CONVERGES TO FLATTER MINIMA", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 506, + 456 + ], + "lines": [ + { + "bbox": [ + 104, + 389, + 507, + 404 + ], + "spans": [ + { + "bbox": [ + 104, + 389, + 507, + 404 + ], + "score": 1.0, + "content": "Figure 4(a) evaluates the spectrum of the Hessian for different local minima. 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We also provide extensive", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "experiments with another variant hierarchical local SGD, showing its adaptivity to available system", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "resources. 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Our local SGD variant, called post-local SGD, not only", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "score": 1.0, + "content": "outperforms large-batch SGD’s generalization performance but also matches that of small-batch SGD.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "score": 1.0, + "content": "In our experiments post-local SGD converged to flatter minima compared to traditional large-batch", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 160, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 174 + ], + "score": 1.0, + "content": "SGD, which partially explains its improved generalization performance. We also provide extensive", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 505, + 185 + ], + "score": 1.0, + "content": "experiments with another variant hierarchical local SGD, showing its adaptivity to available system", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "resources. 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Each consists of a training", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 117, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 117, + 420, + 305, + 433 + ], + "score": 1.0, + "content": "set of 50K and a test set of 10K color images of", + "type": "text" + }, + { + "bbox": [ + 305, + 421, + 337, + 431 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "pixels, as well as 10 and 100 target classes", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 117, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 117, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "respectively. 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ModelCommunication# parametersComputation# flops per imageComputation/Communication scaling ratio
ResNet-20 (CIFAR-10)0.27 million0.041 billion151.85
ResNet-20 (CIFAR-100)0.27 million0.041 billion151.85
ResNet-50 (ImageNet-1k)25.00 million7.7 billion308.00
DenseNet-40-12 (CIFAR-10)1.06 million0.28 billion264.15
DenseNet-40-12 (CIFAR-100)1.10 million0.28 billion254.55
WideResNet-28-10 (CIFAR-10)36.48 million5.24 billion143.64
WideResNet-28-10 (CIFAR-100)36.54 million5.24 billion143.40
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ModelCommunication# parametersComputation# flops per imageComputation/Communication scaling ratio
ResNet-20 (CIFAR-10)0.27 million0.041 billion151.85
ResNet-20 (CIFAR-100)0.27 million0.041 billion151.85
ResNet-50 (ImageNet-1k)25.00 million7.7 billion308.00
DenseNet-40-12 (CIFAR-10)1.06 million0.28 billion264.15
DenseNet-40-12 (CIFAR-100)1.10 million0.28 billion254.55
WideResNet-28-10 (CIFAR-10)36.48 million5.24 billion143.64
WideResNet-28-10 (CIFAR-100)36.54 million5.24 billion143.40
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For a precise definition, this is formalized by the following", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 375, + 186, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 186, + 387 + ], + "score": 1.0, + "content": "two configurations:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 341, + 506, + 387 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 394, + 505, + 491 + ], + "lines": [ + { + "bbox": [ + 131, + 394, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 131, + 394, + 439, + 408 + ], + "score": 1.0, + "content": "• Scaling the learning rate: When the mini-batch size is multiplied by", + "type": "text" + }, + { + "bbox": [ + 440, + 396, + 447, + 406 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 394, + 505, + 408 + ], + "score": 1.0, + "content": ", multiply the", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 407, + 217, + 418 + ], + "spans": [ + { + "bbox": [ + 141, + 407, + 207, + 418 + ], + "score": 1.0, + "content": "learning rate by", + "type": "text" + }, + { + "bbox": [ + 208, + 407, + 214, + 416 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 407, + 217, + 418 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 132, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "• Learning rate gradual warm-up: We gradually ramp up the learning rate from a small", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 433, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 142, + 433, + 430, + 444 + ], + "score": 1.0, + "content": "to a large value. 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More precisely, the incremental step size for each iteration is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 464, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 141, + 466, + 207, + 479 + ], + "score": 1.0, + "content": "calculated from", + "type": "text" + }, + { + "bbox": [ + 203, + 464, + 507, + 482 + ], + "score": 1.0, + "content": "ηˆ\u0001η5N{pknq , where N is the number of total training samples, k is the number", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 479, + 358, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 479, + 237, + 492 + ], + "score": 1.0, + "content": "of computing units and", + "type": "text" + }, + { + "bbox": [ + 237, + 482, + 244, + 489 + ], + "score": 0.79, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 479, + 358, + 492 + ], + "score": 1.0, + "content": "is the local mini-batch size.", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + } + ], + "index": 16.5, + "bbox_fs": [ + 131, + 394, + 507, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 504, + 459, + 526 + ], + "lines": [ + { + "bbox": [ + 107, + 504, + 460, + 516 + ], + "spans": [ + { + "bbox": [ + 107, + 505, + 131, + 515 + ], + "score": 1.0, + "content": "A.4", + "type": "text" + }, + { + "bbox": [ + 131, + 504, + 460, + 516 + ], + "score": 1.0, + "content": "HYPERPARAMETER CHOICES AND TRAINING PROCEDURE, OVER DIFFERENT", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 132, + 515, + 220, + 527 + ], + "spans": [ + { + "bbox": [ + 132, + 515, + 220, + 527 + ], + "score": 1.0, + "content": "MODELS/DATASETS", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 107, + 504, + 460, + 527 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 535, + 244, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 536, + 244, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 244, + 549 + ], + "score": 1.0, + "content": "A.4.1 CIFAR-10/CIFAR-100", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 555, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 506, + 568 + ], + "score": 1.0, + "content": "The experiments follow the common mini-batch SGD training scheme for CIFAR (He et al., 2016a;b;", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Huang et al., 2017b) and all competing methods access the same total amount of data samples", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "regardless of the number of local steps. The training procedure is terminated when the distributed", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 588, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 506, + 600 + ], + "score": 1.0, + "content": "algorithms have accessed the same number of samples as a standalone worker would access. For", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "example, ResNet-20, DensetNet-40-12 and WideResNet-28-10 would access 300, 300 and 250", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 610, + 507, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 507, + 623 + ], + "score": 1.0, + "content": "epochs respectively. The data is partitioned among the GPUs and reshuffled globally every epoch.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "score": 1.0, + "content": "The local mini-batches are then sampled among the local data available on each GPU, and its size is", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 632, + 189, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 139, + 644 + ], + "score": 1.0, + "content": "fixed to", + "type": "text" + }, + { + "bbox": [ + 139, + 632, + 185, + 643 + ], + "score": 0.91, + "content": "B _ { \\mathrm { l o c } } = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 632, + 189, + 644 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 554, + 507, + 644 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "The learning rate scheme follows works (He et al., 2016a; Huang et al., 2017b), where we drop", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 352, + 672 + ], + "score": 1.0, + "content": "the initial learning rate by 10 when the model has accessed", + "type": "text" + }, + { + "bbox": [ + 352, + 660, + 372, + 671 + ], + "score": 0.88, + "content": "5 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 659, + 391, + 672 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 392, + 660, + 411, + 671 + ], + "score": 0.88, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "of the total number of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 671, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 682 + ], + "score": 1.0, + "content": "training samples. The initial learning rates of ResNet-20, DensetNet-40-12 and WideResNet-28-10", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "are fine-tuned on single GPU (which are 0.2, 0.2 and 0.1 respectively), and can be scaled by the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 693, + 366, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 366, + 705 + ], + "score": 1.0, + "content": "global mini-batch size when using large-batch learning schemes.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 648, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In addition to this, we use a Nesterov momentum of 0.9 without dampening, which is applied", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "independently to each local model. For all architectures, following He et al. (2016a), we do not", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "score": 1.0, + "content": "apply weight decay on the learnable Batch Normalization (BN) coefficients. The weight decay", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "of ResNet-20, DensetNet-40-12 and WideResNet-28-10 are 1e-4, 1e-4 and 5e-4 respectively. For", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "the BN for distributed training we again follow Goyal et al. (2017) and compute the BN statistics", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 233, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 233, + 128 + ], + "score": 1.0, + "content": "independently for each worker.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 37.5, + "bbox_fs": [ + 106, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 97 + ], + "score": 1.0, + "content": "apply weight decay on the learnable Batch Normalization (BN) coefficients. The weight decay", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "of ResNet-20, DensetNet-40-12 and WideResNet-28-10 are 1e-4, 1e-4 and 5e-4 respectively. For", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "the BN for distributed training we again follow Goyal et al. (2017) and compute the BN statistics", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 233, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 233, + 128 + ], + "score": 1.0, + "content": "independently for each worker.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 108, + 132, + 504, + 154 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 506, + 145 + ], + "score": 1.0, + "content": "Unless mentioned specifically, local SGD uses the exact same optimization scheme as mini-batch", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 133, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 133, + 156 + ], + "score": 1.0, + "content": "SGD.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "The procedure of fine-tuning. There is no optimal learning rate scaling rule for large-batch SGD", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 179, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 506, + 192 + ], + "score": 1.0, + "content": "across different mini-batch sizes, tasks and architectures, as revealed in the Figure 8 of Shallue et al.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 202 + ], + "score": 1.0, + "content": "(2018). Our tuning procedure is built on the insight of their Figure 8, where we grid-search the optimal", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 214 + ], + "score": 1.0, + "content": "learning rate for each mini-batch size, starting from the linearly scaled learning rate. For example,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 506, + 225 + ], + "score": 1.0, + "content": "compared to mini-batch size 128, mini-batch size 2048 with default large-batch learning schemes", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 506, + 236 + ], + "score": 1.0, + "content": "need to linearly scale the learning rate by the factor of 16. In order to find out its optimal learning rate,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 506, + 246 + ], + "score": 1.0, + "content": "we will evaluate a linear-spaced grid of five different factors (i.e., t15, 15.5, 16, 16.5, 17u). If the best", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 245, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 506, + 257 + ], + "score": 1.0, + "content": "performance was ever at one of the extremes of the grid, we would try new grid points so that the best", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 255, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 505, + 268 + ], + "score": 1.0, + "content": "performance was contained in the middle of the parameters. Please note that this unbounded grid", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "search—even though initialized at the linearly scaled learning rate—does also cover other suggested", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 278, + 485, + 290 + ], + "spans": [ + { + "bbox": [ + 106, + 278, + 485, + 290 + ], + "score": 1.0, + "content": "scaling heuristics, for instance square root scaling, and finds the best scaling for each scenario.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 294, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 308 + ], + "score": 1.0, + "content": "Note that in our experiments of large-batch SGD, either with the default large-batch learning schemes,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 305, + 506, + 318 + ], + "score": 1.0, + "content": "or tuning/using the optimal learning rate, we always warm-up the learning rate for the first 5 epochs.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 330, + 190, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 192, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 192, + 343 + ], + "score": 1.0, + "content": "A.4.2 IMAGENET", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "ResNet-50 training is limited to 90 passes over the data in total, and the data is disjointly partitioned", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "and is re-shuffled globally every epoch. 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Figure 7 shows that local SGD", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "is significantly more communication efficient while guaranteeing the same accuracy and enjoys faster", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 455, + 722 + ], + "score": 1.0, + "content": "convergence speed. In Figure 7, the local models use a fixed local mini-batch size", + "type": "text" + }, + { + "bbox": [ + 455, + 710, + 505, + 721 + ], + "score": 0.9, + "content": "B _ { \\mathrm { l o c } } ~ = ~ 1 2 8", + "type": "inline_equation" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "for all updates. 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The impact", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "of local SGD training upon the total training time is more significant for larger number of local steps", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 107, + 344, + 117, + 354 + ], + "score": 0.75, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 343, + 278, + 357 + ], + "score": 1.0, + "content": "(i.e., Figure 7(b)), resulting in an at least", + "type": "text" + }, + { + "bbox": [ + 279, + 345, + 293, + 355 + ], + "score": 0.86, + "content": "3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 343, + 443, + 357 + ], + "score": 1.0, + "content": "speed-up when comparing mini-batch", + "type": "text" + }, + { + "bbox": [ + 444, + 344, + 472, + 354 + ], + "score": 0.91, + "content": "H = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "to local", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 148, + 368 + ], + "score": 1.0, + "content": "SGD with", + "type": "text" + }, + { + "bbox": [ + 149, + 356, + 182, + 366 + ], + "score": 0.9, + "content": "H = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 354, + 416, + 368 + ], + "score": 1.0, + "content": ". 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We limit", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 424, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 438 + ], + "score": 1.0, + "content": "ResNet-50 training to 90 passes over the data in total, and use the standard training configurations as", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 436, + 230, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 230, + 448 + ], + "score": 1.0, + "content": "mentioned in Appendix A.4.2.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "image", + "bbox": [ + 222, + 464, + 386, + 600 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 222, + 464, + 386, + 600 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 222, + 464, + 386, + 600 + ], + "spans": [ + { + "bbox": [ + 222, + 464, + 386, + 600 + ], + "score": 0.964, + "type": "image", + "image_path": "6062e3f588aa9e0f27dde8c400d7124cc3a94c7fef501d063e5a1f3ca78a02cc.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 222, + 464, + 386, + 476.3636363636364 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 222, + 476.3636363636364, + 386, + 488.72727272727275 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 222, + 488.72727272727275, + 386, + 501.0909090909091 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 222, + 501.0909090909091, + 386, + 513.4545454545455 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 222, + 513.4545454545455, + 386, + 525.8181818181819 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 222, + 525.8181818181819, + 386, + 538.1818181818182 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 222, + 538.1818181818182, + 386, + 550.5454545454546 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 222, + 550.5454545454546, + 386, + 562.909090909091 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 222, + 562.909090909091, + 386, + 575.2727272727274 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 222, + 575.2727272727274, + 386, + 587.6363636363637 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 222, + 587.6363636363637, + 386, + 600.0000000000001 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 609, + 505, + 650 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 609, + 506, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 621 + ], + "score": 1.0, + "content": "Figure 8: The performance of local SGD trained on ImageNet-1k with ResNet-50.We evaluate the model", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 619, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 506, + 631 + ], + "score": 1.0, + "content": "performance on test dataset after each complete accessing of the whole training samples. We apply the large-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 629, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 506, + 641 + ], + "score": 1.0, + "content": "batch learning schemes (Goyal et al., 2017) to the ImageNet for these two methods. 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The impact", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 333, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 505, + 346 + ], + "score": 1.0, + "content": "of local SGD training upon the total training time is more significant for larger number of local steps", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 107, + 344, + 117, + 354 + ], + "score": 0.75, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 343, + 278, + 357 + ], + "score": 1.0, + "content": "(i.e., Figure 7(b)), resulting in an at least", + "type": "text" + }, + { + "bbox": [ + 279, + 345, + 293, + 355 + ], + "score": 0.86, + "content": "3 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 343, + 443, + 357 + ], + "score": 1.0, + "content": "speed-up when comparing mini-batch", + "type": "text" + }, + { + "bbox": [ + 444, + 344, + 472, + 354 + ], + "score": 0.91, + "content": "H = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "to local", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 354, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 148, + 368 + ], + "score": 1.0, + "content": "SGD with", + "type": "text" + }, + { + "bbox": [ + 149, + 356, + 182, + 366 + ], + "score": 0.9, + "content": "H = 1 6", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 354, + 416, + 368 + ], + "score": 1.0, + "content": ". 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For local SGD, the number", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 639, + 217, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 186, + 651 + ], + "score": 1.0, + "content": "of local steps is set to", + "type": "text" + }, + { + "bbox": [ + 186, + 640, + 212, + 649 + ], + "score": 0.9, + "content": "H = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 639, + 217, + 651 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + } + ], + "index": 28.75 + }, + { + "type": "text", + "bbox": [ + 107, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "Moreover, in our ImageNet experiment, the initial phase of local SGD training follows the theoretical", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "assumption mentioned in Subsection 2, and thus we gradually warm up the number of local steps", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 686, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 214, + 700 + ], + "score": 1.0, + "content": "from 1 to the desired value", + "type": "text" + }, + { + "bbox": [ + 214, + 688, + 224, + 698 + ], + "score": 0.75, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 686, + 505, + 700 + ], + "score": 1.0, + "content": "during the first few epochs of the training. We found that exponentially", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "increasing the number of local steps from 1 by the factor of 2 (until reaching the expected number of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 381, + 722 + ], + "score": 1.0, + "content": "local steps) performs well. For example, our ImageNet training uses", + "type": "text" + }, + { + "bbox": [ + 381, + 710, + 410, + 720 + ], + "score": 0.89, + "content": "H = 8", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", so the number of local", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 315, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 237, + 733 + ], + "score": 1.0, + "content": "steps for the first three epochs is", + "type": "text" + }, + { + "bbox": [ + 237, + 721, + 262, + 732 + ], + "score": 0.61, + "content": "1 , 2 , 4", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 721, + 315, + 733 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 666, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 451, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 452, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 452, + 95 + ], + "score": 1.0, + "content": "B.3.3 LOCAL SGD SCALES TO LARGER BATCH SIZES THAN MINI-BATCH SGD", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 101, + 504, + 135 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 114 + ], + "score": 1.0, + "content": "The empirical studies (Shallue et al., 2018; Golmant et al., 2018) reveal the regime of maximal data", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 112, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 506, + 126 + ], + "score": 1.0, + "content": "parallelism across different tasks and models, where the large-batch training would reach the limit", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 123, + 342, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 342, + 136 + ], + "score": 1.0, + "content": "and additional parallelism provides no benefit whatsoever.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "image", + "bbox": [ + 223, + 146, + 387, + 283 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 223, + 146, + 387, + 283 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 223, + 146, + 387, + 283 + ], + "spans": [ + { + "bbox": [ + 223, + 146, + 387, + 283 + ], + "score": 0.97, + "type": "image", + "image_path": "4e126eb32eb72dc647f8c529a1e431ea1a6361df64813ce4cce6bf1d5da1ee85.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 146, + 387, + 159.7 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 223, + 159.7, + 387, + 173.39999999999998 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 223, + 173.39999999999998, + 387, + 187.09999999999997 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 223, + 187.09999999999997, + 387, + 200.79999999999995 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 223, + 200.79999999999995, + 387, + 214.49999999999994 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 223, + 214.49999999999994, + 387, + 228.19999999999993 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 223, + 228.19999999999993, + 387, + 241.89999999999992 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 223, + 241.89999999999992, + 387, + 255.5999999999999 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 223, + 255.5999999999999, + 387, + 269.2999999999999 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 223, + 269.2999999999999, + 387, + 282.9999999999999 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 294, + 505, + 334 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 294, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 305 + ], + "score": 1.0, + "content": "Figure 9: The relationship between steps to top-1 test accuracy and batch size, of training ResNet-20 on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "CIFAR-10. The “step” is equivalent to the number of applying gradients. The global mini-batch size is increased", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 197, + 325 + ], + "score": 1.0, + "content": "by adding more workers", + "type": "text" + }, + { + "bbox": [ + 198, + 315, + 207, + 323 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 313, + 246, + 325 + ], + "score": 1.0, + "content": "with fixed", + "type": "text" + }, + { + "bbox": [ + 247, + 314, + 289, + 324 + ], + "score": 0.89, + "content": "B _ { \\mathrm { l o c } } = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 313, + 505, + 325 + ], + "score": 1.0, + "content": ". Results are averaged over three runs, each with fine-tuned", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 322, + 156, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 156, + 336 + ], + "score": 1.0, + "content": "learning rate.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + } + ], + "index": 12.0 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 505, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "On contrary to standard large-batch training, local SGD scales to larger batch size and provides", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "additional parallelism upon the limitation of current large batch training. Figure 9 shows the example", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 369, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 278, + 381 + ], + "score": 1.0, + "content": "of training ResNet-20 on CIFAR-10 with", + "type": "text" + }, + { + "bbox": [ + 278, + 369, + 308, + 379 + ], + "score": 0.89, + "content": "H = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 369, + 506, + 381 + ], + "score": 1.0, + "content": ", which trains and generalizes better in terms of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 322, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 322, + 392 + ], + "score": 1.0, + "content": "update steps while with reduced communication cost.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 106, + 404, + 486, + 416 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 486, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 486, + 416 + ], + "score": 1.0, + "content": "B.4 PRACTICAL IMPROVEMENT POSSIBILITIES FOR STANDARD LOCAL SGD TRAINING", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 437 + ], + "score": 1.0, + "content": "We investigate different aspects of the training to address the quality when scaling local SGD to the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 434, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 434, + 506, + 450 + ], + "score": 1.0, + "content": "extreme case, e.g., hybrid momentum scheme, warming up the local SGD or fine-tuning the learning", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "rate. In this section, we briefly present how these strategies are, and how they work in practice where", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 458, + 297, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 297, + 470 + ], + "score": 1.0, + "content": "we train ResNet-20 on CIFAR-10 on 16 GPUs.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "title", + "bbox": [ + 107, + 481, + 278, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 279, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 279, + 494 + ], + "score": 1.0, + "content": "B.4.1 LOCAL SGD WITH MOMENTUM", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 504, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "Momentum mini-batch SGD is widely used in place of vanilla SGD. The distributed mini-batch SGD", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 512, + 319, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 216, + 524 + ], + "score": 1.0, + "content": "with vanilla momentum on", + "type": "text" + }, + { + "bbox": [ + 216, + 513, + 227, + 522 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 512, + 319, + 524 + ], + "score": 1.0, + "content": "training nodes follows", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 528, + 430, + 562 + ], + "lines": [ + { + "bbox": [ + 181, + 528, + 430, + 562 + ], + "spans": [ + { + "bbox": [ + 181, + 528, + 430, + 562 + ], + "score": 0.94, + "content": "{ \\pmb u } _ { ( t ) } = m { \\pmb u } _ { ( t - 1 ) } + \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } { \\pmb \\nabla } _ { ( t ) } ^ { k } , \\qquad { \\pmb w } _ { ( t + 1 ) } = { \\pmb w } _ { ( t ) } - \\gamma { \\pmb u } _ { ( t ) }", + "type": "interline_equation", + "image_path": "6511c42a6e15a05d3b688ccfb41f4c319df000a55f4da5af8a6ee5224d0a4db8.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 181, + 528, + 430, + 539.3333333333334 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 181, + 539.3333333333334, + 430, + 550.6666666666667 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 181, + 550.6666666666667, + 430, + 562.0000000000001 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 265, + 586 + ], + "lines": [ + { + "bbox": [ + 133, + 567, + 262, + 587 + ], + "spans": [ + { + "bbox": [ + 133, + 567, + 262, + 587 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb { \\nabla } _ { ( t ) } ^ { k } = \\frac { 1 } { | \\mathcal { T } _ { ( t ) } ^ { k } | } \\sum _ { i \\in \\mathcal { T } _ { ( t ) } ^ { k } } \\nabla f _ { i } \\big ( \\pmb { w } _ { ( t ) } \\big ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 416, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 412, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 129, + 606 + ], + "score": 1.0, + "content": "After", + "type": "text" + }, + { + "bbox": [ + 130, + 591, + 140, + 601 + ], + "score": 0.83, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 587, + 380, + 606 + ], + "score": 1.0, + "content": "updates of mini-batch SGD, we have the following updated", + "type": "text" + }, + { + "bbox": [ + 381, + 593, + 412, + 604 + ], + "score": 0.91, + "content": "{ \\pmb w } _ { ( t + H ) }", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 608, + 491, + 643 + ], + "lines": [ + { + "bbox": [ + 119, + 608, + 491, + 643 + ], + "spans": [ + { + "bbox": [ + 119, + 608, + 491, + 643 + ], + "score": 0.94, + "content": "{ \\pmb w } _ { ( t + H ) } = { \\pmb w } _ { ( t ) } - \\gamma \\left( \\sum _ { \\tau = 1 } ^ { H } m ^ { \\tau } { \\pmb u } _ { ( t - 1 ) } + \\sum _ { \\tau = 0 } ^ { H - 1 } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\pmb { \\nabla } _ { ( t ) } ^ { k } + \\dots + \\sum _ { \\tau = 0 } ^ { 0 } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\pmb { \\nabla } _ { ( t + H - 1 ) } ^ { k } \\right)", + "type": "interline_equation", + "image_path": "194bb0cc6dad1e7c12752e00993d0f87c1bfbdc9a9061b38d40169ebe959bea3.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 119, + 608, + 491, + 619.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 119, + 619.6666666666666, + 491, + 631.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 119, + 631.3333333333333, + 491, + 642.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "score": 1.0, + "content": "Coming back to the setting of local SGD, we can apply momentum acceleration on each local model,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 664, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 506, + 676 + ], + "score": 1.0, + "content": "or on a global level (Chen & Huo, 2016). In the remaining part of this section, we analyze the case of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 675, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 688 + ], + "score": 1.0, + "content": "applying local momentum and global momentum. For ease of understanding, we assume the learning", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 686, + 305, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 123, + 699 + ], + "score": 1.0, + "content": "rate", + "type": "text" + }, + { + "bbox": [ + 124, + 687, + 131, + 698 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 686, + 240, + 699 + ], + "score": 1.0, + "content": "is the same throughout the", + "type": "text" + }, + { + "bbox": [ + 240, + 687, + 250, + 695 + ], + "score": 0.84, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 686, + 305, + 699 + ], + "score": 1.0, + "content": "update steps.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "Local SGD with Local Momentum. When applying local momentum on the local SGD, i.e., using", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "independent identical momentum acceleration for each local model and only globally aggregating the", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + } + ], + "page_idx": 20, + "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, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 82, + 451, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 452, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 452, + 95 + ], + "score": 1.0, + "content": "B.3.3 LOCAL SGD SCALES TO LARGER BATCH SIZES THAN MINI-BATCH SGD", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 101, + 504, + 135 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 505, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 114 + ], + "score": 1.0, + "content": "The empirical studies (Shallue et al., 2018; Golmant et al., 2018) reveal the regime of maximal data", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 112, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 112, + 506, + 126 + ], + "score": 1.0, + "content": "parallelism across different tasks and models, where the large-batch training would reach the limit", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 123, + 342, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 342, + 136 + ], + "score": 1.0, + "content": "and additional parallelism provides no benefit whatsoever.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 101, + 506, + 136 + ] + }, + { + "type": "image", + "bbox": [ + 223, + 146, + 387, + 283 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 223, + 146, + 387, + 283 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 223, + 146, + 387, + 283 + ], + "spans": [ + { + "bbox": [ + 223, + 146, + 387, + 283 + ], + "score": 0.97, + "type": "image", + "image_path": "4e126eb32eb72dc647f8c529a1e431ea1a6361df64813ce4cce6bf1d5da1ee85.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 146, + 387, + 159.7 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 223, + 159.7, + 387, + 173.39999999999998 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 223, + 173.39999999999998, + 387, + 187.09999999999997 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 223, + 187.09999999999997, + 387, + 200.79999999999995 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 223, + 200.79999999999995, + 387, + 214.49999999999994 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 223, + 214.49999999999994, + 387, + 228.19999999999993 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 223, + 228.19999999999993, + 387, + 241.89999999999992 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 223, + 241.89999999999992, + 387, + 255.5999999999999 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 223, + 255.5999999999999, + 387, + 269.2999999999999 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 223, + 269.2999999999999, + 387, + 282.9999999999999 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 294, + 505, + 334 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 294, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 505, + 305 + ], + "score": 1.0, + "content": "Figure 9: The relationship between steps to top-1 test accuracy and batch size, of training ResNet-20 on", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "CIFAR-10. The “step” is equivalent to the number of applying gradients. The global mini-batch size is increased", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 197, + 325 + ], + "score": 1.0, + "content": "by adding more workers", + "type": "text" + }, + { + "bbox": [ + 198, + 315, + 207, + 323 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 313, + 246, + 325 + ], + "score": 1.0, + "content": "with fixed", + "type": "text" + }, + { + "bbox": [ + 247, + 314, + 289, + 324 + ], + "score": 0.89, + "content": "B _ { \\mathrm { l o c } } = 1 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 313, + 505, + 325 + ], + "score": 1.0, + "content": ". Results are averaged over three runs, each with fine-tuned", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 322, + 156, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 156, + 336 + ], + "score": 1.0, + "content": "learning rate.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + } + ], + "index": 12.0 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 505, + 391 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "On contrary to standard large-batch training, local SGD scales to larger batch size and provides", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "additional parallelism upon the limitation of current large batch training. Figure 9 shows the example", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 369, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 278, + 381 + ], + "score": 1.0, + "content": "of training ResNet-20 on CIFAR-10 with", + "type": "text" + }, + { + "bbox": [ + 278, + 369, + 308, + 379 + ], + "score": 0.89, + "content": "H = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 369, + 506, + 381 + ], + "score": 1.0, + "content": ", which trains and generalizes better in terms of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 380, + 322, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 322, + 392 + ], + "score": 1.0, + "content": "update steps while with reduced communication cost.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 346, + 506, + 392 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 404, + 486, + 416 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 486, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 486, + 416 + ], + "score": 1.0, + "content": "B.4 PRACTICAL IMPROVEMENT POSSIBILITIES FOR STANDARD LOCAL SGD TRAINING", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 469 + ], + "lines": [ + { + "bbox": [ + 106, + 426, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 505, + 437 + ], + "score": 1.0, + "content": "We investigate different aspects of the training to address the quality when scaling local SGD to the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 434, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 434, + 506, + 450 + ], + "score": 1.0, + "content": "extreme case, e.g., hybrid momentum scheme, warming up the local SGD or fine-tuning the learning", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "rate. In this section, we briefly present how these strategies are, and how they work in practice where", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 458, + 297, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 297, + 470 + ], + "score": 1.0, + "content": "we train ResNet-20 on CIFAR-10 on 16 GPUs.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 426, + 506, + 470 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 481, + 278, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 279, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 279, + 494 + ], + "score": 1.0, + "content": "B.4.1 LOCAL SGD WITH MOMENTUM", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 504, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "Momentum mini-batch SGD is widely used in place of vanilla SGD. The distributed mini-batch SGD", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 512, + 319, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 512, + 216, + 524 + ], + "score": 1.0, + "content": "with vanilla momentum on", + "type": "text" + }, + { + "bbox": [ + 216, + 513, + 227, + 522 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 512, + 319, + 524 + ], + "score": 1.0, + "content": "training nodes follows", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 500, + 505, + 524 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 528, + 430, + 562 + ], + "lines": [ + { + "bbox": [ + 181, + 528, + 430, + 562 + ], + "spans": [ + { + "bbox": [ + 181, + 528, + 430, + 562 + ], + "score": 0.94, + "content": "{ \\pmb u } _ { ( t ) } = m { \\pmb u } _ { ( t - 1 ) } + \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } { \\pmb \\nabla } _ { ( t ) } ^ { k } , \\qquad { \\pmb w } _ { ( t + 1 ) } = { \\pmb w } _ { ( t ) } - \\gamma { \\pmb u } _ { ( t ) }", + "type": "interline_equation", + "image_path": "6511c42a6e15a05d3b688ccfb41f4c319df000a55f4da5af8a6ee5224d0a4db8.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 181, + 528, + 430, + 539.3333333333334 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 181, + 539.3333333333334, + 430, + 550.6666666666667 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 181, + 550.6666666666667, + 430, + 562.0000000000001 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 567, + 265, + 586 + ], + "lines": [ + { + "bbox": [ + 133, + 567, + 262, + 587 + ], + "spans": [ + { + "bbox": [ + 133, + 567, + 262, + 587 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb { \\nabla } _ { ( t ) } ^ { k } = \\frac { 1 } { | \\mathcal { T } _ { ( t ) } ^ { k } | } \\sum _ { i \\in \\mathcal { T } _ { ( t ) } ^ { k } } \\nabla f _ { i } \\big ( \\pmb { w } _ { ( t ) } \\big ) } \\end{array}", + "type": "inline_equation" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 133, + 567, + 262, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 590, + 416, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 587, + 412, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 129, + 606 + ], + "score": 1.0, + "content": "After", + "type": "text" + }, + { + "bbox": [ + 130, + 591, + 140, + 601 + ], + "score": 0.83, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 587, + 380, + 606 + ], + "score": 1.0, + "content": "updates of mini-batch SGD, we have the following updated", + "type": "text" + }, + { + "bbox": [ + 381, + 593, + 412, + 604 + ], + "score": 0.91, + "content": "{ \\pmb w } _ { ( t + H ) }", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 587, + 412, + 606 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 608, + 491, + 643 + ], + "lines": [ + { + "bbox": [ + 119, + 608, + 491, + 643 + ], + "spans": [ + { + "bbox": [ + 119, + 608, + 491, + 643 + ], + "score": 0.94, + "content": "{ \\pmb w } _ { ( t + H ) } = { \\pmb w } _ { ( t ) } - \\gamma \\left( \\sum _ { \\tau = 1 } ^ { H } m ^ { \\tau } { \\pmb u } _ { ( t - 1 ) } + \\sum _ { \\tau = 0 } ^ { H - 1 } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\pmb { \\nabla } _ { ( t ) } ^ { k } + \\dots + \\sum _ { \\tau = 0 } ^ { 0 } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\pmb { \\nabla } _ { ( t + H - 1 ) } ^ { k } \\right)", + "type": "interline_equation", + "image_path": "194bb0cc6dad1e7c12752e00993d0f87c1bfbdc9a9061b38d40169ebe959bea3.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 119, + 608, + 491, + 619.6666666666666 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 119, + 619.6666666666666, + 491, + 631.3333333333333 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 119, + 631.3333333333333, + 491, + 642.9999999999999 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 653, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 506, + 666 + ], + "score": 1.0, + "content": "Coming back to the setting of local SGD, we can apply momentum acceleration on each local model,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 664, + 506, + 676 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 506, + 676 + ], + "score": 1.0, + "content": "or on a global level (Chen & Huo, 2016). In the remaining part of this section, we analyze the case of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 675, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 688 + ], + "score": 1.0, + "content": "applying local momentum and global momentum. 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When applying local momentum on the local SGD, i.e., using", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "independent identical momentum acceleration for each local model and only globally aggregating the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 81, + 402, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 190, + 97 + ], + "score": 1.0, + "content": "gradients at the time", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 191, + 82, + 224, + 95 + ], + "score": 0.92, + "content": "( t ) + H", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 224, + 81, + 402, + 97 + ], + "score": 1.0, + "content": ", we have the following local update scheme", + "type": "text", + "cross_page": true + } + ], + "index": 0 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 708, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 402, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 402, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 190, + 97 + ], + "score": 1.0, + "content": "gradients at the time", + "type": "text" + }, + { + "bbox": [ + 191, + 82, + 224, + 95 + ], + "score": 0.92, + "content": "( t ) + H", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 81, + 402, + 97 + ], + "score": 1.0, + "content": ", we have the following local update scheme", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 96, + 414, + 113 + ], + "lines": [ + { + "bbox": [ + 195, + 96, + 414, + 113 + ], + "spans": [ + { + "bbox": [ + 195, + 96, + 414, + 113 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\pmb { u } _ { ( t ) } ^ { k } = m \\pmb { u } _ { ( t - 1 ) } ^ { k } + \\pmb { \\nabla } _ { ( t ) } ^ { k } , \\qquad \\pmb { w } _ { ( t ) + 1 } ^ { k } = \\pmb { w } _ { ( t ) } ^ { k } - \\gamma \\pmb { u } _ { ( t ) } ^ { k } , } \\end{array}", + "type": "interline_equation", + "image_path": "6120084ac338ea02919d5d53752e1af956753b262a65d56e1154eb05efcdd30e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 195, + 96, + 414, + 113 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 114, + 405, + 132 + ], + "lines": [ + { + "bbox": [ + 103, + 110, + 406, + 134 + ], + "spans": [ + { + "bbox": [ + 103, + 110, + 189, + 134 + ], + "score": 1.0, + "content": "where ∇kptq \u0010 1|Ik |", + "type": "text" + }, + { + "bbox": [ + 133, + 114, + 263, + 133 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\pmb { \\nabla } _ { ( t ) } ^ { k } = \\frac { 1 } { | \\mathcal { T } _ { ( t ) } ^ { k } | } \\sum _ { i \\in \\mathcal { T } _ { ( t ) } ^ { k } } \\nabla f _ { i } \\big ( \\pmb { w } _ { ( t ) } \\big ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 114, + 346, + 129 + ], + "score": 1.0, + "content": ". Consequently, after", + "type": "text" + }, + { + "bbox": [ + 347, + 116, + 357, + 126 + ], + "score": 0.83, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 114, + 406, + 129 + ], + "score": 1.0, + "content": "local steps,", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 137, + 133, + 473, + 168 + ], + "lines": [ + { + "bbox": [ + 137, + 133, + 473, + 168 + ], + "spans": [ + { + "bbox": [ + 137, + 133, + 473, + 168 + ], + "score": 0.92, + "content": "{ \\pmb w } _ { ( t ) + H } ^ { k } = { \\pmb w } _ { ( t ) } ^ { k } - \\gamma \\left( \\sum _ { \\tau = 1 } ^ { H } m ^ { \\tau } { \\pmb u } _ { ( t - 1 ) } ^ { k } + \\sum _ { \\tau = 0 } ^ { H - 1 } m ^ { \\tau } \\pmb \\nabla _ { ( t ) } ^ { k } + \\dots + \\sum _ { \\tau = 0 } ^ { 0 } m ^ { \\tau } \\pmb \\nabla _ { ( t ) + H - 1 } ^ { k } \\right) .", + "type": "interline_equation", + "image_path": "d6840f4af716bd93dc935d18c1c7aff8ae3c56c5b48c070fca9e2c00088c9c67.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 137, + 133, + 473, + 144.66666666666666 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 137, + 144.66666666666666, + 473, + 156.33333333333331 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 137, + 156.33333333333331, + 473, + 167.99999999999997 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 174, + 362, + 186 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 362, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 362, + 187 + ], + "score": 1.0, + "content": "Substituting the above equation into eq. (2), we have the update", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 186, + 494, + 256 + ], + "lines": [ + { + "bbox": [ + 117, + 186, + 494, + 256 + ], + "spans": [ + { + "bbox": [ + 117, + 186, + 494, + 256 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle { \\pmb w } _ { ( t + 1 ) } = { \\pmb w } _ { ( t ) } - \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\gamma \\left( \\sum _ { \\tau = 1 } ^ { H } m ^ { \\tau } { \\pmb u } _ { ( t - 1 ) } ^ { k } \\sum _ { \\tau = 0 } ^ { H - 1 } m ^ { \\tau } \\pmb \\nabla _ { ( t ) } ^ { k } + \\dots + \\sum _ { \\tau = 0 } ^ { 0 } m ^ { \\tau } \\pmb \\nabla _ { ( t ) + H - 1 } ^ { k } \\right) } } \\\\ { { \\displaystyle ~ = ~ { \\pmb w } _ { ( t ) } - \\gamma \\left( \\sum _ { \\tau = 1 } ^ { H } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } { \\pmb u } _ { ( t - 1 ) } ^ { k } + \\sum _ { \\tau = 0 } ^ { H - 1 } m ^ { \\tau } ( \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\pmb \\nabla _ { ( t ) } ^ { k } ) + \\dots + \\sum _ { \\tau = 0 } ^ { 0 } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\pmb \\nabla _ { ( t ) + H - 1 } ^ { k } \\right) } } \\end{array}", + "type": "interline_equation", + "image_path": "208655ec0e132a311e7579c52c7c2768cecdfd60ca81b45a8938af03bb5a228d.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 117, + 186, + 494, + 209.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 117, + 209.33333333333334, + 494, + 232.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 117, + 232.66666666666669, + 494, + 256.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 260, + 506, + 335 + ], + "lines": [ + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 398, + 272 + ], + "score": 1.0, + "content": "Comparing the mini-batch SGD with local momentum local SGD after", + "type": "text" + }, + { + "bbox": [ + 398, + 261, + 408, + 271 + ], + "score": 0.83, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 260, + 465, + 272 + ], + "score": 1.0, + "content": "update steps", + "type": "text" + }, + { + "bbox": [ + 466, + 261, + 476, + 271 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 260, + 506, + 272 + ], + "score": 1.0, + "content": "global", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 271, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 173, + 283 + ], + "score": 1.0, + "content": "update steps v.s.", + "type": "text" + }, + { + "bbox": [ + 174, + 272, + 184, + 282 + ], + "score": 0.74, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 271, + 506, + 283 + ], + "score": 1.0, + "content": "local update steps and 1 global update step), we witness that the main difference", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 278, + 503, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 323, + 302 + ], + "score": 1.0, + "content": "of these two update schemes is the difference between", + "type": "text" + }, + { + "bbox": [ + 323, + 282, + 389, + 298 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\sum _ { \\tau = 1 } ^ { H } m ^ { \\tau } { \\pmb u } _ { ( t - 1 ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 278, + 408, + 302 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 409, + 282, + 503, + 299 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { \\tau = 1 } ^ { H } \\frac { m ^ { \\tau } } { K } \\sum _ { k = 1 } ^ { K } \\pmb { u } _ { ( t - 1 ) } ^ { k } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 258, + 310 + ], + "score": 1.0, + "content": "where mini-batch SGD holds a global", + "type": "text" + }, + { + "bbox": [ + 258, + 299, + 285, + 311 + ], + "score": 0.89, + "content": "\\pmb { u } _ { ( t - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "while each local model of the local SGD has their own", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 304, + 510, + 329 + ], + "spans": [ + { + "bbox": [ + 107, + 309, + 134, + 325 + ], + "score": 0.92, + "content": "\\pmb { u } _ { ( t - 1 ) } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 304, + 510, + 329 + ], + "score": 1.0, + "content": ". We will soon see the difference between the global momentum of mini-batch SGD and the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 323, + 233, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 233, + 334 + ], + "score": 1.0, + "content": "local momentum of local SGD.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 505, + 379 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "Local SGD with Global Momentum For global momentum local SGD, i.e., a more general variant", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "of block momentum (Chen & Huo, 2016), we would like to apply the momentum factor only to the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 369, + 257, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 257, + 381 + ], + "score": 1.0, + "content": "accumulated/synchronized gradients:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 380, + 477, + 447 + ], + "lines": [ + { + "bbox": [ + 132, + 380, + 477, + 447 + ], + "spans": [ + { + "bbox": [ + 132, + 380, + 477, + 447 + ], + "score": 0.8, + "content": "\\begin{array} { l } { { \\displaystyle { \\pmb u } _ { ( t ) } = m { \\pmb u } _ { ( t - 1 ) } + \\frac { 1 } { \\gamma } \\sum _ { k = 1 } ^ { K } \\frac { 1 } { K } \\big ( { \\pmb w } _ { ( t ) } ^ { k } - { \\pmb w } _ { ( t ) + H } ^ { k } \\big ) = m { \\pmb u } _ { ( t - 1 ) } + \\frac { 1 } { \\gamma } \\sum _ { k = 1 } ^ { K } \\frac { 1 } { K } \\sum _ { l = 0 } ^ { H - 1 } \\gamma \\pmb \\nabla _ { ( t ) + l } ^ { k } , } } \\\\ { { \\displaystyle { \\pmb w } _ { ( t + 1 ) } = { \\pmb w } _ { ( t ) } - \\gamma { \\pmb u } _ { ( t ) } = { \\pmb w } _ { ( t ) } - \\gamma \\big ( m { \\pmb u } _ { ( t - 1 ) } + \\sum _ { l = 0 } ^ { H - 1 } \\sum _ { k = 1 } ^ { K } \\frac { 1 } { K } \\pmb \\nabla _ { ( t ) + l } ^ { k } \\big ) } } \\end{array}", + "type": "interline_equation", + "image_path": "798821934f84793a1c22e251a3bd1664bf67e962d05363b720743226b43d48e1.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 132, + 380, + 477, + 402.3333333333333 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 132, + 402.3333333333333, + 477, + 424.66666666666663 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 132, + 424.66666666666663, + 477, + 446.99999999999994 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 450, + 506, + 498 + ], + "lines": [ + { + "bbox": [ + 102, + 446, + 507, + 470 + ], + "spans": [ + { + "bbox": [ + 102, + 447, + 133, + 470 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 450, + 383, + 466 + ], + "score": 0.84, + "content": "\\begin{array} { r } { \\pmb { w } _ { ( t ) + H } ^ { k } = \\pmb { w } _ { ( t ) } ^ { k } - \\eta \\sum _ { l = 0 } ^ { H - 1 } \\pmb { \\nabla } _ { ( t ) + l } ^ { k } = \\pmb { w } _ { ( t ) } - \\eta \\sum _ { l = 0 } ^ { H - 1 } \\pmb { \\nabla } _ { ( t ) + l } ^ { k } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 446, + 507, + 469 + ], + "score": 1.0, + "content": ". Note that for local SGD, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "consider summing up the gradients from each local update, i.e., the model difference before and after", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "one global synchronization, and then apply the global momentum to the gradients over workers over", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 487, + 219, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 219, + 499 + ], + "score": 1.0, + "content": "previous local update steps.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 503, + 503, + 529 + ], + "lines": [ + { + "bbox": [ + 107, + 504, + 504, + 514 + ], + "spans": [ + { + "bbox": [ + 107, + 504, + 504, + 514 + ], + "score": 1.0, + "content": "Obviously, there exists a significant difference between mini-batch momentum SGD and global", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 101, + 509, + 364, + 535 + ], + "spans": [ + { + "bbox": [ + 101, + 509, + 364, + 535 + ], + "score": 1.0, + "content": "momentum local SGD, at least the term °Hτ\u00100 mτ is cancelled.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 538, + 505, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "Local SGD with Hybrid Momentum. 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Also", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "note that the Figure 2(a) in the main text has already presented one aspect of the trade-off. So the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "exploration below mainly focuses on the other aspects and tries to understand how will the scale of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "the added stochastic noise impact the training stability. 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CIFAR-100
small batch baseline *large batch baseline *post-local SGD (H=8)post-local SGD (H=16)
ResNet-2068.84±0.0667.34±0.3468.38±0.4868.30±0.30
DenseNet-40-1274.85±0.1473.00±0.0474.50±0.3474.96±0.30
WideResNet-28-1079.78±0.1677.82±0.6579.53±0.4579.80±0.39
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CIFAR-100
small batch baseline *large batch baseline *post-local SGD (H=8)post-local SGD (H=16)
ResNet-2068.84±0.0667.34±0.3468.38±0.4868.30±0.30
DenseNet-40-1274.85±0.1473.00±0.0474.50±0.3474.96±0.30
WideResNet-28-1079.78±0.1677.82±0.6579.53±0.4579.80±0.39
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# of epochssmall batch baseline *large batch baseline *post-local SGD (H=8)post-local SGD (H=16)
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# of epochssmall batch baseline *large batch baseline *post-local SGD (H=8)post-local SGD (H=16)
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small batchbaseline*largebatch baseline*large-batch (H=8)large-batch (H=16)
86.50 ±0.3586.90±0.4986.61 ±0.3086.85 ±0.13
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(2015) considers to add time-dependent Gaussian noise to the gradient", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 103, + 232, + 503, + 253 + ], + "spans": [ + { + "bbox": [ + 103, + 232, + 195, + 253 + ], + "score": 1.0, + "content": "at every training step", + "type": "text" + }, + { + "bbox": [ + 195, + 238, + 201, + 247 + ], + "score": 0.53, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 232, + 206, + 253 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 206, + 236, + 364, + 250 + ], + "score": 0.92, + "content": "\\nabla f _ { i } \\big ( \\mathbf { \\dot { w } } _ { ( t ) } \\big ) ^ { \\cdot } \\gets \\nabla f _ { i } \\big ( \\mathbf { w } _ { ( t ) } \\big ) + \\mathcal { N } ( 0 , \\dot { \\pmb { \\sigma } } _ { t } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 232, + 396, + 253 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 396, + 236, + 409, + 249 + ], + "score": 0.9, + "content": "\\pmb { \\sigma } _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 232, + 443, + 253 + ], + "score": 1.0, + "content": "follows", + "type": "text" + }, + { + "bbox": [ + 444, + 236, + 503, + 253 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\sigma _ { t } ^ { 2 } : = \\frac { \\eta } { ( 1 + t ) ^ { \\gamma ^ { \\prime } } } } \\end{array}", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 249, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 264 + ], + "score": 1.0, + "content": "We follow the general hyper-parameter search scheme (as mentioned in Section A.4) and fine-tune", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 487, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 121, + 275 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 121, + 263, + 128, + 273 + ], + "score": 0.79, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 259, + 145, + 275 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 146, + 262, + 156, + 273 + ], + "score": 0.88, + "content": "\\gamma ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 259, + 217, + 275 + ], + "score": 1.0, + "content": "in the range of", + "type": "text" + }, + { + "bbox": [ + 218, + 261, + 321, + 274 + ], + "score": 0.92, + "content": "\\{ 1 e ^ { - 6 } , 5 e ^ { - 6 } , 1 e ^ { - 5 } , 5 e ^ { - 5 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 259, + 340, + 275 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 340, + 262, + 432, + 274 + ], + "score": 0.88, + "content": "\\{ 0 . 5 , 1 , 1 . 5 , 2 , 2 . 5 , 3 . 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 259, + 487, + 275 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "Table 14 below compares the post-local SGD to Neelakantan et al. (2015), where the noise injection", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "scheme in Neelakantan et al. (2015) cannot address the large-batch training issue and could even", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "deteriorate the generalization performance. Note that we also tried to search the hyper-parameters", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "at around the values reported in Neelakantan et al. 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CIFAR-10,KB= 2KCIFAR-100,KB= 4K
Neelakantan et al. (2015)*92.46 ±0.2767.15 ±0.26
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Post-local SGD93.02 ±0.2468.30 ±0.48
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(2015), where the noise injection", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 302 + ], + "score": 1.0, + "content": "scheme in Neelakantan et al. (2015) cannot address the large-batch training issue and could even", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 314 + ], + "score": 1.0, + "content": "deteriorate the generalization performance. Note that we also tried to search the hyper-parameters", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "at around the values reported in Neelakantan et al. 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We revisit the case of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 369 + ], + "score": 1.0, + "content": "ResNet-20 in Table 3 (2K for CIFAR-10) and Table 11 (4K for CIFAR-100), where adding isotropic noise", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 368, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 417, + 380 + ], + "score": 1.0, + "content": "as in Neelakantan et al. (2015) cannot address the large-batch training difficulty. 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CIFAR-10,KB= 2KCIFAR-100,KB= 4K
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Post-local SGD93.02 ±0.2468.30 ±0.48
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The pseudo code can be found in", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 229, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 229, + 117 + ], + "score": 1.0, + "content": "Algorithm 3 and Algorithm 4.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 129, + 505, + 173 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "score": 1.0, + "content": "Post-local SGD with signSGD. We slightly adapt the original signSGD (Bernstein et al., 2018)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "to fit in the local SGD framework. 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CIFAR-10CIFAR-100
signSGD (average over signs)91.61 ±0.2867.15 ±0.10
signSGD (majority vote over signs)91.61 ±0.2667.21 ±0.11
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No learning rate", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 46.5 + } + ], + "page_idx": 32, + "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": "33", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "We use sign-based compression scheme (i.e. signSGD (Bernstein et al., 2018) and EF-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "signSGD (Karimireddy et al., 2019)) for the demonstration. The pseudo code can be found in", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 229, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 229, + 117 + ], + "score": 1.0, + "content": "Algorithm 3 and Algorithm 4.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 129, + 505, + 173 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 506, + 142 + ], + "score": 1.0, + "content": "Post-local SGD with signSGD. We slightly adapt the original signSGD (Bernstein et al., 2018)", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "to fit in the local SGD framework. 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signSGD (average over signs)91.61 ±0.2867.15 ±0.10
signSGD (majority vote over signs)91.61 ±0.2667.21 ±0.11
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In summary, as witnessed by Table 17, hierarchical", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 533, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 506, + 545 + ], + "score": 1.0, + "content": "local SGD outperforms both local SGD and mini-batch SGD in terms of training speed as well as", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "model performance, especially for the training across nodes where inter-node connection is slow but", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 555, + 284, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 284, + 567 + ], + "score": 1.0, + "content": "intra-node communication is more efficient.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 584, + 271, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 273, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 273, + 599 + ], + "score": 1.0, + "content": "E COMMUNICATION SCHEMES", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 504, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 624 + ], + "score": 1.0, + "content": "This section evaluates the communication cost in terms of the number of local steps and block steps,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 620, + 338, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 338, + 633 + ], + "score": 1.0, + "content": "and formalizes the whole communication problem below.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 503, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 504, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 142, + 652 + ], + "score": 1.0, + "content": "Assume", + "type": "text" + }, + { + "bbox": [ + 142, + 638, + 153, + 648 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 635, + 346, + 652 + ], + "score": 1.0, + "content": "computing devices uniformly distributed over", + "type": "text" + }, + { + "bbox": [ + 347, + 638, + 360, + 648 + ], + "score": 0.87, + "content": "K ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 635, + 491, + 652 + ], + "score": 1.0, + "content": "servers, where each server has", + "type": "text" + }, + { + "bbox": [ + 491, + 636, + 504, + 651 + ], + "score": 0.88, + "content": "\\frac { K } { K ^ { \\prime } }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 390, + 662 + ], + "score": 1.0, + "content": "devices. 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The impact of a network of slower communication", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "is further studied in Figure 19(b), where the training is simulated in a realistic scenario and each global", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "communication round comes with an additional delay of 1 second. 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We can observe that for a heterogeneous system with", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 104, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "a sufficient block size, hierarchical local SGD with a sufficient number of block update steps can", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 488, + 469, + 501 + ], + "score": 1.0, + "content": "further improve the generalization performance of local SGD training. More precisely, when", + "type": "text" + }, + { + "bbox": [ + 469, + 488, + 495, + 499 + ], + "score": 0.91, + "content": "{ \\dot { H } } { \\cdot } H ^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 417, + 513 + ], + "score": 1.0, + "content": "fixed, hierarchical local SGD with more frequent inner-node synchronizations", + "type": "text" + }, + { + "bbox": [ + 417, + 499, + 451, + 511 + ], + "score": 0.88, + "content": "\\cdot H ^ { b } > 1 \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 499, + 506, + 513 + ], + "score": 1.0, + "content": ") outperforms", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 104, + 510, + 154, + 524 + ], + "score": 1.0, + "content": "local SGD (", + "type": "text" + }, + { + "bbox": [ + 154, + 510, + 187, + 522 + ], + "score": 0.87, + "content": "\\begin{array} { r } { H ^ { b } = 1 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "), while still maintaining the benefits of significantly reduced communication by", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "the inner synchronizations within each node. In summary, as witnessed by Table 17, hierarchical", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 533, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 506, + 545 + ], + "score": 1.0, + "content": "local SGD outperforms both local SGD and mini-batch SGD in terms of training speed as well as", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "model performance, especially for the training across nodes where inter-node connection is slow but", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 555, + 284, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 284, + 567 + ], + "score": 1.0, + "content": "intra-node communication is more efficient.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 445, + 506, + 567 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 584, + 271, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 273, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 273, + 599 + ], + "score": 1.0, + "content": "E COMMUNICATION SCHEMES", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 504, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 506, + 624 + ], + "score": 1.0, + "content": "This section evaluates the communication cost in terms of the number of local steps and block steps,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 620, + 338, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 338, + 633 + ], + "score": 1.0, + "content": "and formalizes the whole communication problem below.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 607, + 506, + 633 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 503, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 504, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 142, + 652 + ], + "score": 1.0, + "content": "Assume", + "type": "text" + }, + { + "bbox": [ + 142, + 638, + 153, + 648 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 635, + 346, + 652 + ], + "score": 1.0, + "content": "computing devices uniformly distributed over", + "type": "text" + }, + { + "bbox": [ + 347, + 638, + 360, + 648 + ], + "score": 0.87, + "content": "K ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 635, + 491, + 652 + ], + "score": 1.0, + "content": "servers, where each server has", + "type": "text" + }, + { + "bbox": [ + 491, + 636, + 504, + 651 + ], + "score": 0.88, + "content": "\\frac { K } { K ^ { \\prime } }", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 390, + 662 + ], + "score": 1.0, + "content": "devices. The hierarchical local SGD training procedure will access", + "type": "text" + }, + { + "bbox": [ + 391, + 649, + 401, + 659 + ], + "score": 0.83, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "total samples with local", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 659, + 341, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 170, + 673 + ], + "score": 1.0, + "content": "mini-batch size", + "type": "text" + }, + { + "bbox": [ + 170, + 660, + 179, + 670 + ], + "score": 0.78, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 659, + 203, + 673 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 204, + 660, + 214, + 670 + ], + "score": 0.75, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 659, + 276, + 673 + ], + "score": 1.0, + "content": "local steps and", + "type": "text" + }, + { + "bbox": [ + 276, + 659, + 290, + 670 + ], + "score": 0.89, + "content": "H ^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 659, + 341, + 673 + ], + "score": 1.0, + "content": "block steps.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 635, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 107, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "The MPI communication scheme (Gropp et al., 1999) is introduced for communication cost evaluation.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "More precisely, we use general all-reduce, e.g., recursive halving and doubling algorithm (Thakur", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 369, + 712 + ], + "score": 1.0, + "content": "et al., 2005; Rabenseifner, 2004), for gradient aggregation among", + "type": "text" + }, + { + "bbox": [ + 370, + 699, + 380, + 709 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "computation devices. For each", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 707, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 707, + 272, + 724 + ], + "score": 1.0, + "content": "all-reduce communication, it introduces", + "type": "text" + }, + { + "bbox": [ + 272, + 710, + 316, + 722 + ], + "score": 0.92, + "content": "C \\cdot \\log _ { 2 } K", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 707, + 433, + 724 + ], + "score": 1.0, + "content": "communication cost, where", + "type": "text" + }, + { + "bbox": [ + 433, + 710, + 442, + 720 + ], + "score": 0.84, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 707, + 506, + 724 + ], + "score": 1.0, + "content": "is the message", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 266, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 266, + 734 + ], + "score": 1.0, + "content": "transmission time plus network latency.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 677, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 142 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "The communication cost under our hierarchical local SGD setting is mainly determined by the number", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 91, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 91, + 246, + 108 + ], + "score": 1.0, + "content": "of local steps and block steps. The", + "type": "text" + }, + { + "bbox": [ + 246, + 93, + 256, + 106 + ], + "score": 0.89, + "content": "\\textstyle { \\frac { K } { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 91, + 506, + 108 + ], + "score": 1.0, + "content": "models within each server synchronize the gradients for every", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 105, + 504, + 119 + ], + "spans": [ + { + "bbox": [ + 107, + 107, + 117, + 116 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 105, + 403, + 119 + ], + "score": 1.0, + "content": "local mini-batch, and it only performs global gradients aggregation of", + "type": "text" + }, + { + "bbox": [ + 403, + 106, + 414, + 116 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 105, + 489, + 119 + ], + "score": 1.0, + "content": "local models after", + "type": "text" + }, + { + "bbox": [ + 489, + 105, + 504, + 116 + ], + "score": 0.88, + "content": "H ^ { b }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 117, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 131 + ], + "score": 1.0, + "content": "block updates. Thus, the total number of synchronizations among compute devices is reduced to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 128, + 397, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 150, + 143 + ], + "score": 0.92, + "content": "\\Big \\lceil \\frac { N } { K B \\cdot H H ^ { b } } \\Big \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 128, + 370, + 143 + ], + "score": 1.0, + "content": ", and we can formulate the overall communication cost", + "type": "text" + }, + { + "bbox": [ + 371, + 128, + 380, + 140 + ], + "score": 0.86, + "content": "\\tilde { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 128, + 397, + 143 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 148, + 453, + 164 + ], + "lines": [ + { + "bbox": [ + 158, + 148, + 453, + 164 + ], + "spans": [ + { + "bbox": [ + 158, + 148, + 453, + 164 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\tilde { C } \\approx \\left( \\left\\lceil \\frac { N } { K B \\cdot H } \\right\\rceil - \\left\\lceil \\frac { N } { K B \\cdot H H ^ { b } } \\right\\rceil \\right) \\cdot C _ { 1 } \\cdot K ^ { \\prime } \\log _ { 2 } \\frac { K } { K ^ { \\prime } } + \\left\\lceil \\frac { N } { K B \\cdot H H ^ { b } } \\right\\rceil \\cdot C _ { 2 } \\log _ { 2 } K } \\end{array}", + "type": "interline_equation", + "image_path": "722a911482635b9df77720159abae16b40ad3e9440685998d9cda53a0283811d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 158, + 148, + 453, + 164 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 133, + 180 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 168, + 146, + 179 + ], + "score": 0.89, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 168, + 466, + 180 + ], + "score": 1.0, + "content": "is the single message passing cost for compute devices within the same server,", + "type": "text" + }, + { + "bbox": [ + 467, + 169, + 479, + 179 + ], + "score": 0.88, + "content": "C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 179, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 272, + 191 + ], + "score": 1.0, + "content": "cost of that across servers, and obviously", + "type": "text" + }, + { + "bbox": [ + 272, + 179, + 310, + 190 + ], + "score": 0.94, + "content": "C _ { 1 } \\ll C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 179, + 506, + 191 + ], + "score": 1.0, + "content": ". We can easily witness that the number of block", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 129, + 202 + ], + "score": 1.0, + "content": "steps", + "type": "text" + }, + { + "bbox": [ + 129, + 189, + 144, + 200 + ], + "score": 0.9, + "content": "H ^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 189, + 447, + 202 + ], + "score": 1.0, + "content": "is more deterministic in terms of communication reduction than local step", + "type": "text" + }, + { + "bbox": [ + 448, + 190, + 458, + 200 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 189, + 505, + 202 + ], + "score": 1.0, + "content": ". Empirical", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 270, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 270, + 212 + ], + "score": 1.0, + "content": "evaluations can be found in Section D.3.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "Also, note that our hierarchical local SGD is orthogonal to the implementation of gradient aggrega-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "score": 1.0, + "content": "tion (Goyal et al., 2017) optimized for the hardware, but focusing on overcoming the aggregation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "cost of more general distributed scenarios, and can easily be integrated with any optimized all-reduce", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 174, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 174, + 263 + ], + "score": 1.0, + "content": "implementation.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 107, + 278, + 298, + 291 + ], + "lines": [ + { + "bbox": [ + 104, + 276, + 300, + 293 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 300, + 293 + ], + "score": 1.0, + "content": "F DISCUSSION AND FUTURE WORK", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "Data distribution patterns. In our experiments, the dataset is globally shuffled once per epoch and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "each local worker only accesses a disjoint part of the training data. Removing shuffling altogether,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "and instead keeping the disjoint data parts completely local during training might be envisioned for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "extremely large datasets which can not be shared, or also in a federated scenario where data locality", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "is a must for privacy reasons. This scenario is not covered by the current theoretical understanding of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 413, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 413, + 371 + ], + "score": 1.0, + "content": "local SGD, but will be interesting to investigate theoretically and practically.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 506, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "Better learning rate scheduler for local SGD. We have shown in our experiments that local SGD", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "delivers consistent and significant improvements over the state-of-the-art performance of mini-batch", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "SGD. For ImageNet, we simply applied the same configuration of “large-batch learning schemes”", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "score": 1.0, + "content": "by Goyal et al. (2017). However, this set of schemes was specifically developed and tuned for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 104, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "mini-batch SGD only, not for local SGD. For example, scaling the learning rate w.r.t. the global", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 437, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 506, + 448 + ], + "score": 1.0, + "content": "mini-batch size ignores the frequent local updates where each local model only accesses local mini-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "score": 1.0, + "content": "batches for most of the time. Therefore, it is expected that specifically deriving and tuning a learning", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "rate scheduler for local SGD would lead to even more drastic improvements over mini-batch SGD,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 470, + 285, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 285, + 482 + ], + "score": 1.0, + "content": "especially on larger tasks such as ImageNet.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "Adaptive local SGD. As local SGD achieves better generalization than current mini-batch SGD", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 379, + 518 + ], + "score": 1.0, + "content": "approaches, an interesting question is if the number of local steps", + "type": "text" + }, + { + "bbox": [ + 379, + 504, + 389, + 514 + ], + "score": 0.75, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "could be chosen adaptively,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "i.e. change during the training phase. This could potentially eliminate or at least simplify complex", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 538 + ], + "score": 1.0, + "content": "learning rate schedules. Furthermore, recent works (Loshchilov & Hutter, 2017; Huang et al., 2017a)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "leverage cyclic learning rate schedules either improving the general performance of deep neural", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "score": 1.0, + "content": "network training, or using an ensemble multiple neural networks at no additional training cost.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 559, + 453, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 453, + 572 + ], + "score": 1.0, + "content": "Adaptive local SGD could potentially achieve similar goals with reduced training cost.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 596 + ], + "score": 1.0, + "content": "Hierarchical local SGD design. Hierarchical local SGD provides a simple but efficient training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "solution for devices over the complex heterogeneous system. However, its performance might be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 373, + 617 + ], + "score": 1.0, + "content": "impacted by the cluster topology. For example, the topology of", + "type": "text" + }, + { + "bbox": [ + 374, + 605, + 398, + 615 + ], + "score": 0.87, + "content": "8 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "-GPU in Table 17 fails to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "further improve the performance of local SGD by using more frequent inner node synchronizations.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "On contrary, sufficiently large size of the GPU block could easily benefit from the block update of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "hierarchical local SGD, for both communication efficiency and training quality. The design space", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "of hierarchical local SGD for different cluster topologies should be further investigated, e.g., to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "investigate the two levels of model averaging frequency (within and between blocks) in terms of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "convergence, and the interplay of different local minima in the case of very large number of local", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "score": 1.0, + "content": "steps. Another interesting line of work, explores heterogenous systems by allowing for different", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 692, + 428, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 428, + 705 + ], + "score": 1.0, + "content": "number of local steps on different clusters, thus making up for slower machines.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42 + } + ], + "page_idx": 39, + "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": [ + 300, + 752, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 142 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "The communication cost under our hierarchical local SGD setting is mainly determined by the number", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 91, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 91, + 246, + 108 + ], + "score": 1.0, + "content": "of local steps and block steps. The", + "type": "text" + }, + { + "bbox": [ + 246, + 93, + 256, + 106 + ], + "score": 0.89, + "content": "\\textstyle { \\frac { K } { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 91, + 506, + 108 + ], + "score": 1.0, + "content": "models within each server synchronize the gradients for every", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 105, + 504, + 119 + ], + "spans": [ + { + "bbox": [ + 107, + 107, + 117, + 116 + ], + "score": 0.76, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 105, + 403, + 119 + ], + "score": 1.0, + "content": "local mini-batch, and it only performs global gradients aggregation of", + "type": "text" + }, + { + "bbox": [ + 403, + 106, + 414, + 116 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 105, + 489, + 119 + ], + "score": 1.0, + "content": "local models after", + "type": "text" + }, + { + "bbox": [ + 489, + 105, + 504, + 116 + ], + "score": 0.88, + "content": "H ^ { b }", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 117, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 131 + ], + "score": 1.0, + "content": "block updates. Thus, the total number of synchronizations among compute devices is reduced to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 128, + 397, + 143 + ], + "spans": [ + { + "bbox": [ + 106, + 128, + 150, + 143 + ], + "score": 0.92, + "content": "\\Big \\lceil \\frac { N } { K B \\cdot H H ^ { b } } \\Big \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 128, + 370, + 143 + ], + "score": 1.0, + "content": ", and we can formulate the overall communication cost", + "type": "text" + }, + { + "bbox": [ + 371, + 128, + 380, + 140 + ], + "score": 0.86, + "content": "\\tilde { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 128, + 397, + 143 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 104, + 82, + 506, + 143 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 148, + 453, + 164 + ], + "lines": [ + { + "bbox": [ + 158, + 148, + 453, + 164 + ], + "spans": [ + { + "bbox": [ + 158, + 148, + 453, + 164 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\tilde { C } \\approx \\left( \\left\\lceil \\frac { N } { K B \\cdot H } \\right\\rceil - \\left\\lceil \\frac { N } { K B \\cdot H H ^ { b } } \\right\\rceil \\right) \\cdot C _ { 1 } \\cdot K ^ { \\prime } \\log _ { 2 } \\frac { K } { K ^ { \\prime } } + \\left\\lceil \\frac { N } { K B \\cdot H H ^ { b } } \\right\\rceil \\cdot C _ { 2 } \\log _ { 2 } K } \\end{array}", + "type": "interline_equation", + "image_path": "722a911482635b9df77720159abae16b40ad3e9440685998d9cda53a0283811d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 158, + 148, + 453, + 164 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 168, + 505, + 212 + ], + "lines": [ + { + "bbox": [ + 105, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 133, + 180 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 168, + 146, + 179 + ], + "score": 0.89, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 168, + 466, + 180 + ], + "score": 1.0, + "content": "is the single message passing cost for compute devices within the same server,", + "type": "text" + }, + { + "bbox": [ + 467, + 169, + 479, + 179 + ], + "score": 0.88, + "content": "C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 179, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 272, + 191 + ], + "score": 1.0, + "content": "cost of that across servers, and obviously", + "type": "text" + }, + { + "bbox": [ + 272, + 179, + 310, + 190 + ], + "score": 0.94, + "content": "C _ { 1 } \\ll C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 179, + 506, + 191 + ], + "score": 1.0, + "content": ". We can easily witness that the number of block", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 129, + 202 + ], + "score": 1.0, + "content": "steps", + "type": "text" + }, + { + "bbox": [ + 129, + 189, + 144, + 200 + ], + "score": 0.9, + "content": "H ^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 189, + 447, + 202 + ], + "score": 1.0, + "content": "is more deterministic in terms of communication reduction than local step", + "type": "text" + }, + { + "bbox": [ + 448, + 190, + 458, + 200 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 189, + 505, + 202 + ], + "score": 1.0, + "content": ". Empirical", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 270, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 270, + 212 + ], + "score": 1.0, + "content": "evaluations can be found in Section D.3.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 168, + 506, + 212 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 231 + ], + "score": 1.0, + "content": "Also, note that our hierarchical local SGD is orthogonal to the implementation of gradient aggrega-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "score": 1.0, + "content": "tion (Goyal et al., 2017) optimized for the hardware, but focusing on overcoming the aggregation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "cost of more general distributed scenarios, and can easily be integrated with any optimized all-reduce", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 251, + 174, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 174, + 263 + ], + "score": 1.0, + "content": "implementation.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 216, + 506, + 263 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 278, + 298, + 291 + ], + "lines": [ + { + "bbox": [ + 104, + 276, + 300, + 293 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 300, + 293 + ], + "score": 1.0, + "content": "F DISCUSSION AND FUTURE WORK", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "Data distribution patterns. In our experiments, the dataset is globally shuffled once per epoch and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "each local worker only accesses a disjoint part of the training data. Removing shuffling altogether,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "and instead keeping the disjoint data parts completely local during training might be envisioned for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "extremely large datasets which can not be shared, or also in a federated scenario where data locality", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "is a must for privacy reasons. This scenario is not covered by the current theoretical understanding of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 413, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 413, + 371 + ], + "score": 1.0, + "content": "local SGD, but will be interesting to investigate theoretically and practically.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 303, + 506, + 371 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 506, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "Better learning rate scheduler for local SGD. We have shown in our experiments that local SGD", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 405 + ], + "score": 1.0, + "content": "delivers consistent and significant improvements over the state-of-the-art performance of mini-batch", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "SGD. For ImageNet, we simply applied the same configuration of “large-batch learning schemes”", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "score": 1.0, + "content": "by Goyal et al. (2017). However, this set of schemes was specifically developed and tuned for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 104, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "mini-batch SGD only, not for local SGD. For example, scaling the learning rate w.r.t. the global", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 437, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 506, + 448 + ], + "score": 1.0, + "content": "mini-batch size ignores the frequent local updates where each local model only accesses local mini-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 461 + ], + "score": 1.0, + "content": "batches for most of the time. Therefore, it is expected that specifically deriving and tuning a learning", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "rate scheduler for local SGD would lead to even more drastic improvements over mini-batch SGD,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 470, + 285, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 285, + 482 + ], + "score": 1.0, + "content": "especially on larger tasks such as ImageNet.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 381, + 506, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 492, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "Adaptive local SGD. As local SGD achieves better generalization than current mini-batch SGD", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 504, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 379, + 518 + ], + "score": 1.0, + "content": "approaches, an interesting question is if the number of local steps", + "type": "text" + }, + { + "bbox": [ + 379, + 504, + 389, + 514 + ], + "score": 0.75, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 504, + 506, + 518 + ], + "score": 1.0, + "content": "could be chosen adaptively,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 528 + ], + "score": 1.0, + "content": "i.e. change during the training phase. This could potentially eliminate or at least simplify complex", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 538 + ], + "score": 1.0, + "content": "learning rate schedules. Furthermore, recent works (Loshchilov & Hutter, 2017; Huang et al., 2017a)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "leverage cyclic learning rate schedules either improving the general performance of deep neural", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 561 + ], + "score": 1.0, + "content": "network training, or using an ensemble multiple neural networks at no additional training cost.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 559, + 453, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 453, + 572 + ], + "score": 1.0, + "content": "Adaptive local SGD could potentially achieve similar goals with reduced training cost.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 493, + 506, + 572 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 581, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 596 + ], + "score": 1.0, + "content": "Hierarchical local SGD design. Hierarchical local SGD provides a simple but efficient training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "solution for devices over the complex heterogeneous system. However, its performance might be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 373, + 617 + ], + "score": 1.0, + "content": "impacted by the cluster topology. For example, the topology of", + "type": "text" + }, + { + "bbox": [ + 374, + 605, + 398, + 615 + ], + "score": 0.87, + "content": "8 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "-GPU in Table 17 fails to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "further improve the performance of local SGD by using more frequent inner node synchronizations.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "On contrary, sufficiently large size of the GPU block could easily benefit from the block update of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "hierarchical local SGD, for both communication efficiency and training quality. The design space", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "of hierarchical local SGD for different cluster topologies should be further investigated, e.g., to", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "investigate the two levels of model averaging frequency (within and between blocks) in terms of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "convergence, and the interplay of different local minima in the case of very large number of local", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 506, + 694 + ], + "score": 1.0, + "content": "steps. 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1.50 1.25 1.00A1: mini-batch SGD (K=1,B = Bioc) A2: mini-batch SGD (K= 16,B = Boc) A3:mini-batch SGD (K=16,B= 4Bc)Prprerr hraled100 90 (b) Training top-1 accuracy.Waccuracy on test set |top-1 acc.Trpprertststcr 90
Tl lbigiebss 0.75 0.50A4:local SGD (K=16,H=4) A5: post-local SGD (K=16,H = 16)80 70A1: mini-batch SGD (K= 1, B = Boc)85A1: mini-batch SGD (K = 1, B = B(c)
5 0.2560A2: mini-batch SGD (K =16,B = Bioc) A3: mini-batch SGD (K = 16, B = 4B(0c)80A2: mini-batch SGD (K= 16,B =B/0c)A3: mini-batch SGD (K = 16, B = 4B/0c)
0.00 50 0150 100 20025030050 40A4: local SGD (K=16,H=4) A5: post-local SGD (K = 16, H = 16)A5: post-local SGD (K = 16, H = 16)A4: local SGD (K = 16, H = 4)
Epoch050 100150 Epoch250300 200750 50100 150 200 250300 Epoch
(a) Training loss.system performance(c) Test top-1 accuracy.
accuracy on training set loss value top-1 acc.|parallelism communication
Algorithm A1: small mini-batch SGD (K=1,B=Bloc)0.01poor
A2: large mini-batch SGD(K=16,B= Bloc)0.01100% 100%excellent excellent93% 92%excellent×1 ×16
A3: huge mini-batch SGD(K=16,B= 4Bloc)0.1081%poorgood 89% poor×16÷1 ÷4ok good
A4: local SGD(K=16,H=4)|0.0195%ok|92% good×16 ÷4good
A5: post-local SGD(K= 16,H= 16)10.0199%excellent|93%excellent ×16÷1(phase 1),÷16 (phase 2)good
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AlgorithmTop-1 acc.Effect. batch sizeAlgorithmTop-1 acc.Effect. batch size
Mini-batch SGD (K=4)92.6%KB=512Mini-batch SGD (K =16)92.5%KB=2048
Mini-batch SGD (K =4)89.5%KB=8192Mini-batch SGD (K =16)76.3%KB=16384
Mini-batch SGD (K=4)92.5%KB=512→8192Mini-batch SGD(K=16)92.0%KHBloc=2048→ 16384
Local SGD(H=16,K=4)92.5%KHBloc =8192Local SGD(H=8,K=16)92.0%KHBloc =16384
Post-local SGD(H=16,K=4)92.7%KHBloc =512→8192Post-local SGD (H=8,K=16)92.9%KHBloc =2048→ 16384
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CIFAR-10CIFAR-100
Small batch baseline*Large batch baseline * K=2,B=128K=16,B=128 K=16,Bloc=128K=16,Blc =128Post-local SGD (H=16)Post-local SGD (H=32)Small batch baseline *Large batch baseline *Post-local SGD (H=16)Post-local SGD (H=32) K=2,B=128K=16,B= 128K=16,Blo =128 K= 16,Bloc=128
ResNet-2092.63 ±0.2692.48 3±0.1792.80 ±0.1693.02 ±0.2468.84 ±0.0668.17 ±0.1869.24 ±0.2669.38 ±0.20
DenseNet-40-1294.41 ±0.1494.36 ±0.2094.43 ±0.1294.58 ±0.1174.85 ±0.1474.08 3±0.4674.45 ±0.3075.03 ±0.05
WideResNet-28-1095.89 ±0.1095.43 ±0.3795.94 ±0.0695.76 ±0.2579.78 3±0.1679.31 ±0.2380.28 ±0.1380.65 ±0.16
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SGD + Momentum + LARSSGD + Momentum + LARS + Post-local SGD
KBloc=8,19275.9976.13
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CIFAR-10 H=16CIFAR-100
H=1H =32H =64H=1H =16H=32H =64
signSGD (Bernstein etal.,018)91.61 ±0.2891.77±0.0492.22±0.1192.00 ±0.1567.21 ±0.1167.43 ±0.3668.32 ±0.1868.12 ±0.11
EF-signSGD(Karimireddy etal.,2019)92.45 ±0.2892.63±0.1692.72±0.0592.76 ±0.0668.19 ±0.1069.14 ±0.3669.00 ±0.1469.49 ±0.21
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ModelCommunication# parametersComputation# flops per imageComputation/Communication scaling ratio
ResNet-20 (CIFAR-10)0.27 million0.041 billion151.85
ResNet-20 (CIFAR-100)0.27 million0.041 billion151.85
ResNet-50 (ImageNet-1k)25.00 million7.7 billion308.00
DenseNet-40-12 (CIFAR-10)1.06 million0.28 billion264.15
DenseNet-40-12 (CIFAR-100)1.10 million0.28 billion254.55
WideResNet-28-10 (CIFAR-10)36.48 million5.24 billion143.64
WideResNet-28-10 (CIFAR-100)36.54 million5.24 billion143.40
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CIFAR-100
small batch baseline *large batch baseline *post-local SGD (H=8)post-local SGD (H=16)
ResNet-2068.84±0.0667.34±0.3468.38±0.4868.30±0.30
DenseNet-40-1274.85±0.1473.00±0.0474.50±0.3474.96±0.30
WideResNet-28-1079.78±0.1677.82±0.6579.53±0.4579.80±0.39
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# of epochssmall batch baseline *large batch baseline *post-local SGD (H=8)post-local SGD (H=16)
30068.84±0.0667.34±0.3468.38±0.4868.30±0.30
40069.07±0.2767.55±0.2169.06±0.1569.05±0.26
50069.03±0.1067.42±0.6369.02±0.3868.87±0.27
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Neelakantan et al. (2015)*92.46 ±0.2767.15 ±0.26
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CIFAR-10CIFAR-100
signSGD (average over signs)91.61 ±0.2867.15 ±0.10
signSGD (majority vote over signs)91.61 ±0.2667.21 ±0.11
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input: the initial model w(o) E Rd;training data with labels I; mini-batch of size Bloc per local model; step 1: synchronize to have the same initial models wo) := w(o)·size n,and momentum m (optional); number of synchronization steps T,and the first learning rate decay is performed at T'; number of eventual local steps H'; number of nodes K. 5
2: 3:for all k :=1,...,K do in parallel for t :=1,...,T do
4:if t<T'then
5:H(t)=1
6:
7:H(t)=H'
8:
for h := 1,..,H(t) do
9:
10:
11:compute the gradients g(t)+h-1 := 1 BlocVfi(w(t)+h-1).>can involve weight
decay and momentum.
12:update the local model w (t)+h = W(t)+h-1-7(t)sign(gt)+h-1).
end for
13:
14:get model difference △t) :=w(t)
△.
15:d
16:
17:
end for
18:end for
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e(0) = 0 ∈ Rd.input: the initial model w(o) ERd;training data with labels Z; mini-batch of size Bloc per local model; step size n,and momentum m (optional); number of synchronization steps T,and the first learning rate decay is performed at T'; number of eventual local steps H'; number of nodes K. error-compensated memory
3:2: initialize the local error memory eo) := e(0)· forall k :=1,...,K do in parallel
4:for t := 1,...,T do
5:if t<T'then
6:H(t)= 1
7:else
8:H(t)=H' end if
9:
10:for h := 1,...,H(t) do
11:
12:compute the gradient g(t)+h-1 := k 1 BlocVfi(wt)+h-1).> can involve weight
decay and momentum.(t)+h-1
13:
14:end for
15:get model difference△t) ):= Wt)-w(t)+H+e(t).
compess te modeldferece: st）=ig△)）dpt
16:d
17:
K
18:
19:
end for
20: end for
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"html": "
input: the initial model w[(0)]; input: training data with labels Z;
input: mini-batch of size Bloc per local model;
input: step size η,and momentum m (optional);
input: number of synchronization steps T over nodes;
input: number of local update steps H,and block update steps Hb;
input:number of nodes K in total; and nodes K' per GPU-block.
1: synchronize to have the same initial models w[(o)] := w[(0)·
2:for all k := 1,...,K do in parallel
3:for t := 1,...,T do
4:for l := 1,...,H do
5:for h :=1,...,H do
6:
7:compute the gradient k 1
£ Vfi(w(t)+l} +h-1). 9[(t)+l]+h-1:= Bloc ieIk [(t)+l]+h-1
8:update the local model
pt(t)+l]+h-1-7[(t)]9[(t)+l]+h-1
9: 10:end for
inner all-reduce aggregation of the gradients k
-w[(t)+}+H· k [(t)+t]
11:
k k 1
12:
13:end for outer all-reduce aggregation of the gradients
△（）]：=w（）] k k
14:for all K nodes:
15: end for
16:end for
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The approach employs additional modules called local critic networks besides the main network model to be trained, which are used to obtain error gradients without complete feedforward and backward propagation processes. We propose a cascaded learning strategy for these local networks. In addition, the approach is also useful from multi-model perspectives, including structural optimization of neural networks, computationally efficient progressive inference, and ensemble classification for performance improvement. Experimental results show the effectiveness of the proposed approach and suggest guidelines for determining appropriate algorithm parameters. + +# 1 INTRODUCTION + +In recent days, deep learning has been remarkably advanced and successfully applied in numerous fields (LeCun et al., 2015). A key mechanism behind the success of deep neural networks is that they are capable of extracting useful information progressively through their layered structures. It is an increasing trend that more and more complex deep neural network structures are developed in order to solve challenging real-world problems, e.g., He et al. (2016b). Training of deep neural networks is based on backpropagation in most cases, which basically works in a sequential and synchronous manner. During the feedforward pass, the input data is processed through the hidden layers to produce the network output; during the feedback pass, the error gradient is propagated back through the layers to update each layer’s weight parameters. Therefore, training of each layer has dependency on all the other layers, which causes the issue of locking (Jaderberg et al., 2017). This is undesirable in some cases, e.g., a system consisting of several interacting models, a model distributed across multiple computing nodes, etc. + +There have been attempts to remove the locking constraint. In Carreira-Perpinan & Wang (2014), the method of auxiliary coordinates (MAC) is proposed. It replaces the original loss minimization problem with an equality-constrained optimization problem by introducing an auxiliary variable for each data and each hidden unit. Then, solving the problem is formulated as iteratively solving several sub-problems independently. A similar approach using the alternating direction method of multipliers (ADMM) is proposed in Taylor et al. (2016). It also employs an equality-constrained optimization but with different auxiliary variables, so that resulting sub-problems have closed form solutions. However, these methods are not scalable to deep learning architectures such as convolutional neural networks (CNNs). + +The method proposed in Jaderberg et al. (2017), called decoupled neural interface (DNI), directly synthesizes estimated error gradients, called synthetic gradients, using an additional small neural network for training a layer’s weight parameters. As long as the synthetic gradients are close to the actual backpropagated gradients, each layer does not need to wait until the error at the output layer is backpropagated through the preceding layers, which allows independent training of each layer. However, this method suffers from performance degradation when compared to regular backpropagation (Czarnecki et al., 2017a). The idea of having additional modules supporting the layers of the main model is also adopted in Czarnecki et al. (2017a), where the additional modules are trained to approximate the main model’s outputs instead of error gradients. Due to this, however, the method does not resolve the issue of update locking, and in fact, the work does not intend to design a non-sequential learning algorithm. + +![](images/c2f5cb4c44bf704873fe2f46b97b9422343f9e9d1d458c774bb6d5e32319f1f9.jpg) +Figure 1: Learning processes of DNI (Jaderberg et al., 2017) and the proposed local critic training. The black, green, and blue arrows indicate feedforward passes, an error gradient flow, and loss comparison, respectively. + +In this paper, we propose a novel approach for non-sequential learning, called local critic training. The key idea is that additional modules besides the main neural network model are employed, which we call local critics, in order to indirectly deliver error gradients to the main model for training without backpropagation. In other words, a local critic located at a certain layer group is trained in such a way that the derivative of its output serves as the error gradient for training of the corresponding layers’ weight parameters. Thus, the error gradient does not need to be backpropagated, and the feedforward operations and gradient-descent learning can be performed independently. Through extensive experiments, we examine the influences of the network structure, update frequency, and total number of local critics, which provide not only insight into operation characteristics but also guidelines for performance optimization of the proposed method. + +In addition to the capability of implementing training without locking, the proposed approach can be exploited for additional important applications. First, we show that applying the proposed method automatically performs structural optimization of neural networks for a given problem, which has been a challenging issue in the machine learning field. Second, a progressive inference algorithm using the network trained with the proposed method is presented, which can adaptively reduce the computational complexity during the inference process (i.e., test phase) depending on the given data. Third, the network trained by the proposed method naturally enables ensemble inference that can improve the classification performance. + +# 2 PROPOSED APPROACH + +# 2.1 LOCAL CRITIC TRAINING + +The basic idea of the proposed approach is to introduce additional local networks, which we call local critics, besides the main network model, so that they eventually provide estimates of the output of the main network. Each local critic network can serve a group of layers of the main model by being attached to the last layer of the group. The proposed architecture is illustrated in Figure 1, where $f _ { i }$ is the $i$ th layer group (containing one or more layers), $h _ { i }$ is the output of $f _ { i }$ , and $h _ { N }$ is the final output of the main model having $N$ layer groups: + +$$ +h _ { i } = f _ { i } ( h _ { i - 1 } ) +$$ + +$c _ { i }$ is the local critic network for $f _ { i }$ , which is expected to approximate $h _ { N }$ based on $h _ { i }$ , i.e., + +$$ +c _ { i } ( h _ { i } ) \approx h _ { N } +$$ + +Then, this can be used to approximate the loss function of the main network, $L _ { N } = l ( h _ { N } , y )$ , which is used to train $f _ { i }$ , by + +$$ +L _ { i } = l ( c _ { i } ( h _ { i } ) , y ) +$$ + +for $i = 1 , . . . , N - 1$ , i.e., + +$$ +L _ { i } \approx L _ { N } +$$ + +where $y$ is the training target and $l$ is the loss function such as cross-entropy or mean-squared error. Then, the error gradient for training $f _ { i }$ is obtained by differentiating $L _ { i }$ with respect to $h _ { i }$ , i.e., + +$$ +\delta _ { i } = \frac { \partial L _ { i } } { \partial h _ { i } } +$$ + +which can be used to train the weight parameters of $f _ { i }$ , denoted by $\theta _ { i }$ , via a gradient-descent rule: + +$$ +\theta _ { i } \theta _ { i } - \eta \delta _ { i } \frac { \partial h _ { i } } { \partial \theta _ { i } } +$$ + +where $\eta$ is a learning rate. Note that the final layer group $h _ { N }$ does not require a local critic network and can be trained using the regular backproagation because the final output of the main network is directly available. Therefore, the update of $f _ { i }$ does not need to wait until its output $h _ { i }$ propagates till the end of the main network and the error gradient is backpropagated; it can be performed when the operations from (2) to (5) are done. For $c _ { i }$ , we usually use a simple model so that the operations through $c _ { i }$ are simpler than those through $f _ { i + 1 }$ till $f _ { N }$ . + +While the dependency of $f _ { i }$ on $f _ { j } \ ( j > i )$ during training is resolved in this way, there still exists the dependency of $c _ { i }$ on $f _ { j } \ ( j > i )$ , because training $c _ { i }$ requires its ideal target, i.e., $h _ { N }$ , which is available from $f _ { N }$ only after the feedforward pass is complete. In order to resolve this problem, we use an indirect, cascaded approach, where $c _ { i }$ is trained so that its training loss targets the training loss for ${ c _ { i + 1 } } ^ { 1 }$ : + +$$ +L _ { c _ { i } } = l ( L _ { i } , L _ { i + 1 } ) +$$ + +In other words, training of $c _ { i }$ can be performed once the loss for $c _ { i + 1 }$ is available. + +Figure 1 compares the proposed architecture with the existing DNI approach that also employs local networks besides the main network to resolve the issue of locking (Jaderberg et al., 2017). In DNI, the local network $c _ { i }$ directly estimates the error gradient, i.e., + +$$ +c _ { i } ( h _ { i } ) \approx \frac { \partial L _ { N } } { \partial h _ { i } } +$$ + +so that each layer group of the main model can be updated without waiting for the forward and backward propagations in the subsequent layers. And, to update $c _ { i }$ , the error gradient for $f _ { i + 1 }$ estimated by $c _ { i + 1 }$ is backpropagated through $f _ { i + 1 }$ and is used as the (estimated) target for $c _ { i }$ . Therefore, all the necessary computations in the forward and backward passes can be locally confined. The performance of the two methods will be compared in Section 3. + +# 2.2 STRUCTURAL OPTIMIZATION + +In many cases, determining an appropriate structure of neural networks for a given problem is not straightforward. This is usually done through trial-and-error, which is extremely time-consuming. There have been studies to automate the structural optimization process (Cortes et al., 2017; Feng & Darrell, 2015; Kwok & Yeung, 1997; Reed, 1993), but this issue still remains very challenging. + +In deep learning, the problem of structural optimization is even more critical. Large-sized networks may easily show overfitting. Even if large networks may produce high accuracy, they take significantly large amounts of memory and computation, which is undesirable especially for resourceconstrained cases such as embedded and mobile systems. Therefore, it is highly desirable to find an optimal network structure that is sufficiently small while the performance is kept reasonably good. + +During local critic training, each local critic network is trained to estimate the output of the main network eventually. Therefore, once the training of the proposed architecture finishes, we obtain different networks that are supposed to have similar input-output mappings but have different structures and possibly different accuracy, i.e., multiple sub-models and one main model (see Figure 2b). Here, a sub-model is composed of the layers on the path from the input to a certain hidden layer + +# Algorithm 1: Progressive inference + +Input: data $_ x$ , threshold $t$ +Model: sub-model $c _ { i }$ , main-model $f$ +Initialize: classif ication $= 0$ . +for $i = 1$ to $N - 1$ do if max softmax $( c _ { i } ( x ) ) > t$ then classif ication $=$ argmax softmax $( c _ { i } ( x ) )$ ) break end if +end for +if classif ication $= = 0$ then # if all sub-models are not confident classif ication $=$ argmax softmax $( f ( x ) )$ +end if + +and its local critic network. Among the sub-models, we can choose one as a structure-optimized network by considering the trade-off relationship between the complexity and performance. + +It is worth mentioning that our structural optimization approach can be performed instantly after training of the model, whereas many existing methods for structural optimization require iterative search processes, e.g., Zoph & Le (2017). + +# 2.3 PROGRESSIVE INFERENCE + +We propose another simple but effective way to utilize the sub-models obtained by the proposed approach for computational efficiency, which we call progressive inference. Although small submodels (e.g., sub-model 1) tend to show low accuracy, they would still perform well for some data. For such data, we do not need to perform the full feedforward pass but can take the classification decision by the sub-models. Thus, the basic idea of the progressive inference is to finish inference (i.e., classification) with a small sub-model if its confidence on the classification result is high enough, instead of completing the full feedforward pass with the main model, which can reduce the computational complexity. Here, the softmax outputs for all classes are compared and the maximum probability is used as the confidence level. If it is higher than a threshold, we take the decision by the sub-model; otherwise, the feedforward pass continues. The proposed progressive inference method is summarized in Algorithm 12. + +# 2.4 ENSEMBLE INFERENCE + +In recent deep learning systems, it is popular to use ensemble approaches to improve performance in comparison to single models, where multiple networks are combined for producing final results, e.g., He et al. (2016a); Szegedy et al. (2015). The sub-models and main model obtained by applying the proposed local critic training approach can be used for ensemble inference. Figure 4a depicts how the sub-models and the main model can work together to form an ensemble classifier. We take the simplest way to combine them, i.e., summation of the networks’ outputs. + +# 3 EXPERIMENTS + +We conduct extensive experiments to examine the performance of the proposed method in various aspects. We use the CIFAR-10 and CIFAR-100 datasets (Krizhevsky, 2009) with data augmentation. We employ a VGG-like CNN architecture with batch normalization and ReLU activation functions, which is shown in Figure 2a. Note that this structure is the same to that used in Czarnecki et al. (2017a). It has three local critic networks, thus four layer groups that can be trained independently are formed (i.e., $N { = } 4$ ). The local critic networks are also CNNs, and their structures are kept as simple as possible in order to minimize the computational complexity for computing the estimated error gradient given by (5). + +![](images/e7fc0d788432248be4ca47ca6286fe703a2b1d4b34dc6032ad1b70417cbdc6d7.jpg) +Figure 2: (a) Network structure of the proposed approach using three local networks for CIFAR-10. LC1, LC2, and LC3 are local critic networks, each of which contains one convolutional layer. For CIFAR-100, the final fc10 layers of the main network and the local critic networks are replaced with fc100. (b) Sub-models and main model obtained by the proposed approach. + +![](images/cf0f00f590a2351278a4da197976c450ddbde9945826dacb31d04b73235d090e.jpg) +Figure 3: Training loss values of the main model and each sub-model with respect to the training iteration. + +We use the stochastic gradient descent with a momentum of 0.9 for the main network and the Adam optimization with a fixed learning rate of $1 0 ^ { - 4 }$ for the local networks. The L2 regularization is used with $5 \times 1 0 ^ { - 4 }$ for the main network. For the loss functions in (3) and (7), the cross-entropy and the L1 loss are used, respectively, which is determined empirically. The batch size is set to 128, and the maximum training iteration is set to 80,000. The learning rate for the main network is initialized to 0.1 and dropped by an order of magnitude after 40,000 and 60,000 iterations. The Xavier method is used for initialization of the network parameters. All experiments are performed using TensorFlow. We conduct all the experiments five times with different random seeds and report the average accuracy. + +# 3.1 PERFORMANCE EVALUATION + +Figure 3 shows how the loss values of the main network and each local critic network, i.e., $L _ { i }$ in (3), evolve with respect to the training iteration. The graphs show that the local critic networks successfully learn to approximate the main network’s loss with high accuracy during the whole training process. The local critic network farthest from the output side (i.e., $L _ { 1 }$ ) shows larger loss values than the others, which is due to the inaccuracy accumulated through the cascaded approximation. + +The classification performance of the proposed local critic training approach is evaluated in Table 1. For comparison, the performance of the regular backpropagation, DNI (Jaderberg et al., 2017), and critic training (Czarnecki et al., 2017a) is also evaluated. Although the critic training method is not for removing update locking, we include its result because it shares some similarity with our approach, i.e., additional modules to estimate the main network’s output. In all three methods, each additional module is composed of a convolutional layer and an output layer. In the case of the proposed method, we test different numbers of local critic networks. Figure 2a shows the structure with three local critic networks. When only one local network is used, it is located at the place of LC2 in Figure 2a. When five local networks are used, they are placed after every two layers of the main network. + +Table 1: Average test accuracy $( \% )$ of backpropagation (BP), DNI (Jaderberg et al., 2017), critic training (Czarnecki et al., 2017a), and proposed local critic training (LC). The numbers of local networks used are shown in the parentheses. The standard deviation values are also shown. + +
DatasetBPDNI (3)Critic (3)LC(1)LC (3)LC (5)
CIFAR-1093.93 ±0.2064.86 ±0.4291.92 ±0.3092.06 ±0.2092.39 ±0.0991.38 ±0.20
CIFAR-10075.14 ±0.1836.53 ±0.6469.07 ±0.2573.61 ±0.3169.91 ±0.5063.53 ±0.24
+ +Table 2: Average test accuracy $( \% )$ with respect to the number of layers in the local critic networks. $[ a , b , c ]$ means that the numbers of convolutional layers in LC1, LC2, and LC3 are $a , b$ , and $c$ respectively. + +
Dataset[1,1,1] (default)[3,3,3][5,5,5][3,2,1][1,2,3][5,4,3][3,4,5]
CIFAR-1092.39 ±0.0992.36 ±0.2291.72 ±0.1992.07 ±0.2192.20 ±0.1292.10 ±0.1691.90 ±0.16
CIFAR-10069.91 ±0.5070.02 ±0.2970.34 ±0.1670.06 ±0.6469.81 ±0.3370.87 ±0.4069.93 ±0.56
+ +When compared to the result of backpropagation, the proposed approach successfully decouples training of the layer groups at a small expense of accuracy decrease (note that the performance of the proposed method can be made closer to that of backpropagation using different structures, as will be shown in Tables 2 and Figure 4b). The degradation of the accuracy and standard deviation of our method is larger for CIFAR-100, which implies that the influence of gradient estimation is larger for more complex problems. When more local critic networks are used, the accuracy tends to decrease more due to higher reliance on predicted gradients rather than true gradients, while more layer groups can be trained independently. Thus, there exists a trade-off between the accuracy and unlocking effect. The DNI method shows poor performance as in (Czarnecki et al., 2017a). The proposed method shows performance improvement by $0 . 4 \%$ and $0 . 9 \%$ over the critic training method, both with three local networks, for the two datasets, respectively, which are found to be statistically significant using Mann-Whitney tests at a significance level of 0.05. This shows the efficacy of the cascaded learning scheme of the local networks in our method. + +# 3.2 STRUCTURES OF LOCAL CRITIC NETWORKS + +We examine the influence of the structures of the local critic networks in our method. Two aspects are considered, one about the influence of the overall complexity of the local networks and the other about the relative complexities of the local networks for good performance. For this, we change the number of convolutional layers in each local critic network, while keeping the other structural parameters unchanged. + +The results for various structure combinations of the three local critic networks are shown in Table 2. As the number of convolutional layers increase for all local networks (the first three cases in the table), the accuracy for CIFAR-100 slightly increases from $6 9 . 9 1 \%$ (with one convolutional layer) to $7 0 . 0 2 \%$ (three convolutional layers) and $7 0 . 3 4 \%$ (five convolutional layers), whereas for CIFAR-10 the accuracy slightly decreases when five convolutional layers are used. A more complex local network can learn better the target input-output relationship, which leads to the performance improvement for CIFAR-100. For CIFAR-10, on the other hand, the network structure with five convolutional layers seems too complex compared to the data to learn, which causes the performance drop. + +Next, the numbers of layers of the local networks are adjusted differently in order to investigate which local networks should be more complex for good performance. The results are shown in the last four columns of Table 2. Overall, it is more desirable to use more complex structures for the local networks closer to the input side of the main model. For instance, LC1 and LC3 are supposed to learn the relationship from $h _ { 1 }$ to $h _ { 4 }$ and that from $h _ { 3 }$ to $h _ { 4 }$ , respectively. More layers are involved from $h _ { 1 }$ to $h _ { 4 }$ in the main network, so the mapping that LC1 should learn would be more complicated, requiring a network structure with sufficient modeling capability. + +Table 3: Average test accuracy $( \% )$ with respect to the update frequency of local critic networks. + +
Dataset1/11/21/31/41/5
CIFAR-1092.39 ±0.0991.91 ±0.1991.78 ±0.1891.57 ±0.1291.35 ±0.17
CIFAR-10069.91 ±0.5067.99 ±0.4967.76±0.1966.74 ±0.4166.39 ±0.39
+ +Table 4: Average test accuracy $( \% )$ of the sub-models produced by local critic training and the networks trained by regular backpropagation. + +
DatasetBP sub 1LC sub 1BP sub 2LC sub 2BP sub 3LC sub 3
CIFAR-1074.46 ±0.9185.24 ±0.4988.03 ±0.8790.53 ±0.1592.05 ±0.2492.29 ±0.09
CIFAR-10047.58 ±1.1055.39 ±0.5761.79 ±0.9263.62 ±0.3167.81 ±0.2267.54 ±0.70
+ +# 3.3 PERIODIC UPDATE OF LOCAL CRITIC NETWORKS + +A way to increase the efficiency of the proposed approach is to update the local critic networks not at every iteration but only periodically. This may degrade the accuracy but has two benefits. First, the amount of computation required to update the local networks can be reduced. Second, the burden of the communication between the layer groups also can be reduced. These benefits will be more significant when the local networks have larger sizes. + +For the default structure shown in Figure 2a, we compare different update frequency in Table 3. It is noticed that the accuracy only slightly decreases as the frequency decreases. When the update frequency is a half of that for the main network (i.e., 1/2), the accuracy drops by $0 . 4 8 \%$ and $1 . 9 2 \%$ for the two datasets, respectively. Then, the decrease of the accuracy is only $0 . 5 6 \%$ for CIFAR-10 and $1 . 6 0 \%$ for CIFAR-100 when the update frequency decreases from 1/2 to 1/5. + +# 3.4 STRUCTURAL OPTIMIZATION + +Table 4 compares the performance of the sub-models, and Table 5 shows the complexities of the sub-models in terms of the amount of computation for a feedforward pass and the number of weight parameters. A larger network (e.g., sub-model 3) shows better performance than a smaller network (e.g., sub-model 1), which is reasonable due to the difference in learning capability with respect to the model size. The largest sub-model (sub-model 3) shows similar accuracy to the main model $( 9 2 . 2 9 \%$ vs. $9 2 . 3 9 \%$ for CIFAR-10 and $6 7 . 5 4 \%$ vs. $6 9 . 9 1 \%$ for CIFAR-100), while the complexity is significantly reduced. For CIFAR-10, the computational complexity in terms of the number of floating-point operations (FLOPs) and the memory complexity are reduced to only about $30 \%$ (15.72 to 4.52 million FLOPs, and 7.87 to 2.26 million parameters), as shown in Table 5. If an absolute accuracy reduction of $1 . 8 6 \%$ (from $9 2 . 3 9 \%$ to $9 0 . 5 3 \%$ ) is allowed by taking sub-model 2, the reduction of complexity is even more remarkable, up to about one ninth. + +In addition, the table also shows the accuracy of the networks that have the same structures with the sub-models but are trained using regular backpropagation. Surprisingly, such networks do not easily reach accuracy comparable to that of the sub-models obtained by local critic training, particularly for smaller networks (e.g., $7 4 . 4 6 \%$ vs. $8 5 . 2 4 \%$ with sub-model 1 for CIFAR-10). We think that joint training of the sub-models in local critic training helps them to find better solutions than those reached by independent regular backpropagation. + +Therefore, these results demonstrate that a structurally optimized network can be obtained at a cost of a small loss in accuracy by local critic training, which may not be attainable by trial-and-error with backpropagation. + +Table 5: FLOPs required for a feedforward pass and numbers of model parameters in the sub-models and main model for CIFAR-10. Note that sub-model 2 has less FLOPs and parameters than submodel 1 due to the pooling operation in sub-model 2. + +
modelFLOP# of parameters
Sub-model 12.85M1.42M
Sub-model 21.76M0.88M
Sub-model 34.52M2.26M
Main model15.72M7.87M
+ +Table 6: Average FLOPs and accuracy of progressive inference for test data of CIFAR-10 when the threshold is set to 0.9 or 0.95. + +
FLOPAccuracy (%)
Progressive inference (0.9)2.90M91.18 ±0.10
Progressive inference (0.95)3.05M91.75 ±0.16
Main model15.72M92.39 ±0.09
+ +# 3.5 PROGRESSIVE INFERENCE + +We apply the progressive inference algorithm shown in Algorithm 1 to the trained default network for CIFAR-10 with the threshold set to 0.9 or 0.95. The results are shown in Table 6. The feedforward pass ends at different sub-models for different test data, and the average FLOPs over all test data are shown. When the threshold is 0.9, with only a slight loss of accuracy $( 9 2 . 3 9 \%$ to $9 1 . 1 8 \%$ ), the computational complexity is reduced significantly, which is only $1 8 . 4 5 \%$ of that of the main model. When the threshold increases to 0.95, the accuracy loss becomes smaller (only $0 . 6 4 \%$ , while the complexity reduction remains almost the same ( $1 9 . 4 0 \%$ of the main model’s complexity). + +# 3.6 ENSEMBLE INFERENCE + +The results of ensemble inference using the sub-models and main model are shown in Figure 4b. Using an ensemble of the three sub-models, we observe improved classification accuracy $( 9 2 . 6 8 \%$ and $7 0 . 8 6 \%$ for the two datasets, respectively) in comparison to the main model. The performance is further enhanced by an ensemble of both the three sub-models and the main model $( 9 2 . 7 9 \%$ and $7 1 . 8 6 \%$ . The improvement comes from the complementarity among the models, particularly between the models sharing a smaller number of layers. For instance, we found that sub-model 3 and the main model tend to show coincident classification results for a large portion of test data, so their complementarity is not significant; on the other hand, more data are classified differently by sub-model 1 and the main model, where we mainly observe performance improvement. Instead of the simple summation, there could be a better method to combine the models, which is left for future work. + +# 4 CONCLUSION + +In this paper, we proposed the local critic training approach for removing the inter-layer locking constraint in training of deep neural networks. In addition, we proposed three applications of the local critic training method: structural optimization of neural networks, progressive inference, and ensemble classification. It was demonstrated that the proposed method can successfully train CNNs with local critic networks having extremely simple structures. The performance of the method was also evaluated in various aspects, including effects of structures and update intervals of local critic networks and influences of the sizes of layer groups. Finally, it was shown that structural optimization, progressive inference, and ensemble classification can be performed directly using the models trained with the proposed approach without additional procedures. + +![](images/7e438144d79f096a85bdf12b0b985cd481458bf96e698456bea49a86fea6ec05.jpg) +Figure 4: (a) Ensemble inference using the sub-models and main model. (b) Performance of the ensemble inference for an ensemble of the three sub-models $( 1 + 2 + 3 )$ and an ensemble of the submodels and the main model ( $1 + 2 + 3 +$ main). Standard deviation values are also shown as error bars. + +# REFERENCES + +M. A. Carreira-Perpinan and W. Wang. Distributed optimization of deeply nested systems. In International Conference on Artificial Intelligence and Statistics (AISTATS), pp. 10–19, Reykjavik, Iceland, 2014. +C. Cortes, X. Gonzalvo, V. Kuznetsov, M. Mohri, and S. Yang. AdaNet: Adaptive structural learning of artificial neural networks. In International Conference on Machine Learning (ICML), pp. 874– 883, Sydney, Australia, 2017. +W. M. Czarnecki, S. Osindero, M. Jaderberg, G. Swirszcz, and R. Pascanu. Sobolev training for neural networks. In Advances in Neural Information Processing Systems (NIPS), pp. 4278–4287, Long Beach, CA, 2017a. +W. M. Czarnecki, G. Swirszcz, M. Jaderberg, S. Osindero, O. Vinyals, and K. Kavukcuoglu. Understanding synthetic gradients and decoupled neural interfaces. In International Conference on Machine Learning (ICML), pp. 904–912, Sydney, Australia, 2017b. +J. Deng, W. Dong, R. Socher, L. Li, K. Li, and L. Fei-Fei. ImageNet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition (CVPR), pp. 248–255, Miami Beach, FL, 2009. +J. Feng and T. Darrell. Learning the structure of deep convolutional networks. In International Conference on Computer Vision (ICCV), pp. 2749–2757, Santiago, Chile, 2015. +K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770–778, Las Vegas, NV, 2016a. +K. He, X. Zhang, S. Ren, and J. Sun. Identity mappings in deep residual networks. In European Conference on Computer Vision (ECCV), pp. 630–645, Amsterdam, The Netherlands, 2016b. +G. Huang, D. Chen, T. Li, F. Wu, L. Maaten, and K.Q. Weinberger. Multi-scale dense networks for resource efficient image classification. In International Conference on Learning Representations (ICLR), Vancouver, Canada, 2018. +M. Jaderberg, W. M. Czarnecki, S. Osindero, O. Vinyals, A. Graves, D. Silver, and K. Kavukcuoglu. Decoupled neural interfaces using synthetic gradients. In International Conference on Machine Learning (ICML), pp. 1627–1635, Sydney, Australia, 2017. +N. Kriegeskorte, M. Mur, and P. Bandettini. Representational similarity analysis connecting the branches of systems neuroscience. Frontiers in Systems Neuroscience, 2(4), 2008. +A. Krizhevsky. Learning multiple layers of features from tiny images. Master’s thesis, Department of Computer Science, University of Toronto, 2009. +T.-Y. Kwok and D.-Y. Yeung. Constructive algorithms for structure learning in feedforward neural networks for regression problems. IEEE Transactions on Neural Networks, 8(3):630–645, 1997. +G. Larsson, M. Maire, and G. Shakhnarovich. FractalNet: Ultra-deep neural networks without residuals. In International Conference on Learning Representations (ICLR), Toulon, France, 2017. +Y. LeCun, Y. Bengio, and G. Hinton. Deep learning. Nature, 521:436–444, 2015. +R. Reed. Pruning algorithms- a survey. IEEE Transactions on Neural Networks, 4(5):730–747, 1993. +C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1–9, Boston, MA, 2015. +G. Taylor, R. Burmeister, Z. Xu, B. Singh, A. Patel, and T. Goldstein. Training neural networks without gradients: A scalable ADMM approach. In International Conference on Machine Learning (ICML), pp. 2722–2731, New York, NY, 2016. +L. Wang, C. Lee, Z. Tu, and S. Lazebnik. Training deeper convolutional networks with deep supervision. arXiv preprint arXiv:1505.02496v1, 2015. +B. Zoph and Q.V. Le. Neural architecture search with reinforcement learning. In International Conference on Learning Representations (ICLR), Toulon, France, 2017. + +# A ADDITIONAL RESULTS + +# A.1 ADDITIONAL COMPARISON + +In Czarnecki et al. (2017a), a method to minimize not only the loss of the network output but also its derivative is proposed, called Sobolev training, and applied to the critic training algorithm. We also conduct an experiment to use the Sobolev training method in our proposed algorithm. The results are shown in Table 7. In comparison to the performance shown in Table 1, we do not observe significant difference overall. In addition, we test the deep supervision algorithm (Wang et al., 2015), which also has additional modules connected to intermediate layers. The table shows that its performance is not significantly different from that of backpropagation. + +Table 7: Average test accuracy $( \% )$ of Sobolev local critic training (Sob LC), Sobolev critic training (Czarnecki et al., 2017a), and deep supervision (Wang et al., 2015). The numbers of local networks used are shown in the parentheses. The standard deviation values are also shown. + +
DatasetSob LC (1)Sob LC (3)Sob LC (5)Sob critic (3)Deep supervision (3)
CIFAR-1091.83 ±0.2292.32 ±0.1291.42 ±0.2691.96 ±0.2294.08 ±0.11
CIFAR-10073.64 ±0.2269.58 ±0.3263.82 ±1.0468.83 ±0.3175.09 ±0.29
+ +# A.2 RESULTS FOR LARGER NETWORKS + +We examine the effectiveness of the proposed method for larger networks than those used in Section 3. For this, ResNet-50 and ResNet-101 (He et al., 2016a) are trained with backpropagation or the proposed method using three local critic networks. The results shown in Table 8 have a similar trend to those in Table 1 with slight performance improvement in most cases, which confirm that the proposed method works successfully for relatively complex networks. + +Table 8: Test accuracy $( \% )$ of backpropagation (BP), and local critic training (LC) for ResNet-50 and ResNet-101. The numbers of local networks used are shown in the parentheses. + +
DatasetResNet-50 BPResNet-50 LC (3)ResNet-101 BPResNet-101 LC (3)
CIFAR-1094.4292.9594.1893.17
CIFAR-10074.8870.5877.2972.72
+ +In addition, we experiment using the ImageNet dataset (Deng et al., 2009), which is much larger and more complex than CIFAR-10 and CIFAR-100. The results for ResNet-50 in Table 9 show that the proposed method can also work well for large datasets. + +Table 9: Test accuracy $( \% )$ of ResNet-50 trained with backpropagation (BP) and local critic training (LC) for the ImageNet dataset. + +
ImageNetResNet-50 BPResNet-50 LC (3)
Top-5 accuracy92.3886.52
Top-1 accuracy75.0765.41
+ +# B VISUALIZATION OF TRAINED MODELS + +In order to analyze the trained networks, we obtain the representational dissimilarity matrix (RDM) (Kriegeskorte et al., 2008) from each layer of the networks trained by backpropagation and the proposed method. For each of 400 samples from CIFAR-10, the activation of each layer is recorded, and the correlation between two samples is measured, which is shown in Figure 5. In the figure, clear diagonal-blocks indicate that samples from the same class have highly correlated representations (e.g., the last layer). + +![](images/25c4b4bf1fb8d3e89761a9928ade1dad5be347d56a4fa5ebefcc89ee4de14596.jpg) +Figure 5: Representation dissimilarity matrix of each layer of the trained networks (having the structure shown in Figure 2a) for label-ordered samples from CIFAR-10. Red lines indicate the locations of local critic networks. A clear diagonal-block pattern indicates that clear inner-class representation has been trained. + +Overall, block-diagonal patterns become clear at the last layers for all cases. However, the figure also shows that the two training methods result in networks showing qualitatively different characteristics in their internal representations. In particular, the layers at which local critic networks are attached (e.g., layer 5 in LC (1) and LC (3), and layer 6 in LC (5)) show relatively distinguishable blockdiagonal patterns in comparison to those of the network trained by backpropagation. These layers in the proposed method act not only as intermediate layers of the main network but also as near-final layers of the sub-models, and thus are forced to learn class-specific representations to some extent. + +# C LEARNING DYNAMICS + +# C.1 VISUALIZATION OF LOSS EVOLUTION + +As a way to investigate the learning dynamics of the proposed method, the loss values at each local critic network $( L _ { i } )$ for individual data over the training iterations are examined (Czarnecki et al., 2017b). Figure 3 visualizes the loss values for 400 sampled data of CIFAR-10 (arranged in 2D) when three local critic networks are used. For visualization, the same results are shown twice, once sorted by labels (Figure 6a) and once sorted by $L _ { 4 }$ at iteration 20000 (Figure 6b). At the early stage of learning, the loss values at the local critic networks ( $L _ { 1 }$ to $L _ { 3 }$ ) are largely different from those at the main network $( L _ { 4 } )$ , with only slight similarity (e.g., the blue region at iteration 50 in Figure 6a). At the later stage, however, all the losses similarly converged to small values for most of the samples (at iteration 20000 in Figure 6b). + +# C.2 LEARNING CURVES OF SUB-MODELS + +We showed the test performance of sub-models in Table 4. In addition, we show their test accuracy over training iterations for CIFAR-10 in Figure 7. In particular, faster convergence in the case of local critic training than backpropagation is observed in Figure 7a. + +![](images/4d2ec25d549cecb1875fd17273ba16d4340e95ed2b8cf51f67ab792ad5df43f6.jpg) +Figure 6: Evolution of the loss value during the course of the proposed local critic training. + +![](images/86be65d27ac7cde0ef30ef8e27c778c97b88720bda116cd16823e0bacddeb792.jpg) +(c) Sub-model 3 +Figure 7: Test accuracy $( \% )$ of sub-models trained by backpropagation and local critic training over training iterations. \ No newline at end of file diff --git a/parse/train/B1x-LjAcKX/B1x-LjAcKX_content_list.json b/parse/train/B1x-LjAcKX/B1x-LjAcKX_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..99dfc6d12d5202c15e0a777729e10e1455440554 --- /dev/null +++ b/parse/train/B1x-LjAcKX/B1x-LjAcKX_content_list.json @@ -0,0 +1,1250 @@ +[ + { + "type": "text", + "text": "LOCAL CRITIC TRAINING OF DEEP NEURAL NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 236, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This paper proposes a novel approach to train deep neural networks by unlocking the layer-wise dependency of backpropagation training. The approach employs additional modules called local critic networks besides the main network model to be trained, which are used to obtain error gradients without complete feedforward and backward propagation processes. We propose a cascaded learning strategy for these local networks. In addition, the approach is also useful from multi-model perspectives, including structural optimization of neural networks, computationally efficient progressive inference, and ensemble classification for performance improvement. Experimental results show the effectiveness of the proposed approach and suggest guidelines for determining appropriate algorithm parameters. ", + "bbox": [ + 233, + 267, + 764, + 406 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 433, + 336, + 449 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In recent days, deep learning has been remarkably advanced and successfully applied in numerous fields (LeCun et al., 2015). A key mechanism behind the success of deep neural networks is that they are capable of extracting useful information progressively through their layered structures. It is an increasing trend that more and more complex deep neural network structures are developed in order to solve challenging real-world problems, e.g., He et al. (2016b). Training of deep neural networks is based on backpropagation in most cases, which basically works in a sequential and synchronous manner. During the feedforward pass, the input data is processed through the hidden layers to produce the network output; during the feedback pass, the error gradient is propagated back through the layers to update each layer’s weight parameters. Therefore, training of each layer has dependency on all the other layers, which causes the issue of locking (Jaderberg et al., 2017). This is undesirable in some cases, e.g., a system consisting of several interacting models, a model distributed across multiple computing nodes, etc. ", + "bbox": [ + 174, + 464, + 825, + 631 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There have been attempts to remove the locking constraint. In Carreira-Perpinan & Wang (2014), the method of auxiliary coordinates (MAC) is proposed. It replaces the original loss minimization problem with an equality-constrained optimization problem by introducing an auxiliary variable for each data and each hidden unit. Then, solving the problem is formulated as iteratively solving several sub-problems independently. A similar approach using the alternating direction method of multipliers (ADMM) is proposed in Taylor et al. (2016). It also employs an equality-constrained optimization but with different auxiliary variables, so that resulting sub-problems have closed form solutions. However, these methods are not scalable to deep learning architectures such as convolutional neural networks (CNNs). ", + "bbox": [ + 174, + 638, + 825, + 763 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The method proposed in Jaderberg et al. (2017), called decoupled neural interface (DNI), directly synthesizes estimated error gradients, called synthetic gradients, using an additional small neural network for training a layer’s weight parameters. As long as the synthetic gradients are close to the actual backpropagated gradients, each layer does not need to wait until the error at the output layer is backpropagated through the preceding layers, which allows independent training of each layer. However, this method suffers from performance degradation when compared to regular backpropagation (Czarnecki et al., 2017a). The idea of having additional modules supporting the layers of the main model is also adopted in Czarnecki et al. (2017a), where the additional modules are trained to approximate the main model’s outputs instead of error gradients. Due to this, however, the method does not resolve the issue of update locking, and in fact, the work does not intend to design a non-sequential learning algorithm. ", + "bbox": [ + 174, + 770, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/c2f5cb4c44bf704873fe2f46b97b9422343f9e9d1d458c774bb6d5e32319f1f9.jpg", + "image_caption": [ + "Figure 1: Learning processes of DNI (Jaderberg et al., 2017) and the proposed local critic training. The black, green, and blue arrows indicate feedforward passes, an error gradient flow, and loss comparison, respectively. " + ], + "image_footnote": [], + "bbox": [ + 272, + 102, + 723, + 324 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we propose a novel approach for non-sequential learning, called local critic training. The key idea is that additional modules besides the main neural network model are employed, which we call local critics, in order to indirectly deliver error gradients to the main model for training without backpropagation. In other words, a local critic located at a certain layer group is trained in such a way that the derivative of its output serves as the error gradient for training of the corresponding layers’ weight parameters. Thus, the error gradient does not need to be backpropagated, and the feedforward operations and gradient-descent learning can be performed independently. Through extensive experiments, we examine the influences of the network structure, update frequency, and total number of local critics, which provide not only insight into operation characteristics but also guidelines for performance optimization of the proposed method. ", + "bbox": [ + 174, + 405, + 825, + 544 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In addition to the capability of implementing training without locking, the proposed approach can be exploited for additional important applications. First, we show that applying the proposed method automatically performs structural optimization of neural networks for a given problem, which has been a challenging issue in the machine learning field. Second, a progressive inference algorithm using the network trained with the proposed method is presented, which can adaptively reduce the computational complexity during the inference process (i.e., test phase) depending on the given data. Third, the network trained by the proposed method naturally enables ensemble inference that can improve the classification performance. ", + "bbox": [ + 174, + 550, + 825, + 662 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PROPOSED APPROACH", + "text_level": 1, + "bbox": [ + 176, + 681, + 392, + 698 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 LOCAL CRITIC TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 712, + 393, + 727 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The basic idea of the proposed approach is to introduce additional local networks, which we call local critics, besides the main network model, so that they eventually provide estimates of the output of the main network. Each local critic network can serve a group of layers of the main model by being attached to the last layer of the group. The proposed architecture is illustrated in Figure 1, where $f _ { i }$ is the $i$ th layer group (containing one or more layers), $h _ { i }$ is the output of $f _ { i }$ , and $h _ { N }$ is the final output of the main model having $N$ layer groups: ", + "bbox": [ + 173, + 738, + 825, + 821 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/4798629843114ccbf444c93d496df8e33c311412db5fd90183f57df9af2b18b4.jpg", + "text": "$$\nh _ { i } = f _ { i } ( h _ { i - 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 449, + 825, + 549, + 842 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "$c _ { i }$ is the local critic network for $f _ { i }$ , which is expected to approximate $h _ { N }$ based on $h _ { i }$ , i.e., ", + "bbox": [ + 174, + 844, + 764, + 859 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/f7243ea9e293516694889f68f02b53c078ac899af0ef2b3f7f8496f6a6a0647b.jpg", + "text": "$$\nc _ { i } ( h _ { i } ) \\approx h _ { N }\n$$", + "text_format": "latex", + "bbox": [ + 455, + 862, + 542, + 878 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Then, this can be used to approximate the loss function of the main network, $L _ { N } = l ( h _ { N } , y )$ , which is used to train $f _ { i }$ , by ", + "bbox": [ + 168, + 882, + 823, + 909 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/743803046c161cf657886b2305ba61174b50e5c8b0ea6b0183d619abc2b6b94f.jpg", + "text": "$$\nL _ { i } = l ( c _ { i } ( h _ { i } ) , y )\n$$", + "text_format": "latex", + "bbox": [ + 441, + 907, + 557, + 925 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "for $i = 1 , . . . , N - 1$ , i.e., ", + "bbox": [ + 173, + 103, + 338, + 118 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/012f659c556f8f1a7dcae76cb3d4bf0a7348170db6cf552327fe0bc65fc1d1e2.jpg", + "text": "$$\nL _ { i } \\approx L _ { N }\n$$", + "text_format": "latex", + "bbox": [ + 465, + 117, + 531, + 133 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $y$ is the training target and $l$ is the loss function such as cross-entropy or mean-squared error. Then, the error gradient for training $f _ { i }$ is obtained by differentiating $L _ { i }$ with respect to $h _ { i }$ , i.e., ", + "bbox": [ + 171, + 138, + 823, + 166 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a9607d4f17aa62cfa1fcfdfb0d9ece6642dfa5a546a65cf1d8d4ebc392937189.jpg", + "text": "$$\n\\delta _ { i } = \\frac { \\partial L _ { i } } { \\partial h _ { i } }\n$$", + "text_format": "latex", + "bbox": [ + 465, + 172, + 534, + 207 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "which can be used to train the weight parameters of $f _ { i }$ , denoted by $\\theta _ { i }$ , via a gradient-descent rule: ", + "bbox": [ + 171, + 212, + 812, + 227 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/a48258d4afd90677e0829f860dd7f692afa8511ac1fdeaa25a54120f32bcb1af.jpg", + "text": "$$\n\\theta _ { i } \\theta _ { i } - \\eta \\delta _ { i } \\frac { \\partial h _ { i } } { \\partial \\theta _ { i } }\n$$", + "text_format": "latex", + "bbox": [ + 431, + 233, + 566, + 266 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\eta$ is a learning rate. Note that the final layer group $h _ { N }$ does not require a local critic network and can be trained using the regular backproagation because the final output of the main network is directly available. Therefore, the update of $f _ { i }$ does not need to wait until its output $h _ { i }$ propagates till the end of the main network and the error gradient is backpropagated; it can be performed when the operations from (2) to (5) are done. For $c _ { i }$ , we usually use a simple model so that the operations through $c _ { i }$ are simpler than those through $f _ { i + 1 }$ till $f _ { N }$ . ", + "bbox": [ + 173, + 272, + 825, + 358 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "While the dependency of $f _ { i }$ on $f _ { j } \\ ( j > i )$ during training is resolved in this way, there still exists the dependency of $c _ { i }$ on $f _ { j } \\ ( j > i )$ , because training $c _ { i }$ requires its ideal target, i.e., $h _ { N }$ , which is available from $f _ { N }$ only after the feedforward pass is complete. In order to resolve this problem, we use an indirect, cascaded approach, where $c _ { i }$ is trained so that its training loss targets the training loss for ${ c _ { i + 1 } } ^ { 1 }$ : ", + "bbox": [ + 173, + 363, + 825, + 433 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6649e132b0d78475bfab643bdffbdfb783a403a8d86fbbbd243974c850c178ac.jpg", + "text": "$$\nL _ { c _ { i } } = l ( L _ { i } , L _ { i + 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 437, + 433, + 560, + 449 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In other words, training of $c _ { i }$ can be performed once the loss for $c _ { i + 1 }$ is available. ", + "bbox": [ + 174, + 453, + 709, + 468 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1 compares the proposed architecture with the existing DNI approach that also employs local networks besides the main network to resolve the issue of locking (Jaderberg et al., 2017). In DNI, the local network $c _ { i }$ directly estimates the error gradient, i.e., ", + "bbox": [ + 174, + 474, + 825, + 517 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c9b577cb1d09212431e5ca7c9a8a486b25c65d1672992267ddf870c5ef3f1006.jpg", + "text": "$$\nc _ { i } ( h _ { i } ) \\approx \\frac { \\partial L _ { N } } { \\partial h _ { i } }\n$$", + "text_format": "latex", + "bbox": [ + 447, + 523, + 550, + 556 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "so that each layer group of the main model can be updated without waiting for the forward and backward propagations in the subsequent layers. And, to update $c _ { i }$ , the error gradient for $f _ { i + 1 }$ estimated by $c _ { i + 1 }$ is backpropagated through $f _ { i + 1 }$ and is used as the (estimated) target for $c _ { i }$ . Therefore, all the necessary computations in the forward and backward passes can be locally confined. The performance of the two methods will be compared in Section 3. ", + "bbox": [ + 174, + 563, + 825, + 632 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 STRUCTURAL OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 650, + 415, + 665 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In many cases, determining an appropriate structure of neural networks for a given problem is not straightforward. This is usually done through trial-and-error, which is extremely time-consuming. There have been studies to automate the structural optimization process (Cortes et al., 2017; Feng & Darrell, 2015; Kwok & Yeung, 1997; Reed, 1993), but this issue still remains very challenging. ", + "bbox": [ + 173, + 676, + 825, + 733 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In deep learning, the problem of structural optimization is even more critical. Large-sized networks may easily show overfitting. Even if large networks may produce high accuracy, they take significantly large amounts of memory and computation, which is undesirable especially for resourceconstrained cases such as embedded and mobile systems. Therefore, it is highly desirable to find an optimal network structure that is sufficiently small while the performance is kept reasonably good. ", + "bbox": [ + 173, + 739, + 825, + 809 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "During local critic training, each local critic network is trained to estimate the output of the main network eventually. Therefore, once the training of the proposed architecture finishes, we obtain different networks that are supposed to have similar input-output mappings but have different structures and possibly different accuracy, i.e., multiple sub-models and one main model (see Figure 2b). Here, a sub-model is composed of the layers on the path from the input to a certain hidden layer ", + "bbox": [ + 173, + 815, + 825, + 886 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Algorithm 1: Progressive inference ", + "text_level": 1, + "bbox": [ + 174, + 108, + 411, + 122 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Input: data $_ x$ , threshold $t$ \nModel: sub-model $c _ { i }$ , main-model $f$ \nInitialize: classif ication $= 0$ . \nfor $i = 1$ to $N - 1$ do if max softmax $( c _ { i } ( x ) ) > t$ then classif ication $=$ argmax softmax $( c _ { i } ( x ) )$ ) break end if \nend for \nif classif ication $= = 0$ then # if all sub-models are not confident classif ication $=$ argmax softmax $( f ( x ) )$ \nend if ", + "bbox": [ + 179, + 125, + 483, + 290 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "and its local critic network. Among the sub-models, we can choose one as a structure-optimized network by considering the trade-off relationship between the complexity and performance. ", + "bbox": [ + 174, + 323, + 823, + 351 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "It is worth mentioning that our structural optimization approach can be performed instantly after training of the model, whereas many existing methods for structural optimization require iterative search processes, e.g., Zoph & Le (2017). ", + "bbox": [ + 174, + 358, + 825, + 400 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 PROGRESSIVE INFERENCE ", + "text_level": 1, + "bbox": [ + 176, + 417, + 395, + 431 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We propose another simple but effective way to utilize the sub-models obtained by the proposed approach for computational efficiency, which we call progressive inference. Although small submodels (e.g., sub-model 1) tend to show low accuracy, they would still perform well for some data. For such data, we do not need to perform the full feedforward pass but can take the classification decision by the sub-models. Thus, the basic idea of the progressive inference is to finish inference (i.e., classification) with a small sub-model if its confidence on the classification result is high enough, instead of completing the full feedforward pass with the main model, which can reduce the computational complexity. Here, the softmax outputs for all classes are compared and the maximum probability is used as the confidence level. If it is higher than a threshold, we take the decision by the sub-model; otherwise, the feedforward pass continues. The proposed progressive inference method is summarized in Algorithm 12. ", + "bbox": [ + 174, + 443, + 825, + 597 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.4 ENSEMBLE INFERENCE ", + "text_level": 1, + "bbox": [ + 176, + 614, + 375, + 628 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In recent deep learning systems, it is popular to use ensemble approaches to improve performance in comparison to single models, where multiple networks are combined for producing final results, e.g., He et al. (2016a); Szegedy et al. (2015). The sub-models and main model obtained by applying the proposed local critic training approach can be used for ensemble inference. Figure 4a depicts how the sub-models and the main model can work together to form an ensemble classifier. We take the simplest way to combine them, i.e., summation of the networks’ outputs. ", + "bbox": [ + 174, + 640, + 825, + 724 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 746, + 326, + 761 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We conduct extensive experiments to examine the performance of the proposed method in various aspects. We use the CIFAR-10 and CIFAR-100 datasets (Krizhevsky, 2009) with data augmentation. We employ a VGG-like CNN architecture with batch normalization and ReLU activation functions, which is shown in Figure 2a. Note that this structure is the same to that used in Czarnecki et al. (2017a). It has three local critic networks, thus four layer groups that can be trained independently are formed (i.e., $N { = } 4$ ). The local critic networks are also CNNs, and their structures are kept as simple as possible in order to minimize the computational complexity for computing the estimated error gradient given by (5). ", + "bbox": [ + 174, + 777, + 825, + 861 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/e7fc0d788432248be4ca47ca6286fe703a2b1d4b34dc6032ad1b70417cbdc6d7.jpg", + "image_caption": [ + "Figure 2: (a) Network structure of the proposed approach using three local networks for CIFAR-10. LC1, LC2, and LC3 are local critic networks, each of which contains one convolutional layer. For CIFAR-100, the final fc10 layers of the main network and the local critic networks are replaced with fc100. (b) Sub-models and main model obtained by the proposed approach. " + ], + "image_footnote": [], + "bbox": [ + 178, + 101, + 816, + 222 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/cf0f00f590a2351278a4da197976c450ddbde9945826dacb31d04b73235d090e.jpg", + "image_caption": [ + "Figure 3: Training loss values of the main model and each sub-model with respect to the training iteration. " + ], + "image_footnote": [], + "bbox": [ + 187, + 309, + 813, + 474 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 544, + 823, + 573 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We use the stochastic gradient descent with a momentum of 0.9 for the main network and the Adam optimization with a fixed learning rate of $1 0 ^ { - 4 }$ for the local networks. The L2 regularization is used with $5 \\times 1 0 ^ { - 4 }$ for the main network. For the loss functions in (3) and (7), the cross-entropy and the L1 loss are used, respectively, which is determined empirically. The batch size is set to 128, and the maximum training iteration is set to 80,000. The learning rate for the main network is initialized to 0.1 and dropped by an order of magnitude after 40,000 and 60,000 iterations. The Xavier method is used for initialization of the network parameters. All experiments are performed using TensorFlow. We conduct all the experiments five times with different random seeds and report the average accuracy. ", + "bbox": [ + 174, + 579, + 825, + 705 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 PERFORMANCE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 723, + 415, + 737 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Figure 3 shows how the loss values of the main network and each local critic network, i.e., $L _ { i }$ in (3), evolve with respect to the training iteration. The graphs show that the local critic networks successfully learn to approximate the main network’s loss with high accuracy during the whole training process. The local critic network farthest from the output side (i.e., $L _ { 1 }$ ) shows larger loss values than the others, which is due to the inaccuracy accumulated through the cascaded approximation. ", + "bbox": [ + 174, + 750, + 825, + 819 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The classification performance of the proposed local critic training approach is evaluated in Table 1. For comparison, the performance of the regular backpropagation, DNI (Jaderberg et al., 2017), and critic training (Czarnecki et al., 2017a) is also evaluated. Although the critic training method is not for removing update locking, we include its result because it shares some similarity with our approach, i.e., additional modules to estimate the main network’s output. In all three methods, each additional module is composed of a convolutional layer and an output layer. In the case of the proposed method, we test different numbers of local critic networks. Figure 2a shows the structure with three local critic networks. When only one local network is used, it is located at the place of LC2 in Figure 2a. When five local networks are used, they are placed after every two layers of the main network. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/d78ef29c0276fdcb94b71d6c082ad7bac1530b19f008a5382eb1a73095c21bac.jpg", + "table_caption": [ + "Table 1: Average test accuracy $( \\% )$ of backpropagation (BP), DNI (Jaderberg et al., 2017), critic training (Czarnecki et al., 2017a), and proposed local critic training (LC). The numbers of local networks used are shown in the parentheses. The standard deviation values are also shown. " + ], + "table_footnote": [], + "table_body": "
DatasetBPDNI (3)Critic (3)LC(1)LC (3)LC (5)
CIFAR-1093.93 ±0.2064.86 ±0.4291.92 ±0.3092.06 ±0.2092.39 ±0.0991.38 ±0.20
CIFAR-10075.14 ±0.1836.53 ±0.6469.07 ±0.2573.61 ±0.3169.91 ±0.5063.53 ±0.24
", + "bbox": [ + 184, + 154, + 812, + 213 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/9bee6279a18a9c4d65bfadba1f1070eb52102a4773d854928cbaf5f777bc777c.jpg", + "table_caption": [ + "Table 2: Average test accuracy $( \\% )$ with respect to the number of layers in the local critic networks. $[ a , b , c ]$ means that the numbers of convolutional layers in LC1, LC2, and LC3 are $a , b$ , and $c$ respectively. " + ], + "table_footnote": [], + "table_body": "
Dataset[1,1,1] (default)[3,3,3][5,5,5][3,2,1][1,2,3][5,4,3][3,4,5]
CIFAR-1092.39 ±0.0992.36 ±0.2291.72 ±0.1992.07 ±0.2192.20 ±0.1292.10 ±0.1691.90 ±0.16
CIFAR-10069.91 ±0.5070.02 ±0.2970.34 ±0.1670.06 ±0.6469.81 ±0.3370.87 ±0.4069.93 ±0.56
", + "bbox": [ + 186, + 285, + 808, + 332 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 363, + 823, + 405 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "When compared to the result of backpropagation, the proposed approach successfully decouples training of the layer groups at a small expense of accuracy decrease (note that the performance of the proposed method can be made closer to that of backpropagation using different structures, as will be shown in Tables 2 and Figure 4b). The degradation of the accuracy and standard deviation of our method is larger for CIFAR-100, which implies that the influence of gradient estimation is larger for more complex problems. When more local critic networks are used, the accuracy tends to decrease more due to higher reliance on predicted gradients rather than true gradients, while more layer groups can be trained independently. Thus, there exists a trade-off between the accuracy and unlocking effect. The DNI method shows poor performance as in (Czarnecki et al., 2017a). The proposed method shows performance improvement by $0 . 4 \\%$ and $0 . 9 \\%$ over the critic training method, both with three local networks, for the two datasets, respectively, which are found to be statistically significant using Mann-Whitney tests at a significance level of 0.05. This shows the efficacy of the cascaded learning scheme of the local networks in our method. ", + "bbox": [ + 174, + 412, + 825, + 593 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 STRUCTURES OF LOCAL CRITIC NETWORKS ", + "text_level": 1, + "bbox": [ + 178, + 617, + 514, + 631 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We examine the influence of the structures of the local critic networks in our method. Two aspects are considered, one about the influence of the overall complexity of the local networks and the other about the relative complexities of the local networks for good performance. For this, we change the number of convolutional layers in each local critic network, while keeping the other structural parameters unchanged. ", + "bbox": [ + 174, + 645, + 825, + 715 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The results for various structure combinations of the three local critic networks are shown in Table 2. As the number of convolutional layers increase for all local networks (the first three cases in the table), the accuracy for CIFAR-100 slightly increases from $6 9 . 9 1 \\%$ (with one convolutional layer) to $7 0 . 0 2 \\%$ (three convolutional layers) and $7 0 . 3 4 \\%$ (five convolutional layers), whereas for CIFAR-10 the accuracy slightly decreases when five convolutional layers are used. A more complex local network can learn better the target input-output relationship, which leads to the performance improvement for CIFAR-100. For CIFAR-10, on the other hand, the network structure with five convolutional layers seems too complex compared to the data to learn, which causes the performance drop. ", + "bbox": [ + 174, + 722, + 825, + 847 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Next, the numbers of layers of the local networks are adjusted differently in order to investigate which local networks should be more complex for good performance. The results are shown in the last four columns of Table 2. Overall, it is more desirable to use more complex structures for the local networks closer to the input side of the main model. For instance, LC1 and LC3 are supposed to learn the relationship from $h _ { 1 }$ to $h _ { 4 }$ and that from $h _ { 3 }$ to $h _ { 4 }$ , respectively. More layers are involved from $h _ { 1 }$ to $h _ { 4 }$ in the main network, so the mapping that LC1 should learn would be more complicated, requiring a network structure with sufficient modeling capability. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/3259cf77091745a9fdd9e194966ae7486dff2e02e913f51864789835618606f4.jpg", + "table_caption": [ + "Table 3: Average test accuracy $( \\% )$ with respect to the update frequency of local critic networks. " + ], + "table_footnote": [], + "table_body": "
Dataset1/11/21/31/41/5
CIFAR-1092.39 ±0.0991.91 ±0.1991.78 ±0.1891.57 ±0.1291.35 ±0.17
CIFAR-10069.91 ±0.5067.99 ±0.4967.76±0.1966.74 ±0.4166.39 ±0.39
", + "bbox": [ + 232, + 127, + 764, + 185 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/fdb18e1e5dfc65e4f01350a070f6d10e7ca9702498c31d531de4ab408b66e5b6.jpg", + "table_caption": [ + "Table 4: Average test accuracy $( \\% )$ of the sub-models produced by local critic training and the networks trained by regular backpropagation. " + ], + "table_footnote": [], + "table_body": "
DatasetBP sub 1LC sub 1BP sub 2LC sub 2BP sub 3LC sub 3
CIFAR-1074.46 ±0.9185.24 ±0.4988.03 ±0.8790.53 ±0.1592.05 ±0.2492.29 ±0.09
CIFAR-10047.58 ±1.1055.39 ±0.5761.79 ±0.9263.62 ±0.3167.81 ±0.2267.54 ±0.70
", + "bbox": [ + 186, + 247, + 810, + 306 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 340, + 823, + 369 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.3 PERIODIC UPDATE OF LOCAL CRITIC NETWORKS ", + "text_level": 1, + "bbox": [ + 174, + 397, + 549, + 411 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "A way to increase the efficiency of the proposed approach is to update the local critic networks not at every iteration but only periodically. This may degrade the accuracy but has two benefits. First, the amount of computation required to update the local networks can be reduced. Second, the burden of the communication between the layer groups also can be reduced. These benefits will be more significant when the local networks have larger sizes. ", + "bbox": [ + 174, + 426, + 825, + 496 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For the default structure shown in Figure 2a, we compare different update frequency in Table 3. It is noticed that the accuracy only slightly decreases as the frequency decreases. When the update frequency is a half of that for the main network (i.e., 1/2), the accuracy drops by $0 . 4 8 \\%$ and $1 . 9 2 \\%$ for the two datasets, respectively. Then, the decrease of the accuracy is only $0 . 5 6 \\%$ for CIFAR-10 and $1 . 6 0 \\%$ for CIFAR-100 when the update frequency decreases from 1/2 to 1/5. ", + "bbox": [ + 174, + 503, + 825, + 574 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.4 STRUCTURAL OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 176, + 602, + 415, + 614 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 4 compares the performance of the sub-models, and Table 5 shows the complexities of the sub-models in terms of the amount of computation for a feedforward pass and the number of weight parameters. A larger network (e.g., sub-model 3) shows better performance than a smaller network (e.g., sub-model 1), which is reasonable due to the difference in learning capability with respect to the model size. The largest sub-model (sub-model 3) shows similar accuracy to the main model $( 9 2 . 2 9 \\%$ vs. $9 2 . 3 9 \\%$ for CIFAR-10 and $6 7 . 5 4 \\%$ vs. $6 9 . 9 1 \\%$ for CIFAR-100), while the complexity is significantly reduced. For CIFAR-10, the computational complexity in terms of the number of floating-point operations (FLOPs) and the memory complexity are reduced to only about $30 \\%$ (15.72 to 4.52 million FLOPs, and 7.87 to 2.26 million parameters), as shown in Table 5. If an absolute accuracy reduction of $1 . 8 6 \\%$ (from $9 2 . 3 9 \\%$ to $9 0 . 5 3 \\%$ ) is allowed by taking sub-model 2, the reduction of complexity is even more remarkable, up to about one ninth. ", + "bbox": [ + 174, + 631, + 825, + 784 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In addition, the table also shows the accuracy of the networks that have the same structures with the sub-models but are trained using regular backpropagation. Surprisingly, such networks do not easily reach accuracy comparable to that of the sub-models obtained by local critic training, particularly for smaller networks (e.g., $7 4 . 4 6 \\%$ vs. $8 5 . 2 4 \\%$ with sub-model 1 for CIFAR-10). We think that joint training of the sub-models in local critic training helps them to find better solutions than those reached by independent regular backpropagation. ", + "bbox": [ + 174, + 791, + 823, + 875 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Therefore, these results demonstrate that a structurally optimized network can be obtained at a cost of a small loss in accuracy by local critic training, which may not be attainable by trial-and-error with backpropagation. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/42c8c15e4eb185c58b8ec31eda0d1782028bcc19ef82380d38450fb51458b24d.jpg", + "table_caption": [ + "Table 5: FLOPs required for a feedforward pass and numbers of model parameters in the sub-models and main model for CIFAR-10. Note that sub-model 2 has less FLOPs and parameters than submodel 1 due to the pooling operation in sub-model 2. " + ], + "table_footnote": [], + "table_body": "
modelFLOP# of parameters
Sub-model 12.85M1.42M
Sub-model 21.76M0.88M
Sub-model 34.52M2.26M
Main model15.72M7.87M
", + "bbox": [ + 346, + 154, + 650, + 241 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/e0c6d938c2626a77c9239fd0f50c564061e33cd3c57ee05503c3ff07fa43e084.jpg", + "table_caption": [ + "Table 6: Average FLOPs and accuracy of progressive inference for test data of CIFAR-10 when the threshold is set to 0.9 or 0.95. " + ], + "table_footnote": [], + "table_body": "
FLOPAccuracy (%)
Progressive inference (0.9)2.90M91.18 ±0.10
Progressive inference (0.95)3.05M91.75 ±0.16
Main model15.72M92.39 ±0.09
", + "bbox": [ + 312, + 299, + 684, + 372 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.5 PROGRESSIVE INFERENCE ", + "text_level": 1, + "bbox": [ + 174, + 404, + 397, + 417 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We apply the progressive inference algorithm shown in Algorithm 1 to the trained default network for CIFAR-10 with the threshold set to 0.9 or 0.95. The results are shown in Table 6. The feedforward pass ends at different sub-models for different test data, and the average FLOPs over all test data are shown. When the threshold is 0.9, with only a slight loss of accuracy $( 9 2 . 3 9 \\%$ to $9 1 . 1 8 \\%$ ), the computational complexity is reduced significantly, which is only $1 8 . 4 5 \\%$ of that of the main model. When the threshold increases to 0.95, the accuracy loss becomes smaller (only $0 . 6 4 \\%$ , while the complexity reduction remains almost the same ( $1 9 . 4 0 \\%$ of the main model’s complexity). ", + "bbox": [ + 174, + 433, + 825, + 530 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.6 ENSEMBLE INFERENCE ", + "text_level": 1, + "bbox": [ + 176, + 554, + 375, + 568 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The results of ensemble inference using the sub-models and main model are shown in Figure 4b. Using an ensemble of the three sub-models, we observe improved classification accuracy $( 9 2 . 6 8 \\%$ and $7 0 . 8 6 \\%$ for the two datasets, respectively) in comparison to the main model. The performance is further enhanced by an ensemble of both the three sub-models and the main model $( 9 2 . 7 9 \\%$ and $7 1 . 8 6 \\%$ . The improvement comes from the complementarity among the models, particularly between the models sharing a smaller number of layers. For instance, we found that sub-model 3 and the main model tend to show coincident classification results for a large portion of test data, so their complementarity is not significant; on the other hand, more data are classified differently by sub-model 1 and the main model, where we mainly observe performance improvement. Instead of the simple summation, there could be a better method to combine the models, which is left for future work. ", + "bbox": [ + 174, + 583, + 825, + 734 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 762, + 318, + 779 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this paper, we proposed the local critic training approach for removing the inter-layer locking constraint in training of deep neural networks. In addition, we proposed three applications of the local critic training method: structural optimization of neural networks, progressive inference, and ensemble classification. It was demonstrated that the proposed method can successfully train CNNs with local critic networks having extremely simple structures. The performance of the method was also evaluated in various aspects, including effects of structures and update intervals of local critic networks and influences of the sizes of layer groups. Finally, it was shown that structural optimization, progressive inference, and ensemble classification can be performed directly using the models trained with the proposed approach without additional procedures. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/7e438144d79f096a85bdf12b0b985cd481458bf96e698456bea49a86fea6ec05.jpg", + "image_caption": [ + "Figure 4: (a) Ensemble inference using the sub-models and main model. (b) Performance of the ensemble inference for an ensemble of the three sub-models $( 1 + 2 + 3 )$ and an ensemble of the submodels and the main model ( $1 + 2 + 3 +$ main). Standard deviation values are also shown as error bars. " + ], + "image_footnote": [], + "bbox": [ + 250, + 104, + 759, + 487 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 568, + 285, + 583 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "M. A. Carreira-Perpinan and W. Wang. Distributed optimization of deeply nested systems. In International Conference on Artificial Intelligence and Statistics (AISTATS), pp. 10–19, Reykjavik, Iceland, 2014. \nC. Cortes, X. Gonzalvo, V. Kuznetsov, M. Mohri, and S. Yang. AdaNet: Adaptive structural learning of artificial neural networks. In International Conference on Machine Learning (ICML), pp. 874– 883, Sydney, Australia, 2017. \nW. M. Czarnecki, S. Osindero, M. Jaderberg, G. Swirszcz, and R. Pascanu. Sobolev training for neural networks. In Advances in Neural Information Processing Systems (NIPS), pp. 4278–4287, Long Beach, CA, 2017a. \nW. M. Czarnecki, G. Swirszcz, M. Jaderberg, S. Osindero, O. Vinyals, and K. Kavukcuoglu. Understanding synthetic gradients and decoupled neural interfaces. In International Conference on Machine Learning (ICML), pp. 904–912, Sydney, Australia, 2017b. \nJ. Deng, W. Dong, R. Socher, L. Li, K. Li, and L. Fei-Fei. ImageNet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition (CVPR), pp. 248–255, Miami Beach, FL, 2009. \nJ. Feng and T. Darrell. Learning the structure of deep convolutional networks. In International Conference on Computer Vision (ICCV), pp. 2749–2757, Santiago, Chile, 2015. \nK. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 770–778, Las Vegas, NV, 2016a. \nK. He, X. Zhang, S. Ren, and J. Sun. Identity mappings in deep residual networks. In European Conference on Computer Vision (ECCV), pp. 630–645, Amsterdam, The Netherlands, 2016b. \nG. Huang, D. Chen, T. Li, F. Wu, L. Maaten, and K.Q. Weinberger. Multi-scale dense networks for resource efficient image classification. In International Conference on Learning Representations (ICLR), Vancouver, Canada, 2018. \nM. Jaderberg, W. M. Czarnecki, S. Osindero, O. Vinyals, A. Graves, D. Silver, and K. Kavukcuoglu. Decoupled neural interfaces using synthetic gradients. In International Conference on Machine Learning (ICML), pp. 1627–1635, Sydney, Australia, 2017. \nN. Kriegeskorte, M. Mur, and P. Bandettini. Representational similarity analysis connecting the branches of systems neuroscience. Frontiers in Systems Neuroscience, 2(4), 2008. \nA. Krizhevsky. Learning multiple layers of features from tiny images. Master’s thesis, Department of Computer Science, University of Toronto, 2009. \nT.-Y. Kwok and D.-Y. Yeung. Constructive algorithms for structure learning in feedforward neural networks for regression problems. IEEE Transactions on Neural Networks, 8(3):630–645, 1997. \nG. Larsson, M. Maire, and G. Shakhnarovich. FractalNet: Ultra-deep neural networks without residuals. In International Conference on Learning Representations (ICLR), Toulon, France, 2017. \nY. LeCun, Y. Bengio, and G. Hinton. Deep learning. Nature, 521:436–444, 2015. \nR. Reed. Pruning algorithms- a survey. IEEE Transactions on Neural Networks, 4(5):730–747, 1993. \nC. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1–9, Boston, MA, 2015. \nG. Taylor, R. Burmeister, Z. Xu, B. Singh, A. Patel, and T. Goldstein. Training neural networks without gradients: A scalable ADMM approach. In International Conference on Machine Learning (ICML), pp. 2722–2731, New York, NY, 2016. \nL. Wang, C. Lee, Z. Tu, and S. Lazebnik. Training deeper convolutional networks with deep supervision. arXiv preprint arXiv:1505.02496v1, 2015. \nB. Zoph and Q.V. Le. Neural architecture search with reinforcement learning. In International Conference on Learning Representations (ICLR), Toulon, France, 2017. ", + "bbox": [ + 171, + 588, + 826, + 925 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 93, + 826, + 646 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 400, + 118 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.1 ADDITIONAL COMPARISON ", + "text_level": 1, + "bbox": [ + 176, + 133, + 405, + 148 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In Czarnecki et al. (2017a), a method to minimize not only the loss of the network output but also its derivative is proposed, called Sobolev training, and applied to the critic training algorithm. We also conduct an experiment to use the Sobolev training method in our proposed algorithm. The results are shown in Table 7. In comparison to the performance shown in Table 1, we do not observe significant difference overall. In addition, we test the deep supervision algorithm (Wang et al., 2015), which also has additional modules connected to intermediate layers. The table shows that its performance is not significantly different from that of backpropagation. ", + "bbox": [ + 173, + 159, + 825, + 257 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/eff15f586805d6f9d141f48d952efec2accefa91cf7b36c9f2fd1474512cf813.jpg", + "table_caption": [ + "Table 7: Average test accuracy $( \\% )$ of Sobolev local critic training (Sob LC), Sobolev critic training (Czarnecki et al., 2017a), and deep supervision (Wang et al., 2015). The numbers of local networks used are shown in the parentheses. The standard deviation values are also shown. " + ], + "table_footnote": [], + "table_body": "
DatasetSob LC (1)Sob LC (3)Sob LC (5)Sob critic (3)Deep supervision (3)
CIFAR-1091.83 ±0.2292.32 ±0.1291.42 ±0.2691.96 ±0.2294.08 ±0.11
CIFAR-10073.64 ±0.2269.58 ±0.3263.82 ±1.0468.83 ±0.3175.09 ±0.29
", + "bbox": [ + 178, + 324, + 820, + 382 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A.2 RESULTS FOR LARGER NETWORKS ", + "text_level": 1, + "bbox": [ + 176, + 406, + 457, + 420 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We examine the effectiveness of the proposed method for larger networks than those used in Section 3. For this, ResNet-50 and ResNet-101 (He et al., 2016a) are trained with backpropagation or the proposed method using three local critic networks. The results shown in Table 8 have a similar trend to those in Table 1 with slight performance improvement in most cases, which confirm that the proposed method works successfully for relatively complex networks. ", + "bbox": [ + 174, + 431, + 825, + 502 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/90c6150351da7ecaadb4bfeef6aca2dcd762509a5a79da2c885aaa5f49c51f4f.jpg", + "table_caption": [ + "Table 8: Test accuracy $( \\% )$ of backpropagation (BP), and local critic training (LC) for ResNet-50 and ResNet-101. The numbers of local networks used are shown in the parentheses. " + ], + "table_footnote": [], + "table_body": "
DatasetResNet-50 BPResNet-50 LC (3)ResNet-101 BPResNet-101 LC (3)
CIFAR-1094.4292.9594.1893.17
CIFAR-10074.8870.5877.2972.72
", + "bbox": [ + 187, + 554, + 810, + 613 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In addition, we experiment using the ImageNet dataset (Deng et al., 2009), which is much larger and more complex than CIFAR-10 and CIFAR-100. The results for ResNet-50 in Table 9 show that the proposed method can also work well for large datasets. ", + "bbox": [ + 176, + 627, + 823, + 670 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/7910141a55c6f8504704f11cc064c5ec7f4a704a083d8f2e7423750db3f18e34.jpg", + "table_caption": [ + "Table 9: Test accuracy $( \\% )$ of ResNet-50 trained with backpropagation (BP) and local critic training (LC) for the ImageNet dataset. " + ], + "table_footnote": [], + "table_body": "
ImageNetResNet-50 BPResNet-50 LC (3)
Top-5 accuracy92.3886.52
Top-1 accuracy75.0765.41
", + "bbox": [ + 310, + 723, + 684, + 781 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B VISUALIZATION OF TRAINED MODELS ", + "text_level": 1, + "bbox": [ + 174, + 808, + 531, + 825 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In order to analyze the trained networks, we obtain the representational dissimilarity matrix (RDM) (Kriegeskorte et al., 2008) from each layer of the networks trained by backpropagation and the proposed method. For each of 400 samples from CIFAR-10, the activation of each layer is recorded, and the correlation between two samples is measured, which is shown in Figure 5. In the figure, clear diagonal-blocks indicate that samples from the same class have highly correlated representations (e.g., the last layer). ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/25c4b4bf1fb8d3e89761a9928ade1dad5be347d56a4fa5ebefcc89ee4de14596.jpg", + "image_caption": [ + "Figure 5: Representation dissimilarity matrix of each layer of the trained networks (having the structure shown in Figure 2a) for label-ordered samples from CIFAR-10. Red lines indicate the locations of local critic networks. A clear diagonal-block pattern indicates that clear inner-class representation has been trained. " + ], + "image_footnote": [], + "bbox": [ + 179, + 98, + 815, + 290 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Overall, block-diagonal patterns become clear at the last layers for all cases. However, the figure also shows that the two training methods result in networks showing qualitatively different characteristics in their internal representations. In particular, the layers at which local critic networks are attached (e.g., layer 5 in LC (1) and LC (3), and layer 6 in LC (5)) show relatively distinguishable blockdiagonal patterns in comparison to those of the network trained by backpropagation. These layers in the proposed method act not only as intermediate layers of the main network but also as near-final layers of the sub-models, and thus are forced to learn class-specific representations to some extent. ", + "bbox": [ + 174, + 383, + 825, + 481 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C LEARNING DYNAMICS ", + "text_level": 1, + "bbox": [ + 176, + 501, + 397, + 517 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C.1 VISUALIZATION OF LOSS EVOLUTION ", + "text_level": 1, + "bbox": [ + 176, + 534, + 475, + 546 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "As a way to investigate the learning dynamics of the proposed method, the loss values at each local critic network $( L _ { i } )$ for individual data over the training iterations are examined (Czarnecki et al., 2017b). Figure 3 visualizes the loss values for 400 sampled data of CIFAR-10 (arranged in 2D) when three local critic networks are used. For visualization, the same results are shown twice, once sorted by labels (Figure 6a) and once sorted by $L _ { 4 }$ at iteration 20000 (Figure 6b). At the early stage of learning, the loss values at the local critic networks ( $L _ { 1 }$ to $L _ { 3 }$ ) are largely different from those at the main network $( L _ { 4 } )$ , with only slight similarity (e.g., the blue region at iteration 50 in Figure 6a). At the later stage, however, all the losses similarly converged to small values for most of the samples (at iteration 20000 in Figure 6b). ", + "bbox": [ + 174, + 559, + 825, + 684 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C.2 LEARNING CURVES OF SUB-MODELS ", + "text_level": 1, + "bbox": [ + 178, + 702, + 470, + 715 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We showed the test performance of sub-models in Table 4. In addition, we show their test accuracy over training iterations for CIFAR-10 in Figure 7. In particular, faster convergence in the case of local critic training than backpropagation is observed in Figure 7a. ", + "bbox": [ + 176, + 727, + 823, + 770 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/4d2ec25d549cecb1875fd17273ba16d4340e95ed2b8cf51f67ab792ad5df43f6.jpg", + "image_caption": [ + "Figure 6: Evolution of the loss value during the course of the proposed local critic training. " + ], + "image_footnote": [], + "bbox": [ + 187, + 316, + 802, + 684 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/86be65d27ac7cde0ef30ef8e27c778c97b88720bda116cd16823e0bacddeb792.jpg", + "image_caption": [ + "(c) Sub-model 3 ", + "Figure 7: Test accuracy $( \\% )$ of sub-models trained by backpropagation and local critic training over training iterations. 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Therefore, training of each layer", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "has dependency on all the other layers, which causes the issue of locking (Jaderberg et al., 2017).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "This is undesirable in some cases, e.g., a system consisting of several interacting models, a model", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 304, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 304, + 502 + ], + "score": 1.0, + "content": "distributed across multiple computing nodes, etc.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 369, + 506, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "There have been attempts to remove the locking constraint. In Carreira-Perpinan & Wang (2014),", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 516, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 505, + 530 + ], + "score": 1.0, + "content": "the method of auxiliary coordinates (MAC) is proposed. It replaces the original loss minimization", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "problem with an equality-constrained optimization problem by introducing an auxiliary variable", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 537, + 506, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 553 + ], + "score": 1.0, + "content": "for each data and each hidden unit. Then, solving the problem is formulated as iteratively solving", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "several sub-problems independently. A similar approach using the alternating direction method of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "multipliers (ADMM) is proposed in Taylor et al. (2016). It also employs an equality-constrained", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "optimization but with different auxiliary variables, so that resulting sub-problems have closed form", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "solutions. However, these methods are not scalable to deep learning architectures such as convolu-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 595, + 234, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 234, + 606 + ], + "score": 1.0, + "content": "tional neural networks (CNNs).", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 505, + 506, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 504, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 504, + 623 + ], + "score": 1.0, + "content": "The method proposed in Jaderberg et al. (2017), called decoupled neural interface (DNI), directly", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "synthesizes estimated error gradients, called synthetic gradients, using an additional small neural", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "network for training a layer’s weight parameters. 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However, this method suffers from performance degradation when compared to regular back-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "propagation (Czarnecki et al., 2017a). The idea of having additional modules supporting the layers", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "of the main model is also adopted in Czarnecki et al. (2017a), where the additional modules are", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "trained to approximate the main model’s outputs instead of error gradients. Due to this, however,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "the method does not resolve the issue of update locking, and in fact, the work does not intend to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 721, + 282, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 282, + 733 + ], + "score": 1.0, + "content": "design a non-sequential learning algorithm.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 611, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 167, + 81, + 443, + 257 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 167, + 81, + 443, + 257 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 167, + 81, + 443, + 257 + ], + "spans": [ + { + "bbox": [ + 167, + 81, + 443, + 257 + ], + "score": 0.964, + "type": "image", + "image_path": "c2f5cb4c44bf704873fe2f46b97b9422343f9e9d1d458c774bb6d5e32319f1f9.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 167, + 81, + 443, + 139.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 167, + 139.66666666666666, + 443, + 198.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 167, + 198.33333333333331, + 443, + 257.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 268, + 505, + 302 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 267, + 504, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 504, + 282 + ], + "score": 1.0, + "content": "Figure 1: Learning processes of DNI (Jaderberg et al., 2017) and the proposed local critic training.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "The black, green, and blue arrows indicate feedforward passes, an error gradient flow, and loss", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 291, + 210, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 210, + 304 + ], + "score": 1.0, + "content": "comparison, respectively.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 321, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 104, + 320, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 104, + 320, + 505, + 335 + ], + "score": 1.0, + "content": "In this paper, we propose a novel approach for non-sequential learning, called local critic training.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 332, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 332, + 505, + 343 + ], + "score": 1.0, + "content": "The key idea is that additional modules besides the main neural network model are employed, which", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "we call local critics, in order to indirectly deliver error gradients to the main model for training with-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "out backpropagation. In other words, a local critic located at a certain layer group is trained in such", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "a way that the derivative of its output serves as the error gradient for training of the corresponding", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "layers’ weight parameters. Thus, the error gradient does not need to be backpropagated, and the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "feedforward operations and gradient-descent learning can be performed independently. Through", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "extensive experiments, we examine the influences of the network structure, update frequency, and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "total number of local critics, which provide not only insight into operation characteristics but also", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 420, + 369, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 369, + 432 + ], + "score": 1.0, + "content": "guidelines for performance optimization of the proposed method.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 436, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "In addition to the capability of implementing training without locking, the proposed approach can be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "exploited for additional important applications. First, we show that applying the proposed method", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "automatically performs structural optimization of neural networks for a given problem, which has", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 505, + 482 + ], + "score": 1.0, + "content": "been a challenging issue in the machine learning field. Second, a progressive inference algorithm", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "using the network trained with the proposed method is presented, which can adaptively reduce the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "computational complexity during the inference process (i.e., test phase) depending on the given data.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "Third, the network trained by the proposed method naturally enables ensemble inference that can", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 515, + 266, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 266, + 525 + ], + "score": 1.0, + "content": "improve the classification performance.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 108, + 540, + 240, + 553 + ], + "lines": [ + { + "bbox": [ + 104, + 539, + 242, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 539, + 242, + 555 + ], + "score": 1.0, + "content": "2 PROPOSED APPROACH", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 564, + 241, + 576 + ], + "lines": [ + { + "bbox": [ + 106, + 564, + 242, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 242, + 577 + ], + "score": 1.0, + "content": "2.1 LOCAL CRITIC TRAINING", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 585, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "The basic idea of the proposed approach is to introduce additional local networks, which we call", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "local critics, besides the main network model, so that they eventually provide estimates of the output", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 620 + ], + "score": 1.0, + "content": "of the main network. 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The", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 490, + 363, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 363, + 502 + ], + "score": 1.0, + "content": "performance of the two methods will be compared in Section 3.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 515, + 254, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 256, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 256, + 528 + ], + "score": 1.0, + "content": "2.2 STRUCTURAL OPTIMIZATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "In many cases, determining an appropriate structure of neural networks for a given problem is not", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 546, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 561 + ], + "score": 1.0, + "content": "straightforward. 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Therefore,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "all the necessary computations in the forward and backward passes can be locally confined. The", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 490, + 363, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 363, + 502 + ], + "score": 1.0, + "content": "performance of the two methods will be compared in Section 3.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 446, + 506, + 502 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 515, + 254, + 527 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 256, + 528 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 256, + 528 + ], + "score": 1.0, + "content": "2.2 STRUCTURAL OPTIMIZATION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 549 + ], + "score": 1.0, + "content": "In many cases, determining an appropriate structure of neural networks for a given problem is not", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 546, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 505, + 561 + ], + "score": 1.0, + "content": "straightforward. This is usually done through trial-and-error, which is extremely time-consuming.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "There have been studies to automate the structural optimization process (Cortes et al., 2017; Feng &", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 568, + 489, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 489, + 583 + ], + "score": 1.0, + "content": "Darrell, 2015; Kwok & Yeung, 1997; Reed, 1993), but this issue still remains very challenging.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 535, + 506, + 583 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 586, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 106, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "In deep learning, the problem of structural optimization is even more critical. 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Although small sub-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "models (e.g., sub-model 1) tend to show low accuracy, they would still perform well for some data.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "For such data, we do not need to perform the full feedforward pass but can take the classification", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 504, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 504, + 407 + ], + "score": 1.0, + "content": "decision by the sub-models. Thus, the basic idea of the progressive inference is to finish infer-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "ence (i.e., classification) with a small sub-model if its confidence on the classification result is high", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 418, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 429 + ], + "score": 1.0, + "content": "enough, instead of completing the full feedforward pass with the main model, which can reduce the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "computational complexity. Here, the softmax outputs for all classes are compared and the maximum", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "probability is used as the confidence level. If it is higher than a threshold, we take the decision by the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "sub-model; otherwise, the feedforward pass continues. The proposed progressive inference method", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 235, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 235, + 473 + ], + "score": 1.0, + "content": "is summarized in Algorithm 12.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 487, + 230, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 231, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 231, + 499 + ], + "score": 1.0, + "content": "2.4 ENSEMBLE INFERENCE", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "In recent deep learning systems, it is popular to use ensemble approaches to improve performance", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "in comparison to single models, where multiple networks are combined for producing final results,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 528, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 506, + 544 + ], + "score": 1.0, + "content": "e.g., He et al. (2016a); Szegedy et al. (2015). The sub-models and main model obtained by applying", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "the proposed local critic training approach can be used for ensemble inference. Figure 4a depicts", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "how the sub-models and the main model can work together to form an ensemble classifier. We take", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 563, + 413, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 413, + 575 + ], + "score": 1.0, + "content": "the simplest way to combine them, i.e., summation of the networks’ outputs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 108, + 591, + 200, + 603 + ], + "lines": [ + { + "bbox": [ + 104, + 589, + 202, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 202, + 605 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "We conduct extensive experiments to examine the performance of the proposed method in various", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "aspects. We use the CIFAR-10 and CIFAR-100 datasets (Krizhevsky, 2009) with data augmentation.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "We employ a VGG-like CNN architecture with batch normalization and ReLU activation functions,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "which is shown in Figure 2a. Note that this structure is the same to that used in Czarnecki et al.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "(2017a). It has three local critic networks, thus four layer groups that can be trained independently", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 175, + 684 + ], + "score": 1.0, + "content": "are formed (i.e.,", + "type": "text" + }, + { + "bbox": [ + 176, + 671, + 198, + 681 + ], + "score": 0.78, + "content": "N { = } 4", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "). 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However, ours does not require", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "particular network structures (such as multi-scale dense network (Huang et al., 2018) or FractalNet (Larsson", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 266, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 266, + 733 + ], + "score": 1.0, + "content": "et al., 2017)) but works with generic CNNs.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 86, + 252, + 97 + ], + "lines": [ + { + "bbox": [ + 107, + 84, + 252, + 99 + ], + "spans": [ + { + "bbox": [ + 107, + 84, + 252, + 99 + ], + "score": 1.0, + "content": "Algorithm 1: Progressive inference", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "list", + "bbox": [ + 110, + 99, + 296, + 230 + ], + "lines": [ + { + "bbox": [ + 115, + 100, + 210, + 109 + ], + "spans": [ + { + "bbox": [ + 115, + 100, + 160, + 109 + ], + "score": 1.0, + "content": "Input: data", + "type": "text" + }, + { + "bbox": [ + 160, + 101, + 166, + 108 + ], + "score": 0.62, + "content": "_ x", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 100, + 205, + 109 + ], + "score": 1.0, + "content": ", threshold", + "type": "text" + }, + { + "bbox": [ + 205, + 100, + 210, + 108 + ], + "score": 0.54, + "content": "t", + "type": "inline_equation" + } + ], + "index": 1, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 114, + 109, + 250, + 120 + ], + "spans": [ + { + "bbox": [ + 114, + 109, + 185, + 120 + ], + "score": 1.0, + "content": "Model: sub-model", + "type": "text" + }, + { + "bbox": [ + 186, + 110, + 194, + 119 + ], + "score": 0.83, + "content": "c _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 109, + 243, + 120 + ], + "score": 1.0, + "content": ", main-model", + "type": "text" + }, + { + "bbox": [ + 243, + 109, + 250, + 119 + ], + "score": 0.72, + "content": "f", + "type": "inline_equation" + } + ], + "index": 2, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 115, + 119, + 231, + 129 + ], + "spans": [ + { + "bbox": [ + 115, + 119, + 211, + 129 + ], + "score": 1.0, + "content": "Initialize: classif ication", + "type": "text" + }, + { + "bbox": [ + 212, + 119, + 228, + 128 + ], + "score": 0.53, + "content": "= 0", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 119, + 231, + 129 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 115, + 129, + 200, + 140 + ], + "spans": [ + { + "bbox": [ + 115, + 129, + 128, + 140 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 129, + 129, + 150, + 138 + ], + "score": 0.89, + "content": "i = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 129, + 161, + 140 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 161, + 129, + 186, + 138 + ], + "score": 0.86, + "content": "N - 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Among the sub-models, we can choose one as a structure-optimized", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 266, + 473, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 473, + 279 + ], + "score": 1.0, + "content": "network by considering the trade-off relationship between the complexity and performance.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 106, + 255, + 505, + 279 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 506, + 297 + ], + "score": 1.0, + "content": "It is worth mentioning that our structural optimization approach can be performed instantly after", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 505, + 307 + ], + "score": 1.0, + "content": "training of the model, whereas many existing methods for structural optimization require iterative", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 306, + 274, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 274, + 318 + ], + "score": 1.0, + "content": "search processes, e.g., Zoph & Le (2017).", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 282, + 506, + 318 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 331, + 242, + 342 + ], + "lines": [ + { + "bbox": [ + 106, + 331, + 244, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 244, + 343 + ], + "score": 1.0, + "content": "2.3 PROGRESSIVE INFERENCE", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 351, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "We propose another simple but effective way to utilize the sub-models obtained by the proposed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "approach for computational efficiency, which we call progressive inference. Although small sub-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "models (e.g., sub-model 1) tend to show low accuracy, they would still perform well for some data.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "For such data, we do not need to perform the full feedforward pass but can take the classification", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 396, + 504, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 504, + 407 + ], + "score": 1.0, + "content": "decision by the sub-models. Thus, the basic idea of the progressive inference is to finish infer-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "ence (i.e., classification) with a small sub-model if its confidence on the classification result is high", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 418, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 429 + ], + "score": 1.0, + "content": "enough, instead of completing the full feedforward pass with the main model, which can reduce the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 441 + ], + "score": 1.0, + "content": "computational complexity. Here, the softmax outputs for all classes are compared and the maximum", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "probability is used as the confidence level. If it is higher than a threshold, we take the decision by the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "sub-model; otherwise, the feedforward pass continues. The proposed progressive inference method", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 235, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 235, + 473 + ], + "score": 1.0, + "content": "is summarized in Algorithm 12.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 351, + 506, + 473 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 487, + 230, + 498 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 231, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 231, + 499 + ], + "score": 1.0, + "content": "2.4 ENSEMBLE INFERENCE", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 505, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "In recent deep learning systems, it is popular to use ensemble approaches to improve performance", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "in comparison to single models, where multiple networks are combined for producing final results,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 528, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 506, + 544 + ], + "score": 1.0, + "content": "e.g., He et al. (2016a); Szegedy et al. (2015). The sub-models and main model obtained by applying", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "the proposed local critic training approach can be used for ensemble inference. Figure 4a depicts", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "how the sub-models and the main model can work together to form an ensemble classifier. We take", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 563, + 413, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 413, + 575 + ], + "score": 1.0, + "content": "the simplest way to combine them, i.e., summation of the networks’ outputs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 507, + 506, + 575 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 591, + 200, + 603 + ], + "lines": [ + { + "bbox": [ + 104, + 589, + 202, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 202, + 605 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 616, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "We conduct extensive experiments to examine the performance of the proposed method in various", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "aspects. We use the CIFAR-10 and CIFAR-100 datasets (Krizhevsky, 2009) with data augmentation.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "We employ a VGG-like CNN architecture with batch normalization and ReLU activation functions,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "which is shown in Figure 2a. Note that this structure is the same to that used in Czarnecki et al.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "(2017a). It has three local critic networks, thus four layer groups that can be trained independently", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 175, + 684 + ], + "score": 1.0, + "content": "are formed (i.e.,", + "type": "text" + }, + { + "bbox": [ + 176, + 671, + 198, + 681 + ], + "score": 0.78, + "content": "N { = } 4", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "). The local critic networks are also CNNs, and their structures are kept as", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "simple as possible in order to minimize the computational complexity for computing the estimated", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 442, + 217, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 217, + 455 + ], + "score": 1.0, + "content": "error gradient given by (5).", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 615, + 505, + 684 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 109, + 80, + 500, + 176 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 80, + 500, + 176 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 80, + 500, + 176 + ], + "spans": [ + { + "bbox": [ + 109, + 80, + 500, + 176 + ], + "score": 0.966, + "type": "image", + "image_path": "e7fc0d788432248be4ca47ca6286fe703a2b1d4b34dc6032ad1b70417cbdc6d7.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 109, + 80, + 500, + 112.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 109, + 112.0, + 500, + 144.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 109, + 144.0, + 500, + 176.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 185, + 505, + 230 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 505, + 198 + ], + "score": 1.0, + "content": "Figure 2: (a) Network structure of the proposed approach using three local networks for CIFAR-10.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 104, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "LC1, LC2, and LC3 are local critic networks, each of which contains one convolutional layer. For", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "score": 1.0, + "content": "CIFAR-100, the final fc10 layers of the main network and the local critic networks are replaced with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 218, + 410, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 410, + 231 + ], + "score": 1.0, + "content": "fc100. (b) Sub-models and main model obtained by the proposed approach.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "image", + "bbox": [ + 115, + 245, + 498, + 376 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 245, + 498, + 376 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 115, + 245, + 498, + 376 + ], + "spans": [ + { + "bbox": [ + 115, + 245, + 498, + 376 + ], + "score": 0.966, + "type": "image", + "image_path": "cf0f00f590a2351278a4da197976c450ddbde9945826dacb31d04b73235d090e.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 115, + 245, + 498, + 288.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 115, + 288.6666666666667, + 498, + 332.33333333333337 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 115, + 332.33333333333337, + 498, + 376.00000000000006 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 386, + 504, + 408 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 104, + 383, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 383, + 506, + 401 + ], + "score": 1.0, + "content": "Figure 3: Training loss values of the main model and each sub-model with respect to the training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 396, + 146, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 146, + 409 + ], + "score": 1.0, + "content": "iteration.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + } + ], + "index": 9.25 + }, + { + "type": "text", + "bbox": [ + 108, + 431, + 504, + 454 + ], + "lines": [ + { + "bbox": [ + 106, + 431, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 444 + ], + "score": 1.0, + "content": "simple as possible in order to minimize the computational complexity for computing the estimated", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 442, + 217, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 217, + 455 + ], + "score": 1.0, + "content": "error gradient given by (5).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 106, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "We use the stochastic gradient descent with a momentum of 0.9 for the main network and the Adam", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 280, + 482 + ], + "score": 1.0, + "content": "optimization with a fixed learning rate of", + "type": "text" + }, + { + "bbox": [ + 280, + 469, + 302, + 480 + ], + "score": 0.91, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 469, + 506, + 482 + ], + "score": 1.0, + "content": "for the local networks. The L2 regularization is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 479, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 148, + 495 + ], + "score": 1.0, + "content": "used with", + "type": "text" + }, + { + "bbox": [ + 149, + 480, + 188, + 492 + ], + "score": 0.92, + "content": "5 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 479, + 505, + 495 + ], + "score": 1.0, + "content": "for the main network. For the loss functions in (3) and (7), the cross-entropy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "and the L1 loss are used, respectively, which is determined empirically. The batch size is set to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "128, and the maximum training iteration is set to 80,000. The learning rate for the main network", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "is initialized to 0.1 and dropped by an order of magnitude after 40,000 and 60,000 iterations. The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "Xavier method is used for initialization of the network parameters. All experiments are performed", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 548 + ], + "score": 1.0, + "content": "using TensorFlow. We conduct all the experiments five times with different random seeds and report", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 547, + 194, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 194, + 561 + ], + "score": 1.0, + "content": "the average accuracy.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 573, + 254, + 584 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 255, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 255, + 586 + ], + "score": 1.0, + "content": "3.1 PERFORMANCE EVALUATION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 481, + 606 + ], + "score": 1.0, + "content": "Figure 3 shows how the loss values of the main network and each local critic network, i.e.,", + "type": "text" + }, + { + "bbox": [ + 482, + 594, + 493, + 605 + ], + "score": 0.88, + "content": "L _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "(3), evolve with respect to the training iteration. The graphs show that the local critic networks suc-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 614, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 631 + ], + "score": 1.0, + "content": "cessfully learn to approximate the main network’s loss with high accuracy during the whole training", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 386, + 641 + ], + "score": 1.0, + "content": "process. The local critic network farthest from the output side (i.e.,", + "type": "text" + }, + { + "bbox": [ + 386, + 627, + 399, + 638 + ], + "score": 0.84, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 626, + 505, + 641 + ], + "score": 1.0, + "content": ") shows larger loss values", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 637, + 496, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 496, + 651 + ], + "score": 1.0, + "content": "than the others, which is due to the inaccuracy accumulated through the cascaded approximation.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "The classification performance of the proposed local critic training approach is evaluated in Table", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "1. For comparison, the performance of the regular backpropagation, DNI (Jaderberg et al., 2017),", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "and critic training (Czarnecki et al., 2017a) is also evaluated. Although the critic training method", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "is not for removing update locking, we include its result because it shares some similarity with our", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "approach, i.e., additional modules to estimate the main network’s output. In all three methods, each", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "additional module is composed of a convolutional layer and an output layer. In the case of the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "proposed method, we test different numbers of local critic networks. 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For", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 220 + ], + "score": 1.0, + "content": "CIFAR-100, the final fc10 layers of the main network and the local critic networks are replaced with", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 218, + 410, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 410, + 231 + ], + "score": 1.0, + "content": "fc100. 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The L2 regularization is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 479, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 148, + 495 + ], + "score": 1.0, + "content": "used with", + "type": "text" + }, + { + "bbox": [ + 149, + 480, + 188, + 492 + ], + "score": 0.92, + "content": "5 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 479, + 505, + 495 + ], + "score": 1.0, + "content": "for the main network. For the loss functions in (3) and (7), the cross-entropy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "and the L1 loss are used, respectively, which is determined empirically. The batch size is set to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 505, + 515 + ], + "score": 1.0, + "content": "128, and the maximum training iteration is set to 80,000. The learning rate for the main network", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "is initialized to 0.1 and dropped by an order of magnitude after 40,000 and 60,000 iterations. The", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "Xavier method is used for initialization of the network parameters. 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The graphs show that the local critic networks suc-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 614, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 505, + 631 + ], + "score": 1.0, + "content": "cessfully learn to approximate the main network’s loss with high accuracy during the whole training", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 386, + 641 + ], + "score": 1.0, + "content": "process. 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For comparison, the performance of the regular backpropagation, DNI (Jaderberg et al., 2017),", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "and critic training (Czarnecki et al., 2017a) is also evaluated. Although the critic training method", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "is not for removing update locking, we include its result because it shares some similarity with our", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "approach, i.e., additional modules to estimate the main network’s output. 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The numbers of local", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 471, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 471, + 115 + ], + "score": 1.0, + "content": "networks used are shown in the parentheses. The standard deviation values are also shown.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 113, + 122, + 497, + 169 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 122, + 497, + 169 + ], + "spans": [ + { + "bbox": [ + 113, + 122, + 497, + 169 + ], + "score": 0.976, + "html": "
DatasetBPDNI (3)Critic (3)LC(1)LC (3)LC (5)
CIFAR-1093.93 ±0.2064.86 ±0.4291.92 ±0.3092.06 ±0.2092.39 ±0.0991.38 ±0.20
CIFAR-10075.14 ±0.1836.53 ±0.6469.07 ±0.2573.61 ±0.3169.91 ±0.5063.53 ±0.24
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Dataset[1,1,1] (default)[3,3,3][5,5,5][3,2,1][1,2,3][5,4,3][3,4,5]
CIFAR-1092.39 ±0.0992.36 ±0.2291.72 ±0.1992.07 ±0.2192.20 ±0.1292.10 ±0.1691.90 ±0.16
CIFAR-10069.91 ±0.5070.02 ±0.2970.34 ±0.1670.06 ±0.6469.81 ±0.3370.87 ±0.4069.93 ±0.56
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When five local networks are used, they are placed after every two layers of the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 310, + 167, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 167, + 321 + ], + "score": 1.0, + "content": "main network.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 470 + ], + "lines": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "When compared to the result of backpropagation, the proposed approach successfully decouples", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "training of the layer groups at a small expense of accuracy decrease (note that the performance of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 348, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 506, + 363 + ], + "score": 1.0, + "content": "the proposed method can be made closer to that of backpropagation using different structures, as", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 359, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 373 + ], + "score": 1.0, + "content": "will be shown in Tables 2 and Figure 4b). The degradation of the accuracy and standard deviation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "of our method is larger for CIFAR-100, which implies that the influence of gradient estimation is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "larger for more complex problems. When more local critic networks are used, the accuracy tends", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "to decrease more due to higher reliance on predicted gradients rather than true gradients, while", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 403, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 505, + 418 + ], + "score": 1.0, + "content": "more layer groups can be trained independently. Thus, there exists a trade-off between the accuracy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "and unlocking effect. The DNI method shows poor performance as in (Czarnecki et al., 2017a).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 347, + 440 + ], + "score": 1.0, + "content": "The proposed method shows performance improvement by", + "type": "text" + }, + { + "bbox": [ + 347, + 426, + 370, + 436 + ], + "score": 0.87, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 424, + 388, + 440 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 389, + 426, + 411, + 436 + ], + "score": 0.88, + "content": "0 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 424, + 505, + 440 + ], + "score": 1.0, + "content": "over the critic training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "method, both with three local networks, for the two datasets, respectively, which are found to be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "statistically significant using Mann-Whitney tests at a significance level of 0.05. This shows the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 459, + 419, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 419, + 470 + ], + "score": 1.0, + "content": "efficacy of the cascaded learning scheme of the local networks in our method.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 109, + 489, + 315, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 317, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 317, + 502 + ], + "score": 1.0, + "content": "3.2 STRUCTURES OF LOCAL CRITIC NETWORKS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "We examine the influence of the structures of the local critic networks in our method. Two aspects", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "are considered, one about the influence of the overall complexity of the local networks and the other", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "about the relative complexities of the local networks for good performance. For this, we change", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "the number of convolutional layers in each local critic network, while keeping the other structural", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 200, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 200, + 568 + ], + "score": 1.0, + "content": "parameters unchanged.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "The results for various structure combinations of the three local critic networks are shown in Table", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 594 + ], + "score": 1.0, + "content": "2. As the number of convolutional layers increase for all local networks (the first three cases in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 370, + 606 + ], + "score": 1.0, + "content": "the table), the accuracy for CIFAR-100 slightly increases from", + "type": "text" + }, + { + "bbox": [ + 370, + 594, + 403, + 605 + ], + "score": 0.87, + "content": "6 9 . 9 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "(with one convolutional", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 144, + 617 + ], + "score": 1.0, + "content": "layer) to", + "type": "text" + }, + { + "bbox": [ + 144, + 605, + 176, + 615 + ], + "score": 0.88, + "content": "7 0 . 0 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 604, + 309, + 617 + ], + "score": 1.0, + "content": "(three convolutional layers) and", + "type": "text" + }, + { + "bbox": [ + 309, + 605, + 342, + 615 + ], + "score": 0.88, + "content": "7 0 . 3 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "(five convolutional layers), whereas for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "CIFAR-10 the accuracy slightly decreases when five convolutional layers are used. A more complex", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "local network can learn better the target input-output relationship, which leads to the performance", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "improvement for CIFAR-100. For CIFAR-10, on the other hand, the network structure with five", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "convolutional layers seems too complex compared to the data to learn, which causes the performance", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 658, + 131, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 131, + 674 + ], + "score": 1.0, + "content": "drop.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Next, the numbers of layers of the local networks are adjusted differently in order to investigate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "which local networks should be more complex for good performance. The results are shown in", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "the last four columns of Table 2. Overall, it is more desirable to use more complex structures for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "the local networks closer to the input side of the main model. For instance, LC1 and LC3 are", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 267, + 733 + ], + "score": 1.0, + "content": "supposed to learn the relationship from", + "type": "text" + }, + { + "bbox": [ + 267, + 721, + 278, + 732 + ], + "score": 0.88, + "content": "h _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 720, + 290, + 733 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 291, + 721, + 302, + 732 + ], + "score": 0.88, + "content": "h _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 720, + 361, + 733 + ], + "score": 1.0, + "content": "and that from", + "type": "text" + }, + { + "bbox": [ + 361, + 721, + 372, + 732 + ], + "score": 0.88, + "content": "h _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 720, + 384, + 733 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 385, + 721, + 396, + 732 + ], + "score": 0.87, + "content": "h _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ", respectively. 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DatasetBPDNI (3)Critic (3)LC(1)LC (3)LC (5)
CIFAR-1093.93 ±0.2064.86 ±0.4291.92 ±0.3092.06 ±0.2092.39 ±0.0991.38 ±0.20
CIFAR-10075.14 ±0.1836.53 ±0.6469.07 ±0.2573.61 ±0.3169.91 ±0.5063.53 ±0.24
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Dataset[1,1,1] (default)[3,3,3][5,5,5][3,2,1][1,2,3][5,4,3][3,4,5]
CIFAR-1092.39 ±0.0992.36 ±0.2291.72 ±0.1992.07 ±0.2192.20 ±0.1292.10 ±0.1691.90 ±0.16
CIFAR-10069.91 ±0.5070.02 ±0.2970.34 ±0.1670.06 ±0.6469.81 ±0.3370.87 ±0.4069.93 ±0.56
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The degradation of the accuracy and standard deviation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "of our method is larger for CIFAR-100, which implies that the influence of gradient estimation is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "larger for more complex problems. When more local critic networks are used, the accuracy tends", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "to decrease more due to higher reliance on predicted gradients rather than true gradients, while", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 403, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 505, + 418 + ], + "score": 1.0, + "content": "more layer groups can be trained independently. Thus, there exists a trade-off between the accuracy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "and unlocking effect. The DNI method shows poor performance as in (Czarnecki et al., 2017a).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 424, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 347, + 440 + ], + "score": 1.0, + "content": "The proposed method shows performance improvement by", + "type": "text" + }, + { + "bbox": [ + 347, + 426, + 370, + 436 + ], + "score": 0.87, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 424, + 388, + 440 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 389, + 426, + 411, + 436 + ], + "score": 0.88, + "content": "0 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 424, + 505, + 440 + ], + "score": 1.0, + "content": "over the critic training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "method, both with three local networks, for the two datasets, respectively, which are found to be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "statistically significant using Mann-Whitney tests at a significance level of 0.05. This shows the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 459, + 419, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 419, + 470 + ], + "score": 1.0, + "content": "efficacy of the cascaded learning scheme of the local networks in our method.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 326, + 506, + 470 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 489, + 315, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 317, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 317, + 502 + ], + "score": 1.0, + "content": "3.2 STRUCTURES OF LOCAL CRITIC NETWORKS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 511, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "We examine the influence of the structures of the local critic networks in our method. Two aspects", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "are considered, one about the influence of the overall complexity of the local networks and the other", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "about the relative complexities of the local networks for good performance. For this, we change", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "the number of convolutional layers in each local critic network, while keeping the other structural", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 200, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 200, + 568 + ], + "score": 1.0, + "content": "parameters unchanged.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 511, + 506, + 568 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "The results for various structure combinations of the three local critic networks are shown in Table", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 583, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 594 + ], + "score": 1.0, + "content": "2. As the number of convolutional layers increase for all local networks (the first three cases in", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 370, + 606 + ], + "score": 1.0, + "content": "the table), the accuracy for CIFAR-100 slightly increases from", + "type": "text" + }, + { + "bbox": [ + 370, + 594, + 403, + 605 + ], + "score": 0.87, + "content": "6 9 . 9 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "(with one convolutional", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 144, + 617 + ], + "score": 1.0, + "content": "layer) to", + "type": "text" + }, + { + "bbox": [ + 144, + 605, + 176, + 615 + ], + "score": 0.88, + "content": "7 0 . 0 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 604, + 309, + 617 + ], + "score": 1.0, + "content": "(three convolutional layers) and", + "type": "text" + }, + { + "bbox": [ + 309, + 605, + 342, + 615 + ], + "score": 0.88, + "content": "7 0 . 3 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "(five convolutional layers), whereas for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "CIFAR-10 the accuracy slightly decreases when five convolutional layers are used. A more complex", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "local network can learn better the target input-output relationship, which leads to the performance", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "improvement for CIFAR-100. For CIFAR-10, on the other hand, the network structure with five", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "convolutional layers seems too complex compared to the data to learn, which causes the performance", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 658, + 131, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 131, + 674 + ], + "score": 1.0, + "content": "drop.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 572, + 505, + 674 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "Next, the numbers of layers of the local networks are adjusted differently in order to investigate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "which local networks should be more complex for good performance. The results are shown in", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "the last four columns of Table 2. Overall, it is more desirable to use more complex structures for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "the local networks closer to the input side of the main model. For instance, LC1 and LC3 are", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 267, + 733 + ], + "score": 1.0, + "content": "supposed to learn the relationship from", + "type": "text" + }, + { + "bbox": [ + 267, + 721, + 278, + 732 + ], + "score": 0.88, + "content": "h _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 720, + 290, + 733 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 291, + 721, + 302, + 732 + ], + "score": 0.88, + "content": "h _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 720, + 361, + 733 + ], + "score": 1.0, + "content": "and that from", + "type": "text" + }, + { + "bbox": [ + 361, + 721, + 372, + 732 + ], + "score": 0.88, + "content": "h _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 720, + 384, + 733 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 385, + 721, + 396, + 732 + ], + "score": 0.87, + "content": "h _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ", respectively. More layers", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 181, + 282 + ], + "score": 1.0, + "content": "are involved from", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 182, + 271, + 193, + 281 + ], + "score": 0.88, + "content": "h _ { 1 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 194, + 270, + 205, + 282 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 206, + 271, + 218, + 281 + ], + "score": 0.88, + "content": "h _ { 4 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 218, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "in the main network, so the mapping that LC1 should learn would be", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 281, + 444, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 444, + 294 + ], + "score": 1.0, + "content": "more complicated, requiring a network structure with sufficient modeling capability.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 676, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 142, + 101, + 468, + 147 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 109, + 80, + 497, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 78, + 499, + 94 + ], + "spans": [ + { + "bbox": [ + 111, + 78, + 237, + 94 + ], + "score": 1.0, + "content": "Table 3: Average test accuracy", + "type": "text" + }, + { + "bbox": [ + 238, + 81, + 253, + 91 + ], + "score": 0.74, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 78, + 499, + 94 + ], + "score": 1.0, + "content": "with respect to the update frequency of local critic networks.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 142, + 101, + 468, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 142, + 101, + 468, + 147 + ], + "spans": [ + { + "bbox": [ + 142, + 101, + 468, + 147 + ], + "score": 0.977, + "html": "
Dataset1/11/21/31/41/5
CIFAR-1092.39 ±0.0991.91 ±0.1991.78 ±0.1891.57 ±0.1291.35 ±0.17
CIFAR-10069.91 ±0.5067.99 ±0.4967.76±0.1966.74 ±0.4166.39 ±0.39
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DatasetBP sub 1LC sub 1BP sub 2LC sub 2BP sub 3LC sub 3
CIFAR-1074.46 ±0.9185.24 ±0.4988.03 ±0.8790.53 ±0.1592.05 ±0.2492.29 ±0.09
CIFAR-10047.58 ±1.1055.39 ±0.5761.79 ±0.9263.62 ±0.3167.81 ±0.2267.54 ±0.70
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This may degrade the accuracy but has two benefits. First, the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 361, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 372 + ], + "score": 1.0, + "content": "amount of computation required to update the local networks can be reduced. Second, the burden", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "of the communication between the layer groups also can be reduced. These benefits will be more", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 382, + 321, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 321, + 396 + ], + "score": 1.0, + "content": "significant when the local networks have larger sizes.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "For the default structure shown in Figure 2a, we compare different update frequency in Table 3. It", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "is noticed that the accuracy only slightly decreases as the frequency decreases. When the update", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 421, + 504, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 431, + 434 + ], + "score": 1.0, + "content": "frequency is a half of that for the main network (i.e., 1/2), the accuracy drops by", + "type": "text" + }, + { + "bbox": [ + 432, + 421, + 459, + 432 + ], + "score": 0.87, + "content": "0 . 4 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 421, + 477, + 434 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 421, + 504, + 432 + ], + "score": 0.85, + "content": "1 . 9 2 \\%", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 417, + 444 + ], + "score": 1.0, + "content": "for the two datasets, respectively. Then, the decrease of the accuracy is only", + "type": "text" + }, + { + "bbox": [ + 417, + 433, + 444, + 443 + ], + "score": 0.87, + "content": "0 . 5 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "for CIFAR-10", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 442, + 430, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 123, + 455 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 443, + 150, + 454 + ], + "score": 0.85, + "content": "1 . 6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 442, + 430, + 455 + ], + "score": 1.0, + "content": "for CIFAR-100 when the update frequency decreases from 1/2 to 1/5.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 108, + 477, + 254, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 255, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 255, + 489 + ], + "score": 1.0, + "content": "3.4 STRUCTURAL OPTIMIZATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "Table 4 compares the performance of the sub-models, and Table 5 shows the complexities of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 504, + 523 + ], + "score": 1.0, + "content": "sub-models in terms of the amount of computation for a feedforward pass and the number of weight", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "parameters. A larger network (e.g., sub-model 3) shows better performance than a smaller network", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "(e.g., sub-model 1), which is reasonable due to the difference in learning capability with respect to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "the model size. The largest sub-model (sub-model 3) shows similar accuracy to the main model", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 108, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 108, + 555, + 142, + 566 + ], + "score": 0.86, + "content": "( 9 2 . 2 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 555, + 159, + 567 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 159, + 555, + 192, + 566 + ], + "score": 0.88, + "content": "9 2 . 3 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 555, + 270, + 567 + ], + "score": 1.0, + "content": "for CIFAR-10 and", + "type": "text" + }, + { + "bbox": [ + 271, + 555, + 303, + 566 + ], + "score": 0.88, + "content": "6 7 . 5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 555, + 320, + 567 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 320, + 555, + 352, + 566 + ], + "score": 0.88, + "content": "6 9 . 9 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "for CIFAR-100), while the complex-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "ity is significantly reduced. For CIFAR-10, the computational complexity in terms of the number", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 577, + 504, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 484, + 590 + ], + "score": 1.0, + "content": "of floating-point operations (FLOPs) and the memory complexity are reduced to only about", + "type": "text" + }, + { + "bbox": [ + 484, + 577, + 504, + 588 + ], + "score": 0.86, + "content": "30 \\%", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "(15.72 to 4.52 million FLOPs, and 7.87 to 2.26 million parameters), as shown in Table 5. If an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 232, + 612 + ], + "score": 1.0, + "content": "absolute accuracy reduction of", + "type": "text" + }, + { + "bbox": [ + 232, + 599, + 259, + 610 + ], + "score": 0.86, + "content": "1 . 8 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 598, + 285, + 612 + ], + "score": 1.0, + "content": "(from", + "type": "text" + }, + { + "bbox": [ + 285, + 599, + 318, + 610 + ], + "score": 0.86, + "content": "9 2 . 3 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 598, + 329, + 612 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 330, + 599, + 362, + 610 + ], + "score": 0.87, + "content": "9 0 . 5 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 598, + 505, + 612 + ], + "score": 1.0, + "content": ") is allowed by taking sub-model 2,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 610, + 411, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 411, + 623 + ], + "score": 1.0, + "content": "the reduction of complexity is even more remarkable, up to about one ninth.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "In addition, the table also shows the accuracy of the networks that have the same structures with the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "sub-models but are trained using regular backpropagation. Surprisingly, such networks do not easily", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "reach accuracy comparable to that of the sub-models obtained by local critic training, particularly", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 218, + 672 + ], + "score": 1.0, + "content": "for smaller networks (e.g.,", + "type": "text" + }, + { + "bbox": [ + 218, + 660, + 250, + 671 + ], + "score": 0.87, + "content": "7 4 . 4 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 659, + 269, + 672 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 270, + 660, + 302, + 671 + ], + "score": 0.88, + "content": "8 5 . 2 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "with sub-model 1 for CIFAR-10). We think that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "joint training of the sub-models in local critic training helps them to find better solutions than those", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 681, + 305, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 305, + 694 + ], + "score": 1.0, + "content": "reached by independent regular backpropagation.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "Therefore, these results demonstrate that a structurally optimized network can be obtained at a cost", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "of a small loss in accuracy by local critic training, which may not be attainable by trial-and-error", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 198, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 198, + 733 + ], + "score": 1.0, + "content": "with backpropagation.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + } + ], + "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 2019", + "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": "table", + "bbox": [ + 142, + 101, + 468, + 147 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 109, + 80, + 497, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 78, + 499, + 94 + ], + "spans": [ + { + "bbox": [ + 111, + 78, + 237, + 94 + ], + "score": 1.0, + "content": "Table 3: Average test accuracy", + "type": "text" + }, + { + "bbox": [ + 238, + 81, + 253, + 91 + ], + "score": 0.74, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 78, + 499, + 94 + ], + "score": 1.0, + "content": "with respect to the update frequency of local critic networks.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 142, + 101, + 468, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 142, + 101, + 468, + 147 + ], + "spans": [ + { + "bbox": [ + 142, + 101, + 468, + 147 + ], + "score": 0.977, + "html": "
Dataset1/11/21/31/41/5
CIFAR-1092.39 ±0.0991.91 ±0.1991.78 ±0.1891.57 ±0.1291.35 ±0.17
CIFAR-10069.91 ±0.5067.99 ±0.4967.76±0.1966.74 ±0.4166.39 ±0.39
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DatasetBP sub 1LC sub 1BP sub 2LC sub 2BP sub 3LC sub 3
CIFAR-1074.46 ±0.9185.24 ±0.4988.03 ±0.8790.53 ±0.1592.05 ±0.2492.29 ±0.09
CIFAR-10047.58 ±1.1055.39 ±0.5761.79 ±0.9263.62 ±0.3167.81 ±0.2267.54 ±0.70
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This may degrade the accuracy but has two benefits. First, the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 361, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 372 + ], + "score": 1.0, + "content": "amount of computation required to update the local networks can be reduced. Second, the burden", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "of the communication between the layer groups also can be reduced. These benefits will be more", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 382, + 321, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 321, + 396 + ], + "score": 1.0, + "content": "significant when the local networks have larger sizes.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 339, + 505, + 396 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 399, + 505, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 412 + ], + "score": 1.0, + "content": "For the default structure shown in Figure 2a, we compare different update frequency in Table 3. It", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "is noticed that the accuracy only slightly decreases as the frequency decreases. When the update", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 421, + 504, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 431, + 434 + ], + "score": 1.0, + "content": "frequency is a half of that for the main network (i.e., 1/2), the accuracy drops by", + "type": "text" + }, + { + "bbox": [ + 432, + 421, + 459, + 432 + ], + "score": 0.87, + "content": "0 . 4 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 421, + 477, + 434 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 421, + 504, + 432 + ], + "score": 0.85, + "content": "1 . 9 2 \\%", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 417, + 444 + ], + "score": 1.0, + "content": "for the two datasets, respectively. Then, the decrease of the accuracy is only", + "type": "text" + }, + { + "bbox": [ + 417, + 433, + 444, + 443 + ], + "score": 0.87, + "content": "0 . 5 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "for CIFAR-10", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 442, + 430, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 123, + 455 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 443, + 150, + 454 + ], + "score": 0.85, + "content": "1 . 6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 442, + 430, + 455 + ], + "score": 1.0, + "content": "for CIFAR-100 when the update frequency decreases from 1/2 to 1/5.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 398, + 506, + 455 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 477, + 254, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 475, + 255, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 255, + 489 + ], + "score": 1.0, + "content": "3.4 STRUCTURAL OPTIMIZATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "Table 4 compares the performance of the sub-models, and Table 5 shows the complexities of the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 511, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 504, + 523 + ], + "score": 1.0, + "content": "sub-models in terms of the amount of computation for a feedforward pass and the number of weight", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "parameters. A larger network (e.g., sub-model 3) shows better performance than a smaller network", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "(e.g., sub-model 1), which is reasonable due to the difference in learning capability with respect to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "the model size. The largest sub-model (sub-model 3) shows similar accuracy to the main model", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 108, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 108, + 555, + 142, + 566 + ], + "score": 0.86, + "content": "( 9 2 . 2 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 555, + 159, + 567 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 159, + 555, + 192, + 566 + ], + "score": 0.88, + "content": "9 2 . 3 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 555, + 270, + 567 + ], + "score": 1.0, + "content": "for CIFAR-10 and", + "type": "text" + }, + { + "bbox": [ + 271, + 555, + 303, + 566 + ], + "score": 0.88, + "content": "6 7 . 5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 555, + 320, + 567 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 320, + 555, + 352, + 566 + ], + "score": 0.88, + "content": "6 9 . 9 1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "for CIFAR-100), while the complex-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "ity is significantly reduced. For CIFAR-10, the computational complexity in terms of the number", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 577, + 504, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 484, + 590 + ], + "score": 1.0, + "content": "of floating-point operations (FLOPs) and the memory complexity are reduced to only about", + "type": "text" + }, + { + "bbox": [ + 484, + 577, + 504, + 588 + ], + "score": 0.86, + "content": "30 \\%", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "(15.72 to 4.52 million FLOPs, and 7.87 to 2.26 million parameters), as shown in Table 5. If an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 232, + 612 + ], + "score": 1.0, + "content": "absolute accuracy reduction of", + "type": "text" + }, + { + "bbox": [ + 232, + 599, + 259, + 610 + ], + "score": 0.86, + "content": "1 . 8 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 598, + 285, + 612 + ], + "score": 1.0, + "content": "(from", + "type": "text" + }, + { + "bbox": [ + 285, + 599, + 318, + 610 + ], + "score": 0.86, + "content": "9 2 . 3 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 598, + 329, + 612 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 330, + 599, + 362, + 610 + ], + "score": 0.87, + "content": "9 0 . 5 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 598, + 505, + 612 + ], + "score": 1.0, + "content": ") is allowed by taking sub-model 2,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 610, + 411, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 411, + 623 + ], + "score": 1.0, + "content": "the reduction of complexity is even more remarkable, up to about one ninth.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 500, + 506, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 504, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "In addition, the table also shows the accuracy of the networks that have the same structures with the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "sub-models but are trained using regular backpropagation. Surprisingly, such networks do not easily", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "reach accuracy comparable to that of the sub-models obtained by local critic training, particularly", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 218, + 672 + ], + "score": 1.0, + "content": "for smaller networks (e.g.,", + "type": "text" + }, + { + "bbox": [ + 218, + 660, + 250, + 671 + ], + "score": 0.87, + "content": "7 4 . 4 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 659, + 269, + 672 + ], + "score": 1.0, + "content": "vs.", + "type": "text" + }, + { + "bbox": [ + 270, + 660, + 302, + 671 + ], + "score": 0.88, + "content": "8 5 . 2 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 659, + 506, + 672 + ], + "score": 1.0, + "content": "with sub-model 1 for CIFAR-10). We think that", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "joint training of the sub-models in local critic training helps them to find better solutions than those", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 681, + 305, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 305, + 694 + ], + "score": 1.0, + "content": "reached by independent regular backpropagation.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 627, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "Therefore, these results demonstrate that a structurally optimized network can be obtained at a cost", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "of a small loss in accuracy by local critic training, which may not be attainable by trial-and-error", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 721, + 198, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 198, + 733 + ], + "score": 1.0, + "content": "with backpropagation.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 698, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 212, + 122, + 398, + 191 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 505, + 114 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 81, + 505, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 505, + 92 + ], + "score": 1.0, + "content": "Table 5: FLOPs required for a feedforward pass and numbers of model parameters in the sub-models", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 504, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 504, + 104 + ], + "score": 1.0, + "content": "and main model for CIFAR-10. Note that sub-model 2 has less FLOPs and parameters than sub-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 321, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 321, + 115 + ], + "score": 1.0, + "content": "model 1 due to the pooling operation in sub-model 2.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 212, + 122, + 398, + 191 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 212, + 122, + 398, + 191 + ], + "spans": [ + { + "bbox": [ + 212, + 122, + 398, + 191 + ], + "score": 0.978, + "html": "
modelFLOP# of parameters
Sub-model 12.85M1.42M
Sub-model 21.76M0.88M
Sub-model 34.52M2.26M
Main model15.72M7.87M
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FLOPAccuracy (%)
Progressive inference (0.9)2.90M91.18 ±0.10
Progressive inference (0.95)3.05M91.75 ±0.16
Main model15.72M92.39 ±0.09
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The results are shown in Table 6. The feedfor-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "ward pass ends at different sub-models for different test data, and the average FLOPs over all test", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 420, + 388 + ], + "score": 1.0, + "content": "data are shown. When the threshold is 0.9, with only a slight loss of accuracy", + "type": "text" + }, + { + "bbox": [ + 421, + 376, + 454, + 386 + ], + "score": 0.86, + "content": "( 9 2 . 3 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 375, + 465, + 388 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 466, + 376, + 499, + 387 + ], + "score": 0.85, + "content": "9 1 . 1 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 390, + 399 + ], + "score": 1.0, + "content": "the computational complexity is reduced significantly, which is only", + "type": "text" + }, + { + "bbox": [ + 391, + 387, + 423, + 397 + ], + "score": 0.87, + "content": "1 8 . 4 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "of that of the main", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 470, + 410 + ], + "score": 1.0, + "content": "model. When the threshold increases to 0.95, the accuracy loss becomes smaller (only", + "type": "text" + }, + { + "bbox": [ + 471, + 398, + 499, + 408 + ], + "score": 0.84, + "content": "0 . 6 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 397, + 505, + 410 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 337, + 421 + ], + "score": 1.0, + "content": "while the complexity reduction remains almost the same (", + "type": "text" + }, + { + "bbox": [ + 338, + 408, + 369, + 419 + ], + "score": 0.85, + "content": "1 9 . 4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "of the main model’s complexity).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 439, + 230, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 231, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 231, + 451 + ], + "score": 1.0, + "content": "3.6 ENSEMBLE INFERENCE", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "The results of ensemble inference using the sub-models and main model are shown in Figure 4b.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 473, + 504, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 469, + 485 + ], + "score": 1.0, + "content": "Using an ensemble of the three sub-models, we observe improved classification accuracy", + "type": "text" + }, + { + "bbox": [ + 470, + 473, + 504, + 484 + ], + "score": 0.86, + "content": "( 9 2 . 6 8 \\%", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 123, + 496 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 484, + 156, + 494 + ], + "score": 0.88, + "content": "7 0 . 8 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "for the two datasets, respectively) in comparison to the main model. The performance", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 495, + 504, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 469, + 506 + ], + "score": 1.0, + "content": "is further enhanced by an ensemble of both the three sub-models and the main model", + "type": "text" + }, + { + "bbox": [ + 469, + 495, + 504, + 506 + ], + "score": 0.85, + "content": "( 9 2 . 7 9 \\%", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 123, + 518 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 506, + 157, + 516 + ], + "score": 0.87, + "content": "7 1 . 8 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 506, + 505, + 518 + ], + "score": 1.0, + "content": ". 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modelFLOP# of parameters
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FLOPAccuracy (%)
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In International", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 117, + 497, + 405, + 509 + ], + "spans": [ + { + "bbox": [ + 117, + 497, + 405, + 509 + ], + "score": 1.0, + "content": "Conference on Learning Representations (ICLR), Toulon, France, 2017.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 14.5 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "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": "10", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 104, + 74, + 506, + 512 + ], + "lines": [], + "index": 14.5, + "bbox_fs": [ + 104, + 82, + 506, + 509 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 245, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 246, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 246, + 95 + ], + "score": 1.0, + "content": "A ADDITIONAL RESULTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 106, + 248, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 250, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 250, + 118 + ], + "score": 1.0, + "content": "A.1 ADDITIONAL COMPARISON", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 126, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "In Czarnecki et al. (2017a), a method to minimize not only the loss of the network output but also its", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "derivative is proposed, called Sobolev training, and applied to the critic training algorithm. We also", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "conduct an experiment to use the Sobolev training method in our proposed algorithm. The results are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "shown in Table 7. In comparison to the performance shown in Table 1, we do not observe significant", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "difference overall. In addition, we test the deep supervision algorithm (Wang et al., 2015), which", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "also has additional modules connected to intermediate layers. The table shows that its performance", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 340, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 340, + 204 + ], + "score": 1.0, + "content": "is not significantly different from that of backpropagation.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5 + }, + { + "type": "table", + "bbox": [ + 109, + 257, + 502, + 303 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 214, + 505, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 231, + 227 + ], + "score": 1.0, + "content": "Table 7: Average test accuracy", + "type": "text" + }, + { + "bbox": [ + 231, + 215, + 246, + 225 + ], + "score": 0.69, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "of Sobolev local critic training (Sob LC), Sobolev critic training", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "(Czarnecki et al., 2017a), and deep supervision (Wang et al., 2015). The numbers of local networks", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 236, + 432, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 432, + 248 + ], + "score": 1.0, + "content": "used are shown in the parentheses. The standard deviation values are also shown.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "table_body", + "bbox": [ + 109, + 257, + 502, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 257, + 502, + 303 + ], + "spans": [ + { + "bbox": [ + 109, + 257, + 502, + 303 + ], + "score": 0.976, + "html": "
DatasetSob LC (1)Sob LC (3)Sob LC (5)Sob critic (3)Deep supervision (3)
CIFAR-1091.83 ±0.2292.32 ±0.1291.42 ±0.2691.96 ±0.2294.08 ±0.11
CIFAR-10073.64 ±0.2269.58 ±0.3263.82 ±1.0468.83 ±0.3175.09 ±0.29
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DatasetResNet-50 BPResNet-50 LC (3)ResNet-101 BPResNet-101 LC (3)
CIFAR-1094.4292.9594.1893.17
CIFAR-10074.8870.5877.2972.72
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ImageNetResNet-50 BPResNet-50 LC (3)
Top-5 accuracy92.3886.52
Top-1 accuracy75.0765.41
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For each of 400 samples from CIFAR-10, the activation of each layer is recorded,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "and the correlation between two samples is measured, which is shown in Figure 5. In the figure, clear", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "diagonal-blocks indicate that samples from the same class have highly correlated representations", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 720, + 188, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 188, + 734 + ], + "score": 1.0, + "content": "(e.g., the last layer).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + } + ], + "page_idx": 10, + "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 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 245, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 246, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 246, + 95 + ], + "score": 1.0, + "content": "A ADDITIONAL RESULTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 106, + 248, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 250, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 250, + 118 + ], + "score": 1.0, + "content": "A.1 ADDITIONAL COMPARISON", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 126, + 505, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "In Czarnecki et al. (2017a), a method to minimize not only the loss of the network output but also its", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "derivative is proposed, called Sobolev training, and applied to the critic training algorithm. We also", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "conduct an experiment to use the Sobolev training method in our proposed algorithm. The results are", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "shown in Table 7. In comparison to the performance shown in Table 1, we do not observe significant", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "difference overall. In addition, we test the deep supervision algorithm (Wang et al., 2015), which", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "also has additional modules connected to intermediate layers. The table shows that its performance", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 194, + 340, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 340, + 204 + ], + "score": 1.0, + "content": "is not significantly different from that of backpropagation.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 126, + 506, + 204 + ] + }, + { + "type": "table", + "bbox": [ + 109, + 257, + 502, + 303 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 214, + 505, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 231, + 227 + ], + "score": 1.0, + "content": "Table 7: Average test accuracy", + "type": "text" + }, + { + "bbox": [ + 231, + 215, + 246, + 225 + ], + "score": 0.69, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "of Sobolev local critic training (Sob LC), Sobolev critic training", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 238 + ], + "score": 1.0, + "content": "(Czarnecki et al., 2017a), and deep supervision (Wang et al., 2015). The numbers of local networks", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 236, + 432, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 432, + 248 + ], + "score": 1.0, + "content": "used are shown in the parentheses. The standard deviation values are also shown.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "table_body", + "bbox": [ + 109, + 257, + 502, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 257, + 502, + 303 + ], + "spans": [ + { + "bbox": [ + 109, + 257, + 502, + 303 + ], + "score": 0.976, + "html": "
DatasetSob LC (1)Sob LC (3)Sob LC (5)Sob critic (3)Deep supervision (3)
CIFAR-1091.83 ±0.2292.32 ±0.1291.42 ±0.2691.96 ±0.2294.08 ±0.11
CIFAR-10073.64 ±0.2269.58 ±0.3263.82 ±1.0468.83 ±0.3175.09 ±0.29
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DatasetResNet-50 BPResNet-50 LC (3)ResNet-101 BPResNet-101 LC (3)
CIFAR-1094.4292.9594.1893.17
CIFAR-10074.8870.5877.2972.72
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ImageNetResNet-50 BPResNet-50 LC (3)
Top-5 accuracy92.3886.52
Top-1 accuracy75.0765.41
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In the figure, clear", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "diagonal-blocks indicate that samples from the same class have highly correlated representations", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 720, + 188, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 188, + 734 + ], + "score": 1.0, + "content": "(e.g., the last layer).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 665, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 78, + 499, + 230 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 78, + 499, + 230 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 78, + 499, + 230 + ], + "spans": [ + { + "bbox": [ + 110, + 78, + 499, + 230 + ], + "score": 0.954, + "type": "image", + "image_path": "25c4b4bf1fb8d3e89761a9928ade1dad5be347d56a4fa5ebefcc89ee4de14596.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 78, + 499, + 128.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 128.66666666666666, + 499, + 179.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 179.33333333333331, + 499, + 229.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 238, + 505, + 282 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 238, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 505, + 251 + ], + "score": 1.0, + "content": "Figure 5: Representation dissimilarity matrix of each layer of the trained networks (having the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 249, + 504, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 504, + 261 + ], + "score": 1.0, + "content": "structure shown in Figure 2a) for label-ordered samples from CIFAR-10. Red lines indicate the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "locations of local critic networks. A clear diagonal-block pattern indicates that clear inner-class", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 272, + 235, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 235, + 282 + ], + "score": 1.0, + "content": "representation has been trained.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "Overall, block-diagonal patterns become clear at the last layers for all cases. However, the figure also", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "shows that the two training methods result in networks showing qualitatively different characteristics", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "in their internal representations. 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These layers in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 371 + ], + "score": 1.0, + "content": "the proposed method act not only as intermediate layers of the main network but also as near-final", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 370, + 502, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 502, + 383 + ], + "score": 1.0, + "content": "layers of the sub-models, and thus are forced to learn class-specific representations to some extent.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 397, + 243, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 245, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 245, + 412 + ], + "score": 1.0, + "content": "C LEARNING DYNAMICS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 423, + 291, + 433 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 293, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 293, + 435 + ], + "score": 1.0, + "content": "C.1 VISUALIZATION OF LOSS EVOLUTION", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 443, + 505, + 542 + ], + "lines": [ + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "As a way to investigate the learning dynamics of the proposed method, the loss values at each local", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 166, + 466 + ], + "score": 1.0, + "content": "critic network", + "type": "text" + }, + { + "bbox": [ + 167, + 454, + 184, + 465 + ], + "score": 0.85, + "content": "( L _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "for individual data over the training iterations are examined (Czarnecki et al.,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "2017b). Figure 3 visualizes the loss values for 400 sampled data of CIFAR-10 (arranged in 2D)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "when three local critic networks are used. For visualization, the same results are shown twice, once", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 485, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 485, + 295, + 501 + ], + "score": 1.0, + "content": "sorted by labels (Figure 6a) and once sorted by", + "type": "text" + }, + { + "bbox": [ + 296, + 487, + 308, + 498 + ], + "score": 0.89, + "content": "L _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 485, + 506, + 501 + ], + "score": 1.0, + "content": "at iteration 20000 (Figure 6b). At the early stage", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 333, + 510 + ], + "score": 1.0, + "content": "of learning, the loss values at the local critic networks (", + "type": "text" + }, + { + "bbox": [ + 333, + 498, + 345, + 509 + ], + "score": 0.85, + "content": "L _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 498, + 358, + 510 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 358, + 498, + 371, + 509 + ], + "score": 0.81, + "content": "L _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 498, + 505, + 510 + ], + "score": 1.0, + "content": ") are largely different from those", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 190, + 522 + ], + "score": 1.0, + "content": "at the main network", + "type": "text" + }, + { + "bbox": [ + 190, + 509, + 209, + 520 + ], + "score": 0.83, + "content": "( L _ { 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 509, + 505, + 522 + ], + "score": 1.0, + "content": ", with only slight similarity (e.g., the blue region at iteration 50 in Figure", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "6a). 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Input:data x,threshold t Model: sub-model ci,main-model f Initialize:classification = 0. fori=1toN-1do if max softmax(ci(x))>t then classification = argmax softmax(ci(x)) break
end if
end for
if classification== O then
#if all sub-models are not confident
classification= argmax softmax(f(x)) end if
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DatasetBPDNI (3)Critic (3)LC(1)LC (3)LC (5)
CIFAR-1093.93 ±0.2064.86 ±0.4291.92 ±0.3092.06 ±0.2092.39 ±0.0991.38 ±0.20
CIFAR-10075.14 ±0.1836.53 ±0.6469.07 ±0.2573.61 ±0.3169.91 ±0.5063.53 ±0.24
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Dataset[1,1,1] (default)[3,3,3][5,5,5][3,2,1][1,2,3][5,4,3][3,4,5]
CIFAR-1092.39 ±0.0992.36 ±0.2291.72 ±0.1992.07 ±0.2192.20 ±0.1292.10 ±0.1691.90 ±0.16
CIFAR-10069.91 ±0.5070.02 ±0.2970.34 ±0.1670.06 ±0.6469.81 ±0.3370.87 ±0.4069.93 ±0.56
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DatasetBP sub 1LC sub 1BP sub 2LC sub 2BP sub 3LC sub 3
CIFAR-1074.46 ±0.9185.24 ±0.4988.03 ±0.8790.53 ±0.1592.05 ±0.2492.29 ±0.09
CIFAR-10047.58 ±1.1055.39 ±0.5761.79 ±0.9263.62 ±0.3167.81 ±0.2267.54 ±0.70
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Dataset1/11/21/31/41/5
CIFAR-1092.39 ±0.0991.91 ±0.1991.78 ±0.1891.57 ±0.1291.35 ±0.17
CIFAR-10069.91 ±0.5067.99 ±0.4967.76±0.1966.74 ±0.4166.39 ±0.39
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modelFLOP# of parameters
Sub-model 12.85M1.42M
Sub-model 21.76M0.88M
Sub-model 34.52M2.26M
Main model15.72M7.87M
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FLOPAccuracy (%)
Progressive inference (0.9)2.90M91.18 ±0.10
Progressive inference (0.95)3.05M91.75 ±0.16
Main model15.72M92.39 ±0.09
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DatasetResNet-50 BPResNet-50 LC (3)ResNet-101 BPResNet-101 LC (3)
CIFAR-1094.4292.9594.1893.17
CIFAR-10074.8870.5877.2972.72
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DatasetSob LC (1)Sob LC (3)Sob LC (5)Sob critic (3)Deep supervision (3)
CIFAR-1091.83 ±0.2292.32 ±0.1291.42 ±0.2691.96 ±0.2294.08 ±0.11
CIFAR-10073.64 ±0.2269.58 ±0.3263.82 ±1.0468.83 ±0.3175.09 ±0.29
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ImageNetResNet-50 BPResNet-50 LC (3)
Top-5 accuracy92.3886.52
Top-1 accuracy75.0765.41
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15, + "poly": [ + 297.0, + 73.0, + 858.0, + 73.0, + 858.0, + 108.0, + 297.0, + 108.0 + ], + "score": 1.0, + "text": "" + } + ], + "page_info": { + "page_no": 13, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/BJypUGZ0Z/BJypUGZ0Z.md b/parse/train/BJypUGZ0Z/BJypUGZ0Z.md new file mode 100644 index 0000000000000000000000000000000000000000..d82f3190742a4d8c61ff32db689448a9948d0894 --- /dev/null +++ b/parse/train/BJypUGZ0Z/BJypUGZ0Z.md @@ -0,0 +1,308 @@ +# ACCELERATING NEURAL ARCHITECTURE SEARCH USING PERFORMANCE PREDICTION + +Bowen Baker∗, Otkrist Gupta∗, Ramesh Raskar + +# Nikhil Naik + +Media Laboratory Massachusetts Institute of Technology Cambridge, MA 02142, USA {bowen,otkrist,raskar}@mit.edu + +Faculty of Arts and Sciences Harvard University Cambridge, MA 02138, USA naik@fas.harvard.edu + +# ABSTRACT + +Methods for neural network hyperparameter optimization and architecture search are computationally expensive due to the need to train a large number of model configurations. In this paper, we show that simple regression models can predict the final performance of partially trained model configurations using features based on network architectures, hyperparameters, and time-series validation performance data. We empirically show that our performance prediction models are much more accurate than prominent Bayesian counterparts, are simpler to implement, and are faster to train. Our models can predict final performance in both visual classification and language modeling domains, are effective for predicting performance of drastically varying model architectures, and can even generalize between model classes. Using these prediction models, we also implement an early stopping method for hyperparameter optimization and architecture search, which obtains a speedup of a factor up to 6x in both hyperparameter optimization and architecture search. Finally, we empirically show that our early stopping method can be seamlessly incorporated into both reinforcement learning-based architecture selection algorithms and bandit based search methods. Through extensive experimentation, we empirically show our performance prediction models and early stopping algorithm are state-of-the-art in terms of prediction accuracy and speedup achieved while still identifying the optimal model configurations. + +# 1 INTRODUCTION + +Significant human expertise and labor is required for designing high-performing neural network architectures and successfully training them for different applications. Ongoing research in two areas—architecture search and hyperparameter optimization—attempts to reduce the amount of human intervention required for these tasks. Hyperparameter optimization methods (e.g., Hutter et al. (2011); Snoek et al. (2015); Li et al. (2017)) focus primarily on obtaining good optimization hyperparameter configurations for training human-designed networks, whereas architecture search algorithms (Bergstra et al., 2013; Verbancsics & Harguess, 2013; Baker et al., 2017; Zoph & Le, 2017) aim to design neural network architectures from scratch. Both sets of algorithms require training a large number of neural network configurations for identifying the right set of hyperparameters or the right network architecture and are hence computationally expensive. + +When sampling many different model configurations, it is likely that many subpar configurations will be explored. Human experts are quite adept at recognizing and terminating suboptimal model configurations by inspecting their partial learning curves. In this paper we seek to emulate this behavior and automatically identify and terminate subpar model configurations in order to speedup both architecture search and hyperparameter optimization methods. Our method parameterizes learning curve trajectories with simple features derived from model architectures, training hyperparameters, and early time-series measurements from the learning curve. We use these features to train a set of simple regression models that predict the final validation performance of partially trained neural network configurations using a small training set of fully trained curves, and we empirically validate our method in both image classification and language modeling domains. We use these predictions along with uncertainty estimates obtained from small model ensembles to construct a simple early stopping algorithm that can speedup both architecture search and hyperparameter optimization methods. + +![](images/25baeeba5fe8e2faa370c9042f700f36b0568f1611f8d41291f30bb7417b7ef6.jpg) +Figure 1: Example Learning Curves: Example learning curves from experiments considered in this paper. Note the diversity in convergence times and overall learning curve shapes. + +While there is some prior work on neural network performance prediction using Bayesian methods (Domhan et al., 2015; Klein et al., 2017), our proposed method is significantly more accurate, accessible, and efficient. We hope that our work leads to inclusion of neural network performance prediction and early stopping in the practical neural network training pipeline. + +# 2 RELATED WORK + +Neural Network Performance Prediction: There has been limited work on predicting neural network performance during training. Domhan et al. (2015) introduce a weighted probabilistic model for learning curves and utilize this model for speeding up hyperparameter search in small convolutional neural networks (CNNs) and fully-connected networks (FCNs). Building on Domhan et al. (2015), Klein et al. (2017) train Bayesian neural networks for predicting unobserved learning curves using a training set of fully and partially observed learning curves. Both methods rely on expensive Markov chain Monte Carlo (MCMC) sampling procedures and handcrafted learning curve basis functions. We also note that Swersky et al. (2014) develop a Gaussian Process kernel for predicting individual learning curves, which they use to automatically stop and restart configurations. + +Architecture Search: We define architecture search as an algorithmic approach for designing neural network architectures from scratch. The earliest architecture search approaches were based on genetic algorithms (Schaffer et al., 1992; Stanley & Miikkulainen, 2002; Verbancsics & Harguess, 2013) or Bayesian optimization (Bergstra et al., 2013; Shahriari et al., 2016). More recently, reinforcement learning methods have become popular. Baker et al. (2017) use Q-learning to design competitive CNNs for image classification. Zoph & Le (2017) use policy gradients to design state-of-the-art CNNs and recurrent cell architectures. Several methods for architecture search (Cortes et al., 2017; Negrinho & Gordon, 2017; Zoph et al., 2017; Brock et al., 2017; Suganuma et al., 2017) have been proposed this year since the publication of Baker et al. (2017) and Zoph & Le (2017). + +Hyperparameter Optimization: We define hyperparameter optimization as an algorithmic approach for finding optimal values of architecture independent hyperparameters such as learning rate and batch size, along with a limited search through the network design space. Bayesian hyperparameter optimization methods include those based on sequential model-based optimization (SMAC) (Hutter et al., 2011), Gaussian processes (GP) (Snoek et al., 2012), TPE (Bergstra et al., 2013), and neural networks Snoek et al. (2015). However, random search or grid search is most commonly used in practical settings (Bergstra & Bengio, 2012). Recently, Li et al. (2017) introduced Hyperband, a multiarmed bandit-based efficient random search technique that outperforms state-of-the-art Bayesian optimization methods. + +# 3 NEURAL NETWORK PERFORMANCE PREDICTION + +We first describe our model for neural network performance prediction, followed by a description of the datasets used to evaluate our model, and finally present experimental results. + +# 3.1 MODELING LEARNING CURVES + +Our goal is to model the validation performance $y _ { T }$ of a neural network configuration $\mathbf { x } \in \mathcal { X } \subset \mathbb { R } ^ { d }$ at epoch $T \in \mathbb { Z } ^ { + }$ using previous performance observations $y ( t )$ . For each configuration $\mathbf { x }$ trained for $T$ epochs, we record a time-series $y ( T ) = y _ { 1 } , y _ { 2 } , \dotsc , y _ { T }$ of validation performances. We train a population of $n$ configurations, obtaining a set $\tilde { \mathcal { S } } = \{ ( \mathbf { x } ^ { 1 } , y ^ { 1 } ( T ) ) , ( \mathbf { x } ^ { 2 } , y ^ { 2 } ( T ) ) , \ldots , ( \mathbf { x } ^ { n } , y ^ { n } ( T ) ) \} .$ This problem formulation is very similar to Klein et al. (2017). Note that most architecture and hyperparameter search methods naturally collect $s$ . + +We propose to use a set of features $u _ { \mathbf { x } }$ , derived from the neural network configuration $\mathbf { x }$ , along with a subset of time-series performances $y ( \tau ) = ( y _ { t } ) _ { t = 1 , 2 , . . . , \tau }$ (where $1 \leq \tau < T )$ ) from $s$ to train a regression model for estimating $y _ { T }$ . We use models without natural support for variable size input, so we train $T - 1$ regression models, where each successive model uses one more point of the time-series validation data. As we shall see in subsequent sections, this use of sequential regression models (SRM) is more computationally efficient and more precise than methods that train a single model. + +Features: We use features based on time-series (TS) validation performances, architecture parameters (AP), and hyperparameters (HP). (1) TS: Assume we are training the $\tau ^ { \mathrm { { t h } } }$ model in the SRM. TS features include the validation performances $y ( \tau ) = ( y _ { t } ) _ { t = 1 , 2 , . . . , \tau }$ , the first-order differences of validation performances, i.e. $y _ { t } ^ { \prime } = \left( y _ { t } - y _ { t - 1 } \right)$ , the second-order differences of validation performances, i.e. ${ y _ { t } } ^ { \prime \bar { \prime } } = ( { y _ { t } } ^ { \prime } - { y _ { t - 1 } ^ { \prime } } )$ , and mean and standard deviation of performances. (2) AP: These include total number of weights and number of layers. (3) HP: These include all hyperparameters used for training the neural networks, e.g. initial learning rate and learning rate decay (full list in Appendix Table 4). + +# 3.2 DATASETS AND TRAINING PROCEDURES + +Figure 1 shows example learning curves from three of the datasets considered in our experiments. We provide brief summary of the datasets below. Please see Appendix Section A for further details on the search space, preprocessing, hyperparameters and training settings of all datasets. + +DATASETS WITH VARYING ARCHITECTURES: + +Deep Resnets (TinyImageNet): We sample 500 ResNet architectures and train them on the TinyImageNet\* dataset (containing 200 classes with 500 training images of $3 2 \times 3 2$ pixels) for 140 epochs. We vary depths, filter sizes and number of convolutional filter block outputs. The network depths vary between 14 and 110. + +Deep Resnets (CIFAR-10): We sample 500 39-layer ResNet architectures from a search space similar to Zoph & Le (2017), varying kernel width, kernel height, and number of kernels. We train these models for 50 epochs on CIFAR-10. + +MetaQNN CNNs (CIFAR-10 and SVHN): We sample 1,000 model architectures from the search space detailed by Baker et al. (2017), which allows for varying the numbers and orderings of convolution, pooling, and fully connected layers. The models are between 1 and 12 layers for the SVHN experiment and between 1 and 18 layers for the CIFAR-10 experiment. Each architecture is trained on SVHN and CIFAR-10 datasets for 20 epochs. + +LSTM (PTB): We sample 300 LSTM models and train them on the Penn Treebank dataset for 60 epochs, evaluating perplexity on the validation set. We vary number of LSTM cells and hidden layer inputs between 10-1400. + +# DATASETS WITH VARYING HYPERPARAMETERS: + +Cuda-Convnet (CIFAR-10 and SVHN): We train Cuda-Convnet architecture (Krizhevsky, 2012) with varying values of initial learning rate, learning rate reduction step size, weight decay for convolutional and fully connected layers, and scale and power of local response normalization layers. We train models with CIFAR-10 for 60 epochs and with SVHN for 12 epochs. + +
Datasetv-SVROLSBLRRF
MetaQNN (CIFAR-10)97.22 ± 0.3197.04± 0.4297.11 ± 0.3294.50 ± 0.78
Resnet (TinyImageNet)87.12 ± 4.6588.41 ± 3.7682.40± 1.6586.39 ± 3.58
LSTM (Penn Treebank)98.03± 1.4687.83 ± 21.5397.99 ± 1.9057.28± 24.98
+ +Table 1: Performance Prediction Model Comparison: We report the coefficient of determination $R ^ { 2 } * 1 0 0$ for four standard methods. Each model is trained with 100 samples on $2 5 \%$ of the learning curve. We find that $\nu$ -SVR works best on average, though not by a large margin. + +![](images/e1e08d693f58b2451fc566e9d907921036494e900edf6b2beed895ec69970c9e.jpg) +Figure 2: Predicted vs True Values of Final Performance: We show the shape of the predictive distribution on three experiments: MetaQNN models, Deep Resnets, and LSTMs. Each $\nu$ -SVR model is trained with 100 configurations with data from $2 5 \%$ of the learning curve. We predict validation set classification accuracy for MetaQNN and Deep ResNets, and perplexity for LSTMs. + +# 3.3 PREDICTION PERFORMANCE + +Choice of Regression Method: We now describe our results for predicting final neural network performance. For all experiments, we train our SRMs on 100 randomly sampled neural network configurations. We obtain the best performing method using random hyperparameter search over 3-fold cross-validation. We then compute the regression performance over the remainder of the dataset using the coefficient of determination $R ^ { 2 }$ . We repeat each experiment 10 times and report the results with standard errors. We experiment with a few simple regression models, including ordinary least squares (OLS), random forests (RF), Bayesian linear regression (BLR), and $\nu$ -support vector machine regression ( $\nu$ -SVR). For both OLS and $\nu \cdot$ -SVR, we choose between the linear and RBF kernels based on prediction performance—after hyperparameter search, in case of $\nu$ -SVR. We chose 100 as the number of sampled configurations somewhat arbitrarily, though we found that prediction performance was quite poor below 50 and improves steadily after. As seen in Table 1, $\nu$ -SVR performs the best on most datasets, though not by a large margin. For the rest of this paper, we use $\nu$ -SVR unless otherwise specified. + +Ablation Study on Feature Sets: In Table 2, we compare the predictive ability of different feature sets, training SVR with time-series (TS) features obtained from $2 5 \%$ of the learning curve, along with features of architecture parameters (AP), and hyperparameters (HP). TS features explain the largest fraction of the variance in all cases and found that in general AP are more important that HP. AP features almost match TS on the ResNet (TinyImageNet) dataset, indicating that choice of architecture has a large influence on accuracy for ResNets. Figure 2 shows the true vs. predicted performance for all test points in three datasets, trained with TS, AP, and HP features. + +Generalization To Out Of Distribution Configurations: We also test to see whether our SRMs can accurately predict the performance of out-of-distribution neural networks. For instance, if we train a $\nu$ -SVR model on the Deep Resnet (TinyImagenet) dataset at $2 5 \%$ of the learning curve observed on configurations below median depth (60 layers), we can achieve $R ^ { 2 } = 0 . 8 6$ on configurations above median depth. The corresponding number for prediction below median depth is 0.89. We repeated this experiment all datasets with different hyperparameter values below/above the median and found that SRMs generally showed consistent performance across such splits. Full results can be found in Appendix Section $_ \mathrm { H }$ . + +
Feature SetMetaQNN(CIFAR-10)ResNets(TinyImageNet)LSTM(Penn Treebank)Cuda-Convnet(CIFAR-10)
HP-2.96 ± 2.710.48 ± 7.24-1.72 ± 1.620.48 ± 7.24
AP15.38 ± 4.6667.66 ± 9.0813.76 ± 12.0867.66 ± 9.08
TS97.00 ± 0.5282.71 ± 4.3197.38 ± 1.4982.71 ± 4.31
AP+HP14.89 ± 5.1270.25 ± 6.4213.76 ± 12.0870.25 ± 6.42
TS+HP97.02 ± 0.5282.67 ± 6.8197.38 ± 1.4982.67± 6.81
TS+AP97.12 ± 0.3787.36± 4.2397.87 ± 1.6887.36± 4.23
TS+AP+HP97.15 ± 0.4886.05 ± 8.2497.87 ± 1.6886.05 ± 8.24
+ +Table 2: Ablation Study on Feature Sets: We report the $R ^ { 2 } * 1 0 0$ metric for different combinations of features. Time-series features (TS) refers to the partially observed learning curves, architecture parameters (AP) refer to the number of layers and number of weights in a deep model, and hyperparameters (HP) refer to the optimization parameters such as learning rate. All results are with $\nu$ -SVR. $2 5 \%$ of learning curve used for models including TS features. + +# 3.3.1 COMPARISON WITH EXISTING METHODS: + +We now compare the neural network performance prediction ability of SRMs with three existing learning curve prediction methods: (1) Bayesian Neural Network (BNN) (Klein et al., 2017), (2) the learning curve extrapolation (LCE) method (Domhan et al., 2015), and (3) the last seen value (LastSeenValue) heuristic (Li et al., 2017). When training the BNN, we not only present it with the subset of fully observed learning curves but also all other partially observed learning curves from the training set. While we do not present the partially observed curves to the SRMs for training, we felt this was a fair comparison as an SRM uses the entire partially observed learning curve during inference. Methods (2) and (3) do not incorporate prior learning curves during training. Figure 3 shows the $R ^ { 2 }$ obtained by each method for predicting the final performance versus the percent of the learning curve used for training the model. We see that in all neural network configuration spaces and across all datasets, $\nu { - } \mathbf { S } \mathbf { V } \mathbf { R }$ outperform the competing methods. Bayesian linear regression (BLR) also performs comparably well to $\nu$ -SVR and better than the more complex Bayesian methods (BNN and LCE). The LastSeenValue heuristic only becomes viable when the configurations are near convergence, and its performance is worse than an SRM for very deep models. We also find that the SRMs outperform the LCE method in all experiments, even after we remove a few extreme prediction outliers produced by LCE. Finally, while BNN outperforms the LastSeenValue and LCE methods when only a few iterations have been observed, it does worse than our proposed method. In summary, we show that our simple SRMs outperforms existing Bayesian approaches on predicting neural network performance on modern, very deep models in computer vision and language modeling tasks. + +Since most of our experiments perform stepwise learning rate decay; it is conceivable that the performance gap between SRMs and both LCE and BNN results from a lack of sharp jumps in their basis functions. For completeness, we also experimented with exponential learning rate decay (ELRD), which the basis functions in LCE are designed for. We trained 630 random nets with ELRD, from the 1000 MetaQNN-CIFAR10 nets. Predicting from $2 5 \%$ of the learning curve, the $R ^ { 2 }$ is 0.95 for $\nu$ -SVR, 0.48 for LCE (with extreme outlier removal, negative without), and 0.31 for BNN. This comparison illuminates another benefit of our method: we do not require handcrafted basis functions to model new learning curve types. + +Moreover, SRMs are faster to train and do inference in than LCE and BNN. On 1 core of a Intel $6 7 0 0 \mathrm { k }$ CPU, an $\nu$ -SVR with 100 training points trains in 0.006 seconds, and each inference takes 0.00006 seconds. In comparison, the LCE code takes 60 seconds and BNN code takes 0.024 seconds on the same hardware for each inference. + +# 4 APPLYING PERFORMANCE PREDICTION FOR EARLY STOPPING + +To speed up hyperparameter optimization and architecture search methods, we develop an algorithm to determine whether to continue training a partially trained model configuration using our sequential regression models. This approach follows the idea proposed by Domhan et al. (2015) and further expanded on by Klein et al. (2017). Concretely, if we would like to sample $N$ total neural network configurations, we begin by sampling and training $n \ll N$ configurations to create a training set $s$ . We then train a model $f ( x _ { f } )$ , where in our case $f$ is an SRM with $T - 1$ individual models, to predict $y _ { T }$ Now, given the current best performance observed $y _ { \mathrm { B E S T } }$ , we would like to terminate training a new configuration $\mathbf { x } _ { \mathrm { N E W } }$ at iteration $\tau$ given its partial learning curve $y _ { \mathrm { N E W } } ( \tau )$ if $f ( x _ { \mathrm { { N E W } } } f ) = \hat { y } _ { T } \overset { \cdot } { \leq } y _ { \mathrm { { B E S T } } }$ so as to not waste computational resources exploring a suboptimal configuration. + +![](images/b307684c90185a1bcebe23b1c8b3d267a9a5c28ed548460d497c2352fbae0626.jpg) +Figure 3: Performance Prediction Results: We plot the performance of each method versus the percent of learning curve observed. For BNN and $\nu$ -SVR, we sample 10 different training sets, plot the mean $R ^ { 2 }$ , and shade the corresponding standard error. We compare our method against BNN (Klein et al., 2017), LCE (Domhan et al., 2015), and a “last seen value” heuristic (Li et al., 2017). Absent results for a model indicate that it did not achieve a positive $R ^ { 2 }$ . The results for Cuda-Convnet on the SVHN dataset are shown in Appendix Figure 11. + +However, in the case $f$ has poor out-of-sample generalization, we may mistakenly terminate the optimal configuration. If we assume that our estimate can be modeled as a Gaussian perturbation of the true value $\hat { y } _ { T } \sim \mathcal { N } ( y _ { T } , \sigma ( \mathbf { x } , \tau ) )$ , then we can find the probability $\begin{array} { r } { p ( \hat { y } _ { T } \leq y _ { \mathrm { B E S T } } \mathbf { \bar { \Phi } } | \sigma ( \mathbf { x } , \tau ) ) = } \end{array}$ $\varPhi ( y _ { \mathrm { B E S T } } ; y _ { T } , \sigma )$ , where $\varPhi ( \cdot ; \mu , \sigma )$ is the CDF of ${ \mathcal { N } } ( \mu , \sigma )$ . Note that in general the uncertainty will depend on both the configuration and $\tau$ , the number of points observed from the learning curve. Because frequentist models do not admit a natural estimate of uncertainty, we assume that $\sigma$ is independent of $\mathbf { x }$ yet still dependent on $\tau$ and estimate it via Leave One Out Cross Validation (LOOCV). In addition, we show results using a Bayesian linear regression SRM, which has natural uncertainty estimates, removing the need for LOOCV. + +Now that we can estimate the model uncertainty, given a new configuration xNEW and an observed learning curve $y _ { \mathrm { N E W } } ( \tau )$ , we may set our termination criteria to be $p ( \hat { y } _ { T } \le y _ { \mathrm { B E S T } } ) \ge \varDelta$ . $\varDelta$ balances the trade-off between increased speedups and risk of prematurely terminating good configurations. In many cases, one may want several configurations that are close to optimal, for the purpose of ensembling. We offer two modifications in this case. First, one may relax the termination criterion to $p ( \hat { y } _ { T } \le \bar { y } _ { \mathtt { B E S T } } - \delta ) \ge \Delta$ , which will allow configurations within $\delta$ of optimal performance to complete training. One can alternatively set the criterion based on the $n ^ { \mathrm { t h } }$ best configuration observed, guaranteeing that with high probability the top $n$ configurations will be fully trained. + +# 4.1 EARLY STOPPING FOR ARCHITECTURE SEARCH + +Baker et al. (2017) train a $Q$ -learning agent to design convolutional neural networks. In this method, the agent samples architectures from a large, finite space by traversing a path from input layer to termination layer. However, the MetaQNN method uses 100 GPU-days to train 2700 neural architectures and the similar experiment by Zoph & Le (2017) utilized 10,000 GPU-days to train 12,800 models on CIFAR-10. The amount of computing resources required for these approaches makes them prohibitively expensive for large datasets (e.g., Imagenet) and larger search spaces. The main computational expense of reinforcement learning-based architecture search methods is training the neural network configuration to $T$ epochs (where $T$ is typically a large number at which the network stabilizes to peak accuracy). + +![](images/dc85c447f26791182cfb27434b6a9360fc11cf05a77d5f67d68ebe85227a11e5.jpg) +Figure 4: Simulated Speedup in MetaQNN Search Space: We compare the three variants of the early stopping algorithm presented in Section 4 for both a $\nu$ -SVR and BLR SRM. Each SRM is trained using the first 100 learning curves, and each algorithm is tested on 10 independent orderings of the model configurations. Triangles indicate an algorithm that successfully recovered the optimal model for more than half of the 10 orderings, and X’s indicate those that did not. + +We now detail the performance of $\nu$ -SVR and BLR SRMs in speeding up architecture search. First, we take 1,000 random models from the MetaQNN (Baker et al., 2017) search space. We simulate the MetaQNN algorithm by taking 10 random orderings of each set and running our early stopping algorithm. We compare against the LCE early stopping algorithm (Domhan et al., 2015) as a baseline, which has a similar probability threshold termination criterion. Our SRM trains off of the first 100 fully observed curves, while the LCE model trains from each individual partial curve and can begin early termination immediately. Despite this “burn in” time needed by an SRM, it is still able to outperform the LCE model (Figure 4). In addition, fitting the LCE model to a learning curve takes between 1-3 minutes on a modern CPU due to expensive MCMC sampling, and it is necessary to fit a new LCE model each time a new point on the learning curve is observed. However, the difference in computing time between the methods is not as important as the difference in prediction accuracies. + +We furthermore simulate early stopping for ResNets trained on CIFAR-10. We found that only the probability threshold $\varDelta = 0 . 9 9$ resulted in recovering the top model consistently. However, even with such a conservative threshold, the search was sped up by a factor of 3.4 over the baseline. While we do not have the computational resources to run the full experiment from Zoph & Le (2017), our method could provide similar gains in large scale architecture searches. + +It is not enough, however, to simply simulate the speedup because architecture search algorithms typically use the observed performance in order to update an acquisition function to inform future sampling. In the reinforcement learning setting, the performance is given to the agent as a reward, so we also empirically verify that substituting $\hat { y } _ { T }$ for $y _ { T }$ does not cause the MetaQNN agent to converge to a subpar policy. Replicating the MetaQNN experiment on CIFAR-10 (see Figure 5), we find that integrating early stopping with the $Q$ -learning procedure does not disrupt learning and resulted in a speedup of $3 . 8 \mathrm { x }$ with $\varDelta = 0 . 9 9$ . After training the top models to 300 epochs, we also find that the resulting performance (just under $9 3 \%$ ) is on par with original results of Baker et al. (2017). + +![](images/aab0e4a5f68df68fc6dd3b814ef2564c865449f63f59a2bb14bd5810251a37ab.jpg) +Figure 5: MetaQNN on CIFAR-10 with Early Stopping: A full run of the MetaQNN algorithm (Baker et al., 2017) on the CIFAR10 dataset with early stopping. We use the $\nu { - } \mathbf { S } \mathbf { V } \mathbf { R }$ SRM with a probability threshold $\varDelta = 0 . 9 9$ . Light blue bars indicate the average model accuracy per decrease in $\epsilon$ , which represents the shift to a more greedy policy. We also plot the cumulative best, top 5, and top 15 performance to show that the agent continues to find better architectures. + +![](images/1cbaded586a36c427e79abfbd1d1e03d5fd12763217a0e9b703aa0b5e71656a0.jpg) +Figure 6: Simulated Max Accuracy vs SGD Iterations for Hyperband: We show the trajectories of the maximum performance so far versus total computational resources used for 40 consecutive Hyperband runs with $\eta = 4 . 0$ and $\varDelta = 0 . 9 5$ . Small vertical lines indicate the point at which each f-Hyperband run has searched over the same number of models as vanilla Hyperband. On Cifar-10, both BLR and $\nu$ -SVR f-Hyperbands outperform vanilla Hyperband, but on SVHN only $\nu$ -SVR f-Hyperbands do. Each triangle marks the completion of full Hyperband algorithm. + +# 4.2 EARLY STOPPING FOR HYPERPARAMETER OPTIMIZATION + +Recently, Li et al. (2017) introduced Hyperband, a random search technique based on multi-armed bandits that obtains state-of-the-art performance in hyperparameter optimization in a variety of settings. The Hyperband algorithm trains a population of models with different hyperparameter configurations and iteratively discards models below a certain percentile in performance among the population until the computational budget is exhausted or satisfactory results are obtained. Among previous work, Klein et al. (2017) have improved Hyperband by algorithmically choosing the next models to evaluate. + +# 4.2.1 FAST HYPERBAND + +We present a Fast Hyperband (f-Hyperband) algorithm based on our early stopping scheme. During each iteration of successive halving, Hyperband trains $n _ { i }$ configurations to $r _ { i }$ epochs. In f-Hyperband, we train an SRM to predict $y _ { r _ { i } }$ and do early stopping within each iteration of successive halving. We initialize f-Hyperband in exactly the same way as vanilla Hyperband, except once we have trained 100 models to $r _ { i }$ iterations, we begin early stopping for all future successive halving iterations that train to $r _ { i }$ iterations. By doing this, we exhibit no computational overhead over Hyperband other than training the SRMs, which we have shown to be cheap. We also introduce a parameter $\kappa$ which denotes the proportion of the $n _ { i }$ models in each iteration that must be trained to the full $r _ { i }$ iterations. This is similar to setting the criterion based on the $n ^ { \mathrm { t h } }$ best model in the previous section. See Appendix section C for the full algorithmic representation of f-Hyperband. + +We empirically evaluate f-Hyperband using Cuda-Convnet trained on CIFAR-10 and SVHN datasets. Figure 6 shows that f-Hyperband evaluates the same number of unique configurations as Hyperband within half the compute time, while achieving the same final accuracy within standard error. Moreover, when we run f-Hyperband for the same amount of time as Hyperband, f-Hyperband with a $\nu$ -SVR SRM outperforms f-Hyperband. When reinitializing hyperparameter searches, one can use previouslytrained set of SRMs to achieve even larger speedups. Figure 12 in Appendix shows that one can safely achieve up to a $4 \mathbf { x }$ speedup in such cases. + +# 5 CONCLUSION + +In this paper we introduce a simple, fast, and accurate model for predicting future neural network performance using features derived from network architectures, hyperparameters, and time-series performance data. We show that the performance of drastically different network architectures can be jointly learned and predicted on both image classification and language models. Using our simple algorithm, we can speedup hyperparameter search techniques with complex acquisition functions, such as a $Q$ -learning agent, by a factor of $3 \mathbf { x }$ to 6x and Hyperband—a state-of-the-art hyperparameter search method—by a factor of $2 \mathbf { x }$ , without disturbing the search procedure. We outperform all competing methods for performance prediction in terms of accuracy, train and test time, and speedups obtained on hyperparameter search methods. We hope that the simplicity and success of our method will allow it to be easily incorporated into current hyperparameter optimization pipelines for deep neural networks. With the advent of large scale automated architecture search (Baker et al., 2017; Zoph & Le, 2017), methods such as ours will be vital in exploring even larger and more complex search spaces. + +# REFERENCES + +Bowen Baker, Otkrist Gupta, Nikhil Naik, and Ramesh Raskar. Designing neural network architectures using reinforcement learning. International Conference on Learning Representations, 2017. + +James Bergstra and Yoshua Bengio. Random search for hyper-parameter optimization. JMLR, 13 (Feb):281–305, 2012. + +James Bergstra, Daniel Yamins, and David D Cox. Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures. ICML (1), 28:115–123, 2013. + +Andrew Brock, Theodore Lim, JM Ritchie, and Nick Weston. Smash: One-shot model architecture search through hypernetworks. arXiv preprint arXiv:1708.05344, 2017. + +Corinna Cortes, Xavier Gonzalvo, Vitaly Kuznetsov, Mehryar Mohri, and Scott Yang. AdaNet: Adaptive structural learning of artificial neural networks. International Conference on Machine Learning, 70:874–883, 2017. + +Tobias Domhan, Jost Tobias Springenberg, and Frank Hutter. Speeding up automatic hyperparameter optimization of deep neural networks by extrapolation of learning curves. IJCAI, 2015. + +Frank Hutter, Holger H Hoos, and Kevin Leyton-Brown. Sequential model-based optimization for general algorithm configuration. In International Conference on Learning and Intelligent Optimization, pp. 507–523. Springer, 2011. + +Aaron Klein, Stefan Falkner, Jost Tobias Springenberg, and Frank Hutter. Learning curve prediction with bayesian neural networks. International Conference on Learning Representations, 17, 2017. + +Alex Krizhevsky. Cuda-convnet. https://code.google.com/p/cuda-convnet/, 2012. + +Lisha Li, Kevin Jamieson, Giulia DeSalvo, Afshin Rostamizadeh, and Ameet Talwalkar. Hyperband: A novel bandit-based approach to hyperparameter optimization. International Conference on Learning Representations, 2017. + +Renato Negrinho and Geoff Gordon. Deeparchitect: Automatically designing and training deep architectures. arXiv preprint arXiv:1704.08792, 2017. + +J David Schaffer, Darrell Whitley, and Larry J Eshelman. Combinations of genetic algorithms and neural networks: A survey of the state of the art. International Workshop on Combinations of Genetic Algorithms and Neural Networks, pp. 1–37, 1992. + +Bobak Shahriari, Kevin Swersky, Ziyu Wang, Ryan P Adams, and Nando de Freitas. Taking the human out of the loop: A review of bayesian optimization. Proceedings of the IEEE, 104(1): 148–175, 2016. + +Jasper Snoek, Hugo Larochelle, and Ryan P Adams. Practical bayesian optimization of machine learning algorithms. NIPS, pp. 2951–2959, 2012. +Jasper Snoek, Oren Rippel, Kevin Swersky, Ryan Kiros, Nadathur Satish, Narayanan Sundaram, Mostofa Patwary, Mr Prabhat, and Ryan Adams. Scalable bayesian optimization using deep neural networks. In International Conference on Machine Learning, pp. 2171–2180, 2015. +Kenneth O Stanley and Risto Miikkulainen. Evolving neural networks through augmenting topologies. Evolutionary Computation, 10(2):99–127, 2002. +Masanori Suganuma, Shinichi Shirakawa, and Tomoharu Nagao. A genetic programming approach to designing convolutional neural network architectures. arXiv preprint arXiv:1704.00764, 2017. +Kevin Swersky, Jasper Snoek, and Ryan Prescott Adams. Freeze-thaw bayesian optimization. arXiv preprint arXiv:1406.3896, 2014. +Phillip Verbancsics and Josh Harguess. Generative neuroevolution for deep learning. arXiv preprint arXiv:1312.5355, 2013. +Barret Zoph and Quoc V Le. Neural architecture search with reinforcement learning. International Conference on Learning Representations, 2017. +Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V Le. Learning transferable architectures for scalable image recognition. arXiv preprint arXiv:1707.07012, 2017. + +# APPENDIX + +# A DATASETS AND ARCHITECTURES + +Deep Resnets (TinyImageNet): We sample 500 ResNet architectures and train them on the TinyImageNet† dataset (containing 200 classes with 500 training images of $3 2 \times 3 2$ pixels) for 140 epochs. We vary depths, filter sizes and number of convolutional filter block outputs. Filter sizes are sampled from $\{ 3 , 5 , 7 \}$ and number of filters is sampled from $\{ 2 , 3 , 4 , . . . , 2 2 \}$ . Each ResNet block is composed of three convolutional layers followed by batch normalization and summation layers. We vary the number of blocks from 2 to 18, giving us networks with depths varying between 14 and 110. Each network is trained for 140 epochs, using Nesterov optimizer. The learning rate is set to 0.1 and learning rate reduction and momentum are set to 0.1 and 0.9 respectively. + +Deep Resnets (CIFAR-10): We sample 500 39-layer ResNet architectures from a search space similar to Zoph & Le (2017), varying kernel width, kernel height, and number of kernels. We train these models for 50 epochs on CIFAR-10. Each architecture consists of 39 layers: $1 2 \ : c o n \nu$ , a $2 \mathbf { x } 2$ max pool, 9 conv, a 2x2 max pool, $1 5 \ c o n \nu$ , and softmax. Each conv layer is followed by batch normalization and a ReLU nonlinearity. Each block of 3 conv layers are densely connected via residual connections and also share the same kernel width, kernel height, and number of learnable kernels. Kernel height and width are independently sampled from $\{ 1 , 3 , 5 , 7 \}$ and number of kernels is sampled from $\{ 6 , 1 2 , 2 4 , 3 6 \}$ . Finally, we randomly sample residual connections between each block of conv layers. Each network is trained for 50 epochs using the RMSProp optimizer, with weight decay $1 0 ^ { - 4 }$ , initial learning rate 0.001, and a learning rate reduction to $1 0 ^ { - 5 }$ at epoch 30 on the CIFAR-10 dataset. + +MetaQNN CNNs (CIFAR-10 and SVHN): We sample 1,000 model architectures from the search space detailed by Baker et al. (2017), which allows for varying the numbers and orderings of convolution, pooling, and fully connected layers. The models are between 1 and 12 layers for the SVHN experiment and between 1 and 18 layers for the CIFAR-10 experiment. Each architecture is trained on SVHN and CIFAR-10 datasets for 20 epochs. Table 3 displays the state space of the MetaQNN algorithm. + +Table 3: Experimental State Space For MetaQNN. For each layer type, we list the relevant parameters and the values each parameter is allowed to take. The networks are sampled beginning from the starting layer. Convolutional layers are allowed to transition to any other layer. Pooling layers are allowed to transition to any layer other than pooling layers. Fully connected layers are only allowed to transition to fully connected or softmax layers. A convolutional or pooling layer may only go to a fully connected layer if the current image representation size is below 8. We use this space to both randomly sample and simulate the behavior of a MetaQNN run as well as directly run the MetaQNN with early stopping. + +
Layer TypeLayerParametersParameterValues
Convolution (C)i~Layer depthf~ Receptive field sizel~Strided ~ # receptive fieldsn ~ Representation size<12Square. ∈ {1,3,5}Square.Always equal to 1∈{64,128,256,512}∈{(∞,8],(8,4],(4,1]}
Pooling (P)i~Layer depth(f,l)~ (Receptive field size,Strides)n ~Representation size<12Square.∈{(5,3),(3,2),(2,2)}∈{(∞,8],(8,4] and (4,1]}
Fully Connected (FC)i~Layerdepthn ~ # consecutive FC layersd ～# neurons<12<3∈ {512,256,128}
Termination StateS~Previous Statet~TypeGlobal Avg.Pooling/Softmax
+ +LSTM (PTB): We sample 300 LSTM models and train them on the Penn Treebank dataset for 60 epochs. Number of hidden layer inputs and lstm cells was varied from 10 to 1400 in steps of 20. Each network was trained for 60 epochs with batch size of 50 and trained the models using stochastic gradient descent. Dropout ratio of 0.5 was used to prevent overfitting. Dictionary size of 400 words was used to generate embeddings when vectorizing the data. + +Cuda-Convnet (CIFAR-10 and SVHN): We train Cuda-Convnet architecture (Krizhevsky, 2012) with varying values of initial learning rate, learning rate reduction step size, weight decay for convolutional and fully connected layers, and scale and power of local response normalization layers. We train models with CIFAR-10 for 60 epochs and with SVHN for 12 epochs. Table 4 show the hyperparameter ranges for the Cuda Convnet experiments. + +Table 4: Range of hyperparameter settings used for the Hyperband experiment (Section 4.1) + +
ExperimentHyperparameterScaleMinMax
CIFAR-10, Imagenet, SVHNInitial Learning RateLog5×10-55
Learning Rate ReductionsInteger03
CIFAR-10, SVHNConv1 L2 PenaltyLog5×10-55
Conv2 L2 PenaltyLog5×10-55
Conv3 L2 PenaltyLog5×10-55
FC4 L2 PenaltyLog5×10-55
Response Normalization ScaleLog5×10-65
Response Normalization PowerLinear1×10-23
+ +# B HYPERPARAMETER SELECTION IN PERFORMANCE PREDICTION MODELS + +When training performance prediction models we divided the data into training and validation and used 3-fold cross validation to select optimal hyperparameters. The models were then trained on full training data using the best hyperparameters. For random forests we varied number of trees between 10 and 800, and varied ratio of number of features from 0.1 to 0.5. For $\nu$ -SVR, we perform a random search over 1000 hyperparameter configurations from the space $C \sim$ LogUniform $( 1 0 ^ { - 5 }$ , 10), $\nu \sim$ Uniform $( 0 , 1 )$ , and $\gamma \sim \mathrm { L o g U n i f o r m } ( 1 0 ^ { - 5 } , 1 0$ ) (when using the RBF kernel). For Bayesian linear regression, we perform random search over the hyperparameters of the gamma prior distribution over both noise and weights in the range $[ 1 0 ^ { - 7 } , 1 0 ^ { - 5 } ]$ . + +# C F-HYPERBAND + +Algorithm 1 of this text replicates Algorithm 1 from Li et al. (2017), except we initialize two dictionaries: $D$ to store training data and $M$ to store performance prediction models. $D [ r ]$ will correspond to a dictionary containing all datasets with prediction target epoch $r$ . $D [ r ] [ \bar { \tau } ]$ will correspond to the dataset for predicting $y _ { r }$ based on the observed $y ( t ) _ { 1 - \tau }$ , and $M [ r ] [ \tau ]$ will hold the corresponding performance prediction model. We will assume that the performance prediction model will have a train function, and a predict function that will return the prediction and standard deviation of the prediction. In addition to the standard Hyperband hyperparameters $R$ and $\eta$ , we include $\varDelta$ and $\delta$ described in Section 4 and $\kappa$ . During each iteration of successive halving, we train $n _ { i }$ configurations to $r _ { i }$ epochs; $\kappa$ denotes the fraction of the top $n _ { i }$ models that should be run to the full $r _ { i }$ iterations. This is similar to setting the criterion based on the $n ^ { \mathrm { t h } }$ best model in the previous section. + +We also detail the run_then_return_validation_loss function in Algorithm 2. This algorithm runs a set of configurations, adds training data from observed learning curves, trains the performance prediction models when there is enough training data present, and then uses the models to terminate poor configurations. It assumes we have a function max_k, which returns the $k ^ { \mathrm { { t h } } }$ max value or $- \infty$ if the list has less than $k$ values. + +# Algorithm 1: f-Hyperband + +input : $R$ (Max resources allocated to any configuration) η (default $\eta = 3 ,$ ) $\varDelta$ (Probability threshold for early termination) $\delta$ (Performance offset for early termination) $d$ (# points required to train performance predictors) $\kappa$ (Proportion of models to train) initialize : $D = \mathrm { d i c t ( ) }$ M = dict() smax = blogη(R)c $B = ( s _ { \mathrm { m a x } } + 1 ) R$ 1 for 2 T $\begin{array} { r } { n = \lceil \frac { B } { R } \frac { \eta ^ { s } } { s + 1 } \rceil , \quad r = R \eta ^ { - s } } \end{array}$ $s \in \{ s _ { m a x } , \ldots , 0 \}$ do 3 // begin SUCCESSIVEHALVING with $\left( n , \begin{array} { l l } { r } \end{array} \right)$ inner loop 4 $T = { \mathfrak { g e t } }$ _hyperparameter_configuration( $: n \mid$ ) for $i \in \{ 0 , \ldots , s \}$ do 6 $n _ { i } = \lfloor n \eta ^ { - i } \rfloor$ , $r _ { i } = r \eta ^ { i }$ 7 $\begin{array} { r } { n _ { \mathrm { n e x t } } = \lfloor \frac { n _ { i } } { \eta } \rfloor } \end{array}$ if $i ! = s$ else 1 8 L = run_then_return_validation_loss $( T , r _ { i } , n _ { \mathrm { n e x t } } , D , M )$ 9 $\begin{array} { r } { T = \mathrm { t o p \_ k } \left( T , L , \lfloor \frac { n _ { i } } { \eta } \rfloor \right) } \end{array}$ ) 0 end + +11 end + +input : T hyperparameter configurations +r resources to use for training +$n$ # configurations in next iteration of successive halving +$D$ dictionary storing training data +$M$ – dictionary storing performance prediction models +initialize : $L = [ ]$ +1 for $t \in T$ do +2 $\ell = \left[ \ \right]$ +3 for $i \in \{ 0 , \ldots , r - 1 \}$ do +4 $\ell _ { i } =$ run_one_epoch_return_validation_loss(t) +5 $\ell$ .append $( \ell _ { i }$ ) +6 if $M [ r ] [ i ]$ .trained() then +7 ${ \hat { y } } _ { r }$ , $\dot { \sigma } = M [ r ] [ i ]$ .predict $( \ell )$ +8 if $\begin{array} { r } { \varPhi ( \mathrm { m a x \_ k } ( \bar { L } , \kappa n ) - \delta ; \hat { y } _ { r } , \sigma ) \geq \varDelta } \end{array}$ then +9 $L$ .append $( \hat { y } _ { r }$ ) +10 break +11 end +12 end +13 else if $i = = r - 1$ then +14 $L$ .append $( \ell _ { i }$ ) +15 end +16 end +17 if length $( D [ r ] [ 0 ] ) < d$ and length $( { \boldsymbol { \ell } } ) \ = = r$ then +18 $\{ D [ r ] [ i ]$ .append $\langle \ell [ 0 , \ldots , i ] , \ell [ r ] \} \rangle : i \in \{ 0 , \ldots , r - 1 \} \}$ +19 if not $M [ r ] [ i ]$ .trained() then +20 $M [ r ] [ i ]$ .train $( D [ r ] [ i ] ,$ ) +21 end +22 end +23 end +24 return L + +# D F-HYPERBAND WITH SVR ACQUISITION FUNCTION + +Similar to Klein et al. (2017), it is possible to use our simple models as an acquisition function for Hyperband. As shown in Table 2, our models have reasonable performance with only AP and HP features used, meaning we can use these models to rank hyperparameter settings before training the configurations. In particular, for each Hyperband iteration, we randomly select 10,000 configurations, weight them by their predicted performance and sample from this weighted distribution. In Figure 7, we compare f-Hyperband with this $\nu$ -SVR acquisition function to f-Hyperband with the standard uniform acquisition function and also to vanilla Hyperband. It seems that the acquisition function could help slightly, but it does not improve performance across all experiments, and where it does improve performance it does so only slightly. + +![](images/cd7670b52afd22f5572cddfea1b647b61e4c33d65e54d6765e5cede1a1f1cd49.jpg) +Figure 7: f-Hyperband with SVR Acquisition Function: We compare f-Hyperband with an SVR acquisition function to f-Hyperband with the uniform acquisition function (this is the standard version described in the main text) and to vanilla Hyperband. The results shown are inconclusive; it seems that there could be some benefit of incorporating the SVR acquisition function, but it isn’t significant enough to make a strong statement. + +# E GAUSSIAN ERROR ASSUMPTION + +In order to have uncertainty estimates with frequentist models, we estimated the mean and variance of the error from leave-one-out cross validation and assumed that the error was Gaussian. In Figure 8 we compare example error distributions between training and validation from a $\nu$ -SVR SRM. Visually one can see that the assumption holds very well for both visual classification tasks and holds slightly less well for the language modeling task. In Figure 9, we show the mean log likelihood of held out prediction errors being drawn from a Gaussian parameterized by the mean and variance of the error on the training set is very close to the mean log likelihood of samples drawn from the same distribution, which shows that our Gaussian error assumption is relatively strong. + +# F ANALYZING IMPORTANCE OF FEATURES IN THE PREDICTION MODEL + +We used a linear $\nu$ -SVR model to compare the weights of all features, which are normalized before training. Figure 10 shows the weight trends for time-series features, and Tables 5, 6, and 7 compare weights of some architecture features and hyperparameter features to statistics of time-series features for the MetaQNN (CIFAR-10), LSTMs (Penn Treebank), and Deep Resnets (TinyImagenet) experiments, respectively. We found the following main insights across datasets: + +• The time-series (TS) features are on average have higher weights than HP and AP features (which is confirmed by our ablation studies in Table 2 as well). The original validation accuracies $( y _ { t } )$ have higher weights on average than the first-order differences $( { y } _ { t } ^ { \prime } )$ and second-order differences $( { y } _ { t } ^ { \prime \prime } )$ . • In general, later epochs in $y _ { t }$ have higher weights. In other words, the latest performance of the model available for prediction is much more important for performance prediction than initial performance, when the model has just started training. However, the weights for earlier performance metrics are non-zero, indicating there are high order autoregressive functions being learned. + +![](images/6ab4eee391e270952d9a993a07231dd2f41ebb64efe1a472221b93e477a96b83.jpg) +Figure 8: Example Error Distributions: Here we compare example error distributions between training and validation from a $\nu$ -SVR SRM. From left to right we see plots for models trained with increasing amounts of the learning curve. The blue line shows the Gaussian probability density function estimated from the training set with LOOCV. The orange histogram shows the error distribution of the held out validation set. + +![](images/10a7774b17a92c4e92355b15a9b04bb03ddd75fa3f8c268d6042658e62a1507b.jpg) +Figure 9: Error Log Likelihood: In orange we show the mean log likelihood of prediction errors measured from validation sets being drawn from the Gaussian distribution parameterized by the mean and variance of the training error, again measured with LOOCV. For a baseline, we show the mean log likelihood for 1,000,000 samples drawn from the same Gaussian. We calculate these metrics over 10 data splits and report the mean and standard error. + +• Early values of $( { y } _ { t } ^ { \prime } )$ also have high weights, which indicates learning quickly in the beginning of training is a predictor of better final performance in our datasets. + +• Among AP and HP features, the total number of parameters and depth have non-zero weight for the CNNs, and they are assigned weights comparable or higher than late epoch accuracies $( y _ { t } )$ . However, they have much lower weight in the LSTM experiment. The number of filters in each convolutional layer also has a high positive weight for CNNs. In general, architectural features are much more important for CNNs as compared to LSTMs. Hyperparameters like initial learning rate and step size were generally not as important as the architectural features, which is corroborated by the ablation study in Table 1 in the main text. + +![](images/b2971eb1174e30d287fd59999deb67712ba2523822f222b2dc7573d44e9acc4c.jpg) +Figure 10: Time Series Weights: We show the trends for weights for time-series features, along with first- and second-order differences. + +Table 5: Summary of weights for a linear kernel $\nu$ -SVR SRM trained on configurations from MetaQNN (CIFAR-10). + +
PercentObservedMeanPerfStd DevPerfMean 1stOrder DiffMean 2ndOrder DiffDepth#ParamsLearningRate
100.0229890.0138450.0170690.0090210.0146380.0055410.005355
200.0169710.0108820.0101170.0034940.0081530.0034080.005335
300.0137330.0073960.0069210.0025500.0041990.0020980.003310
400.0111250.0070230.0050310.0019030.0028470.0030410.002968
500.0094090.0058360.0045940.0012340.0005780.0002230.001107
600.0080990.0047680.0033700.0009230.001306-0.0009060.000287
700.0070160.0042460.0025730.0006270.000458-0.000777-0.000324
800.0061900.0034170.0021590.0004940.0004540.0003550.000583
900.0055530.0031410.0017920.0003710.0003820.0000200.000259
+ +Table 6: Summary of weights for a linear kernel $\nu$ -SVR SRM trained on configurations from LSTMs (Penn Treebank). + +
PercentObservedMeanPerfStd DevPerfMean 1stOrder DiffMean 2ndOrder DiffDepth# ParamsLearningRate Step
101.129178-0.9527122.5625840.822748-2.6087090.599052-1.940918
200.388479-0.1687980.8283430.287950-1.1486190.390338-0.667870
300.202656-0.2540580.2821630.132048-1.134044-0.033193-0.301467
400.149271-0.2058680.1839590.093839-0.2806460.050759-0.057207
500.129166-0.1600910.1272300.078865-0.219122-0.044931-0.119674
600.116550-0.1604100.1225570.085949-0.1288210.040022-0.097907
700.112243-0.1442780.1042090.078584-0.082918-0.077808-0.057439
800.102105-0.1274180.0783280.068932-0.0625060.054454-0.006713
900.106653-0.0673690.1418870.0745150.079686-0.1695340.158086
+ +Table 7: Summary of weights for a linear kernel $\nu$ -SVR SRM trained on configurations from Deep Resnets (TinyImagenet). + +
PercentObservedMeanPerfStd DevPerfMean 1stOrder DiffMean 2ndOrder DiffDepth#ParamsKernelSize
100.0084860.0085760.0018610.0015020.0059850.0109870.000267
200.0047230.0048510.0010920.0009960.0017090.0052390.002484
300.0032290.0028480.0007890.0006190.0045330.002553-0.001138
400.0023940.0029850.0006600.0005570.0034430.001092-0.000302
500.0017540.0019930.0004610.0003960.0013180.0010810.000072
600.0014560.0019460.0003700.0003090.0009680.0005590.000036
700.0014280.0026780.0004990.0003440.001161-0.000590-0.000971
800.0011680.0018950.0003520.0002560.0004550.000131-0.001092
900.0010150.0016400.0002850.0002150.000208-0.000091-0.000958
+ +# G ADDITIONAL PLOTS + +Figure 11 shows the performance prediction results for the Cuda Convnet SVHN experiment. The instability of BLR with low percentage of learning curve observed may be what caused suboptimal performance of BLR f-Hyperband shown in Figure 6. Figure 12 shows the relative speedup of f-Hyperband over vanilla Hyperband for each consecutive iteration of Hyperband. However, Figure 13, which shows results for f-Hyperband with pretrained $\nu$ -SVR SRMs, shows that more aggressive settings of f-Hyperband result in suboptimal performance. We see that we can safely get a $3 { - } 4 \mathbf { x }$ speedup over vanilla Hyperband with pretrained SRMs. + +![](images/c750bab18bbb20f6ff9821a6eabe408b08df4b123f9375788a4aa1f50624cd94.jpg) +Figure 11: Cuda Convnet SVHN Performance Prediction Results: We plot the performance of each method versus the percent of learning curve observed for the Cuda Convnet SVHN experiment. For BNN, $\nu$ -SVR, and BLR we sample 10 different training sets, plot the mean $R ^ { 2 }$ , and shade the corresponding standard error. We compare our method against BNN (Klein et al., 2017), LCE (Domhan et al., 2015), and a “last seen value” heuristic (Li et al., 2017). Absent results for a model indicate that it did not achieve a positive $R ^ { 2 }$ . + +![](images/893b61c3e8dee896d83890cfd6f959ee2023596c2286f47916b4990e7a40b892.jpg) +Figure 12: Simulated Speedup on Hyperband vs Hyperband Iteration: We show the speedup using the f-Hyperband algorithm over Hyperband on 40 consecutive runs with $\eta = 4 . 0$ and $\varDelta = 0 . 9 5$ . The major jump in speedup comes at iteration 10, where we have trained more than 100 models to the full $R$ iterations. + +![](images/e9cd3871458632935a702159971de0403ccd0159d69e2a3b0451acad4716e165.jpg) +Figure 13: Simulated f-Hyperband with Pretrained SRMs: Each f-Hyperband model is initialized with a pretrained SRM, which simulates the case where one reinitializes an experiment. Every triangle indicates a full run of Hyperband, and small vertical lines indicate the point at which the f-Hyperband runs have completed the same number of full Hyperband runs as shown for vanilla Hyperband. Top 1 indicates only training the best model to $r _ { i }$ iterations, and $\delta$ represents an offset off the threshold used for early stopping (See Section 4).More aggressive f-Hyperband settings lead to suboptimal performance, indicating that there is a limit to how much speedup f-Hyperband can achieve. + +# H SRM ROBUSTNESS TO OUT OF DISTRIBUTION CONFIGURATIONS + +Table 8 shows that SRMs are relatively robust when training below or above the median of a specific architecture or training hyperparameter and testing on the other split. + +
Cuda Convnet (CIFAR-10)Learning RateWeightDecayLRN ScaleLRN Power
Test<Median93.06 ± 0.1895.93 ± 0.2197.27 ± 0.006896.96 ± 0.13
Test>Median81.97 ± 1.2696.19± 0.1393.37 ± 0.2395.05 ± 0.19
+ +
LSTMs (Penn Treebank)Number of NodesDepth
Test<Median98.08 ± 0.2698.58 ± 0.04
Test>Median22.83 ± 5.5596.96 ± 0.91
+ +
DeepResnets (TinyImagenet)DepthNumber ofKernelsKernel Size
Test<Median89.49 ± 1.8485.12 ± 2.2481.83 ± 0.59
Test >Median85.92 ± 1.7862.37 ± 0.8289.61 ± 0.31
+ +Table 8: Accuracy when splitting test and train set based on median values for hyperparameters for different datasets. + +
MetaQNN (CIFAR-10)Depth
Test<Median96.69 ± 0.16
Test>Median79.31 ± 5.66
\ No newline at end of file diff --git a/parse/train/BJypUGZ0Z/BJypUGZ0Z_content_list.json b/parse/train/BJypUGZ0Z/BJypUGZ0Z_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..4e81a670d09afc1c1d9a50e6249e3b3dd95c4fda --- /dev/null +++ b/parse/train/BJypUGZ0Z/BJypUGZ0Z_content_list.json @@ -0,0 +1,1556 @@ +[ + { + "type": "text", + "text": "ACCELERATING NEURAL ARCHITECTURE SEARCH USING PERFORMANCE PREDICTION ", + "text_level": 1, + "bbox": [ + 174, + 99, + 825, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Bowen Baker∗, Otkrist Gupta∗, Ramesh Raskar ", + "bbox": [ + 183, + 169, + 526, + 184 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Nikhil Naik ", + "text_level": 1, + "bbox": [ + 580, + 170, + 661, + 183 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Media Laboratory Massachusetts Institute of Technology Cambridge, MA 02142, USA {bowen,otkrist,raskar}@mit.edu ", + "bbox": [ + 184, + 185, + 478, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Faculty of Arts and Sciences Harvard University Cambridge, MA 02138, USA naik@fas.harvard.edu ", + "bbox": [ + 578, + 185, + 776, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 276, + 544, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Methods for neural network hyperparameter optimization and architecture search are computationally expensive due to the need to train a large number of model configurations. In this paper, we show that simple regression models can predict the final performance of partially trained model configurations using features based on network architectures, hyperparameters, and time-series validation performance data. We empirically show that our performance prediction models are much more accurate than prominent Bayesian counterparts, are simpler to implement, and are faster to train. Our models can predict final performance in both visual classification and language modeling domains, are effective for predicting performance of drastically varying model architectures, and can even generalize between model classes. Using these prediction models, we also implement an early stopping method for hyperparameter optimization and architecture search, which obtains a speedup of a factor up to 6x in both hyperparameter optimization and architecture search. Finally, we empirically show that our early stopping method can be seamlessly incorporated into both reinforcement learning-based architecture selection algorithms and bandit based search methods. Through extensive experimentation, we empirically show our performance prediction models and early stopping algorithm are state-of-the-art in terms of prediction accuracy and speedup achieved while still identifying the optimal model configurations. ", + "bbox": [ + 233, + 309, + 766, + 573 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 602, + 336, + 617 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Significant human expertise and labor is required for designing high-performing neural network architectures and successfully training them for different applications. Ongoing research in two areas—architecture search and hyperparameter optimization—attempts to reduce the amount of human intervention required for these tasks. Hyperparameter optimization methods (e.g., Hutter et al. (2011); Snoek et al. (2015); Li et al. (2017)) focus primarily on obtaining good optimization hyperparameter configurations for training human-designed networks, whereas architecture search algorithms (Bergstra et al., 2013; Verbancsics & Harguess, 2013; Baker et al., 2017; Zoph & Le, 2017) aim to design neural network architectures from scratch. Both sets of algorithms require training a large number of neural network configurations for identifying the right set of hyperparameters or the right network architecture and are hence computationally expensive. ", + "bbox": [ + 174, + 627, + 825, + 767 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "When sampling many different model configurations, it is likely that many subpar configurations will be explored. Human experts are quite adept at recognizing and terminating suboptimal model configurations by inspecting their partial learning curves. In this paper we seek to emulate this behavior and automatically identify and terminate subpar model configurations in order to speedup both architecture search and hyperparameter optimization methods. Our method parameterizes learning curve trajectories with simple features derived from model architectures, training hyperparameters, and early time-series measurements from the learning curve. We use these features to train a set of simple regression models that predict the final validation performance of partially trained neural network configurations using a small training set of fully trained curves, and we empirically validate our method in both image classification and language modeling domains. We use these predictions along with uncertainty estimates obtained from small model ensembles to construct a simple early stopping algorithm that can speedup both architecture search and hyperparameter optimization methods. ", + "bbox": [ + 174, + 773, + 825, + 898 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/25baeeba5fe8e2faa370c9042f700f36b0568f1611f8d41291f30bb7417b7ef6.jpg", + "image_caption": [ + "Figure 1: Example Learning Curves: Example learning curves from experiments considered in this paper. Note the diversity in convergence times and overall learning curve shapes. " + ], + "image_footnote": [], + "bbox": [ + 178, + 103, + 818, + 231 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 303, + 820, + 344 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While there is some prior work on neural network performance prediction using Bayesian methods (Domhan et al., 2015; Klein et al., 2017), our proposed method is significantly more accurate, accessible, and efficient. We hope that our work leads to inclusion of neural network performance prediction and early stopping in the practical neural network training pipeline. ", + "bbox": [ + 174, + 352, + 825, + 407 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 431, + 344, + 446 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Neural Network Performance Prediction: There has been limited work on predicting neural network performance during training. Domhan et al. (2015) introduce a weighted probabilistic model for learning curves and utilize this model for speeding up hyperparameter search in small convolutional neural networks (CNNs) and fully-connected networks (FCNs). Building on Domhan et al. (2015), Klein et al. (2017) train Bayesian neural networks for predicting unobserved learning curves using a training set of fully and partially observed learning curves. Both methods rely on expensive Markov chain Monte Carlo (MCMC) sampling procedures and handcrafted learning curve basis functions. We also note that Swersky et al. (2014) develop a Gaussian Process kernel for predicting individual learning curves, which they use to automatically stop and restart configurations. ", + "bbox": [ + 174, + 458, + 825, + 583 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Architecture Search: We define architecture search as an algorithmic approach for designing neural network architectures from scratch. The earliest architecture search approaches were based on genetic algorithms (Schaffer et al., 1992; Stanley & Miikkulainen, 2002; Verbancsics & Harguess, 2013) or Bayesian optimization (Bergstra et al., 2013; Shahriari et al., 2016). More recently, reinforcement learning methods have become popular. Baker et al. (2017) use Q-learning to design competitive CNNs for image classification. Zoph & Le (2017) use policy gradients to design state-of-the-art CNNs and recurrent cell architectures. Several methods for architecture search (Cortes et al., 2017; Negrinho & Gordon, 2017; Zoph et al., 2017; Brock et al., 2017; Suganuma et al., 2017) have been proposed this year since the publication of Baker et al. (2017) and Zoph & Le (2017). ", + "bbox": [ + 173, + 590, + 825, + 715 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Hyperparameter Optimization: We define hyperparameter optimization as an algorithmic approach for finding optimal values of architecture independent hyperparameters such as learning rate and batch size, along with a limited search through the network design space. Bayesian hyperparameter optimization methods include those based on sequential model-based optimization (SMAC) (Hutter et al., 2011), Gaussian processes (GP) (Snoek et al., 2012), TPE (Bergstra et al., 2013), and neural networks Snoek et al. (2015). However, random search or grid search is most commonly used in practical settings (Bergstra & Bengio, 2012). Recently, Li et al. (2017) introduced Hyperband, a multiarmed bandit-based efficient random search technique that outperforms state-of-the-art Bayesian optimization methods. ", + "bbox": [ + 174, + 722, + 825, + 848 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 NEURAL NETWORK PERFORMANCE PREDICTION ", + "text_level": 1, + "bbox": [ + 174, + 871, + 612, + 887 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We first describe our model for neural network performance prediction, followed by a description of the datasets used to evaluate our model, and finally present experimental results. ", + "bbox": [ + 173, + 898, + 823, + 928 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3.1 MODELING LEARNING CURVES ", + "text_level": 1, + "bbox": [ + 176, + 103, + 434, + 117 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our goal is to model the validation performance $y _ { T }$ of a neural network configuration $\\mathbf { x } \\in \\mathcal { X } \\subset \\mathbb { R } ^ { d }$ at epoch $T \\in \\mathbb { Z } ^ { + }$ using previous performance observations $y ( t )$ . For each configuration $\\mathbf { x }$ trained for $T$ epochs, we record a time-series $y ( T ) = y _ { 1 } , y _ { 2 } , \\dotsc , y _ { T }$ of validation performances. We train a population of $n$ configurations, obtaining a set $\\tilde { \\mathcal { S } } = \\{ ( \\mathbf { x } ^ { 1 } , y ^ { 1 } ( T ) ) , ( \\mathbf { x } ^ { 2 } , y ^ { 2 } ( T ) ) , \\ldots , ( \\mathbf { x } ^ { n } , y ^ { n } ( T ) ) \\} .$ This problem formulation is very similar to Klein et al. (2017). Note that most architecture and hyperparameter search methods naturally collect $s$ . ", + "bbox": [ + 173, + 126, + 825, + 209 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We propose to use a set of features $u _ { \\mathbf { x } }$ , derived from the neural network configuration $\\mathbf { x }$ , along with a subset of time-series performances $y ( \\tau ) = ( y _ { t } ) _ { t = 1 , 2 , . . . , \\tau }$ (where $1 \\leq \\tau < T )$ ) from $s$ to train a regression model for estimating $y _ { T }$ . We use models without natural support for variable size input, so we train $T - 1$ regression models, where each successive model uses one more point of the time-series validation data. As we shall see in subsequent sections, this use of sequential regression models (SRM) is more computationally efficient and more precise than methods that train a single model. ", + "bbox": [ + 173, + 217, + 825, + 300 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Features: We use features based on time-series (TS) validation performances, architecture parameters (AP), and hyperparameters (HP). (1) TS: Assume we are training the $\\tau ^ { \\mathrm { { t h } } }$ model in the SRM. TS features include the validation performances $y ( \\tau ) = ( y _ { t } ) _ { t = 1 , 2 , . . . , \\tau }$ , the first-order differences of validation performances, i.e. $y _ { t } ^ { \\prime } = \\left( y _ { t } - y _ { t - 1 } \\right)$ , the second-order differences of validation performances, i.e. ${ y _ { t } } ^ { \\prime \\bar { \\prime } } = ( { y _ { t } } ^ { \\prime } - { y _ { t - 1 } ^ { \\prime } } )$ , and mean and standard deviation of performances. (2) AP: These include total number of weights and number of layers. (3) HP: These include all hyperparameters used for training the neural networks, e.g. initial learning rate and learning rate decay (full list in Appendix Table 4). ", + "bbox": [ + 173, + 306, + 825, + 419 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 DATASETS AND TRAINING PROCEDURES ", + "text_level": 1, + "bbox": [ + 174, + 441, + 495, + 457 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Figure 1 shows example learning curves from three of the datasets considered in our experiments. We provide brief summary of the datasets below. Please see Appendix Section A for further details on the search space, preprocessing, hyperparameters and training settings of all datasets. ", + "bbox": [ + 174, + 462, + 825, + 505 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "DATASETS WITH VARYING ARCHITECTURES: ", + "bbox": [ + 178, + 526, + 486, + 540 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Deep Resnets (TinyImageNet): We sample 500 ResNet architectures and train them on the TinyImageNet\\* dataset (containing 200 classes with 500 training images of $3 2 \\times 3 2$ pixels) for 140 epochs. We vary depths, filter sizes and number of convolutional filter block outputs. The network depths vary between 14 and 110. ", + "bbox": [ + 174, + 553, + 825, + 608 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Deep Resnets (CIFAR-10): We sample 500 39-layer ResNet architectures from a search space similar to Zoph & Le (2017), varying kernel width, kernel height, and number of kernels. We train these models for 50 epochs on CIFAR-10. ", + "bbox": [ + 176, + 616, + 823, + 659 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "MetaQNN CNNs (CIFAR-10 and SVHN): We sample 1,000 model architectures from the search space detailed by Baker et al. (2017), which allows for varying the numbers and orderings of convolution, pooling, and fully connected layers. The models are between 1 and 12 layers for the SVHN experiment and between 1 and 18 layers for the CIFAR-10 experiment. Each architecture is trained on SVHN and CIFAR-10 datasets for 20 epochs. ", + "bbox": [ + 174, + 665, + 825, + 736 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "LSTM (PTB): We sample 300 LSTM models and train them on the Penn Treebank dataset for 60 epochs, evaluating perplexity on the validation set. We vary number of LSTM cells and hidden layer inputs between 10-1400. ", + "bbox": [ + 174, + 742, + 823, + 784 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "DATASETS WITH VARYING HYPERPARAMETERS: ", + "text_level": 1, + "bbox": [ + 174, + 806, + 509, + 820 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Cuda-Convnet (CIFAR-10 and SVHN): We train Cuda-Convnet architecture (Krizhevsky, 2012) with varying values of initial learning rate, learning rate reduction step size, weight decay for convolutional and fully connected layers, and scale and power of local response normalization layers. We train models with CIFAR-10 for 60 epochs and with SVHN for 12 epochs. ", + "bbox": [ + 174, + 833, + 825, + 888 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/860755a4c53e29ddffe1564634ed76c1a0fb098fc5cd9dd145716f7f85631c91.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Datasetv-SVROLSBLRRF
MetaQNN (CIFAR-10)97.22 ± 0.3197.04± 0.4297.11 ± 0.3294.50 ± 0.78
Resnet (TinyImageNet)87.12 ± 4.6588.41 ± 3.7682.40± 1.6586.39 ± 3.58
LSTM (Penn Treebank)98.03± 1.4687.83 ± 21.5397.99 ± 1.9057.28± 24.98
", + "bbox": [ + 187, + 101, + 810, + 161 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Table 1: Performance Prediction Model Comparison: We report the coefficient of determination $R ^ { 2 } * 1 0 0$ for four standard methods. Each model is trained with 100 samples on $2 5 \\%$ of the learning curve. We find that $\\nu$ -SVR works best on average, though not by a large margin. ", + "bbox": [ + 176, + 171, + 825, + 214 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/e1e08d693f58b2451fc566e9d907921036494e900edf6b2beed895ec69970c9e.jpg", + "image_caption": [ + "Figure 2: Predicted vs True Values of Final Performance: We show the shape of the predictive distribution on three experiments: MetaQNN models, Deep Resnets, and LSTMs. Each $\\nu$ -SVR model is trained with 100 configurations with data from $2 5 \\%$ of the learning curve. We predict validation set classification accuracy for MetaQNN and Deep ResNets, and perplexity for LSTMs. " + ], + "image_footnote": [], + "bbox": [ + 209, + 224, + 789, + 400 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 PREDICTION PERFORMANCE ", + "text_level": 1, + "bbox": [ + 176, + 493, + 413, + 507 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Choice of Regression Method: We now describe our results for predicting final neural network performance. For all experiments, we train our SRMs on 100 randomly sampled neural network configurations. We obtain the best performing method using random hyperparameter search over 3-fold cross-validation. We then compute the regression performance over the remainder of the dataset using the coefficient of determination $R ^ { 2 }$ . We repeat each experiment 10 times and report the results with standard errors. We experiment with a few simple regression models, including ordinary least squares (OLS), random forests (RF), Bayesian linear regression (BLR), and $\\nu$ -support vector machine regression ( $\\nu$ -SVR). For both OLS and $\\nu \\cdot$ -SVR, we choose between the linear and RBF kernels based on prediction performance—after hyperparameter search, in case of $\\nu$ -SVR. We chose 100 as the number of sampled configurations somewhat arbitrarily, though we found that prediction performance was quite poor below 50 and improves steadily after. As seen in Table 1, $\\nu$ -SVR performs the best on most datasets, though not by a large margin. For the rest of this paper, we use $\\nu$ -SVR unless otherwise specified. ", + "bbox": [ + 174, + 520, + 825, + 700 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Ablation Study on Feature Sets: In Table 2, we compare the predictive ability of different feature sets, training SVR with time-series (TS) features obtained from $2 5 \\%$ of the learning curve, along with features of architecture parameters (AP), and hyperparameters (HP). TS features explain the largest fraction of the variance in all cases and found that in general AP are more important that HP. AP features almost match TS on the ResNet (TinyImageNet) dataset, indicating that choice of architecture has a large influence on accuracy for ResNets. Figure 2 shows the true vs. predicted performance for all test points in three datasets, trained with TS, AP, and HP features. ", + "bbox": [ + 173, + 708, + 825, + 805 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Generalization To Out Of Distribution Configurations: We also test to see whether our SRMs can accurately predict the performance of out-of-distribution neural networks. For instance, if we train a $\\nu$ -SVR model on the Deep Resnet (TinyImagenet) dataset at $2 5 \\%$ of the learning curve observed on configurations below median depth (60 layers), we can achieve $R ^ { 2 } = 0 . 8 6$ on configurations above median depth. The corresponding number for prediction below median depth is 0.89. We repeated this experiment all datasets with different hyperparameter values below/above the median and found that SRMs generally showed consistent performance across such splits. Full results can be found in Appendix Section $_ \\mathrm { H }$ . ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/7436f624fbd0b0b90cbd5af810fd267f1d00fd523372aff399deedbb30b95960.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Feature SetMetaQNN(CIFAR-10)ResNets(TinyImageNet)LSTM(Penn Treebank)Cuda-Convnet(CIFAR-10)
HP-2.96 ± 2.710.48 ± 7.24-1.72 ± 1.620.48 ± 7.24
AP15.38 ± 4.6667.66 ± 9.0813.76 ± 12.0867.66 ± 9.08
TS97.00 ± 0.5282.71 ± 4.3197.38 ± 1.4982.71 ± 4.31
AP+HP14.89 ± 5.1270.25 ± 6.4213.76 ± 12.0870.25 ± 6.42
TS+HP97.02 ± 0.5282.67 ± 6.8197.38 ± 1.4982.67± 6.81
TS+AP97.12 ± 0.3787.36± 4.2397.87 ± 1.6887.36± 4.23
TS+AP+HP97.15 ± 0.4886.05 ± 8.2497.87 ± 1.6886.05 ± 8.24
", + "bbox": [ + 207, + 101, + 789, + 232 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Table 2: Ablation Study on Feature Sets: We report the $R ^ { 2 } * 1 0 0$ metric for different combinations of features. Time-series features (TS) refers to the partially observed learning curves, architecture parameters (AP) refer to the number of layers and number of weights in a deep model, and hyperparameters (HP) refer to the optimization parameters such as learning rate. All results are with $\\nu$ -SVR. $2 5 \\%$ of learning curve used for models including TS features. ", + "bbox": [ + 173, + 242, + 828, + 313 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3.1 COMPARISON WITH EXISTING METHODS: ", + "text_level": 1, + "bbox": [ + 174, + 340, + 517, + 354 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now compare the neural network performance prediction ability of SRMs with three existing learning curve prediction methods: (1) Bayesian Neural Network (BNN) (Klein et al., 2017), (2) the learning curve extrapolation (LCE) method (Domhan et al., 2015), and (3) the last seen value (LastSeenValue) heuristic (Li et al., 2017). When training the BNN, we not only present it with the subset of fully observed learning curves but also all other partially observed learning curves from the training set. While we do not present the partially observed curves to the SRMs for training, we felt this was a fair comparison as an SRM uses the entire partially observed learning curve during inference. Methods (2) and (3) do not incorporate prior learning curves during training. Figure 3 shows the $R ^ { 2 }$ obtained by each method for predicting the final performance versus the percent of the learning curve used for training the model. We see that in all neural network configuration spaces and across all datasets, $\\nu { - } \\mathbf { S } \\mathbf { V } \\mathbf { R }$ outperform the competing methods. Bayesian linear regression (BLR) also performs comparably well to $\\nu$ -SVR and better than the more complex Bayesian methods (BNN and LCE). The LastSeenValue heuristic only becomes viable when the configurations are near convergence, and its performance is worse than an SRM for very deep models. We also find that the SRMs outperform the LCE method in all experiments, even after we remove a few extreme prediction outliers produced by LCE. Finally, while BNN outperforms the LastSeenValue and LCE methods when only a few iterations have been observed, it does worse than our proposed method. In summary, we show that our simple SRMs outperforms existing Bayesian approaches on predicting neural network performance on modern, very deep models in computer vision and language modeling tasks. ", + "bbox": [ + 173, + 362, + 825, + 640 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Since most of our experiments perform stepwise learning rate decay; it is conceivable that the performance gap between SRMs and both LCE and BNN results from a lack of sharp jumps in their basis functions. For completeness, we also experimented with exponential learning rate decay (ELRD), which the basis functions in LCE are designed for. We trained 630 random nets with ELRD, from the 1000 MetaQNN-CIFAR10 nets. Predicting from $2 5 \\%$ of the learning curve, the $R ^ { 2 }$ is 0.95 for $\\nu$ -SVR, 0.48 for LCE (with extreme outlier removal, negative without), and 0.31 for BNN. This comparison illuminates another benefit of our method: we do not require handcrafted basis functions to model new learning curve types. ", + "bbox": [ + 174, + 647, + 825, + 758 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Moreover, SRMs are faster to train and do inference in than LCE and BNN. On 1 core of a Intel $6 7 0 0 \\mathrm { k }$ CPU, an $\\nu$ -SVR with 100 training points trains in 0.006 seconds, and each inference takes 0.00006 seconds. In comparison, the LCE code takes 60 seconds and BNN code takes 0.024 seconds on the same hardware for each inference. ", + "bbox": [ + 174, + 766, + 825, + 821 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 APPLYING PERFORMANCE PREDICTION FOR EARLY STOPPING ", + "text_level": 1, + "bbox": [ + 176, + 842, + 727, + 858 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To speed up hyperparameter optimization and architecture search methods, we develop an algorithm to determine whether to continue training a partially trained model configuration using our sequential regression models. This approach follows the idea proposed by Domhan et al. (2015) and further expanded on by Klein et al. (2017). Concretely, if we would like to sample $N$ total neural network configurations, we begin by sampling and training $n \\ll N$ configurations to create a training set $s$ . We then train a model $f ( x _ { f } )$ , where in our case $f$ is an SRM with $T - 1$ individual models, to predict $y _ { T }$ Now, given the current best performance observed $y _ { \\mathrm { B E S T } }$ , we would like to terminate training a new configuration $\\mathbf { x } _ { \\mathrm { N E W } }$ at iteration $\\tau$ given its partial learning curve $y _ { \\mathrm { N E W } } ( \\tau )$ if $f ( x _ { \\mathrm { { N E W } } } f ) = \\hat { y } _ { T } \\overset { \\cdot } { \\leq } y _ { \\mathrm { { B E S T } } }$ so as to not waste computational resources exploring a suboptimal configuration. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/b307684c90185a1bcebe23b1c8b3d267a9a5c28ed548460d497c2352fbae0626.jpg", + "image_caption": [ + "Figure 3: Performance Prediction Results: We plot the performance of each method versus the percent of learning curve observed. For BNN and $\\nu$ -SVR, we sample 10 different training sets, plot the mean $R ^ { 2 }$ , and shade the corresponding standard error. We compare our method against BNN (Klein et al., 2017), LCE (Domhan et al., 2015), and a “last seen value” heuristic (Li et al., 2017). Absent results for a model indicate that it did not achieve a positive $R ^ { 2 }$ . The results for Cuda-Convnet on the SVHN dataset are shown in Appendix Figure 11. " + ], + "image_footnote": [], + "bbox": [ + 174, + 104, + 818, + 361 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 489, + 825, + 560 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "However, in the case $f$ has poor out-of-sample generalization, we may mistakenly terminate the optimal configuration. If we assume that our estimate can be modeled as a Gaussian perturbation of the true value $\\hat { y } _ { T } \\sim \\mathcal { N } ( y _ { T } , \\sigma ( \\mathbf { x } , \\tau ) )$ , then we can find the probability $\\begin{array} { r } { p ( \\hat { y } _ { T } \\leq y _ { \\mathrm { B E S T } } \\mathbf { \\bar { \\Phi } } | \\sigma ( \\mathbf { x } , \\tau ) ) = } \\end{array}$ $\\varPhi ( y _ { \\mathrm { B E S T } } ; y _ { T } , \\sigma )$ , where $\\varPhi ( \\cdot ; \\mu , \\sigma )$ is the CDF of ${ \\mathcal { N } } ( \\mu , \\sigma )$ . Note that in general the uncertainty will depend on both the configuration and $\\tau$ , the number of points observed from the learning curve. Because frequentist models do not admit a natural estimate of uncertainty, we assume that $\\sigma$ is independent of $\\mathbf { x }$ yet still dependent on $\\tau$ and estimate it via Leave One Out Cross Validation (LOOCV). In addition, we show results using a Bayesian linear regression SRM, which has natural uncertainty estimates, removing the need for LOOCV. ", + "bbox": [ + 173, + 568, + 825, + 693 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Now that we can estimate the model uncertainty, given a new configuration xNEW and an observed learning curve $y _ { \\mathrm { N E W } } ( \\tau )$ , we may set our termination criteria to be $p ( \\hat { y } _ { T } \\le y _ { \\mathrm { B E S T } } ) \\ge \\varDelta$ . $\\varDelta$ balances the trade-off between increased speedups and risk of prematurely terminating good configurations. In many cases, one may want several configurations that are close to optimal, for the purpose of ensembling. We offer two modifications in this case. First, one may relax the termination criterion to $p ( \\hat { y } _ { T } \\le \\bar { y } _ { \\mathtt { B E S T } } - \\delta ) \\ge \\Delta$ , which will allow configurations within $\\delta$ of optimal performance to complete training. One can alternatively set the criterion based on the $n ^ { \\mathrm { t h } }$ best configuration observed, guaranteeing that with high probability the top $n$ configurations will be fully trained. ", + "bbox": [ + 174, + 699, + 825, + 810 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 EARLY STOPPING FOR ARCHITECTURE SEARCH ", + "text_level": 1, + "bbox": [ + 174, + 832, + 547, + 847 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baker et al. (2017) train a $Q$ -learning agent to design convolutional neural networks. In this method, the agent samples architectures from a large, finite space by traversing a path from input layer to termination layer. However, the MetaQNN method uses 100 GPU-days to train 2700 neural architectures and the similar experiment by Zoph & Le (2017) utilized 10,000 GPU-days to train 12,800 models on CIFAR-10. The amount of computing resources required for these approaches makes them prohibitively expensive for large datasets (e.g., Imagenet) and larger search spaces. The main computational expense of reinforcement learning-based architecture search methods is training the neural network configuration to $T$ epochs (where $T$ is typically a large number at which the network stabilizes to peak accuracy). ", + "bbox": [ + 174, + 854, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/dc85c447f26791182cfb27434b6a9360fc11cf05a77d5f67d68ebe85227a11e5.jpg", + "image_caption": [ + "Figure 4: Simulated Speedup in MetaQNN Search Space: We compare the three variants of the early stopping algorithm presented in Section 4 for both a $\\nu$ -SVR and BLR SRM. Each SRM is trained using the first 100 learning curves, and each algorithm is tested on 10 independent orderings of the model configurations. Triangles indicate an algorithm that successfully recovered the optimal model for more than half of the 10 orderings, and X’s indicate those that did not. " + ], + "image_footnote": [], + "bbox": [ + 209, + 102, + 787, + 268 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 388, + 825, + 444 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now detail the performance of $\\nu$ -SVR and BLR SRMs in speeding up architecture search. First, we take 1,000 random models from the MetaQNN (Baker et al., 2017) search space. We simulate the MetaQNN algorithm by taking 10 random orderings of each set and running our early stopping algorithm. We compare against the LCE early stopping algorithm (Domhan et al., 2015) as a baseline, which has a similar probability threshold termination criterion. Our SRM trains off of the first 100 fully observed curves, while the LCE model trains from each individual partial curve and can begin early termination immediately. Despite this “burn in” time needed by an SRM, it is still able to outperform the LCE model (Figure 4). In addition, fitting the LCE model to a learning curve takes between 1-3 minutes on a modern CPU due to expensive MCMC sampling, and it is necessary to fit a new LCE model each time a new point on the learning curve is observed. However, the difference in computing time between the methods is not as important as the difference in prediction accuracies. ", + "bbox": [ + 173, + 450, + 826, + 604 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We furthermore simulate early stopping for ResNets trained on CIFAR-10. We found that only the probability threshold $\\varDelta = 0 . 9 9$ resulted in recovering the top model consistently. However, even with such a conservative threshold, the search was sped up by a factor of 3.4 over the baseline. While we do not have the computational resources to run the full experiment from Zoph & Le (2017), our method could provide similar gains in large scale architecture searches. ", + "bbox": [ + 174, + 611, + 517, + 736 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "It is not enough, however, to simply simulate the speedup because architecture search algorithms typically use the observed performance in order to update an acquisition function to inform future sampling. In the reinforcement learning setting, the performance is given to the agent as a reward, so we also empirically verify that substituting $\\hat { y } _ { T }$ for $y _ { T }$ does not cause the MetaQNN agent to converge to a subpar policy. Replicating the MetaQNN experiment on CIFAR-10 (see Figure 5), we find that integrating early stopping with the $Q$ -learning procedure does not disrupt learning and resulted in a speedup of $3 . 8 \\mathrm { x }$ with $\\varDelta = 0 . 9 9$ . After training the top models to 300 epochs, we also find that the resulting performance (just under $9 3 \\%$ ) is on par with original results of Baker et al. (2017). ", + "bbox": [ + 174, + 743, + 517, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/aab0e4a5f68df68fc6dd3b814ef2564c865449f63f59a2bb14bd5810251a37ab.jpg", + "image_caption": [ + "Figure 5: MetaQNN on CIFAR-10 with Early Stopping: A full run of the MetaQNN algorithm (Baker et al., 2017) on the CIFAR10 dataset with early stopping. We use the $\\nu { - } \\mathbf { S } \\mathbf { V } \\mathbf { R }$ SRM with a probability threshold $\\varDelta = 0 . 9 9$ . Light blue bars indicate the average model accuracy per decrease in $\\epsilon$ , which represents the shift to a more greedy policy. We also plot the cumulative best, top 5, and top 15 performance to show that the agent continues to find better architectures. " + ], + "image_footnote": [], + "bbox": [ + 532, + 630, + 821, + 791 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/1cbaded586a36c427e79abfbd1d1e03d5fd12763217a0e9b703aa0b5e71656a0.jpg", + "image_caption": [ + "Figure 6: Simulated Max Accuracy vs SGD Iterations for Hyperband: We show the trajectories of the maximum performance so far versus total computational resources used for 40 consecutive Hyperband runs with $\\eta = 4 . 0$ and $\\varDelta = 0 . 9 5$ . Small vertical lines indicate the point at which each f-Hyperband run has searched over the same number of models as vanilla Hyperband. On Cifar-10, both BLR and $\\nu$ -SVR f-Hyperbands outperform vanilla Hyperband, but on SVHN only $\\nu$ -SVR f-Hyperbands do. Each triangle marks the completion of full Hyperband algorithm. " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 808, + 286 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 424, + 516, + 452 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 EARLY STOPPING FOR HYPERPARAMETER OPTIMIZATION ", + "text_level": 1, + "bbox": [ + 173, + 483, + 449, + 512 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Recently, Li et al. (2017) introduced Hyperband, a random search technique based on multi-armed bandits that obtains state-of-the-art performance in hyperparameter optimization in a variety of settings. The Hyperband algorithm trains a population of models with different hyperparameter configurations and iteratively discards models below a certain percentile in performance among the population until the computational budget is exhausted or satisfactory results are obtained. Among previous work, Klein et al. (2017) have improved Hyperband by algorithmically choosing the next models to evaluate. ", + "bbox": [ + 174, + 529, + 825, + 626 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2.1 FAST HYPERBAND ", + "text_level": 1, + "bbox": [ + 176, + 656, + 356, + 671 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We present a Fast Hyperband (f-Hyperband) algorithm based on our early stopping scheme. During each iteration of successive halving, Hyperband trains $n _ { i }$ configurations to $r _ { i }$ epochs. In f-Hyperband, we train an SRM to predict $y _ { r _ { i } }$ and do early stopping within each iteration of successive halving. We initialize f-Hyperband in exactly the same way as vanilla Hyperband, except once we have trained 100 models to $r _ { i }$ iterations, we begin early stopping for all future successive halving iterations that train to $r _ { i }$ iterations. By doing this, we exhibit no computational overhead over Hyperband other than training the SRMs, which we have shown to be cheap. We also introduce a parameter $\\kappa$ which denotes the proportion of the $n _ { i }$ models in each iteration that must be trained to the full $r _ { i }$ iterations. This is similar to setting the criterion based on the $n ^ { \\mathrm { t h } }$ best model in the previous section. See Appendix section C for the full algorithmic representation of f-Hyperband. ", + "bbox": [ + 173, + 680, + 825, + 819 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We empirically evaluate f-Hyperband using Cuda-Convnet trained on CIFAR-10 and SVHN datasets. Figure 6 shows that f-Hyperband evaluates the same number of unique configurations as Hyperband within half the compute time, while achieving the same final accuracy within standard error. Moreover, when we run f-Hyperband for the same amount of time as Hyperband, f-Hyperband with a $\\nu$ -SVR SRM outperforms f-Hyperband. When reinitializing hyperparameter searches, one can use previouslytrained set of SRMs to achieve even larger speedups. Figure 12 in Appendix shows that one can safely achieve up to a $4 \\mathbf { x }$ speedup in such cases. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 102, + 318, + 118 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper we introduce a simple, fast, and accurate model for predicting future neural network performance using features derived from network architectures, hyperparameters, and time-series performance data. We show that the performance of drastically different network architectures can be jointly learned and predicted on both image classification and language models. Using our simple algorithm, we can speedup hyperparameter search techniques with complex acquisition functions, such as a $Q$ -learning agent, by a factor of $3 \\mathbf { x }$ to 6x and Hyperband—a state-of-the-art hyperparameter search method—by a factor of $2 \\mathbf { x }$ , without disturbing the search procedure. We outperform all competing methods for performance prediction in terms of accuracy, train and test time, and speedups obtained on hyperparameter search methods. We hope that the simplicity and success of our method will allow it to be easily incorporated into current hyperparameter optimization pipelines for deep neural networks. With the advent of large scale automated architecture search (Baker et al., 2017; Zoph & Le, 2017), methods such as ours will be vital in exploring even larger and more complex search spaces. ", + "bbox": [ + 174, + 128, + 825, + 308 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 337, + 285, + 353 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Bowen Baker, Otkrist Gupta, Nikhil Naik, and Ramesh Raskar. Designing neural network architectures using reinforcement learning. International Conference on Learning Representations, 2017. ", + "bbox": [ + 174, + 361, + 826, + 404 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "James Bergstra and Yoshua Bengio. Random search for hyper-parameter optimization. JMLR, 13 (Feb):281–305, 2012. ", + "bbox": [ + 173, + 414, + 823, + 443 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "James Bergstra, Daniel Yamins, and David D Cox. Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures. ICML (1), 28:115–123, 2013. ", + "bbox": [ + 173, + 452, + 823, + 481 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Andrew Brock, Theodore Lim, JM Ritchie, and Nick Weston. Smash: One-shot model architecture search through hypernetworks. arXiv preprint arXiv:1708.05344, 2017. ", + "bbox": [ + 171, + 491, + 823, + 520 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Corinna Cortes, Xavier Gonzalvo, Vitaly Kuznetsov, Mehryar Mohri, and Scott Yang. AdaNet: Adaptive structural learning of artificial neural networks. International Conference on Machine Learning, 70:874–883, 2017. ", + "bbox": [ + 176, + 529, + 823, + 571 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Tobias Domhan, Jost Tobias Springenberg, and Frank Hutter. Speeding up automatic hyperparameter optimization of deep neural networks by extrapolation of learning curves. 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Hyperband: A novel bandit-based approach to hyperparameter optimization. International Conference on Learning Representations, 2017. ", + "bbox": [ + 176, + 737, + 825, + 780 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Renato Negrinho and Geoff Gordon. Deeparchitect: Automatically designing and training deep architectures. arXiv preprint arXiv:1704.08792, 2017. ", + "bbox": [ + 168, + 790, + 825, + 819 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "J David Schaffer, Darrell Whitley, and Larry J Eshelman. Combinations of genetic algorithms and neural networks: A survey of the state of the art. International Workshop on Combinations of Genetic Algorithms and Neural Networks, pp. 1–37, 1992. ", + "bbox": [ + 178, + 829, + 823, + 872 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Bobak Shahriari, Kevin Swersky, Ziyu Wang, Ryan P Adams, and Nando de Freitas. Taking the human out of the loop: A review of bayesian optimization. Proceedings of the IEEE, 104(1): 148–175, 2016. ", + "bbox": [ + 173, + 881, + 825, + 922 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Jasper Snoek, Hugo Larochelle, and Ryan P Adams. Practical bayesian optimization of machine learning algorithms. NIPS, pp. 2951–2959, 2012. \nJasper Snoek, Oren Rippel, Kevin Swersky, Ryan Kiros, Nadathur Satish, Narayanan Sundaram, Mostofa Patwary, Mr Prabhat, and Ryan Adams. Scalable bayesian optimization using deep neural networks. In International Conference on Machine Learning, pp. 2171–2180, 2015. \nKenneth O Stanley and Risto Miikkulainen. Evolving neural networks through augmenting topologies. Evolutionary Computation, 10(2):99–127, 2002. \nMasanori Suganuma, Shinichi Shirakawa, and Tomoharu Nagao. A genetic programming approach to designing convolutional neural network architectures. arXiv preprint arXiv:1704.00764, 2017. \nKevin Swersky, Jasper Snoek, and Ryan Prescott Adams. Freeze-thaw bayesian optimization. arXiv preprint arXiv:1406.3896, 2014. \nPhillip Verbancsics and Josh Harguess. Generative neuroevolution for deep learning. arXiv preprint arXiv:1312.5355, 2013. \nBarret Zoph and Quoc V Le. Neural architecture search with reinforcement learning. International Conference on Learning Representations, 2017. \nBarret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V Le. Learning transferable architectures for scalable image recognition. arXiv preprint arXiv:1707.07012, 2017. ", + "bbox": [ + 171, + 102, + 828, + 412 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 263, + 117 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A DATASETS AND ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 178, + 136, + 482, + 151 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Deep Resnets (TinyImageNet): We sample 500 ResNet architectures and train them on the TinyImageNet† dataset (containing 200 classes with 500 training images of $3 2 \\times 3 2$ pixels) for 140 epochs. We vary depths, filter sizes and number of convolutional filter block outputs. Filter sizes are sampled from $\\{ 3 , 5 , 7 \\}$ and number of filters is sampled from $\\{ 2 , 3 , 4 , . . . , 2 2 \\}$ . Each ResNet block is composed of three convolutional layers followed by batch normalization and summation layers. We vary the number of blocks from 2 to 18, giving us networks with depths varying between 14 and 110. Each network is trained for 140 epochs, using Nesterov optimizer. The learning rate is set to 0.1 and learning rate reduction and momentum are set to 0.1 and 0.9 respectively. ", + "bbox": [ + 173, + 166, + 825, + 279 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Deep Resnets (CIFAR-10): We sample 500 39-layer ResNet architectures from a search space similar to Zoph & Le (2017), varying kernel width, kernel height, and number of kernels. We train these models for 50 epochs on CIFAR-10. Each architecture consists of 39 layers: $1 2 \\ : c o n \\nu$ , a $2 \\mathbf { x } 2$ max pool, 9 conv, a 2x2 max pool, $1 5 \\ c o n \\nu$ , and softmax. Each conv layer is followed by batch normalization and a ReLU nonlinearity. Each block of 3 conv layers are densely connected via residual connections and also share the same kernel width, kernel height, and number of learnable kernels. Kernel height and width are independently sampled from $\\{ 1 , 3 , 5 , 7 \\}$ and number of kernels is sampled from $\\{ 6 , 1 2 , 2 4 , 3 6 \\}$ . Finally, we randomly sample residual connections between each block of conv layers. Each network is trained for 50 epochs using the RMSProp optimizer, with weight decay $1 0 ^ { - 4 }$ , initial learning rate 0.001, and a learning rate reduction to $1 0 ^ { - 5 }$ at epoch 30 on the CIFAR-10 dataset. ", + "bbox": [ + 173, + 285, + 825, + 438 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "MetaQNN CNNs (CIFAR-10 and SVHN): We sample 1,000 model architectures from the search space detailed by Baker et al. (2017), which allows for varying the numbers and orderings of convolution, pooling, and fully connected layers. The models are between 1 and 12 layers for the SVHN experiment and between 1 and 18 layers for the CIFAR-10 experiment. Each architecture is trained on SVHN and CIFAR-10 datasets for 20 epochs. Table 3 displays the state space of the MetaQNN algorithm. ", + "bbox": [ + 173, + 444, + 825, + 529 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/26f3fde9c37c7c849040c2bd1c43665efee8beea051e9f510d95f30b9c314288.jpg", + "table_caption": [ + "Table 3: Experimental State Space For MetaQNN. For each layer type, we list the relevant parameters and the values each parameter is allowed to take. The networks are sampled beginning from the starting layer. Convolutional layers are allowed to transition to any other layer. Pooling layers are allowed to transition to any layer other than pooling layers. Fully connected layers are only allowed to transition to fully connected or softmax layers. A convolutional or pooling layer may only go to a fully connected layer if the current image representation size is below 8. We use this space to both randomly sample and simulate the behavior of a MetaQNN run as well as directly run the MetaQNN with early stopping. " + ], + "table_footnote": [], + "table_body": "
Layer TypeLayerParametersParameterValues
Convolution (C)i~Layer depthf~ Receptive field sizel~Strided ~ # receptive fieldsn ~ Representation size<12Square. ∈ {1,3,5}Square.Always equal to 1∈{64,128,256,512}∈{(∞,8],(8,4],(4,1]}
Pooling (P)i~Layer depth(f,l)~ (Receptive field size,Strides)n ~Representation size<12Square.∈{(5,3),(3,2),(2,2)}∈{(∞,8],(8,4] and (4,1]}
Fully Connected (FC)i~Layerdepthn ~ # consecutive FC layersd ～# neurons<12<3∈ {512,256,128}
Termination StateS~Previous Statet~TypeGlobal Avg.Pooling/Softmax
", + "bbox": [ + 199, + 540, + 799, + 722 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "LSTM (PTB): We sample 300 LSTM models and train them on the Penn Treebank dataset for 60 epochs. Number of hidden layer inputs and lstm cells was varied from 10 to 1400 in steps of 20. Each network was trained for 60 epochs with batch size of 50 and trained the models using stochastic gradient descent. Dropout ratio of 0.5 was used to prevent overfitting. Dictionary size of 400 words was used to generate embeddings when vectorizing the data. ", + "bbox": [ + 174, + 858, + 825, + 900 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Cuda-Convnet (CIFAR-10 and SVHN): We train Cuda-Convnet architecture (Krizhevsky, 2012) with varying values of initial learning rate, learning rate reduction step size, weight decay for convolutional and fully connected layers, and scale and power of local response normalization layers. We train models with CIFAR-10 for 60 epochs and with SVHN for 12 epochs. Table 4 show the hyperparameter ranges for the Cuda Convnet experiments. ", + "bbox": [ + 174, + 138, + 825, + 209 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/18e99fa9c88e515b7324338bafda81471c0ef1084a1df6b1a5f908026192c3fe.jpg", + "table_caption": [ + "Table 4: Range of hyperparameter settings used for the Hyperband experiment (Section 4.1) " + ], + "table_footnote": [], + "table_body": "
ExperimentHyperparameterScaleMinMax
CIFAR-10, Imagenet, SVHNInitial Learning RateLog5×10-55
Learning Rate ReductionsInteger03
CIFAR-10, SVHNConv1 L2 PenaltyLog5×10-55
Conv2 L2 PenaltyLog5×10-55
Conv3 L2 PenaltyLog5×10-55
FC4 L2 PenaltyLog5×10-55
Response Normalization ScaleLog5×10-65
Response Normalization PowerLinear1×10-23
", + "bbox": [ + 181, + 232, + 813, + 386 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B HYPERPARAMETER SELECTION IN PERFORMANCE PREDICTION MODELS ", + "text_level": 1, + "bbox": [ + 171, + 472, + 812, + 489 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "When training performance prediction models we divided the data into training and validation and used 3-fold cross validation to select optimal hyperparameters. The models were then trained on full training data using the best hyperparameters. For random forests we varied number of trees between 10 and 800, and varied ratio of number of features from 0.1 to 0.5. For $\\nu$ -SVR, we perform a random search over 1000 hyperparameter configurations from the space $C \\sim$ LogUniform $( 1 0 ^ { - 5 }$ , 10), $\\nu \\sim$ Uniform $( 0 , 1 )$ , and $\\gamma \\sim \\mathrm { L o g U n i f o r m } ( 1 0 ^ { - 5 } , 1 0$ ) (when using the RBF kernel). For Bayesian linear regression, we perform random search over the hyperparameters of the gamma prior distribution over both noise and weights in the range $[ 1 0 ^ { - 7 } , 1 0 ^ { - 5 } ]$ . ", + "bbox": [ + 173, + 511, + 825, + 623 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "C F-HYPERBAND ", + "text_level": 1, + "bbox": [ + 176, + 655, + 333, + 671 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 1 of this text replicates Algorithm 1 from Li et al. (2017), except we initialize two dictionaries: $D$ to store training data and $M$ to store performance prediction models. $D [ r ]$ will correspond to a dictionary containing all datasets with prediction target epoch $r$ . $D [ r ] [ \\bar { \\tau } ]$ will correspond to the dataset for predicting $y _ { r }$ based on the observed $y ( t ) _ { 1 - \\tau }$ , and $M [ r ] [ \\tau ]$ will hold the corresponding performance prediction model. We will assume that the performance prediction model will have a train function, and a predict function that will return the prediction and standard deviation of the prediction. In addition to the standard Hyperband hyperparameters $R$ and $\\eta$ , we include $\\varDelta$ and $\\delta$ described in Section 4 and $\\kappa$ . During each iteration of successive halving, we train $n _ { i }$ configurations to $r _ { i }$ epochs; $\\kappa$ denotes the fraction of the top $n _ { i }$ models that should be run to the full $r _ { i }$ iterations. This is similar to setting the criterion based on the $n ^ { \\mathrm { t h } }$ best model in the previous section. ", + "bbox": [ + 173, + 694, + 825, + 847 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We also detail the run_then_return_validation_loss function in Algorithm 2. This algorithm runs a set of configurations, adds training data from observed learning curves, trains the performance prediction models when there is enough training data present, and then uses the models to terminate poor configurations. It assumes we have a function max_k, which returns the $k ^ { \\mathrm { { t h } } }$ max value or $- \\infty$ if the list has less than $k$ values. ", + "bbox": [ + 174, + 854, + 823, + 922 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 1: f-Hyperband ", + "text_level": 1, + "bbox": [ + 174, + 111, + 354, + 126 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "input : $R$ (Max resources allocated to any configuration) η (default $\\eta = 3 ,$ ) $\\varDelta$ (Probability threshold for early termination) $\\delta$ (Performance offset for early termination) $d$ (# points required to train performance predictors) $\\kappa$ (Proportion of models to train) initialize : $D = \\mathrm { d i c t ( ) }$ M = dict() smax = blogη(R)c $B = ( s _ { \\mathrm { m a x } } + 1 ) R$ 1 for 2 T $\\begin{array} { r } { n = \\lceil \\frac { B } { R } \\frac { \\eta ^ { s } } { s + 1 } \\rceil , \\quad r = R \\eta ^ { - s } } \\end{array}$ $s \\in \\{ s _ { m a x } , \\ldots , 0 \\}$ do 3 // begin SUCCESSIVEHALVING with $\\left( n , \\begin{array} { l l } { r } \\end{array} \\right)$ inner loop 4 $T = { \\mathfrak { g e t } }$ _hyperparameter_configuration( $: n \\mid$ ) for $i \\in \\{ 0 , \\ldots , s \\}$ do 6 $n _ { i } = \\lfloor n \\eta ^ { - i } \\rfloor$ , $r _ { i } = r \\eta ^ { i }$ 7 $\\begin{array} { r } { n _ { \\mathrm { n e x t } } = \\lfloor \\frac { n _ { i } } { \\eta } \\rfloor } \\end{array}$ if $i ! = s$ else 1 8 L = run_then_return_validation_loss $( T , r _ { i } , n _ { \\mathrm { n e x t } } , D , M )$ 9 $\\begin{array} { r } { T = \\mathrm { t o p \\_ k } \\left( T , L , \\lfloor \\frac { n _ { i } } { \\eta } \\rfloor \\right) } \\end{array}$ ) 0 end ", + "bbox": [ + 165, + 128, + 705, + 426 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "11 end ", + "bbox": [ + 156, + 422, + 202, + 435 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "input : T hyperparameter configurations \nr resources to use for training \n$n$ # configurations in next iteration of successive halving \n$D$ dictionary storing training data \n$M$ – dictionary storing performance prediction models \ninitialize : $L = [ ]$ \n1 for $t \\in T$ do \n2 $\\ell = \\left[ \\ \\right]$ \n3 for $i \\in \\{ 0 , \\ldots , r - 1 \\}$ do \n4 $\\ell _ { i } =$ run_one_epoch_return_validation_loss(t) \n5 $\\ell$ .append $( \\ell _ { i }$ ) \n6 if $M [ r ] [ i ]$ .trained() then \n7 ${ \\hat { y } } _ { r }$ , $\\dot { \\sigma } = M [ r ] [ i ]$ .predict $( \\ell )$ \n8 if $\\begin{array} { r } { \\varPhi ( \\mathrm { m a x \\_ k } ( \\bar { L } , \\kappa n ) - \\delta ; \\hat { y } _ { r } , \\sigma ) \\geq \\varDelta } \\end{array}$ then \n9 $L$ .append $( \\hat { y } _ { r }$ ) \n10 break \n11 end \n12 end \n13 else if $i = = r - 1$ then \n14 $L$ .append $( \\ell _ { i }$ ) \n15 end \n16 end \n17 if length $( D [ r ] [ 0 ] ) < d$ and length $( { \\boldsymbol { \\ell } } ) \\ = = r$ then \n18 $\\{ D [ r ] [ i ]$ .append $\\langle \\ell [ 0 , \\ldots , i ] , \\ell [ r ] \\} \\rangle : i \\in \\{ 0 , \\ldots , r - 1 \\} \\}$ \n19 if not $M [ r ] [ i ]$ .trained() then \n20 $M [ r ] [ i ]$ .train $( D [ r ] [ i ] ,$ ) \n21 end \n22 end \n23 end \n24 return L ", + "bbox": [ + 156, + 486, + 691, + 911 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "D F-HYPERBAND WITH SVR ACQUISITION FUNCTION", + "text_level": 1, + "bbox": [ + 174, + 102, + 642, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Similar to Klein et al. (2017), it is possible to use our simple models as an acquisition function for Hyperband. As shown in Table 2, our models have reasonable performance with only AP and HP features used, meaning we can use these models to rank hyperparameter settings before training the configurations. In particular, for each Hyperband iteration, we randomly select 10,000 configurations, weight them by their predicted performance and sample from this weighted distribution. In Figure 7, we compare f-Hyperband with this $\\nu$ -SVR acquisition function to f-Hyperband with the standard uniform acquisition function and also to vanilla Hyperband. It seems that the acquisition function could help slightly, but it does not improve performance across all experiments, and where it does improve performance it does so only slightly. ", + "bbox": [ + 173, + 133, + 826, + 260 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/cd7670b52afd22f5572cddfea1b647b61e4c33d65e54d6765e5cede1a1f1cd49.jpg", + "image_caption": [ + "Figure 7: f-Hyperband with SVR Acquisition Function: We compare f-Hyperband with an SVR acquisition function to f-Hyperband with the uniform acquisition function (this is the standard version described in the main text) and to vanilla Hyperband. The results shown are inconclusive; it seems that there could be some benefit of incorporating the SVR acquisition function, but it isn’t significant enough to make a strong statement. " + ], + "image_footnote": [], + "bbox": [ + 176, + 273, + 816, + 457 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "E GAUSSIAN ERROR ASSUMPTION ", + "text_level": 1, + "bbox": [ + 176, + 570, + 477, + 587 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "In order to have uncertainty estimates with frequentist models, we estimated the mean and variance of the error from leave-one-out cross validation and assumed that the error was Gaussian. In Figure 8 we compare example error distributions between training and validation from a $\\nu$ -SVR SRM. Visually one can see that the assumption holds very well for both visual classification tasks and holds slightly less well for the language modeling task. In Figure 9, we show the mean log likelihood of held out prediction errors being drawn from a Gaussian parameterized by the mean and variance of the error on the training set is very close to the mean log likelihood of samples drawn from the same distribution, which shows that our Gaussian error assumption is relatively strong. ", + "bbox": [ + 173, + 602, + 825, + 713 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "F ANALYZING IMPORTANCE OF FEATURES IN THE PREDICTION MODEL ", + "text_level": 1, + "bbox": [ + 173, + 734, + 781, + 751 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We used a linear $\\nu$ -SVR model to compare the weights of all features, which are normalized before training. Figure 10 shows the weight trends for time-series features, and Tables 5, 6, and 7 compare weights of some architecture features and hyperparameter features to statistics of time-series features for the MetaQNN (CIFAR-10), LSTMs (Penn Treebank), and Deep Resnets (TinyImagenet) experiments, respectively. We found the following main insights across datasets: ", + "bbox": [ + 173, + 765, + 825, + 835 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• The time-series (TS) features are on average have higher weights than HP and AP features (which is confirmed by our ablation studies in Table 2 as well). The original validation accuracies $( y _ { t } )$ have higher weights on average than the first-order differences $( { y } _ { t } ^ { \\prime } )$ and second-order differences $( { y } _ { t } ^ { \\prime \\prime } )$ . • In general, later epochs in $y _ { t }$ have higher weights. In other words, the latest performance of the model available for prediction is much more important for performance prediction than initial performance, when the model has just started training. However, the weights for earlier performance metrics are non-zero, indicating there are high order autoregressive functions being learned. ", + "bbox": [ + 173, + 848, + 826, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/6ab4eee391e270952d9a993a07231dd2f41ebb64efe1a472221b93e477a96b83.jpg", + "image_caption": [ + "Figure 8: Example Error Distributions: Here we compare example error distributions between training and validation from a $\\nu$ -SVR SRM. From left to right we see plots for models trained with increasing amounts of the learning curve. The blue line shows the Gaussian probability density function estimated from the training set with LOOCV. The orange histogram shows the error distribution of the held out validation set. " + ], + "image_footnote": [], + "bbox": [ + 176, + 101, + 821, + 488 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/10a7774b17a92c4e92355b15a9b04bb03ddd75fa3f8c268d6042658e62a1507b.jpg", + "image_caption": [ + "Figure 9: Error Log Likelihood: In orange we show the mean log likelihood of prediction errors measured from validation sets being drawn from the Gaussian distribution parameterized by the mean and variance of the training error, again measured with LOOCV. For a baseline, we show the mean log likelihood for 1,000,000 samples drawn from the same Gaussian. We calculate these metrics over 10 data splits and report the mean and standard error. " + ], + "image_footnote": [], + "bbox": [ + 176, + 588, + 816, + 750 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 179, + 862, + 823, + 890 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• Early values of $( { y } _ { t } ^ { \\prime } )$ also have high weights, which indicates learning quickly in the beginning of training is a predictor of better final performance in our datasets. ", + "bbox": [ + 171, + 895, + 826, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• Among AP and HP features, the total number of parameters and depth have non-zero weight for the CNNs, and they are assigned weights comparable or higher than late epoch accuracies $( y _ { t } )$ . However, they have much lower weight in the LSTM experiment. The number of filters in each convolutional layer also has a high positive weight for CNNs. In general, architectural features are much more important for CNNs as compared to LSTMs. Hyperparameters like initial learning rate and step size were generally not as important as the architectural features, which is corroborated by the ablation study in Table 1 in the main text. ", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/b2971eb1174e30d287fd59999deb67712ba2523822f222b2dc7573d44e9acc4c.jpg", + "image_caption": [ + "Figure 10: Time Series Weights: We show the trends for weights for time-series features, along with first- and second-order differences. " + ], + "image_footnote": [], + "bbox": [ + 176, + 220, + 820, + 719 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/ec62dc41e1b03cda3e1ab2a1004c86755be32b5eb082f09ae2195245695b6b12.jpg", + "table_caption": [ + "Table 5: Summary of weights for a linear kernel $\\nu$ -SVR SRM trained on configurations from MetaQNN (CIFAR-10). " + ], + "table_footnote": [], + "table_body": "
PercentObservedMeanPerfStd DevPerfMean 1stOrder DiffMean 2ndOrder DiffDepth#ParamsLearningRate
100.0229890.0138450.0170690.0090210.0146380.0055410.005355
200.0169710.0108820.0101170.0034940.0081530.0034080.005335
300.0137330.0073960.0069210.0025500.0041990.0020980.003310
400.0111250.0070230.0050310.0019030.0028470.0030410.002968
500.0094090.0058360.0045940.0012340.0005780.0002230.001107
600.0080990.0047680.0033700.0009230.001306-0.0009060.000287
700.0070160.0042460.0025730.0006270.000458-0.000777-0.000324
800.0061900.0034170.0021590.0004940.0004540.0003550.000583
900.0055530.0031410.0017920.0003710.0003820.0000200.000259
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PercentObservedMeanPerfStd DevPerfMean 1stOrder DiffMean 2ndOrder DiffDepth# ParamsLearningRate Step
101.129178-0.9527122.5625840.822748-2.6087090.599052-1.940918
200.388479-0.1687980.8283430.287950-1.1486190.390338-0.667870
300.202656-0.2540580.2821630.132048-1.134044-0.033193-0.301467
400.149271-0.2058680.1839590.093839-0.2806460.050759-0.057207
500.129166-0.1600910.1272300.078865-0.219122-0.044931-0.119674
600.116550-0.1604100.1225570.085949-0.1288210.040022-0.097907
700.112243-0.1442780.1042090.078584-0.082918-0.077808-0.057439
800.102105-0.1274180.0783280.068932-0.0625060.054454-0.006713
900.106653-0.0673690.1418870.0745150.079686-0.1695340.158086
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PercentObservedMeanPerfStd DevPerfMean 1stOrder DiffMean 2ndOrder DiffDepth#ParamsKernelSize
100.0084860.0085760.0018610.0015020.0059850.0109870.000267
200.0047230.0048510.0010920.0009960.0017090.0052390.002484
300.0032290.0028480.0007890.0006190.0045330.002553-0.001138
400.0023940.0029850.0006600.0005570.0034430.001092-0.000302
500.0017540.0019930.0004610.0003960.0013180.0010810.000072
600.0014560.0019460.0003700.0003090.0009680.0005590.000036
700.0014280.0026780.0004990.0003440.001161-0.000590-0.000971
800.0011680.0018950.0003520.0002560.0004550.000131-0.001092
900.0010150.0016400.0002850.0002150.000208-0.000091-0.000958
", + "bbox": [ + 173, + 686, + 854, + 847 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "G ADDITIONAL PLOTS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 379, + 117 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Figure 11 shows the performance prediction results for the Cuda Convnet SVHN experiment. The instability of BLR with low percentage of learning curve observed may be what caused suboptimal performance of BLR f-Hyperband shown in Figure 6. Figure 12 shows the relative speedup of f-Hyperband over vanilla Hyperband for each consecutive iteration of Hyperband. However, Figure 13, which shows results for f-Hyperband with pretrained $\\nu$ -SVR SRMs, shows that more aggressive settings of f-Hyperband result in suboptimal performance. We see that we can safely get a $3 { - } 4 \\mathbf { x }$ speedup over vanilla Hyperband with pretrained SRMs. ", + "bbox": [ + 173, + 133, + 826, + 231 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/c750bab18bbb20f6ff9821a6eabe408b08df4b123f9375788a4aa1f50624cd94.jpg", + "image_caption": [ + "Figure 11: Cuda Convnet SVHN Performance Prediction Results: We plot the performance of each method versus the percent of learning curve observed for the Cuda Convnet SVHN experiment. For BNN, $\\nu$ -SVR, and BLR we sample 10 different training sets, plot the mean $R ^ { 2 }$ , and shade the corresponding standard error. We compare our method against BNN (Klein et al., 2017), LCE (Domhan et al., 2015), and a “last seen value” heuristic (Li et al., 2017). Absent results for a model indicate that it did not achieve a positive $R ^ { 2 }$ . " + ], + "image_footnote": [], + "bbox": [ + 176, + 248, + 813, + 457 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/893b61c3e8dee896d83890cfd6f959ee2023596c2286f47916b4990e7a40b892.jpg", + "image_caption": [ + "Figure 12: Simulated Speedup on Hyperband vs Hyperband Iteration: We show the speedup using the f-Hyperband algorithm over Hyperband on 40 consecutive runs with $\\eta = 4 . 0$ and $\\varDelta = 0 . 9 5$ . The major jump in speedup comes at iteration 10, where we have trained more than 100 models to the full $R$ iterations. " + ], + "image_footnote": [], + "bbox": [ + 176, + 583, + 823, + 804 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/e9cd3871458632935a702159971de0403ccd0159d69e2a3b0451acad4716e165.jpg", + "image_caption": [ + "Figure 13: Simulated f-Hyperband with Pretrained SRMs: Each f-Hyperband model is initialized with a pretrained SRM, which simulates the case where one reinitializes an experiment. Every triangle indicates a full run of Hyperband, and small vertical lines indicate the point at which the f-Hyperband runs have completed the same number of full Hyperband runs as shown for vanilla Hyperband. Top 1 indicates only training the best model to $r _ { i }$ iterations, and $\\delta$ represents an offset off the threshold used for early stopping (See Section 4).More aggressive f-Hyperband settings lead to suboptimal performance, indicating that there is a limit to how much speedup f-Hyperband can achieve. " + ], + "image_footnote": [], + "bbox": [ + 176, + 102, + 807, + 287 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "H SRM ROBUSTNESS TO OUT OF DISTRIBUTION CONFIGURATIONS ", + "text_level": 1, + "bbox": [ + 173, + 431, + 754, + 448 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Table 8 shows that SRMs are relatively robust when training below or above the median of a specific architecture or training hyperparameter and testing on the other split. ", + "bbox": [ + 173, + 462, + 825, + 492 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/81e34eb3e70308337741521ba7b7b1f582128bec8e017a83500e4b7b7e91837b.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Cuda Convnet (CIFAR-10)Learning RateWeightDecayLRN ScaleLRN Power
Test<Median93.06 ± 0.1895.93 ± 0.2197.27 ± 0.006896.96 ± 0.13
Test>Median81.97 ± 1.2696.19± 0.1393.37 ± 0.2395.05 ± 0.19
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LSTMs (Penn Treebank)Number of NodesDepth
Test<Median98.08 ± 0.2698.58 ± 0.04
Test>Median22.83 ± 5.5596.96 ± 0.91
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DeepResnets (TinyImagenet)DepthNumber ofKernelsKernel Size
Test<Median89.49 ± 1.8485.12 ± 2.2481.83 ± 0.59
Test >Median85.92 ± 1.7862.37 ± 0.8289.61 ± 0.31
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MetaQNN (CIFAR-10)Depth
Test<Median96.69 ± 0.16
Test>Median79.31 ± 5.66
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Ongoing research in two", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "areas—architecture search and hyperparameter optimization—attempts to reduce the amount of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "human intervention required for these tasks. Hyperparameter optimization methods (e.g., Hutter", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "et al. (2011); Snoek et al. (2015); Li et al. (2017)) focus primarily on obtaining good optimization", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 564 + ], + "score": 1.0, + "content": "hyperparameter configurations for training human-designed networks, whereas architecture search", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "algorithms (Bergstra et al., 2013; Verbancsics & Harguess, 2013; Baker et al., 2017; Zoph & Le, 2017)", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 588 + ], + "score": 1.0, + "content": "aim to design neural network architectures from scratch. Both sets of algorithms require training a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "large number of neural network configurations for identifying the right set of hyperparameters or the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 596, + 381, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 381, + 609 + ], + "score": 1.0, + "content": "right network architecture and are hence computationally expensive.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 613, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "When sampling many different model configurations, it is likely that many subpar configurations", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "will be explored. 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We empirically show that our performance prediction models are much more", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 312, + 469, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 312, + 469, + 323 + ], + "score": 1.0, + "content": "accurate than prominent Bayesian counterparts, are simpler to implement, and are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 322, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 470, + 334 + ], + "score": 1.0, + "content": "faster to train. 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Ongoing research in two", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "areas—architecture search and hyperparameter optimization—attempts to reduce the amount of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "human intervention required for these tasks. Hyperparameter optimization methods (e.g., Hutter", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "et al. (2011); Snoek et al. (2015); Li et al. 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As we shall see in subsequent sections, this use of sequential regression models", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 227, + 498, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 498, + 239 + ], + "score": 1.0, + "content": "(SRM) is more computationally efficient and more precise than methods that train a single model.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 243, + 505, + 332 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "Features: We use features based on time-series (TS) validation performances, architecture parame-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 398, + 267 + ], + "score": 1.0, + "content": "ters (AP), and hyperparameters (HP). 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Please see Appendix Section A for further details", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 461, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 461, + 403 + ], + "score": 1.0, + "content": "on the search space, preprocessing, hyperparameters and training settings of all datasets.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 109, + 417, + 298, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 417, + 299, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 299, + 429 + ], + "score": 1.0, + "content": "DATASETS WITH VARYING ARCHITECTURES:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 438, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "Deep Resnets (TinyImageNet): We sample 500 ResNet architectures and train them on the TinyIm-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 450, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 379, + 462 + ], + "score": 1.0, + "content": "ageNet* dataset (containing 200 classes with 500 training images of", + "type": "text" + }, + { + "bbox": [ + 379, + 450, + 412, + 460 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 450, + 506, + 462 + ], + "score": 1.0, + "content": "pixels) for 140 epochs.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "We vary depths, filter sizes and number of convolutional filter block outputs. The network depths", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 472, + 212, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 212, + 483 + ], + "score": 1.0, + "content": "vary between 14 and 110.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 108, + 488, + 504, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "Deep Resnets (CIFAR-10): We sample 500 39-layer ResNet architectures from a search space", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 499, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 505, + 511 + ], + "score": 1.0, + "content": "similar to Zoph & Le (2017), varying kernel width, kernel height, and number of kernels. 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(2017), which allows for varying the numbers and orderings of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 561 + ], + "score": 1.0, + "content": "convolution, pooling, and fully connected layers. The models are between 1 and 12 layers for the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "SVHN experiment and between 1 and 18 layers for the CIFAR-10 experiment. Each architecture is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 570, + 333, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 333, + 585 + ], + "score": 1.0, + "content": "trained on SVHN and CIFAR-10 datasets for 20 epochs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 528, + 506, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 504, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "score": 1.0, + "content": "LSTM (PTB): We sample 300 LSTM models and train them on the Penn Treebank dataset for 60", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "epochs, evaluating perplexity on the validation set. We vary number of LSTM cells and hidden layer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 207, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 207, + 621 + ], + "score": 1.0, + "content": "inputs between 10-1400.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 588, + 506, + 621 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 639, + 312, + 650 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 313, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 313, + 651 + ], + "score": 1.0, + "content": "DATASETS WITH VARYING HYPERPARAMETERS:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "Cuda-Convnet (CIFAR-10 and SVHN): We train Cuda-Convnet architecture (Krizhevsky, 2012)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "with varying values of initial learning rate, learning rate reduction step size, weight decay for", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 683, + 507, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 507, + 695 + ], + "score": 1.0, + "content": "convolutional and fully connected layers, and scale and power of local response normalization layers.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 693, + 421, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 421, + 705 + ], + "score": 1.0, + "content": "We train models with CIFAR-10 for 60 epochs and with SVHN for 12 epochs.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 659, + 507, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 115, + 80, + 496, + 128 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 115, + 80, + 496, + 128 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 80, + 496, + 128 + ], + "spans": [ + { + "bbox": [ + 115, + 80, + 496, + 128 + ], + "score": 0.975, + "html": "
Datasetv-SVROLSBLRRF
MetaQNN (CIFAR-10)97.22 ± 0.3197.04± 0.4297.11 ± 0.3294.50 ± 0.78
Resnet (TinyImageNet)87.12 ± 4.6588.41 ± 3.7682.40± 1.6586.39 ± 3.58
LSTM (Penn Treebank)98.03± 1.4687.83 ± 21.5397.99 ± 1.9057.28± 24.98
", + "type": "table", + "image_path": "860755a4c53e29ddffe1564634ed76c1a0fb098fc5cd9dd145716f7f85631c91.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 80, + 496, + 96.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 96.0, + 496, + 112.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 112.0, + 496, + 128.0 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 108, + 136, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 106, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "Table 1: Performance Prediction Model Comparison: We report the coefficient of determination", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 145, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 107, + 146, + 145, + 157 + ], + "score": 0.9, + "content": "R ^ { 2 } * 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 145, + 424, + 160 + ], + "score": 1.0, + "content": "for four standard methods. Each model is trained with 100 samples on", + "type": "text" + }, + { + "bbox": [ + 424, + 147, + 444, + 158 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 145, + 505, + 160 + ], + "score": 1.0, + "content": "of the learning", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 428, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 185, + 171 + ], + "score": 1.0, + "content": "curve. We find that", + "type": "text" + }, + { + "bbox": [ + 185, + 160, + 192, + 168 + ], + "score": 0.69, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 158, + 428, + 171 + ], + "score": 1.0, + "content": "-SVR works best on average, though not by a large margin.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "image", + "bbox": [ + 128, + 178, + 483, + 317 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 128, + 178, + 483, + 317 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 178, + 483, + 317 + ], + "spans": [ + { + "bbox": [ + 128, + 178, + 483, + 317 + ], + "score": 0.97, + "type": "image", + "image_path": "e1e08d693f58b2451fc566e9d907921036494e900edf6b2beed895ec69970c9e.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 128, + 178, + 483, + 224.33333333333334 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 128, + 224.33333333333334, + 483, + 270.6666666666667 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 128, + 270.6666666666667, + 483, + 317.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 328, + 506, + 373 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "Figure 2: Predicted vs True Values of Final Performance: We show the shape of the predictive", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 448, + 352 + ], + "score": 1.0, + "content": "distribution on three experiments: MetaQNN models, Deep Resnets, and LSTMs. Each", + "type": "text" + }, + { + "bbox": [ + 448, + 341, + 455, + 349 + ], + "score": 0.67, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "-SVR model", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 306, + 363 + ], + "score": 1.0, + "content": "is trained with 100 configurations with data from", + "type": "text" + }, + { + "bbox": [ + 306, + 351, + 326, + 361 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "of the learning curve. We predict validation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 362, + 459, + 374 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 459, + 374 + ], + "score": 1.0, + "content": "set classification accuracy for MetaQNN and Deep ResNets, and perplexity for LSTMs.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "title", + "bbox": [ + 108, + 391, + 253, + 402 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 254, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 254, + 403 + ], + "score": 1.0, + "content": "3.3 PREDICTION PERFORMANCE", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "Choice of Regression Method: We now describe our results for predicting final neural network", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "score": 1.0, + "content": "performance. For all experiments, we train our SRMs on 100 randomly sampled neural network", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "configurations. We obtain the best performing method using random hyperparameter search over", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "3-fold cross-validation. We then compute the regression performance over the remainder of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 292, + 469 + ], + "score": 1.0, + "content": "dataset using the coefficient of determination", + "type": "text" + }, + { + "bbox": [ + 293, + 456, + 306, + 466 + ], + "score": 0.87, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 456, + 506, + 469 + ], + "score": 1.0, + "content": ". We repeat each experiment 10 times and report", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "the results with standard errors. We experiment with a few simple regression models, including", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 464, + 491 + ], + "score": 1.0, + "content": "ordinary least squares (OLS), random forests (RF), Bayesian linear regression (BLR), and", + "type": "text" + }, + { + "bbox": [ + 464, + 480, + 471, + 488 + ], + "score": 0.75, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "-support", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 219, + 502 + ], + "score": 1.0, + "content": "vector machine regression (", + "type": "text" + }, + { + "bbox": [ + 219, + 491, + 225, + 499 + ], + "score": 0.53, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 489, + 331, + 502 + ], + "score": 1.0, + "content": "-SVR). For both OLS and", + "type": "text" + }, + { + "bbox": [ + 332, + 491, + 339, + 500 + ], + "score": 0.68, + "content": "\\nu \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "-SVR, we choose between the linear and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 456, + 514 + ], + "score": 1.0, + "content": "RBF kernels based on prediction performance—after hyperparameter search, in case of", + "type": "text" + }, + { + "bbox": [ + 456, + 502, + 463, + 510 + ], + "score": 0.69, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "-SVR. We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "chose 100 as the number of sampled configurations somewhat arbitrarily, though we found that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 522, + 507, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 507, + 535 + ], + "score": 1.0, + "content": "prediction performance was quite poor below 50 and improves steadily after. As seen in Table 1,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 533, + 507, + 547 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 113, + 543 + ], + "score": 0.65, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 533, + 507, + 547 + ], + "score": 1.0, + "content": "-SVR performs the best on most datasets, though not by a large margin. For the rest of this paper,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 545, + 275, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 136, + 556 + ], + "score": 1.0, + "content": "we use", + "type": "text" + }, + { + "bbox": [ + 136, + 546, + 143, + 554 + ], + "score": 0.7, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 545, + 275, + 556 + ], + "score": 1.0, + "content": "-SVR unless otherwise specified.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "Ablation Study on Feature Sets: In Table 2, we compare the predictive ability of different feature", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 570, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 368, + 587 + ], + "score": 1.0, + "content": "sets, training SVR with time-series (TS) features obtained from", + "type": "text" + }, + { + "bbox": [ + 368, + 572, + 388, + 583 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 570, + 506, + 587 + ], + "score": 1.0, + "content": "of the learning curve, along", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "with features of architecture parameters (AP), and hyperparameters (HP). TS features explain the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "largest fraction of the variance in all cases and found that in general AP are more important that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "HP. AP features almost match TS on the ResNet (TinyImageNet) dataset, indicating that choice of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "architecture has a large influence on accuracy for ResNets. Figure 2 shows the true vs. predicted", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 627, + 451, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 451, + 639 + ], + "score": 1.0, + "content": "performance for all test points in three datasets, trained with TS, AP, and HP features.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Generalization To Out Of Distribution Configurations: We also test to see whether our SRMs can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "accurately predict the performance of out-of-distribution neural networks. For instance, if we train a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 107, + 667, + 113, + 676 + ], + "score": 0.66, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 666, + 348, + 678 + ], + "score": 1.0, + "content": "-SVR model on the Deep Resnet (TinyImagenet) dataset at", + "type": "text" + }, + { + "bbox": [ + 349, + 666, + 368, + 676 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "of the learning curve observed on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 361, + 690 + ], + "score": 1.0, + "content": "configurations below median depth (60 layers), we can achieve", + "type": "text" + }, + { + "bbox": [ + 361, + 676, + 406, + 687 + ], + "score": 0.91, + "content": "R ^ { 2 } = 0 . 8 6", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "on configurations above", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "median depth. The corresponding number for prediction below median depth is 0.89. We repeated", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "this experiment all datasets with different hyperparameter values below/above the median and found", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "that SRMs generally showed consistent performance across such splits. 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Datasetv-SVROLSBLRRF
MetaQNN (CIFAR-10)97.22 ± 0.3197.04± 0.4297.11 ± 0.3294.50 ± 0.78
Resnet (TinyImageNet)87.12 ± 4.6588.41 ± 3.7682.40± 1.6586.39 ± 3.58
LSTM (Penn Treebank)98.03± 1.4687.83 ± 21.5397.99 ± 1.9057.28± 24.98
", + "type": "table", + "image_path": "860755a4c53e29ddffe1564634ed76c1a0fb098fc5cd9dd145716f7f85631c91.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 80, + 496, + 96.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 96.0, + 496, + 112.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 112.0, + 496, + 128.0 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 108, + 136, + 505, + 170 + ], + "lines": [ + { + "bbox": [ + 106, + 136, + 505, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 505, + 148 + ], + "score": 1.0, + "content": "Table 1: Performance Prediction Model Comparison: We report the coefficient of determination", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 145, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 107, + 146, + 145, + 157 + ], + "score": 0.9, + "content": "R ^ { 2 } * 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 145, + 424, + 160 + ], + "score": 1.0, + "content": "for four standard methods. Each model is trained with 100 samples on", + "type": "text" + }, + { + "bbox": [ + 424, + 147, + 444, + 158 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 145, + 505, + 160 + ], + "score": 1.0, + "content": "of the learning", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 158, + 428, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 185, + 171 + ], + "score": 1.0, + "content": "curve. 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We predict validation", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 362, + 459, + 374 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 459, + 374 + ], + "score": 1.0, + "content": "set classification accuracy for MetaQNN and Deep ResNets, and perplexity for LSTMs.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "title", + "bbox": [ + 108, + 391, + 253, + 402 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 254, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 254, + 403 + ], + "score": 1.0, + "content": "3.3 PREDICTION PERFORMANCE", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "Choice of Regression Method: We now describe our results for predicting final neural network", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "score": 1.0, + "content": "performance. For all experiments, we train our SRMs on 100 randomly sampled neural network", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "configurations. We obtain the best performing method using random hyperparameter search over", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "3-fold cross-validation. We then compute the regression performance over the remainder of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 292, + 469 + ], + "score": 1.0, + "content": "dataset using the coefficient of determination", + "type": "text" + }, + { + "bbox": [ + 293, + 456, + 306, + 466 + ], + "score": 0.87, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 456, + 506, + 469 + ], + "score": 1.0, + "content": ". We repeat each experiment 10 times and report", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "the results with standard errors. We experiment with a few simple regression models, including", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 464, + 491 + ], + "score": 1.0, + "content": "ordinary least squares (OLS), random forests (RF), Bayesian linear regression (BLR), and", + "type": "text" + }, + { + "bbox": [ + 464, + 480, + 471, + 488 + ], + "score": 0.75, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "-support", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 219, + 502 + ], + "score": 1.0, + "content": "vector machine regression (", + "type": "text" + }, + { + "bbox": [ + 219, + 491, + 225, + 499 + ], + "score": 0.53, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 489, + 331, + 502 + ], + "score": 1.0, + "content": "-SVR). For both OLS and", + "type": "text" + }, + { + "bbox": [ + 332, + 491, + 339, + 500 + ], + "score": 0.68, + "content": "\\nu \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "-SVR, we choose between the linear and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 456, + 514 + ], + "score": 1.0, + "content": "RBF kernels based on prediction performance—after hyperparameter search, in case of", + "type": "text" + }, + { + "bbox": [ + 456, + 502, + 463, + 510 + ], + "score": 0.69, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "-SVR. We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "chose 100 as the number of sampled configurations somewhat arbitrarily, though we found that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 522, + 507, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 507, + 535 + ], + "score": 1.0, + "content": "prediction performance was quite poor below 50 and improves steadily after. As seen in Table 1,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 533, + 507, + 547 + ], + "spans": [ + { + "bbox": [ + 107, + 535, + 113, + 543 + ], + "score": 0.65, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 533, + 507, + 547 + ], + "score": 1.0, + "content": "-SVR performs the best on most datasets, though not by a large margin. For the rest of this paper,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 545, + 275, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 136, + 556 + ], + "score": 1.0, + "content": "we use", + "type": "text" + }, + { + "bbox": [ + 136, + 546, + 143, + 554 + ], + "score": 0.7, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 545, + 275, + 556 + ], + "score": 1.0, + "content": "-SVR unless otherwise specified.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 412, + 507, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 561, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "Ablation Study on Feature Sets: In Table 2, we compare the predictive ability of different feature", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 570, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 368, + 587 + ], + "score": 1.0, + "content": "sets, training SVR with time-series (TS) features obtained from", + "type": "text" + }, + { + "bbox": [ + 368, + 572, + 388, + 583 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 570, + 506, + 587 + ], + "score": 1.0, + "content": "of the learning curve, along", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "with features of architecture parameters (AP), and hyperparameters (HP). TS features explain the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "largest fraction of the variance in all cases and found that in general AP are more important that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "HP. AP features almost match TS on the ResNet (TinyImageNet) dataset, indicating that choice of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "architecture has a large influence on accuracy for ResNets. Figure 2 shows the true vs. predicted", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 627, + 451, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 451, + 639 + ], + "score": 1.0, + "content": "performance for all test points in three datasets, trained with TS, AP, and HP features.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 561, + 506, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Generalization To Out Of Distribution Configurations: We also test to see whether our SRMs can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "accurately predict the performance of out-of-distribution neural networks. For instance, if we train a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 107, + 667, + 113, + 676 + ], + "score": 0.66, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 666, + 348, + 678 + ], + "score": 1.0, + "content": "-SVR model on the Deep Resnet (TinyImagenet) dataset at", + "type": "text" + }, + { + "bbox": [ + 349, + 666, + 368, + 676 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "of the learning curve observed on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 361, + 690 + ], + "score": 1.0, + "content": "configurations below median depth (60 layers), we can achieve", + "type": "text" + }, + { + "bbox": [ + 361, + 676, + 406, + 687 + ], + "score": 0.91, + "content": "R ^ { 2 } = 0 . 8 6", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "on configurations above", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "median depth. The corresponding number for prediction below median depth is 0.89. We repeated", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "this experiment all datasets with different hyperparameter values below/above the median and found", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "that SRMs generally showed consistent performance across such splits. 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Feature SetMetaQNN(CIFAR-10)ResNets(TinyImageNet)LSTM(Penn Treebank)Cuda-Convnet(CIFAR-10)
HP-2.96 ± 2.710.48 ± 7.24-1.72 ± 1.620.48 ± 7.24
AP15.38 ± 4.6667.66 ± 9.0813.76 ± 12.0867.66 ± 9.08
TS97.00 ± 0.5282.71 ± 4.3197.38 ± 1.4982.71 ± 4.31
AP+HP14.89 ± 5.1270.25 ± 6.4213.76 ± 12.0870.25 ± 6.42
TS+HP97.02 ± 0.5282.67 ± 6.8197.38 ± 1.4982.67± 6.81
TS+AP97.12 ± 0.3787.36± 4.2397.87 ± 1.6887.36± 4.23
TS+AP+HP97.15 ± 0.4886.05 ± 8.2497.87 ± 1.6886.05 ± 8.24
", + "type": "table", + "image_path": "7436f624fbd0b0b90cbd5af810fd267f1d00fd523372aff399deedbb30b95960.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 127, + 80, + 483, + 114.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 127, + 114.66666666666666, + 483, + 149.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 127, + 149.33333333333331, + 483, + 183.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 192, + 507, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 334, + 205 + ], + "score": 1.0, + "content": "Table 2: Ablation Study on Feature Sets: We report the", + "type": "text" + }, + { + "bbox": [ + 335, + 192, + 372, + 203 + ], + "score": 0.9, + "content": "R ^ { 2 } * 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "metric for different combinations", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "of features. Time-series features (TS) refers to the partially observed learning curves, architecture", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 214, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 507, + 228 + ], + "score": 1.0, + "content": "parameters (AP) refer to the number of layers and number of weights in a deep model, and hyperpa-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 474, + 238 + ], + "score": 1.0, + "content": "rameters (HP) refer to the optimization parameters such as learning rate. All results are with", + "type": "text" + }, + { + "bbox": [ + 474, + 227, + 480, + 236 + ], + "score": 0.66, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 226, + 506, + 238 + ], + "score": 1.0, + "content": "-SVR.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 237, + 354, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 126, + 247 + ], + "score": 0.86, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 237, + 354, + 249 + ], + "score": 1.0, + "content": "of learning curve used for models including TS features.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 270, + 317, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 318, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 318, + 282 + ], + "score": 1.0, + "content": "3.3.1 COMPARISON WITH EXISTING METHODS:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 287, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 302 + ], + "score": 1.0, + "content": "We now compare the neural network performance prediction ability of SRMs with three existing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "learning curve prediction methods: (1) Bayesian Neural Network (BNN) (Klein et al., 2017), (2)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "the learning curve extrapolation (LCE) method (Domhan et al., 2015), and (3) the last seen value", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "(LastSeenValue) heuristic (Li et al., 2017). When training the BNN, we not only present it with the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 331, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 331, + 506, + 346 + ], + "score": 1.0, + "content": "subset of fully observed learning curves but also all other partially observed learning curves from", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "the training set. While we do not present the partially observed curves to the SRMs for training, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "felt this was a fair comparison as an SRM uses the entire partially observed learning curve during", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "inference. Methods (2) and (3) do not incorporate prior learning curves during training. Figure", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 158, + 389 + ], + "score": 1.0, + "content": "3 shows the", + "type": "text" + }, + { + "bbox": [ + 158, + 375, + 171, + 386 + ], + "score": 0.87, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "obtained by each method for predicting the final performance versus the percent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "of the learning curve used for training the model. We see that in all neural network configuration", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 104, + 398, + 226, + 411 + ], + "score": 1.0, + "content": "spaces and across all datasets,", + "type": "text" + }, + { + "bbox": [ + 226, + 398, + 255, + 408 + ], + "score": 0.29, + "content": "\\nu { - } \\mathbf { S } \\mathbf { V } \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "outperform the competing methods. Bayesian linear regression", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 267, + 421 + ], + "score": 1.0, + "content": "(BLR) also performs comparably well to", + "type": "text" + }, + { + "bbox": [ + 268, + 410, + 275, + 419 + ], + "score": 0.66, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "-SVR and better than the more complex Bayesian methods", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "(BNN and LCE). The LastSeenValue heuristic only becomes viable when the configurations are", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 432, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 506, + 443 + ], + "score": 1.0, + "content": "near convergence, and its performance is worse than an SRM for very deep models. We also find", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "that the SRMs outperform the LCE method in all experiments, even after we remove a few extreme", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "prediction outliers produced by LCE. Finally, while BNN outperforms the LastSeenValue and LCE", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "score": 1.0, + "content": "methods when only a few iterations have been observed, it does worse than our proposed method. In", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "summary, we show that our simple SRMs outperforms existing Bayesian approaches on predicting", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "neural network performance on modern, very deep models in computer vision and language modeling", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 497, + 132, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 132, + 509 + ], + "score": 1.0, + "content": "tasks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "Since most of our experiments perform stepwise learning rate decay; it is conceivable that the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "performance gap between SRMs and both LCE and BNN results from a lack of sharp jumps in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 549 + ], + "score": 1.0, + "content": "their basis functions. For completeness, we also experimented with exponential learning rate decay", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "(ELRD), which the basis functions in LCE are designed for. We trained 630 random nets with ELRD,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 338, + 569 + ], + "score": 1.0, + "content": "from the 1000 MetaQNN-CIFAR10 nets. Predicting from", + "type": "text" + }, + { + "bbox": [ + 338, + 558, + 358, + 568 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 557, + 461, + 569 + ], + "score": 1.0, + "content": "of the learning curve, the", + "type": "text" + }, + { + "bbox": [ + 461, + 557, + 474, + 568 + ], + "score": 0.87, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "is 0.95", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 120, + 581 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 120, + 570, + 127, + 578 + ], + "score": 0.62, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "-SVR, 0.48 for LCE (with extreme outlier removal, negative without), and 0.31 for BNN. This", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "comparison illuminates another benefit of our method: we do not require handcrafted basis functions", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 590, + 249, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 249, + 603 + ], + "score": 1.0, + "content": "to model new learning curve types.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 618 + ], + "score": 1.0, + "content": "Moreover, SRMs are faster to train and do inference in than LCE and BNN. On 1 core of a Intel", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 133, + 628 + ], + "score": 0.3, + "content": "6 7 0 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 618, + 172, + 630 + ], + "score": 1.0, + "content": "CPU, an", + "type": "text" + }, + { + "bbox": [ + 172, + 619, + 179, + 628 + ], + "score": 0.63, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "-SVR with 100 training points trains in 0.006 seconds, and each inference takes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "0.00006 seconds. In comparison, the LCE code takes 60 seconds and BNN code takes 0.024 seconds", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 639, + 273, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 273, + 653 + ], + "score": 1.0, + "content": "on the same hardware for each inference.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 108, + 667, + 445, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 447, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 447, + 682 + ], + "score": 1.0, + "content": "4 APPLYING PERFORMANCE PREDICTION FOR EARLY STOPPING", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "To speed up hyperparameter optimization and architecture search methods, we develop an algorithm", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "to determine whether to continue training a partially trained model configuration using our sequential", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 711, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 721 + ], + "score": 1.0, + "content": "regression models. This approach follows the idea proposed by Domhan et al. (2015) and further", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 410, + 732 + ], + "score": 1.0, + "content": "expanded on by Klein et al. (2017). Concretely, if we would like to sample", + "type": "text" + }, + { + "bbox": [ + 411, + 721, + 420, + 730 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "total neural network", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 224, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 225, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 225, + 38 + ], + "score": 1.0, + "content": "Workshop track - 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": "table", + "bbox": [ + 127, + 80, + 483, + 184 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 127, + 80, + 483, + 184 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 80, + 483, + 184 + ], + "spans": [ + { + "bbox": [ + 127, + 80, + 483, + 184 + ], + "score": 0.982, + "html": "
Feature SetMetaQNN(CIFAR-10)ResNets(TinyImageNet)LSTM(Penn Treebank)Cuda-Convnet(CIFAR-10)
HP-2.96 ± 2.710.48 ± 7.24-1.72 ± 1.620.48 ± 7.24
AP15.38 ± 4.6667.66 ± 9.0813.76 ± 12.0867.66 ± 9.08
TS97.00 ± 0.5282.71 ± 4.3197.38 ± 1.4982.71 ± 4.31
AP+HP14.89 ± 5.1270.25 ± 6.4213.76 ± 12.0870.25 ± 6.42
TS+HP97.02 ± 0.5282.67 ± 6.8197.38 ± 1.4982.67± 6.81
TS+AP97.12 ± 0.3787.36± 4.2397.87 ± 1.6887.36± 4.23
TS+AP+HP97.15 ± 0.4886.05 ± 8.2497.87 ± 1.6886.05 ± 8.24
", + "type": "table", + "image_path": "7436f624fbd0b0b90cbd5af810fd267f1d00fd523372aff399deedbb30b95960.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 127, + 80, + 483, + 114.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 127, + 114.66666666666666, + 483, + 149.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 127, + 149.33333333333331, + 483, + 183.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 192, + 507, + 248 + ], + "lines": [ + { + "bbox": [ + 106, + 192, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 334, + 205 + ], + "score": 1.0, + "content": "Table 2: Ablation Study on Feature Sets: We report the", + "type": "text" + }, + { + "bbox": [ + 335, + 192, + 372, + 203 + ], + "score": 0.9, + "content": "R ^ { 2 } * 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 193, + 505, + 205 + ], + "score": 1.0, + "content": "metric for different combinations", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 506, + 217 + ], + "score": 1.0, + "content": "of features. Time-series features (TS) refers to the partially observed learning curves, architecture", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 214, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 507, + 228 + ], + "score": 1.0, + "content": "parameters (AP) refer to the number of layers and number of weights in a deep model, and hyperpa-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 226, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 474, + 238 + ], + "score": 1.0, + "content": "rameters (HP) refer to the optimization parameters such as learning rate. All results are with", + "type": "text" + }, + { + "bbox": [ + 474, + 227, + 480, + 236 + ], + "score": 0.66, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 226, + 506, + 238 + ], + "score": 1.0, + "content": "-SVR.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 237, + 354, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 126, + 247 + ], + "score": 0.86, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 237, + 354, + 249 + ], + "score": 1.0, + "content": "of learning curve used for models including TS features.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 192, + 507, + 249 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 270, + 317, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 318, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 318, + 282 + ], + "score": 1.0, + "content": "3.3.1 COMPARISON WITH EXISTING METHODS:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 287, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 287, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 506, + 302 + ], + "score": 1.0, + "content": "We now compare the neural network performance prediction ability of SRMs with three existing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 312 + ], + "score": 1.0, + "content": "learning curve prediction methods: (1) Bayesian Neural Network (BNN) (Klein et al., 2017), (2)", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 311, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 323 + ], + "score": 1.0, + "content": "the learning curve extrapolation (LCE) method (Domhan et al., 2015), and (3) the last seen value", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "score": 1.0, + "content": "(LastSeenValue) heuristic (Li et al., 2017). When training the BNN, we not only present it with the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 331, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 331, + 506, + 346 + ], + "score": 1.0, + "content": "subset of fully observed learning curves but also all other partially observed learning curves from", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "the training set. While we do not present the partially observed curves to the SRMs for training, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "score": 1.0, + "content": "felt this was a fair comparison as an SRM uses the entire partially observed learning curve during", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "inference. Methods (2) and (3) do not incorporate prior learning curves during training. Figure", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 375, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 158, + 389 + ], + "score": 1.0, + "content": "3 shows the", + "type": "text" + }, + { + "bbox": [ + 158, + 375, + 171, + 386 + ], + "score": 0.87, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 375, + 505, + 389 + ], + "score": 1.0, + "content": "obtained by each method for predicting the final performance versus the percent", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "of the learning curve used for training the model. We see that in all neural network configuration", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 104, + 398, + 226, + 411 + ], + "score": 1.0, + "content": "spaces and across all datasets,", + "type": "text" + }, + { + "bbox": [ + 226, + 398, + 255, + 408 + ], + "score": 0.29, + "content": "\\nu { - } \\mathbf { S } \\mathbf { V } \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "outperform the competing methods. Bayesian linear regression", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 267, + 421 + ], + "score": 1.0, + "content": "(BLR) also performs comparably well to", + "type": "text" + }, + { + "bbox": [ + 268, + 410, + 275, + 419 + ], + "score": 0.66, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "-SVR and better than the more complex Bayesian methods", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 505, + 432 + ], + "score": 1.0, + "content": "(BNN and LCE). The LastSeenValue heuristic only becomes viable when the configurations are", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 432, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 506, + 443 + ], + "score": 1.0, + "content": "near convergence, and its performance is worse than an SRM for very deep models. We also find", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 506, + 455 + ], + "score": 1.0, + "content": "that the SRMs outperform the LCE method in all experiments, even after we remove a few extreme", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 453, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 506, + 465 + ], + "score": 1.0, + "content": "prediction outliers produced by LCE. Finally, while BNN outperforms the LastSeenValue and LCE", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "score": 1.0, + "content": "methods when only a few iterations have been observed, it does worse than our proposed method. In", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 104, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "summary, we show that our simple SRMs outperforms existing Bayesian approaches on predicting", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 499 + ], + "score": 1.0, + "content": "neural network performance on modern, very deep models in computer vision and language modeling", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 497, + 132, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 132, + 509 + ], + "score": 1.0, + "content": "tasks.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 287, + 506, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 526 + ], + "score": 1.0, + "content": "Since most of our experiments perform stepwise learning rate decay; it is conceivable that the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 537 + ], + "score": 1.0, + "content": "performance gap between SRMs and both LCE and BNN results from a lack of sharp jumps in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 549 + ], + "score": 1.0, + "content": "their basis functions. For completeness, we also experimented with exponential learning rate decay", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "(ELRD), which the basis functions in LCE are designed for. We trained 630 random nets with ELRD,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 557, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 338, + 569 + ], + "score": 1.0, + "content": "from the 1000 MetaQNN-CIFAR10 nets. Predicting from", + "type": "text" + }, + { + "bbox": [ + 338, + 558, + 358, + 568 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 557, + 461, + 569 + ], + "score": 1.0, + "content": "of the learning curve, the", + "type": "text" + }, + { + "bbox": [ + 461, + 557, + 474, + 568 + ], + "score": 0.87, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "is 0.95", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 569, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 120, + 581 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 120, + 570, + 127, + 578 + ], + "score": 0.62, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 569, + 505, + 581 + ], + "score": 1.0, + "content": "-SVR, 0.48 for LCE (with extreme outlier removal, negative without), and 0.31 for BNN. This", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "comparison illuminates another benefit of our method: we do not require handcrafted basis functions", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 590, + 249, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 249, + 603 + ], + "score": 1.0, + "content": "to model new learning curve types.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 513, + 506, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 607, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 618 + ], + "score": 1.0, + "content": "Moreover, SRMs are faster to train and do inference in than LCE and BNN. On 1 core of a Intel", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 133, + 628 + ], + "score": 0.3, + "content": "6 7 0 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 618, + 172, + 630 + ], + "score": 1.0, + "content": "CPU, an", + "type": "text" + }, + { + "bbox": [ + 172, + 619, + 179, + 628 + ], + "score": 0.63, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "-SVR with 100 training points trains in 0.006 seconds, and each inference takes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "0.00006 seconds. In comparison, the LCE code takes 60 seconds and BNN code takes 0.024 seconds", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 639, + 273, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 273, + 653 + ], + "score": 1.0, + "content": "on the same hardware for each inference.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 607, + 505, + 653 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 667, + 445, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 447, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 447, + 682 + ], + "score": 1.0, + "content": "4 APPLYING PERFORMANCE PREDICTION FOR EARLY STOPPING", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "To speed up hyperparameter optimization and architecture search methods, we develop an algorithm", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "to determine whether to continue training a partially trained model configuration using our sequential", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 711, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 721 + ], + "score": 1.0, + "content": "regression models. This approach follows the idea proposed by Domhan et al. (2015) and further", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 506, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 410, + 732 + ], + "score": 1.0, + "content": "expanded on by Klein et al. (2017). Concretely, if we would like to sample", + "type": "text" + }, + { + "bbox": [ + 411, + 721, + 420, + 730 + ], + "score": 0.8, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 721, + 506, + 732 + ], + "score": 1.0, + "content": "total neural network", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 301, + 402 + ], + "score": 1.0, + "content": "configurations, we begin by sampling and training", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 301, + 389, + 333, + 399 + ], + "score": 0.9, + "content": "n \\ll N", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 334, + 388, + 477, + 402 + ], + "score": 1.0, + "content": "configurations to create a training set", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 478, + 389, + 486, + 399 + ], + "score": 0.77, + "content": "s", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 486, + 388, + 505, + 402 + ], + "score": 1.0, + "content": ". 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For BNN and", + "type": "text" + }, + { + "bbox": [ + 318, + 309, + 325, + 318 + ], + "score": 0.6, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 308, + 507, + 322 + ], + "score": 1.0, + "content": "-SVR, we sample 10 different training sets,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 166, + 332 + ], + "score": 1.0, + "content": "plot the mean", + "type": "text" + }, + { + "bbox": [ + 166, + 319, + 179, + 330 + ], + "score": 0.86, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 318, + 506, + 332 + ], + "score": 1.0, + "content": ", and shade the corresponding standard error. We compare our method against", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "BNN (Klein et al., 2017), LCE (Domhan et al., 2015), and a “last seen value” heuristic (Li et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 422, + 353 + ], + "score": 1.0, + "content": "2017). Absent results for a model indicate that it did not achieve a positive", + "type": "text" + }, + { + "bbox": [ + 423, + 341, + 435, + 352 + ], + "score": 0.88, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 340, + 506, + 353 + ], + "score": 1.0, + "content": ". The results for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 352, + 392, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 392, + 365 + ], + "score": 1.0, + "content": "Cuda-Convnet on the SVHN dataset are shown in Appendix Figure 11.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 301, + 402 + ], + "score": 1.0, + "content": "configurations, we begin by sampling and training", + "type": "text" + }, + { + "bbox": [ + 301, + 389, + 333, + 399 + ], + "score": 0.9, + "content": "n \\ll N", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 388, + 477, + 402 + ], + "score": 1.0, + "content": "configurations to create a training set", + "type": "text" + }, + { + "bbox": [ + 478, + 389, + 486, + 399 + ], + "score": 0.77, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 388, + 505, + 402 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 398, + 503, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 179, + 413 + ], + "score": 1.0, + "content": "then train a model", + "type": "text" + }, + { + "bbox": [ + 180, + 400, + 205, + 412 + ], + "score": 0.93, + "content": "f ( x _ { f } )", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 398, + 279, + 413 + ], + "score": 1.0, + "content": ", where in our case", + "type": "text" + }, + { + "bbox": [ + 279, + 400, + 286, + 411 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 398, + 351, + 413 + ], + "score": 1.0, + "content": "is an SRM with", + "type": "text" + }, + { + "bbox": [ + 351, + 400, + 375, + 410 + ], + "score": 0.88, + "content": "T - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 398, + 490, + 413 + ], + "score": 1.0, + "content": "individual models, to predict", + "type": "text" + }, + { + "bbox": [ + 491, + 402, + 503, + 411 + ], + "score": 0.82, + "content": "y _ { T }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 309, + 424 + ], + "score": 1.0, + "content": "Now, given the current best performance observed", + "type": "text" + }, + { + "bbox": [ + 309, + 412, + 333, + 423 + ], + "score": 0.51, + "content": "y _ { \\mathrm { B E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 410, + 505, + 424 + ], + "score": 1.0, + "content": ", we would like to terminate training a new", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 421, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 161, + 436 + ], + "score": 1.0, + "content": "configuration", + "type": "text" + }, + { + "bbox": [ + 161, + 424, + 185, + 433 + ], + "score": 0.25, + "content": "\\mathbf { x } _ { \\mathrm { N E W } }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 421, + 230, + 436 + ], + "score": 1.0, + "content": "at iteration", + "type": "text" + }, + { + "bbox": [ + 231, + 424, + 238, + 432 + ], + "score": 0.73, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 421, + 360, + 436 + ], + "score": 1.0, + "content": "given its partial learning curve", + "type": "text" + }, + { + "bbox": [ + 361, + 422, + 396, + 434 + ], + "score": 0.91, + "content": "y _ { \\mathrm { N E W } } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 421, + 406, + 436 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 407, + 421, + 504, + 434 + ], + "score": 0.88, + "content": "f ( x _ { \\mathrm { { N E W } } } f ) = \\hat { y } _ { T } \\overset { \\cdot } { \\leq } y _ { \\mathrm { { B E S T } } }", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 433, + 431, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 431, + 445 + ], + "score": 1.0, + "content": "so as to not waste computational resources exploring a suboptimal configuration.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 450, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 195, + 462 + ], + "score": 1.0, + "content": "However, in the case", + "type": "text" + }, + { + "bbox": [ + 195, + 450, + 203, + 461 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "has poor out-of-sample generalization, we may mistakenly terminate the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "optimal configuration. If we assume that our estimate can be modeled as a Gaussian perturbation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 471, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 175, + 486 + ], + "score": 1.0, + "content": "of the true value", + "type": "text" + }, + { + "bbox": [ + 175, + 471, + 263, + 483 + ], + "score": 0.91, + "content": "\\hat { y } _ { T } \\sim \\mathcal { N } ( y _ { T } , \\sigma ( \\mathbf { x } , \\tau ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 471, + 399, + 486 + ], + "score": 1.0, + "content": ", then we can find the probability", + "type": "text" + }, + { + "bbox": [ + 400, + 471, + 505, + 483 + ], + "score": 0.9, + "content": "\\begin{array} { r } { p ( \\hat { y } _ { T } \\leq y _ { \\mathrm { B E S T } } \\mathbf { \\bar { \\Phi } } | \\sigma ( \\mathbf { x } , \\tau ) ) = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 170, + 495 + ], + "score": 0.91, + "content": "\\varPhi ( y _ { \\mathrm { B E S T } } ; y _ { T } , \\sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 482, + 202, + 496 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 202, + 483, + 241, + 495 + ], + "score": 0.92, + "content": "\\varPhi ( \\cdot ; \\mu , \\sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 482, + 300, + 496 + ], + "score": 1.0, + "content": "is the CDF of", + "type": "text" + }, + { + "bbox": [ + 300, + 482, + 335, + 494 + ], + "score": 0.93, + "content": "{ \\mathcal { N } } ( \\mu , \\sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 482, + 506, + 496 + ], + "score": 1.0, + "content": ". Note that in general the uncertainty will", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 493, + 507, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 264, + 506 + ], + "score": 1.0, + "content": "depend on both the configuration and", + "type": "text" + }, + { + "bbox": [ + 264, + 495, + 271, + 504 + ], + "score": 0.76, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 493, + 507, + 506 + ], + "score": 1.0, + "content": ", the number of points observed from the learning curve.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 486, + 517 + ], + "score": 1.0, + "content": "Because frequentist models do not admit a natural estimate of uncertainty, we assume that", + "type": "text" + }, + { + "bbox": [ + 486, + 506, + 494, + 514 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 171, + 528 + ], + "score": 1.0, + "content": "independent of", + "type": "text" + }, + { + "bbox": [ + 172, + 517, + 180, + 525 + ], + "score": 0.46, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 515, + 275, + 528 + ], + "score": 1.0, + "content": "yet still dependent on", + "type": "text" + }, + { + "bbox": [ + 276, + 517, + 283, + 525 + ], + "score": 0.74, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "and estimate it via Leave One Out Cross Validation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "(LOOCV). In addition, we show results using a Bayesian linear regression SRM, which has natural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 537, + 324, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 324, + 549 + ], + "score": 1.0, + "content": "uncertainty estimates, removing the need for LOOCV.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "Now that we can estimate the model uncertainty, given a new configuration xNEW and an observed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 166, + 578 + ], + "score": 1.0, + "content": "learning curve", + "type": "text" + }, + { + "bbox": [ + 166, + 565, + 201, + 577 + ], + "score": 0.9, + "content": "y _ { \\mathrm { N E W } } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 565, + 370, + 578 + ], + "score": 1.0, + "content": ", we may set our termination criteria to be", + "type": "text" + }, + { + "bbox": [ + 371, + 565, + 453, + 577 + ], + "score": 0.88, + "content": "p ( \\hat { y } _ { T } \\le y _ { \\mathrm { B E S T } } ) \\ge \\varDelta", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 565, + 457, + 578 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 457, + 565, + 467, + 575 + ], + "score": 0.72, + "content": "\\varDelta", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "balances", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 576, + 507, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 507, + 589 + ], + "score": 1.0, + "content": "the trade-off between increased speedups and risk of prematurely terminating good configurations.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "In many cases, one may want several configurations that are close to optimal, for the purpose of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "ensembling. We offer two modifications in this case. First, one may relax the termination criterion", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 608, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 117, + 622 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 609, + 221, + 621 + ], + "score": 0.92, + "content": "p ( \\hat { y } _ { T } \\le \\bar { y } _ { \\mathtt { B E S T } } - \\delta ) \\ge \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 608, + 387, + 622 + ], + "score": 1.0, + "content": ", which will allow configurations within", + "type": "text" + }, + { + "bbox": [ + 387, + 609, + 393, + 619 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 608, + 506, + 622 + ], + "score": 1.0, + "content": "of optimal performance to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 379, + 633 + ], + "score": 1.0, + "content": "complete training. One can alternatively set the criterion based on the", + "type": "text" + }, + { + "bbox": [ + 380, + 620, + 393, + 630 + ], + "score": 0.87, + "content": "n ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "best configuration observed,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 631, + 446, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 294, + 643 + ], + "score": 1.0, + "content": "guaranteeing that with high probability the top", + "type": "text" + }, + { + "bbox": [ + 294, + 633, + 302, + 641 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 631, + 446, + 643 + ], + "score": 1.0, + "content": "configurations will be fully trained.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 659, + 335, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 336, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 336, + 672 + ], + "score": 1.0, + "content": "4.1 EARLY STOPPING FOR ARCHITECTURE SEARCH", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 210, + 689 + ], + "score": 1.0, + "content": "Baker et al. (2017) train a", + "type": "text" + }, + { + "bbox": [ + 211, + 677, + 219, + 688 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "-learning agent to design convolutional neural networks. In this method,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "the agent samples architectures from a large, finite space by traversing a path from input layer", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "to termination layer. However, the MetaQNN method uses 100 GPU-days to train 2700 neural", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "architectures and the similar experiment by Zoph & Le (2017) utilized 10,000 GPU-days to train", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "12,800 models on CIFAR-10. The amount of computing resources required for these approaches", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 224, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 225, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 225, + 38 + ], + "score": 1.0, + "content": "Workshop track - ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 83, + 501, + 286 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 83, + 501, + 286 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 83, + 501, + 286 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 501, + 286 + ], + "score": 0.968, + "type": "image", + "image_path": "b307684c90185a1bcebe23b1c8b3d267a9a5c28ed548460d497c2352fbae0626.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 83, + 501, + 150.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 150.66666666666669, + 501, + 218.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 218.33333333333337, + 501, + 286.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 297, + 505, + 363 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 297, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 310 + ], + "score": 1.0, + "content": "Figure 3: Performance Prediction Results: We plot the performance of each method versus the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 308, + 507, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 318, + 322 + ], + "score": 1.0, + "content": "percent of learning curve observed. For BNN and", + "type": "text" + }, + { + "bbox": [ + 318, + 309, + 325, + 318 + ], + "score": 0.6, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 308, + 507, + 322 + ], + "score": 1.0, + "content": "-SVR, we sample 10 different training sets,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 166, + 332 + ], + "score": 1.0, + "content": "plot the mean", + "type": "text" + }, + { + "bbox": [ + 166, + 319, + 179, + 330 + ], + "score": 0.86, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 318, + 506, + 332 + ], + "score": 1.0, + "content": ", and shade the corresponding standard error. We compare our method against", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "BNN (Klein et al., 2017), LCE (Domhan et al., 2015), and a “last seen value” heuristic (Li et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 422, + 353 + ], + "score": 1.0, + "content": "2017). Absent results for a model indicate that it did not achieve a positive", + "type": "text" + }, + { + "bbox": [ + 423, + 341, + 435, + 352 + ], + "score": 0.88, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 340, + 506, + 353 + ], + "score": 1.0, + "content": ". The results for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 352, + 392, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 392, + 365 + ], + "score": 1.0, + "content": "Cuda-Convnet on the SVHN dataset are shown in Appendix Figure 11.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + } + ], + "index": 3.25 + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 444 + ], + "lines": [], + "index": 11, + "bbox_fs": [ + 105, + 388, + 505, + 445 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 450, + 505, + 549 + ], + "lines": [ + { + "bbox": [ + 105, + 449, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 195, + 462 + ], + "score": 1.0, + "content": "However, in the case", + "type": "text" + }, + { + "bbox": [ + 195, + 450, + 203, + 461 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 449, + 505, + 462 + ], + "score": 1.0, + "content": "has poor out-of-sample generalization, we may mistakenly terminate the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "optimal configuration. If we assume that our estimate can be modeled as a Gaussian perturbation", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 471, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 175, + 486 + ], + "score": 1.0, + "content": "of the true value", + "type": "text" + }, + { + "bbox": [ + 175, + 471, + 263, + 483 + ], + "score": 0.91, + "content": "\\hat { y } _ { T } \\sim \\mathcal { N } ( y _ { T } , \\sigma ( \\mathbf { x } , \\tau ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 471, + 399, + 486 + ], + "score": 1.0, + "content": ", then we can find the probability", + "type": "text" + }, + { + "bbox": [ + 400, + 471, + 505, + 483 + ], + "score": 0.9, + "content": "\\begin{array} { r } { p ( \\hat { y } _ { T } \\leq y _ { \\mathrm { B E S T } } \\mathbf { \\bar { \\Phi } } | \\sigma ( \\mathbf { x } , \\tau ) ) = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 170, + 495 + ], + "score": 0.91, + "content": "\\varPhi ( y _ { \\mathrm { B E S T } } ; y _ { T } , \\sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 482, + 202, + 496 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 202, + 483, + 241, + 495 + ], + "score": 0.92, + "content": "\\varPhi ( \\cdot ; \\mu , \\sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 482, + 300, + 496 + ], + "score": 1.0, + "content": "is the CDF of", + "type": "text" + }, + { + "bbox": [ + 300, + 482, + 335, + 494 + ], + "score": 0.93, + "content": "{ \\mathcal { N } } ( \\mu , \\sigma )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 482, + 506, + 496 + ], + "score": 1.0, + "content": ". Note that in general the uncertainty will", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 493, + 507, + 506 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 264, + 506 + ], + "score": 1.0, + "content": "depend on both the configuration and", + "type": "text" + }, + { + "bbox": [ + 264, + 495, + 271, + 504 + ], + "score": 0.76, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 493, + 507, + 506 + ], + "score": 1.0, + "content": ", the number of points observed from the learning curve.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 504, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 486, + 517 + ], + "score": 1.0, + "content": "Because frequentist models do not admit a natural estimate of uncertainty, we assume that", + "type": "text" + }, + { + "bbox": [ + 486, + 506, + 494, + 514 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 504, + 506, + 517 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 515, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 171, + 528 + ], + "score": 1.0, + "content": "independent of", + "type": "text" + }, + { + "bbox": [ + 172, + 517, + 180, + 525 + ], + "score": 0.46, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 515, + 275, + 528 + ], + "score": 1.0, + "content": "yet still dependent on", + "type": "text" + }, + { + "bbox": [ + 276, + 517, + 283, + 525 + ], + "score": 0.74, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 515, + 506, + 528 + ], + "score": 1.0, + "content": "and estimate it via Leave One Out Cross Validation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "(LOOCV). In addition, we show results using a Bayesian linear regression SRM, which has natural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 537, + 324, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 324, + 549 + ], + "score": 1.0, + "content": "uncertainty estimates, removing the need for LOOCV.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 449, + 507, + 549 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 554, + 505, + 642 + ], + "lines": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "Now that we can estimate the model uncertainty, given a new configuration xNEW and an observed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 565, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 166, + 578 + ], + "score": 1.0, + "content": "learning curve", + "type": "text" + }, + { + "bbox": [ + 166, + 565, + 201, + 577 + ], + "score": 0.9, + "content": "y _ { \\mathrm { N E W } } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 565, + 370, + 578 + ], + "score": 1.0, + "content": ", we may set our termination criteria to be", + "type": "text" + }, + { + "bbox": [ + 371, + 565, + 453, + 577 + ], + "score": 0.88, + "content": "p ( \\hat { y } _ { T } \\le y _ { \\mathrm { B E S T } } ) \\ge \\varDelta", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 565, + 457, + 578 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 457, + 565, + 467, + 575 + ], + "score": 0.72, + "content": "\\varDelta", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 565, + 506, + 578 + ], + "score": 1.0, + "content": "balances", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 576, + 507, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 507, + 589 + ], + "score": 1.0, + "content": "the trade-off between increased speedups and risk of prematurely terminating good configurations.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "In many cases, one may want several configurations that are close to optimal, for the purpose of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "ensembling. We offer two modifications in this case. First, one may relax the termination criterion", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 608, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 117, + 622 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 118, + 609, + 221, + 621 + ], + "score": 0.92, + "content": "p ( \\hat { y } _ { T } \\le \\bar { y } _ { \\mathtt { B E S T } } - \\delta ) \\ge \\Delta", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 608, + 387, + 622 + ], + "score": 1.0, + "content": ", which will allow configurations within", + "type": "text" + }, + { + "bbox": [ + 387, + 609, + 393, + 619 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 608, + 506, + 622 + ], + "score": 1.0, + "content": "of optimal performance to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 379, + 633 + ], + "score": 1.0, + "content": "complete training. One can alternatively set the criterion based on the", + "type": "text" + }, + { + "bbox": [ + 380, + 620, + 393, + 630 + ], + "score": 0.87, + "content": "n ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "best configuration observed,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 631, + 446, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 294, + 643 + ], + "score": 1.0, + "content": "guaranteeing that with high probability the top", + "type": "text" + }, + { + "bbox": [ + 294, + 633, + 302, + 641 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 631, + 446, + 643 + ], + "score": 1.0, + "content": "configurations will be fully trained.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 554, + 507, + 643 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 659, + 335, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 336, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 336, + 672 + ], + "score": 1.0, + "content": "4.1 EARLY STOPPING FOR ARCHITECTURE SEARCH", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 210, + 689 + ], + "score": 1.0, + "content": "Baker et al. (2017) train a", + "type": "text" + }, + { + "bbox": [ + 211, + 677, + 219, + 688 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "-learning agent to design convolutional neural networks. In this method,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "the agent samples architectures from a large, finite space by traversing a path from input layer", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "to termination layer. However, the MetaQNN method uses 100 GPU-days to train 2700 neural", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "architectures and the similar experiment by Zoph & Le (2017) utilized 10,000 GPU-days to train", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "12,800 models on CIFAR-10. The amount of computing resources required for these approaches", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "makes them prohibitively expensive for large datasets (e.g., Imagenet) and larger search spaces. 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Each SRM is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "score": 1.0, + "content": "trained using the first 100 learning curves, and each algorithm is tested on 10 independent orderings", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "of the model configurations. 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First,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "we take 1,000 random models from the MetaQNN (Baker et al., 2017) search space. We simulate", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 378, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 394 + ], + "score": 1.0, + "content": "the MetaQNN algorithm by taking 10 random orderings of each set and running our early stopping", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 391, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 506, + 403 + ], + "score": 1.0, + "content": "algorithm. 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Light blue bars indicate the aver-", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 323, + 704, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 323, + 704, + 469, + 714 + ], + "score": 1.0, + "content": "age model accuracy per decrease in", + "type": "text" + }, + { + "bbox": [ + 469, + 705, + 474, + 713 + ], + "score": 0.34, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 704, + 506, + 714 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 323, + 714, + 506, + 726 + ], + "spans": [ + { + "bbox": [ + 323, + 714, + 506, + 726 + ], + "score": 1.0, + "content": "represents the shift to a more greedy policy.", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 323, + 725, + 505, + 737 + ], + "spans": [ + { + "bbox": [ + 323, + 725, + 505, + 737 + ], + "score": 1.0, + "content": "We also plot the cumulative best, top 5, and", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 324, + 736, + 505, + 748 + ], + "spans": [ + { + "bbox": [ + 324, + 736, + 505, + 748 + ], + "score": 1.0, + "content": "top 15 performance to show that the agent", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 324, + 747, + 474, + 758 + ], + "spans": [ + { + "bbox": [ + 324, + 747, + 474, + 758 + ], + "score": 1.0, + "content": "continues to find better architectures.", + "type": "text" + } + ], + "index": 65 + } + ], + "index": 60 + } + ], + "index": 54.75 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 224, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 225, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 225, + 38 + ], + "score": 1.0, + "content": "Workshop track - ICLR 2018", + "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": [ + 128, + 81, + 482, + 213 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 128, + 81, + 482, + 213 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 128, + 81, + 482, + 213 + ], + "spans": [ + { + "bbox": [ + 128, + 81, + 482, + 213 + ], + "score": 0.973, + "type": "image", + "image_path": "dc85c447f26791182cfb27434b6a9360fc11cf05a77d5f67d68ebe85227a11e5.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 128, + 81, + 482, + 125.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 128, + 125.0, + 482, + 169.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 128, + 169.0, + 482, + 213.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 225, + 505, + 280 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 226, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 505, + 237 + ], + "score": 1.0, + "content": "Figure 4: Simulated Speedup in MetaQNN Search Space: We compare the three variants of the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 348, + 249 + ], + "score": 1.0, + "content": "early stopping algorithm presented in Section 4 for both a", + "type": "text" + }, + { + "bbox": [ + 349, + 238, + 356, + 246 + ], + "score": 0.63, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 235, + 506, + 249 + ], + "score": 1.0, + "content": "-SVR and BLR SRM. 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Small vertical lines indicate the point at which each", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 271, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 506, + 283 + ], + "score": 1.0, + "content": "f-Hyperband run has searched over the same number of models as vanilla Hyperband. 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During", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 550, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 318, + 562 + ], + "score": 1.0, + "content": "each iteration of successive halving, Hyperband trains", + "type": "text" + }, + { + "bbox": [ + 319, + 552, + 329, + 561 + ], + "score": 0.85, + "content": "n _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 550, + 398, + 562 + ], + "score": 1.0, + "content": "configurations to", + "type": "text" + }, + { + "bbox": [ + 398, + 552, + 407, + 561 + ], + "score": 0.85, + "content": "r _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 550, + 505, + 562 + ], + "score": 1.0, + "content": "epochs. 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International", + "type": "text" + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 284, + 310, + 298 + ], + "spans": [ + { + "bbox": [ + 115, + 284, + 310, + 298 + ], + "score": 1.0, + "content": "Conference on Learning Representations, 2017.", + "type": "text" + } + ], + "index": 14, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 302, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 317 + ], + "score": 1.0, + "content": "Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V Le. Learning transferable architectures", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 314, + 406, + 328 + ], + "spans": [ + { + "bbox": [ + 115, + 314, + 406, + 328 + ], + "score": 1.0, + "content": "for scalable image recognition. arXiv preprint arXiv:1707.07012, 2017.", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + } + ], + "index": 8, + "bbox_fs": [ + 104, + 82, + 507, + 328 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 161, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "score": 1.0, + "content": "APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 109, + 108, + 295, + 120 + ], + "lines": [ + { + "bbox": [ + 106, + 107, + 298, + 122 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 298, + 122 + ], + "score": 1.0, + "content": "A DATASETS AND ARCHITECTURES", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "Deep Resnets (TinyImageNet): We sample 500 ResNet architectures and train them on the TinyIm-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 143, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 379, + 155 + ], + "score": 1.0, + "content": "ageNet† dataset (containing 200 classes with 500 training images of", + "type": "text" + }, + { + "bbox": [ + 379, + 144, + 412, + 154 + ], + "score": 0.91, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 143, + 506, + 155 + ], + "score": 1.0, + "content": "pixels) for 140 epochs.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "We vary depths, filter sizes and number of convolutional filter block outputs. Filter sizes are sampled", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 128, + 178 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 128, + 165, + 163, + 177 + ], + "score": 0.91, + "content": "\\{ 3 , 5 , 7 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 165, + 312, + 178 + ], + "score": 1.0, + "content": "and number of filters is sampled from", + "type": "text" + }, + { + "bbox": [ + 312, + 165, + 373, + 177 + ], + "score": 0.92, + "content": "\\{ 2 , 3 , 4 , . . . , 2 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 165, + 506, + 178 + ], + "score": 1.0, + "content": ". Each ResNet block is composed", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "of three convolutional layers followed by batch normalization and summation layers. We vary the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "number of blocks from 2 to 18, giving us networks with depths varying between 14 and 110. Each", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "network is trained for 140 epochs, using Nesterov optimizer. The learning rate is set to 0.1 and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 402, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 402, + 223 + ], + "score": 1.0, + "content": "learning rate reduction and momentum are set to 0.1 and 0.9 respectively.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "Deep Resnets (CIFAR-10): We sample 500 39-layer ResNet architectures from a search space", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "similar to Zoph & Le (2017), varying kernel width, kernel height, and number of kernels. We train", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 248, + 504, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 444, + 260 + ], + "score": 1.0, + "content": "these models for 50 epochs on CIFAR-10. Each architecture consists of 39 layers:", + "type": "text" + }, + { + "bbox": [ + 444, + 248, + 477, + 259 + ], + "score": 0.31, + "content": "1 2 \\ : c o n \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 248, + 487, + 260 + ], + "score": 1.0, + "content": ", a", + "type": "text" + }, + { + "bbox": [ + 487, + 248, + 504, + 259 + ], + "score": 0.71, + "content": "2 \\mathbf { x } 2", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 254, + 271 + ], + "score": 1.0, + "content": "max pool, 9 conv, a 2x2 max pool,", + "type": "text" + }, + { + "bbox": [ + 254, + 259, + 288, + 270 + ], + "score": 0.33, + "content": "1 5 \\ c o n \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 259, + 505, + 271 + ], + "score": 1.0, + "content": ", and softmax. Each conv layer is followed by batch", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "normalization and a ReLU nonlinearity. Each block of 3 conv layers are densely connected via", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "residual connections and also share the same kernel width, kernel height, and number of learnable", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 369, + 303 + ], + "score": 1.0, + "content": "kernels. Kernel height and width are independently sampled from", + "type": "text" + }, + { + "bbox": [ + 369, + 292, + 413, + 304 + ], + "score": 0.93, + "content": "\\{ 1 , 3 , 5 , 7 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "and number of kernels", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 175, + 316 + ], + "score": 1.0, + "content": "is sampled from", + "type": "text" + }, + { + "bbox": [ + 176, + 303, + 234, + 315 + ], + "score": 0.9, + "content": "\\{ 6 , 1 2 , 2 4 , 3 6 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 303, + 505, + 316 + ], + "score": 1.0, + "content": ". Finally, we randomly sample residual connections between each", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "block of conv layers. Each network is trained for 50 epochs using the RMSProp optimizer, with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 162, + 337 + ], + "score": 1.0, + "content": "weight decay", + "type": "text" + }, + { + "bbox": [ + 162, + 324, + 183, + 335 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 324, + 421, + 337 + ], + "score": 1.0, + "content": ", initial learning rate 0.001, and a learning rate reduction to", + "type": "text" + }, + { + "bbox": [ + 421, + 324, + 442, + 335 + ], + "score": 0.9, + "content": "1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 324, + 505, + 337 + ], + "score": 1.0, + "content": "at epoch 30 on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 198, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 198, + 347 + ], + "score": 1.0, + "content": "the CIFAR-10 dataset.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "MetaQNN CNNs (CIFAR-10 and SVHN): We sample 1,000 model architectures from the search", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "space detailed by Baker et al. (2017), which allows for varying the numbers and orderings of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "convolution, pooling, and fully connected layers. The models are between 1 and 12 layers for the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "SVHN experiment and between 1 and 18 layers for the CIFAR-10 experiment. Each architecture", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "is trained on SVHN and CIFAR-10 datasets for 20 epochs. Table 3 displays the state space of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 194, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 194, + 420 + ], + "score": 1.0, + "content": "MetaQNN algorithm.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "table", + "bbox": [ + 122, + 428, + 489, + 572 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 122, + 428, + 489, + 572 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 428, + 489, + 572 + ], + "spans": [ + { + "bbox": [ + 122, + 428, + 489, + 572 + ], + "score": 0.984, + "html": "
Layer TypeLayerParametersParameterValues
Convolution (C)i~Layer depthf~ Receptive field sizel~Strided ~ # receptive fieldsn ~ Representation size<12Square. ∈ {1,3,5}Square.Always equal to 1∈{64,128,256,512}∈{(∞,8],(8,4],(4,1]}
Pooling (P)i~Layer depth(f,l)~ (Receptive field size,Strides)n ~Representation size<12Square.∈{(5,3),(3,2),(2,2)}∈{(∞,8],(8,4] and (4,1]}
Fully Connected (FC)i~Layerdepthn ~ # consecutive FC layersd ～# neurons<12<3∈ {512,256,128}
Termination StateS~Previous Statet~TypeGlobal Avg.Pooling/Softmax
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For each layer type, we list the relevant", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "parameters and the values each parameter is allowed to take. The networks are sampled beginning", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "from the starting layer. Convolutional layers are allowed to transition to any other layer. Pooling", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "layers are allowed to transition to any layer other than pooling layers. Fully connected layers are only", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "allowed to transition to fully connected or softmax layers. A convolutional or pooling layer may only", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "go to a fully connected layer if the current image representation size is below 8. We use this space", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 645, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 659 + ], + "score": 1.0, + "content": "to both randomly sample and simulate the behavior of a MetaQNN run as well as directly run the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 656, + 233, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 233, + 670 + ], + "score": 1.0, + "content": "MetaQNN with early stopping.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + } + ], + "index": 30.75 + }, + { + "type": "text", + "bbox": [ + 107, + 680, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 680, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 505, + 691 + ], + "score": 1.0, + "content": "LSTM (PTB): We sample 300 LSTM models and train them on the Penn Treebank dataset for 60", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "epochs. Number of hidden layer inputs and lstm cells was varied from 10 to 1400 in steps of 20.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 701, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 714 + ], + "score": 1.0, + "content": "Each network was trained for 60 epochs with batch size of 50 and trained the models using stochastic", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39 + } + ], + "page_idx": 10, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 120, + 722, + 259, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 719, + 260, + 734 + ], + "spans": [ + { + "bbox": [ + 119, + 719, + 260, + 734 + ], + "score": 1.0, + "content": "†https://tiny-imagenet.herokuapp.com/", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 224, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 225, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 225, + 38 + ], + "score": 1.0, + "content": "Workshop track - ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 161, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 163, + 96 + ], + "score": 1.0, + "content": "APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 109, + 108, + 295, + 120 + ], + "lines": [ + { + "bbox": [ + 106, + 107, + 298, + 122 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 298, + 122 + ], + "score": 1.0, + "content": "A DATASETS AND ARCHITECTURES", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 506, + 145 + ], + "score": 1.0, + "content": "Deep Resnets (TinyImageNet): We sample 500 ResNet architectures and train them on the TinyIm-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 143, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 379, + 155 + ], + "score": 1.0, + "content": "ageNet† dataset (containing 200 classes with 500 training images of", + "type": "text" + }, + { + "bbox": [ + 379, + 144, + 412, + 154 + ], + "score": 0.91, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 143, + 506, + 155 + ], + "score": 1.0, + "content": "pixels) for 140 epochs.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "We vary depths, filter sizes and number of convolutional filter block outputs. Filter sizes are sampled", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 128, + 178 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 128, + 165, + 163, + 177 + ], + "score": 0.91, + "content": "\\{ 3 , 5 , 7 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 165, + 312, + 178 + ], + "score": 1.0, + "content": "and number of filters is sampled from", + "type": "text" + }, + { + "bbox": [ + 312, + 165, + 373, + 177 + ], + "score": 0.92, + "content": "\\{ 2 , 3 , 4 , . . . , 2 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 165, + 506, + 178 + ], + "score": 1.0, + "content": ". Each ResNet block is composed", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "of three convolutional layers followed by batch normalization and summation layers. We vary the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "number of blocks from 2 to 18, giving us networks with depths varying between 14 and 110. Each", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "network is trained for 140 epochs, using Nesterov optimizer. The learning rate is set to 0.1 and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 402, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 402, + 223 + ], + "score": 1.0, + "content": "learning rate reduction and momentum are set to 0.1 and 0.9 respectively.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 133, + 506, + 223 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 226, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "Deep Resnets (CIFAR-10): We sample 500 39-layer ResNet architectures from a search space", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "similar to Zoph & Le (2017), varying kernel width, kernel height, and number of kernels. We train", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 248, + 504, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 444, + 260 + ], + "score": 1.0, + "content": "these models for 50 epochs on CIFAR-10. Each architecture consists of 39 layers:", + "type": "text" + }, + { + "bbox": [ + 444, + 248, + 477, + 259 + ], + "score": 0.31, + "content": "1 2 \\ : c o n \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 248, + 487, + 260 + ], + "score": 1.0, + "content": ", a", + "type": "text" + }, + { + "bbox": [ + 487, + 248, + 504, + 259 + ], + "score": 0.71, + "content": "2 \\mathbf { x } 2", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 254, + 271 + ], + "score": 1.0, + "content": "max pool, 9 conv, a 2x2 max pool,", + "type": "text" + }, + { + "bbox": [ + 254, + 259, + 288, + 270 + ], + "score": 0.33, + "content": "1 5 \\ c o n \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 259, + 505, + 271 + ], + "score": 1.0, + "content": ", and softmax. Each conv layer is followed by batch", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "normalization and a ReLU nonlinearity. Each block of 3 conv layers are densely connected via", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 293 + ], + "score": 1.0, + "content": "residual connections and also share the same kernel width, kernel height, and number of learnable", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 369, + 303 + ], + "score": 1.0, + "content": "kernels. Kernel height and width are independently sampled from", + "type": "text" + }, + { + "bbox": [ + 369, + 292, + 413, + 304 + ], + "score": 0.93, + "content": "\\{ 1 , 3 , 5 , 7 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "and number of kernels", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 175, + 316 + ], + "score": 1.0, + "content": "is sampled from", + "type": "text" + }, + { + "bbox": [ + 176, + 303, + 234, + 315 + ], + "score": 0.9, + "content": "\\{ 6 , 1 2 , 2 4 , 3 6 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 303, + 505, + 316 + ], + "score": 1.0, + "content": ". Finally, we randomly sample residual connections between each", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "block of conv layers. Each network is trained for 50 epochs using the RMSProp optimizer, with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 324, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 162, + 337 + ], + "score": 1.0, + "content": "weight decay", + "type": "text" + }, + { + "bbox": [ + 162, + 324, + 183, + 335 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 324, + 421, + 337 + ], + "score": 1.0, + "content": ", initial learning rate 0.001, and a learning rate reduction to", + "type": "text" + }, + { + "bbox": [ + 421, + 324, + 442, + 335 + ], + "score": 0.9, + "content": "1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 324, + 505, + 337 + ], + "score": 1.0, + "content": "at epoch 30 on", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 336, + 198, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 198, + 347 + ], + "score": 1.0, + "content": "the CIFAR-10 dataset.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 226, + 506, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 352, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "MetaQNN CNNs (CIFAR-10 and SVHN): We sample 1,000 model architectures from the search", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "space detailed by Baker et al. (2017), which allows for varying the numbers and orderings of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "convolution, pooling, and fully connected layers. The models are between 1 and 12 layers for the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "SVHN experiment and between 1 and 18 layers for the CIFAR-10 experiment. Each architecture", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "is trained on SVHN and CIFAR-10 datasets for 20 epochs. Table 3 displays the state space of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 408, + 194, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 194, + 420 + ], + "score": 1.0, + "content": "MetaQNN algorithm.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 352, + 506, + 420 + ] + }, + { + "type": "table", + "bbox": [ + 122, + 428, + 489, + 572 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 122, + 428, + 489, + 572 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 428, + 489, + 572 + ], + "spans": [ + { + "bbox": [ + 122, + 428, + 489, + 572 + ], + "score": 0.984, + "html": "
Layer TypeLayerParametersParameterValues
Convolution (C)i~Layer depthf~ Receptive field sizel~Strided ~ # receptive fieldsn ~ Representation size<12Square. ∈ {1,3,5}Square.Always equal to 1∈{64,128,256,512}∈{(∞,8],(8,4],(4,1]}
Pooling (P)i~Layer depth(f,l)~ (Receptive field size,Strides)n ~Representation size<12Square.∈{(5,3),(3,2),(2,2)}∈{(∞,8],(8,4] and (4,1]}
Fully Connected (FC)i~Layerdepthn ~ # consecutive FC layersd ～# neurons<12<3∈ {512,256,128}
Termination StateS~Previous Statet~TypeGlobal Avg.Pooling/Softmax
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For each layer type, we list the relevant", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "parameters and the values each parameter is allowed to take. The networks are sampled beginning", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 104, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "from the starting layer. Convolutional layers are allowed to transition to any other layer. Pooling", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "layers are allowed to transition to any layer other than pooling layers. Fully connected layers are only", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "allowed to transition to fully connected or softmax layers. A convolutional or pooling layer may only", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "score": 1.0, + "content": "go to a fully connected layer if the current image representation size is below 8. We use this space", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 645, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 659 + ], + "score": 1.0, + "content": "to both randomly sample and simulate the behavior of a MetaQNN run as well as directly run the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 656, + 233, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 233, + 670 + ], + "score": 1.0, + "content": "MetaQNN with early stopping.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5 + } + ], + "index": 30.75 + }, + { + "type": "text", + "bbox": [ + 107, + 680, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 680, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 505, + 691 + ], + "score": 1.0, + "content": "LSTM (PTB): We sample 300 LSTM models and train them on the Penn Treebank dataset for 60", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "epochs. Number of hidden layer inputs and lstm cells was varied from 10 to 1400 in steps of 20.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 701, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 714 + ], + "score": 1.0, + "content": "Each network was trained for 60 epochs with batch size of 50 and trained the models using stochastic", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "gradient descent. Dropout ratio of 0.5 was used to prevent overfitting. Dictionary size of 400 words", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 349, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 349, + 106 + ], + "score": 1.0, + "content": "was used to generate embeddings when vectorizing the data.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 680, + 506, + 714 + ] + } + ] + }, + { + "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": "gradient descent. Dropout ratio of 0.5 was used to prevent overfitting. Dictionary size of 400 words", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 349, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 349, + 106 + ], + "score": 1.0, + "content": "was used to generate embeddings when vectorizing the data.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "Cuda-Convnet (CIFAR-10 and SVHN): We train Cuda-Convnet architecture (Krizhevsky, 2012)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "with varying values of initial learning rate, learning rate reduction step size, weight decay for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 131, + 507, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 507, + 146 + ], + "score": 1.0, + "content": "convolutional and fully connected layers, and scale and power of local response normalization layers.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "We train models with CIFAR-10 for 60 epochs and with SVHN for 12 epochs. Table 4 show the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 341, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 341, + 168 + ], + "score": 1.0, + "content": "hyperparameter ranges for the Cuda Convnet experiments.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "table", + "bbox": [ + 111, + 184, + 498, + 306 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 111, + 184, + 498, + 306 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 184, + 498, + 306 + ], + "spans": [ + { + "bbox": [ + 111, + 184, + 498, + 306 + ], + "score": 0.983, + "html": "
ExperimentHyperparameterScaleMinMax
CIFAR-10, Imagenet, SVHNInitial Learning RateLog5×10-55
Learning Rate ReductionsInteger03
CIFAR-10, SVHNConv1 L2 PenaltyLog5×10-55
Conv2 L2 PenaltyLog5×10-55
Conv3 L2 PenaltyLog5×10-55
FC4 L2 PenaltyLog5×10-55
Response Normalization ScaleLog5×10-65
Response Normalization PowerLinear1×10-23
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For", + "type": "text" + }, + { + "bbox": [ + 385, + 439, + 392, + 448 + ], + "score": 0.68, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 439, + 504, + 450 + ], + "score": 1.0, + "content": "-SVR, we perform a random", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 449, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 449, + 366, + 461 + ], + "score": 1.0, + "content": "search over 1000 hyperparameter configurations from the space", + "type": "text" + }, + { + "bbox": [ + 366, + 450, + 387, + 460 + ], + "score": 0.43, + "content": "C \\sim", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 449, + 440, + 461 + ], + "score": 1.0, + "content": "LogUniform", + "type": "text" + }, + { + "bbox": [ + 441, + 449, + 464, + 460 + ], + "score": 0.8, + "content": "( 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 449, + 486, + 461 + ], + "score": 1.0, + "content": ", 10),", + "type": "text" + }, + { + "bbox": [ + 486, + 450, + 505, + 460 + ], + "score": 0.75, + "content": "\\nu \\sim", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 142, + 474 + ], + "score": 1.0, + "content": "Uniform", + "type": "text" + }, + { + "bbox": [ + 142, + 461, + 164, + 472 + ], + "score": 0.35, + "content": "( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 459, + 185, + 474 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 185, + 460, + 297, + 473 + ], + "score": 0.67, + "content": "\\gamma \\sim \\mathrm { L o g U n i f o r m } ( 1 0 ^ { - 5 } , 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 459, + 506, + 474 + ], + "score": 1.0, + "content": ") (when using the RBF kernel). 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(2017), except we initialize two", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 161, + 574 + ], + "score": 1.0, + "content": "dictionaries:", + "type": "text" + }, + { + "bbox": [ + 161, + 562, + 171, + 571 + ], + "score": 0.76, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 561, + 280, + 574 + ], + "score": 1.0, + "content": "to store training data and", + "type": "text" + }, + { + "bbox": [ + 280, + 562, + 292, + 571 + ], + "score": 0.75, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 561, + 464, + 574 + ], + "score": 1.0, + "content": "to store performance prediction models.", + "type": "text" + }, + { + "bbox": [ + 465, + 561, + 484, + 573 + ], + "score": 0.88, + "content": "D [ r ]", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 438, + 585 + ], + "score": 1.0, + "content": "correspond to a dictionary containing all datasets with prediction target epoch", + "type": "text" + }, + { + "bbox": [ + 439, + 574, + 444, + 582 + ], + "score": 0.63, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 571, + 453, + 585 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 453, + 572, + 484, + 584 + ], + "score": 0.89, + "content": "D [ r ] [ \\bar { \\tau } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 263, + 596 + ], + "score": 1.0, + "content": "correspond to the dataset for predicting", + "type": "text" + }, + { + "bbox": [ + 263, + 584, + 274, + 595 + ], + "score": 0.84, + "content": "y _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 583, + 365, + 596 + ], + "score": 1.0, + "content": "based on the observed", + "type": "text" + }, + { + "bbox": [ + 365, + 583, + 397, + 595 + ], + "score": 0.92, + "content": "y ( t ) _ { 1 - \\tau }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 583, + 418, + 596 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 418, + 583, + 451, + 595 + ], + "score": 0.93, + "content": "M [ r ] [ \\tau ]", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "will hold the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "corresponding performance prediction model. 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The results shown are inconclusive; it seems", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "that there could be some benefit of incorporating the SVR acquisition function, but it isn’t significant", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 417, + 250, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 250, + 430 + ], + "score": 1.0, + "content": "enough to make a strong statement.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + } + ], + "index": 13.0 + }, + { + "type": "title", + "bbox": [ + 108, + 452, + 292, + 465 + ], + "lines": [ + { + "bbox": [ + 104, + 450, + 294, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 294, + 467 + ], + "score": 1.0, + "content": "E GAUSSIAN ERROR ASSUMPTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 477, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 489 + ], + "score": 1.0, + "content": "In order to have uncertainty estimates with frequentist models, we estimated the mean and variance of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 489, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 500 + ], + "score": 1.0, + "content": "the error from leave-one-out cross validation and assumed that the error was Gaussian. In Figure 8 we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 500, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 412, + 511 + ], + "score": 1.0, + "content": "compare example error distributions between training and validation from a", + "type": "text" + }, + { + "bbox": [ + 412, + 501, + 419, + 509 + ], + "score": 0.66, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 500, + 505, + 511 + ], + "score": 1.0, + "content": "-SVR SRM. Visually", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "one can see that the assumption holds very well for both visual classification tasks and holds slightly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "less well for the language modeling task. In Figure 9, we show the mean log likelihood of held", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "out prediction errors being drawn from a Gaussian parameterized by the mean and variance of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "error on the training set is very close to the mean log likelihood of samples drawn from the same", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 553, + 432, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 432, + 568 + ], + "score": 1.0, + "content": "distribution, which shows that our Gaussian error assumption is relatively strong.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 106, + 582, + 478, + 595 + ], + "lines": [ + { + "bbox": [ + 104, + 581, + 480, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 480, + 596 + ], + "score": 1.0, + "content": "F ANALYZING IMPORTANCE OF FEATURES IN THE PREDICTION MODEL", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 606, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 181, + 619 + ], + "score": 1.0, + "content": "We used a linear", + "type": "text" + }, + { + "bbox": [ + 181, + 608, + 188, + 617 + ], + "score": 0.69, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "-SVR model to compare the weights of all features, which are normalized", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "before training. Figure 10 shows the weight trends for time-series features, and Tables 5, 6, and 7", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "compare weights of some architecture features and hyperparameter features to statistics of time-series", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "features for the MetaQNN (CIFAR-10), LSTMs (Penn Treebank), and Deep Resnets (TinyImagenet)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 651, + 430, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 430, + 664 + ], + "score": 1.0, + "content": "experiments, respectively. We found the following main insights across datasets:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 672, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "score": 1.0, + "content": "• The time-series (TS) features are on average have higher weights than HP and AP features (which", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 114, + 684, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 114, + 684, + 467, + 695 + ], + "score": 1.0, + "content": "is confirmed by our ablation studies in Table 2 as well). The original validation accuracies", + "type": "text" + }, + { + "bbox": [ + 467, + 684, + 483, + 695 + ], + "score": 0.82, + "content": "( y _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 684, + 505, + 695 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 113, + 694, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 113, + 694, + 344, + 707 + ], + "score": 1.0, + "content": "higher weights on average than the first-order differences", + "type": "text" + }, + { + "bbox": [ + 344, + 695, + 363, + 706 + ], + "score": 0.87, + "content": "( { y } _ { t } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 694, + 482, + 707 + ], + "score": 1.0, + "content": "and second-order differences", + "type": "text" + }, + { + "bbox": [ + 482, + 695, + 502, + 706 + ], + "score": 0.89, + "content": "( { y } _ { t } ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 694, + 506, + 707 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 222, + 722 + ], + "score": 1.0, + "content": "• In general, later epochs in", + "type": "text" + }, + { + "bbox": [ + 223, + 711, + 233, + 721 + ], + "score": 0.84, + "content": "y _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "have higher weights. In other words, the latest performance of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 114, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 114, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "model available for prediction is much more important for performance prediction than initial", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 224, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 225, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 225, + 38 + ], + "score": 1.0, + "content": "Workshop track - ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 393, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 394, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 394, + 97 + ], + "score": 1.0, + "content": "D F-HYPERBAND WITH SVR ACQUISITION FUNCTION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 506, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "Similar to Klein et al. (2017), it is possible to use our simple models as an acquisition function for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "Hyperband. As shown in Table 2, our models have reasonable performance with only AP and HP", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "features used, meaning we can use these models to rank hyperparameter settings before training the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 139, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 106, + 139, + 506, + 152 + ], + "score": 1.0, + "content": "configurations. In particular, for each Hyperband iteration, we randomly select 10,000 configurations,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 506, + 163 + ], + "score": 1.0, + "content": "weight them by their predicted performance and sample from this weighted distribution. In Figure 7,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 251, + 173 + ], + "score": 1.0, + "content": "we compare f-Hyperband with this", + "type": "text" + }, + { + "bbox": [ + 252, + 162, + 258, + 171 + ], + "score": 0.64, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 161, + 506, + 173 + ], + "score": 1.0, + "content": "-SVR acquisition function to f-Hyperband with the standard", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 505, + 185 + ], + "score": 1.0, + "content": "uniform acquisition function and also to vanilla Hyperband. It seems that the acquisition function", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "could help slightly, but it does not improve performance across all experiments, and where it does", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 194, + 289, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 289, + 207 + ], + "score": 1.0, + "content": "improve performance it does so only slightly.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 105, + 506, + 207 + ] + }, + { + "type": "image", + "bbox": [ + 108, + 217, + 500, + 362 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 217, + 500, + 362 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 217, + 500, + 362 + ], + "spans": [ + { + "bbox": [ + 108, + 217, + 500, + 362 + ], + "score": 0.963, + "type": "image", + "image_path": "cd7670b52afd22f5572cddfea1b647b61e4c33d65e54d6765e5cede1a1f1cd49.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 108, + 217, + 500, + 265.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 108, + 265.3333333333333, + 500, + 313.66666666666663 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 108, + 313.66666666666663, + 500, + 361.99999999999994 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 373, + 505, + 429 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 373, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 387 + ], + "score": 1.0, + "content": "Figure 7: f-Hyperband with SVR Acquisition Function: We compare f-Hyperband with an SVR", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "acquisition function to f-Hyperband with the uniform acquisition function (this is the standard version", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "described in the main text) and to vanilla Hyperband. The results shown are inconclusive; it seems", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "that there could be some benefit of incorporating the SVR acquisition function, but it isn’t significant", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 417, + 250, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 250, + 430 + ], + "score": 1.0, + "content": "enough to make a strong statement.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + } + ], + "index": 13.0 + }, + { + "type": "title", + "bbox": [ + 108, + 452, + 292, + 465 + ], + "lines": [ + { + "bbox": [ + 104, + 450, + 294, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 294, + 467 + ], + "score": 1.0, + "content": "E GAUSSIAN ERROR ASSUMPTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 477, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 489 + ], + "score": 1.0, + "content": "In order to have uncertainty estimates with frequentist models, we estimated the mean and variance of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 489, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 500 + ], + "score": 1.0, + "content": "the error from leave-one-out cross validation and assumed that the error was Gaussian. In Figure 8 we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 500, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 412, + 511 + ], + "score": 1.0, + "content": "compare example error distributions between training and validation from a", + "type": "text" + }, + { + "bbox": [ + 412, + 501, + 419, + 509 + ], + "score": 0.66, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 500, + 505, + 511 + ], + "score": 1.0, + "content": "-SVR SRM. Visually", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "one can see that the assumption holds very well for both visual classification tasks and holds slightly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "less well for the language modeling task. In Figure 9, we show the mean log likelihood of held", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 532, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 545 + ], + "score": 1.0, + "content": "out prediction errors being drawn from a Gaussian parameterized by the mean and variance of the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "error on the training set is very close to the mean log likelihood of samples drawn from the same", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 553, + 432, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 432, + 568 + ], + "score": 1.0, + "content": "distribution, which shows that our Gaussian error assumption is relatively strong.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 478, + 506, + 568 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 582, + 478, + 595 + ], + "lines": [ + { + "bbox": [ + 104, + 581, + 480, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 480, + 596 + ], + "score": 1.0, + "content": "F ANALYZING IMPORTANCE OF FEATURES IN THE PREDICTION MODEL", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 606, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 181, + 619 + ], + "score": 1.0, + "content": "We used a linear", + "type": "text" + }, + { + "bbox": [ + 181, + 608, + 188, + 617 + ], + "score": 0.69, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "-SVR model to compare the weights of all features, which are normalized", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "before training. Figure 10 shows the weight trends for time-series features, and Tables 5, 6, and 7", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "compare weights of some architecture features and hyperparameter features to statistics of time-series", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 653 + ], + "score": 1.0, + "content": "features for the MetaQNN (CIFAR-10), LSTMs (Penn Treebank), and Deep Resnets (TinyImagenet)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 651, + 430, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 430, + 664 + ], + "score": 1.0, + "content": "experiments, respectively. We found the following main insights across datasets:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 607, + 506, + 664 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 672, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "score": 1.0, + "content": "• The time-series (TS) features are on average have higher weights than HP and AP features (which", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 114, + 684, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 114, + 684, + 467, + 695 + ], + "score": 1.0, + "content": "is confirmed by our ablation studies in Table 2 as well). The original validation accuracies", + "type": "text" + }, + { + "bbox": [ + 467, + 684, + 483, + 695 + ], + "score": 0.82, + "content": "( y _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 684, + 505, + 695 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 113, + 694, + 506, + 707 + ], + "spans": [ + { + "bbox": [ + 113, + 694, + 344, + 707 + ], + "score": 1.0, + "content": "higher weights on average than the first-order differences", + "type": "text" + }, + { + "bbox": [ + 344, + 695, + 363, + 706 + ], + "score": 0.87, + "content": "( { y } _ { t } ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 694, + 482, + 707 + ], + "score": 1.0, + "content": "and second-order differences", + "type": "text" + }, + { + "bbox": [ + 482, + 695, + 502, + 706 + ], + "score": 0.89, + "content": "( { y } _ { t } ^ { \\prime \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 694, + 506, + 707 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 222, + 722 + ], + "score": 1.0, + "content": "• In general, later epochs in", + "type": "text" + }, + { + "bbox": [ + 223, + 711, + 233, + 721 + ], + "score": 0.84, + "content": "y _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "have higher weights. In other words, the latest performance of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 114, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 114, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "model available for prediction is much more important for performance prediction than initial", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 114, + 682, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 114, + 682, + 506, + 696 + ], + "score": 1.0, + "content": "performance, when the model has just started training. However, the weights for earlier performance", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 114, + 694, + 482, + 706 + ], + "spans": [ + { + "bbox": [ + 114, + 694, + 482, + 706 + ], + "score": 1.0, + "content": "metrics are non-zero, indicating there are high order autoregressive functions being learned.", + "type": "text", + "cross_page": true + } + ], + "index": 17 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 671, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 80, + 503, + 387 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 80, + 503, + 387 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 80, + 503, + 387 + ], + "spans": [ + { + "bbox": [ + 108, + 80, + 503, + 387 + ], + "score": 0.968, + "type": "image", + "image_path": "6ab4eee391e270952d9a993a07231dd2f41ebb64efe1a472221b93e477a96b83.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 80, + 503, + 182.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 182.33333333333331, + 503, + 284.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 284.66666666666663, + 503, + 386.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 397, + 505, + 452 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "Figure 8: Example Error Distributions: Here we compare example error distributions between", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 409, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 234, + 421 + ], + "score": 1.0, + "content": "training and validation from a", + "type": "text" + }, + { + "bbox": [ + 235, + 410, + 242, + 419 + ], + "score": 0.67, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 409, + 505, + 421 + ], + "score": 1.0, + "content": "-SVR SRM. From left to right we see plots for models trained", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 505, + 433 + ], + "score": 1.0, + "content": "with increasing amounts of the learning curve. The blue line shows the Gaussian probability", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 429, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 444 + ], + "score": 1.0, + "content": "density function estimated from the training set with LOOCV. 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Absent results for a model indicate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 428, + 252, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 235, + 441 + ], + "score": 1.0, + "content": "that it did not achieve a positive", + "type": "text" + }, + { + "bbox": [ + 236, + 428, + 249, + 439 + ], + "score": 0.88, + "content": "R ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 428, + 252, + 441 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + } + ], + "index": 11.25 + }, + { + "type": "image", + "bbox": [ + 108, + 462, + 504, + 637 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 462, + 504, + 637 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 462, + 504, + 637 + ], + "spans": [ + { + "bbox": [ + 108, + 462, + 504, + 637 + ], + "score": 0.971, + "type": "image", + "image_path": "893b61c3e8dee896d83890cfd6f959ee2023596c2286f47916b4990e7a40b892.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 108, + 462, + 504, + 520.3333333333334 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 108, + 520.3333333333334, + 504, + 578.6666666666667 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 108, + 578.6666666666667, + 504, + 637.0000000000001 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 648, + 506, + 693 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 104, + 646, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 104, + 646, + 506, + 662 + ], + "score": 1.0, + "content": "Figure 12: Simulated Speedup on Hyperband vs Hyperband Iteration: We show the speedup", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 659, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 412, + 671 + ], + "score": 1.0, + "content": "using the f-Hyperband algorithm over Hyperband on 40 consecutive runs with", + "type": "text" + }, + { + "bbox": [ + 412, + 659, + 445, + 671 + ], + "score": 0.91, + "content": "\\eta = 4 . 0", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 659, + 462, + 671 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 463, + 659, + 503, + 669 + ], + "score": 0.9, + "content": "\\varDelta = 0 . 9 5", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 659, + 506, + 671 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "The major jump in speedup comes at iteration 10, where we have trained more than 100 models to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 680, + 190, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 137, + 693 + ], + "score": 1.0, + "content": "the full", + "type": "text" + }, + { + "bbox": [ + 137, + 682, + 147, + 691 + ], + "score": 0.79, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 680, + 190, + 693 + ], + "score": 1.0, + "content": "iterations.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5 + } + ], + "index": 19.75 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 81, + 494, + 228 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 81, + 494, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 81, + 494, + 228 + ], + "spans": [ + { + "bbox": [ + 108, + 81, + 494, + 228 + ], + "score": 0.958, + "type": "image", + "image_path": "e9cd3871458632935a702159971de0403ccd0159d69e2a3b0451acad4716e165.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 81, + 494, + 130.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 130.0, + 494, + 179.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 179.0, + 494, + 228.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 239, + 505, + 317 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 505, + 252 + ], + "score": 1.0, + "content": "Figure 13: Simulated f-Hyperband with Pretrained SRMs: Each f-Hyperband model is initialized", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 263 + ], + "score": 1.0, + "content": "with a pretrained SRM, which simulates the case where one reinitializes an experiment. 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Cuda Convnet (CIFAR-10)Learning RateWeightDecayLRN ScaleLRN Power
Test<Median93.06 ± 0.1895.93 ± 0.2197.27 ± 0.006896.96 ± 0.13
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LSTMs (Penn Treebank)Number of NodesDepth
Test<Median98.08 ± 0.2698.58 ± 0.04
Test>Median22.83 ± 5.5596.96 ± 0.91
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DeepResnets (TinyImagenet)DepthNumber ofKernelsKernel Size
Test<Median89.49 ± 1.8485.12 ± 2.2481.83 ± 0.59
Test >Median85.92 ± 1.7862.37 ± 0.8289.61 ± 0.31
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MetaQNN (CIFAR-10)Depth
Test<Median96.69 ± 0.16
Test>Median79.31 ± 5.66
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Cuda Convnet (CIFAR-10)Learning RateWeightDecayLRN ScaleLRN Power
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LSTMs (Penn Treebank)Number of NodesDepth
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Test<Median89.49 ± 1.8485.12 ± 2.2481.83 ± 0.59
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MetaQNN (CIFAR-10)Depth
Test<Median96.69 ± 0.16
Test>Median79.31 ± 5.66
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Datasetv-SVROLSBLRRF
MetaQNN (CIFAR-10)97.22 ± 0.3197.04± 0.4297.11 ± 0.3294.50 ± 0.78
Resnet (TinyImageNet)87.12 ± 4.6588.41 ± 3.7682.40± 1.6586.39 ± 3.58
LSTM (Penn Treebank)98.03± 1.4687.83 ± 21.5397.99 ± 1.9057.28± 24.98
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Feature SetMetaQNN(CIFAR-10)ResNets(TinyImageNet)LSTM(Penn Treebank)Cuda-Convnet(CIFAR-10)
HP-2.96 ± 2.710.48 ± 7.24-1.72 ± 1.620.48 ± 7.24
AP15.38 ± 4.6667.66 ± 9.0813.76 ± 12.0867.66 ± 9.08
TS97.00 ± 0.5282.71 ± 4.3197.38 ± 1.4982.71 ± 4.31
AP+HP14.89 ± 5.1270.25 ± 6.4213.76 ± 12.0870.25 ± 6.42
TS+HP97.02 ± 0.5282.67 ± 6.8197.38 ± 1.4982.67± 6.81
TS+AP97.12 ± 0.3787.36± 4.2397.87 ± 1.6887.36± 4.23
TS+AP+HP97.15 ± 0.4886.05 ± 8.2497.87 ± 1.6886.05 ± 8.24
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Layer TypeLayerParametersParameterValues
Convolution (C)i~Layer depthf~ Receptive field sizel~Strided ~ # receptive fieldsn ~ Representation size<12Square. ∈ {1,3,5}Square.Always equal to 1∈{64,128,256,512}∈{(∞,8],(8,4],(4,1]}
Pooling (P)i~Layer depth(f,l)~ (Receptive field size,Strides)n ~Representation size<12Square.∈{(5,3),(3,2),(2,2)}∈{(∞,8],(8,4] and (4,1]}
Fully Connected (FC)i~Layerdepthn ~ # consecutive FC layersd ～# neurons<12<3∈ {512,256,128}
Termination StateS~Previous Statet~TypeGlobal Avg.Pooling/Softmax
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ExperimentHyperparameterScaleMinMax
CIFAR-10, Imagenet, SVHNInitial Learning RateLog5×10-55
Learning Rate ReductionsInteger03
CIFAR-10, SVHNConv1 L2 PenaltyLog5×10-55
Conv2 L2 PenaltyLog5×10-55
Conv3 L2 PenaltyLog5×10-55
FC4 L2 PenaltyLog5×10-55
Response Normalization ScaleLog5×10-65
Response Normalization PowerLinear1×10-23
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@@ -0,0 +1,511 @@ +# EXPRESSIVE YET TRACTABLE BAYESIAN DEEP LEARNING VIA SUBNETWORK INFERENCE + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +The Bayesian paradigm has the potential to solve some of the core issues in modern deep learning, such as poor calibration, data inefficiency, and catastrophic forgetting. However, scaling Bayesian inference to the high-dimensional parameter spaces of deep neural networks requires restrictive approximations. In this paper, we propose performing inference over only a small subset of the model parameters while keeping all others as point estimates. This enables us to use expressive posterior approximations that would otherwise be intractable for the full model. In particular, we develop a practical and scalable Bayesian deep learning method that first trains a point estimate, and then infers a full covariance Gaussian posterior approximation over a subnetwork. We propose a subnetwork selection procedure which aims to maximally preserve posterior uncertainty. We empirically demonstrate the effectiveness of our approach compared to point-estimated networks and methods that use less expressive posterior approximations over the full network. + +# 1 INTRODUCTION + +Deep neural networks (DNNs) still suffer from critical shortcomings that make them unfit for important applications. For instance, DNNs tend to be poorly calibrated and overconfident in their predictions, especially when there is a shift in the train and test distributions (Nguyen et al., 2015; Guo et al., 2017). To reliably inform decision making, DNNs must be able to robustly quantify the uncertainty in their predictions, which is particularly important in safety-critical areas such as healthcare or autonomous driving (Amodei et al., 2016; Filos et al., 2019a; Fridman et al., 2019). + +Bayesian modeling (Ghahramani, 2015; Gal, 2016) presents a principled way to capture predictive uncertainty via the posterior distribution over model parameters. Unfortunately, due to their nonlinearities, exact posterior inference is intractable in DNNs. Despite recent successes in the field of Bayesian deep learning (Blundell et al., 2015; Gal & Ghahramani, 2016; Osawa et al., 2019; Maddox et al., 2019; Dusenberry et al., 2020), existing methods are only made scalable to modern DNNs with large numbers of parameters by invoking unrealistic assumptions. This severely limits the expressiveness of the inferred posterior and thus deteriorates the quality of the induced uncertainty estimates (Ovadia et al., 2019; Fort et al., 2019; Foong et al., 2019a; Ashukha et al., 2020a). + +Due to the heavy overparameterization of DNNs, their accuracy is well-preserved by a small subnetwork (Cheng et al., 2017). Additionally, recent work by Izmailov et al. (2019) has shown how performing inference over a low dimensional subspace of the weights can result in accurate uncertainty quantification. These observations prompt the following question for a DNN’s uncertainty: Can a full DNN’s model uncertainty be well-preserved by a small subnetwork’s model uncertainty? We answer this question in the affirmative. We show both theoretically and empirically that the full network posterior can be well represented by a subnetwork’s posterior. As a result, we can use more expensive but faithful posterior approximations over just that subnetwork. We show that this achieves better uncertainty quantification than if we use cheaper, but more crude, posterior approximations over the full network. + +The contributions of this paper are as follows: + +1. We propose a new Bayesian deep learning approach that performs Bayesian inference over only a small subset of the model weights and keeps all other weights deterministic. This allows us to use expressive posterior approximations that are typically intractable in DNNs. + +![](images/69ba0d7682cd0284507b2c8b65f0296f13b556dfc570af6f9e27b927c22c918d.jpg) +Figure 1: Schematic illustration of our proposed approach. (a) We train a neural network using standard techniques to obtain a point estimate of the weights. (b) We identify a small subset of the weights. (c) We estimate a posterior distribution over the selected subnetwork via Bayesian inference techniques. (d) We make predictions using the full network of mixed Bayesian/deterministic weights. + +2. As a concrete instantiation of this framework, we develop a practical and scalable Bayesian deep learning method that uses the linearized Laplace approximation to infer a fullcovariance Gaussian posterior over a subnetwork within a point-estimated neural network. +3. We formally characterize the discrepancy between the posterior distributions over a subnetwork and the full network (in terms of their Wasserstein distance) in the linearized model, and derive a theoretically motivated strategy to select a subnetwork that minimizes this discrepancy under certain assumptions. +4. We empirically show, on various benchmarks, that our method compares favourably against point-estimated networks and other Bayesian deep learning methods, experimentally confirming that expressive subnetwork inference is superior to crude inference over full networks. + +# 2 SUBNETWORK POSTERIOR APPROXIMATION + +Bayesian neural networks (BNNs) aim to capture model uncertainty, i.e., uncertainty about the choice of weights $W$ which arises due to multiple plausible explanations of the training data $\{ y , X \}$ . Here, $\textbf { { y } }$ is the dependent variable (e.g. classification label) and $\boldsymbol { X }$ is the feature matrix. A prior distribution $p ( \mathbf { W } )$ is specified over the BNN’s weights. We wish to infer their full posterior distribution + +$$ +\begin{array} { r l r } { p ( { \bf W } | { \bf y } , { \pmb X } ) } & { \propto } & { p ( { \pmb y } | { \pmb X } , { \bf W } ) p ( { \bf W } ) . } \end{array} +$$ + +To make predictions, we then estimate the posterior predictive distribution that averages the network’s predictions across all possible settings of the weights, weighted by their posterior probability, i.e. + +$$ +p ( { \mathbf { y } ^ { * } } | X ^ { * } , y , X ) = \int _ { \mathbf { W } } p ( { \mathbf { y } ^ { * } } | X ^ { * } , \mathbf { W } ) p ( \mathbf { W } | y , X ) d \mathbf { W } . +$$ + +Unfortunately, due to the size of modern deep neural networks, it is not only intractable to infer the exact posterior distribution $p ( \mathbf { W } | \mathbf { \boldsymbol { y } } , X )$ in Eq. (1), but it is even computationally challenging to properly approximate it. As a consequence, crude posterior approximations such as complete factorization are commonly employed (Blundell et al., 2015; Hernandez-Lobato & Adams, 2015; ´ Kingma et al., 2015; Khan et al., 2018; Osawa et al., 2019), i.e. $\begin{array} { r } { p ( \mathbf { W } | \pmb { y } , \pmb { X } ) \approx \prod _ { d = 1 } ^ { D } q ( \mathbf { w } _ { d } ) } \end{array}$ where $\mathbf { W } _ { d }$ denotes the $d$ -th weight in the $D$ -dimensional neural network weight vector $\mathbf { W _ { \lambda } } \in \mathbb { R } ^ { D }$ (the concatenation and flattening of all layers’ weight matrices). Clearly, this is a very wishful assumption; In practise, it suffers from severe pathologies (Foong et al., 2019a;b). + +In this work, we question the implicit assumption that a good posterior approximation needs to include all BNN parameters. Instead, we aim to perform inference only over a small subset of the weights. This approach is well-motivated for two reasons: + +1. Overparameterization: Maddox et al. (2020) have shown that, in the neighborhood of local optima, there are many directions that leave the NN’s predictions unchanged. Moreover, NNs can be heavily pruned without sacrificing test-set accuracy (Frankle & Carbin, 2019). Thus, the majority of a NN’s predictive power might be isolated to a small subnetwork. + +2. Inference over submodels: Previous work1 has provided evidence that inference can be effective even when not done on the full parameter space. Izmailov et al. (2019) performed inference over a low-dimensional projection of the weights. Neural-linear models, which give a Bayesian treatment to only the last layer of a DNN, have shown to be competitive with full-network approaches (Riquelme et al., 2018; Ober & Rasmussen, 2019). + +We thus combine these ideas, making the following two-step approximation of the posterior in Eq. (1): + +$$ +p ( \mathbf { W } | \pmb { y } , \pmb { X } ) \ \approx \ p ( \mathbf { W } _ { S } | \pmb { y } , \pmb { X } ) \prod _ { r } \delta ( \mathbf { w } _ { r } - \mathbf { w } _ { r } ^ { * } ) \ \approx \ q ( \mathbf { W } _ { S } ) \prod _ { r } \delta ( \mathbf { w } _ { r } - \mathbf { w } _ { r } ^ { * } ) . +$$ + +The first approximation decomposes the full neural network posterior $p ( \mathbf { W } | \mathbf { \boldsymbol { y } } , X )$ into a posterior $p ( \mathbf { W } _ { S } | \pmb { y } , \pmb { X } )$ over the subnetwork $\mathbf { W } _ { S }$ and delta functions $\delta \big ( \mathbf { w } _ { r } - \mathbf { w } _ { r } ^ { * } \big )$ over all remaining weights $\left\{ \mathbf { w } _ { r } \right\} _ { r }$ , keeping them at fixed values $\mathbf { w } _ { r } ^ { * } \in \mathbb { R }$ . This can be viewed as pruning the variances of the weights $\left\{ \mathbf { w } _ { r } \right\} _ { r }$ to zero, which is in contrast to ordinary weight pruning methods that set the weights $\left\{ \mathbf { w } _ { r } \right\} _ { r }$ themselves to zero. The second approximation is a result of posterior inference over the subnetwork still being intractable. In turn, we introduce the approximate distribution $q ( \mathbf { W } _ { S } )$ . Yet, as the subnetwork is much smaller than the full network, we can afford to make $q ( \mathbf { W } _ { S } )$ expressive and able to capture rich dependencies across the weights within the subnetwork. + +# 3 SUBNETWORK INFERENCE VIA LAPLACE APPROXIMATION + +To obtain a method that is as practical as possible, we propose to use inference techniques that can estimate a posterior distribution post-hoc from a point-estimated network. The Laplace approximation (MacKay, 1992) is well-suited to this task as it derives the approximate posterior from the local optimization landscape. Other inference procedures, such as SWAG (Maddox et al., 2019), could also be used. Nevertheless, we focus on Laplace due to it being a well-studied, fundamental technique. + +Step #1: Point Estimation. The first step of the procedure is to train a neural network to obtain a point estimate of the weights, denoted $W _ { M A P }$ . This estimate should respect the Bayesian model given in Eq. (1), and therefore we optimize the maximum $a$ -posteriori (MAP) objective: + +$$ +\begin{array} { r } { { \cal W } _ { M A P } = \arg \operatorname* { m a x } _ { W } \left[ \log p ( { \bf y } | X , W ) + \log p ( W ) \right] . } \end{array} +$$ + +This can be done using standard stochastic gradient-based optimization methods commonly-used in modern deep learning (Goodfellow et al., 2016). This step is illustrated in Fig. 1 (a). + +Step #2: Subnetwork Selection. The second step is to identify a small subnetwork $W _ { S }$ . Ideally, we would like to identify the subnetwork whose posterior is ‘closest’ to the full-network posterior. We formalize this argument in Section 4 and describe a principled strategy that, under certain conditions, minimizes the 2-Wasserstein distance between the sub- and full-network posteriors. All other weights not belonging to that subnetwork are then assigned fixed values: the MAP estimates obtained in Step #1. See Fig. 1 (b) for an illustration of this step. + +Step #3: Bayesian Inference. Given the subnetwork point estimate $W _ { M A P } ^ { S }$ , we use the Laplace approximation to infer a full-covariance Gaussian posterior distribution over the subnetwork $\mathbf { W } _ { S }$ : + +$$ +p ( \mathbf { W } _ { S } | \mathbf { \boldsymbol { y } } , \boldsymbol { X } ) \approx \ q ( \mathbf { W } _ { S } ) \ = \ N \left( \mathbf { W } _ { S } ; \mathbf { \boldsymbol { W } } _ { M A P } ^ { S } , H ^ { - 1 } \right) +$$ + +where the posterior covariance matrix $H ^ { - 1 } \in \mathbb { R } ^ { D \times D }$ corresponds to the inverse of the average Hessian of the negative log posterior, i.e. $H = N \mathbb { E } \left[ - \partial ^ { 2 } \log \hat { p } ( \mathbf { y } | X , W ) / \partial W ^ { 2 } \right] + \lambda I$ . Here, the expectation is w.r.t. the data generating distribution and $\lambda$ is the precision of a zero-mean factorized Gaussian prior $p ( \mathbf { W } ) = \tilde { \mathcal { N } } ( \mathbf { W } ; \mathbf { 0 } , \lambda ^ { \top } I )$ . In practice, we approximate the Hessian $H$ with the generalized Gauss-Newton (GGN) matrix $\widetilde { H }$ (Schraudolph, 2002), i.e. + +$$ +\widetilde { H } = \sum _ { n = 1 } ^ { N } J _ { n } ^ { \top } H _ { n } J _ { n } + \lambda I , \mathrm { w i t h } J _ { n } = \frac { \partial f ( x _ { n } , W ) } { \partial W } \mathrm { a n d } H _ { n } = \frac { \partial ^ { 2 } L ( y _ { n } , f ( x _ { n } , W ) ) } { \partial ^ { 2 } f ( x _ { n } , W ) } +$$ + +where $J _ { n } \in \mathbb { R } ^ { O \times D }$ is the Jacobian of the neural network features $\pmb { f } ( \pmb { x } _ { n } , \pmb { W } ) \in \mathbb { R } ^ { O }$ w.r.t. the weights $W$ , and $\pmb { H } _ { n } \in \mathbb { R } ^ { O \times O }$ is the Hessian of the loss $L ( { \pmb y } _ { n } , { \pmb f } ( { \pmb x } _ { n } , { \pmb W } ) )$ w.r.t. the features $\mathbf { { \mathit { f } } } ( \mathbf { { \mathit { x } } } _ { n } , W )$ .The GGN $\widetilde { H }$ has clear practical advantages over the Hessian $H$ ; see Martens $\&$ Sutskever (2011) and Martens (2016). Using the Laplace approximation with the GGN Hessian can be viewed as an implicit local linearization of the underlying neural network $\mathbf { \Delta } f ( \mathbf { \Delta } x , W )$ at its MAP estimate $W _ { M A P }$ , + +$$ +f _ { l i n } ^ { M A P } ( { \pmb x } , { \pmb W } ) = { \pmb f } ( { \pmb x } , { \pmb W } _ { M A P } ) + J _ { { \pmb W } _ { M A P } } ( { \pmb x } ) ( { \pmb W } - { \pmb W } _ { M A P } ) +$$ + +where $J _ { W _ { M A P } } ( \pmb { x } ) = \partial \pmb { f } ( \pmb { x } , \pmb { W } _ { M A P } ) / \partial \pmb { W } _ { M A P } \in \mathbb { R } ^ { O \times D }$ (Immer et al., 2020). Note that the model in Eq. (7) is linear in $W$ , as only the term $J _ { W _ { M A P } } ( { \pmb x } ) { \pmb W }$ depends linearly on $W$ , while the other terms are constant w.r.t. $W$ and can thus be subsumed into an additive bias term (Khan et al., 2019). The GGN approximation thus locally turns the underlying probabilistic model from a Bayesian neural network into a (generalized) linear model, with basis function expansion $J _ { W _ { M A P } } ( { \pmb x } )$ of covariate $_ { \textbf { \em x } }$ (Immer et al., 2020). Put differently, linearized Laplace in the neural network $\mathbf { \Delta } f ( \mathbf { \Delta } x , W )$ is equivalent to ordinary Laplace in the linear model $\pmb { f } _ { l i n } ^ { M A P } ( \pmb { x } , \pmb { W } )$ in Eq. (7), as the GGN $\widetilde { H }$ corresponding to in Eq. (6) is equivalent to the Hessian 9). This is a useful property that will $H$ corresponding to ow us to derive a $\pmb { f } _ { l i n } ^ { M A P } ( \pmb { x } , \pmb { W } )$ in Eq. (7) (Khanetwork selection strategy in Section 4. This step is illustrated in Fig. 1 (c). We emphasize that this whole procedure (i.e. Steps #1-#3) is a perfectly valid mixed inference strategy, performing full Laplace inference over the selected subnetwork and MAP inference over all remaining weights. + +Step #4: Prediction. Given the linearized Laplace approximation over the subnetwork $\mathbf { W } _ { S }$ in Eqs. (5) and (6), i.e. $q ( \mathbf { W } _ { S } ) = \mathcal { N } ( \mathbf { W } _ { S } ; { W _ { M A P } ^ { S } } , \mathbf { \widetilde { H } } ^ { - 1 } )$ , we can then compute the posterior predictive distribution. While, traditionally, one would compute the predictive distribution using the original Bayesian neural network likelihood, i.e. $p ( \mathbf { y } | X , \bar { \mathbf { W } } ) = p ( \bar { \mathbf { y } } | f ( \mathbf { x } , W ) )$ , Immer et al. (2020) recently suggested that, since inference was (implicitly) done in the GGpled to instead predict using the linearized likelihood Eq. (7), i.e. $p ( \mathbf { y } \vert X , \mathbf { W } ) = p ( \mathbf { y } \vert \pmb { f } _ { l i n } ^ { M A P } ( \pmb { \dot { x } } , \pmb { W } ) )$ -. This provides a formal justification for the empirical superiority of this approach observed previously (Lawrence, 2001; Foong et al., 2019b). We thus obtain the linearized predictive distribution + +$$ +p ( \mathbf { y } ^ { * } | \boldsymbol { X } ^ { * } , \boldsymbol { y } , \boldsymbol { X } ) \approx \int _ { \mathbf { W } } p ( \mathbf { y } ^ { * } | f _ { l i n } ^ { M A P } ( \boldsymbol { X } ^ { * } , \boldsymbol { W } ) ) \mathcal { N } ( \mathbf { W } _ { S } ; \boldsymbol { W } _ { M A P } ^ { S } , \tilde { H } ^ { - 1 } ) \prod _ { r } \delta ( \mathbf { w } _ { r } - \mathbf { w } _ { r } ^ { * } ) d \mathbf { W } . +$$ + +There are two ways to compute Eq. (8): Firstly, via a Monte Carlo approximation $p ( \mathbf { y } ^ { * } | X ^ { * } , y , X ) \simeq$ $\begin{array} { r } { \frac { 1 } { M } \sum _ { m = 1 } ^ { M } p ( \mathbf { y } ^ { * } | \pmb { f } _ { l i n } ^ { M A P } ( \pmb { X } ^ { * } , \widetilde { W } _ { m } ) ) } \end{array}$ by sampling $\widetilde { W } _ { m }$ from $\mathcal { N } ( W _ { M A P } ^ { S } , \widetilde { H } ^ { - 1 } )$ and $\begin{array} { r } { \prod _ { r } \delta ( \mathbf { w } _ { r } - \mathbf { w } _ { r } ^ { * } ) } \end{array}$ , M m=1 | lin f f N MAP e r − the latter of which is trivial. Secondly, due to linearity of p(y∗|f MAPlin (X∗, W )), there are closed- form expressions which are exact for Gaussian likelihoods (i.e. regression) and approximate for categorical ones (i.e. classification) (Bishop, 2006; Gibbs, 1998). This step is illustrated in Fig. 1 (d). + +# 4 PRINCIPLED SUBNETWORK SELECTION FOR LINEAR(IZED) MODELS + +We next analyze the subnetwork inference procedure described in Section 3 for the case of a generalized linear model (GLM) (Nelder & Baker, 1972), which models the expected response $y _ { n }$ given the basis function expansion of the covariates $\phi _ { n } = \phi ( { \mathbf { { x } } } _ { n } )$ as + +$$ +\mathbb { E } [ y _ { n } | \phi _ { n } ] = g ^ { - 1 } ( \mathbf { w } ^ { T } \phi _ { n } ) . +$$ + +Here, $\mathbf { w } \in \mathbb { R } ^ { D }$ is the vector of model parameters (which subsumes a scalar bias $\beta _ { 0 }$ for notational convenience) and $g ^ { - 1 } ( \cdot )$ denotes a link function such that $g ^ { - 1 } : \mathbb { R } \mapsto \mu _ { y | \phi }$ . In particular, we consider a Bayesian GLM, by specifying a prior distribution $p ( \mathbf { w } )$ over model parameters and aiming to infer the posterior distribution $p ( \mathbf { w } | \mathbf { y } , \pmb { \Phi } ) \propto p ( \mathbf { y } | \pmb { \Phi } , \mathbf { w } ) p ( \mathbf { w } )$ , where $\begin{array} { r } { \pmb { \Phi } = [ \phi _ { 1 } , \dots \phi _ { N } ] ^ { T } } \end{array}$ . + +1. Point Estimation. Obtain the MAP estimate, $\begin{array} { r } { \mathbf { w } _ { M A P } = \arg \operatorname* { m a x } _ { \mathbf { w } } \log p ( \pmb { y } | \Phi , \mathbf { w } ) + \log p ( \mathbf { w } ) } \end{array}$ . For commonly-used link functions (e.g. the identity in case of a Gaussian likelihood for regression, or the sigmoid/softmax function in case of a categorical likelihood for classification) and commonly-used priors (e.g. a Gaussian), the log-posterior $\propto \log p ( \pmb { y } | \pmb { \Phi } , \mathbf { w } ) + \log p ( \mathbf { w } )$ is concave. This allows for simple gradient-based MAP optimisation. It also makes a full-covariance Gaussian, estimated via Laplace, a faithful approximation to the true, uni-modal posterior, i.e. + +$$ +p ( \mathbf { w } | \mathbf { y } , \boldsymbol { \Phi } ) \approx \widetilde { p } ( \mathbf { w } | \mathbf { y } , \boldsymbol { \Phi } ) = \mathcal { N } ( \mathbf { w } ; \mathbf { w } _ { M A P } , H ^ { - 1 } ) +$$ + +where $H$ is the Hessian defined in Section 3. Note that for the GLM we consider, the Hessian $H$ is equivalent to the GGN $\widetilde { H }$ defined in Eq. (6), meaning that an ordinary Laplace approximation is equivalent to a linearized Laplace approximation (Martens, 2016). For the case of an identity link function (i.e. a Gaussian likelihood with noise variance $\sigma _ { 0 } ^ { 2 }$ ) and a Gaussian prior w $\sim \mathcal { N } ( \mathbf { 0 } , \mathbf { \dot { N } } _ { 0 } ^ { - 1 } )$ , the MAP estimate even has a closed-form expression, $\begin{array} { r } { { \bf w } _ { M A P } = ( \Phi ^ { T } \Phi + \sigma _ { 0 } ^ { 2 } \Lambda _ { 0 } ) ^ { - 1 } \Phi ^ { T } { \pmb y } . } \end{array}$ . Here, the Laplace approximation in Eq. (10) exactly corresponds to the true posterior, i.e. $\widetilde { p } ( \mathbf { w } | \mathbf { y } , \Phi ) =$ $p ( \mathbf { w } | \mathbf { y } , \Phi )$ . We will thus refer to the posterior $\widetilde { p } ( \mathbf { w } | \mathbf { y } , \Phi )$ e in Eq. (10) as the full posterior. + +2. Subnetwork Selection. Select a subset of $S$ model weights via a method of choice, yielding a binary vector $\pmb { m } \in \mathbb { R } ^ { D }$ where $m _ { d } = 1$ if the $d$ -th weight is part of the subset, and $m _ { d } = 0$ otherwise. For convenience, we define the binary mask matrix $\mathbf { M } _ { S } = \pmb { m } \pmb { m } ^ { \top } \in \mathbb { R } ^ { D \times D }$ which contains 1s in the rows/columns corresponding to the $S$ subnetwork weights2, and 0s otherwise. + +3. Bayesian Inference. Compute the posterior over the subnetwork via a Laplace approximation: + +$$ +p _ { S } ( \mathbf { w } | \pmb { y } , \pmb { \Phi } ) = \mathcal { N } ( \mathbf { w } ; \mathbf { w } _ { M A P } , M _ { S } \odot H ^ { - 1 } ) \ . +$$ + +Firstly, note that the mean of the subnetwork posterior in Eq. (11) is the MAP estimate $\mathbf { w } _ { M A P }$ and thus equal to the mean of the full posterior $\widetilde { p } ( \mathbf { w } | \mathbf { y } , \Phi )$ in Eq. (10). Secondly, note that the ecovariance matrix of the subnetwork posterior in Eq. (11) is the element-wise product $M _ { S } \odot H ^ { - 1 }$ , which masks the (co-)variances of all weights not belonging to the subnetwork to zero, effectively making them deterministic. More precisely, the subnetwork covariance matrix, $M _ { S } \odot H ^ { - 1 }$ , is a $D \times D$ matrix that is equal to the full posterior covariance matrix $H ^ { - 1 }$ in the rows/columns of the $S$ weights in the subnetwork, and zero in the rows/columns of all other $D - S$ weights. + +We consider what we term the posterior gap—the Wasserstein distance3 (in particular the squared 2-Wasserstein distance) between the posterior distribution over the full network and the posterior distribution over the subnetwork. The proofs for all results below will be presented in Appendix A. + +Proposition 1 (Posterior Gap). For a subnetwork of size $S < D$ , the Wasserstein gap between the full posterior $\widetilde { p } ( \mathbf { w } | \mathbf { y } , \Phi )$ in Eq. (10) and the subnetwork posterior $p _ { \mathrm { S } } ( { \bf w } | { \pmb y } , { \pmb \Phi } )$ in Eq. (11) is: + +$$ +\begin{array} { r } { W \left[ \widetilde { p } ( \mathbf { w } | y , \Phi ) \mid | p _ { \mathrm { S } } ( \mathbf { w } | y , \Phi ) \right] = \sum _ { d = 1 } ^ { D } \left( 1 + m _ { d d } \right) \sigma _ { d } ^ { 2 } - \mathrm { t r a c e } ( 2 ( H ^ { - 1 } ( M _ { S } \odot H ^ { - 1 } ) ) ^ { 1 / 2 } ) . } \end{array} +$$ + +The optimal subnetwork should then minimize the posterior gap in Eq. (12). However, for full covariance matrices $H ^ { - 1 }$ and a large number of weights $D$ , this will generally be infeasible as Eq. (12) depends on all entries of the $D \times D$ -matrix $H ^ { - 1 }$ , which is intractable to compute/store. To derive a practical subnetwork selection strategy, we assume the covariance matrix to be diagonal. + +Corollary 1.1 (Optimality of Maximum Variance Subnetwork Selection under Decorrelation). For $a$ generalized linear model with posterior covariance matrix $H ^ { - 1 } = d i a g ( \sigma _ { 1 } ^ { 2 } , . . . , \sigma _ { D } ^ { 2 } )$ , the optimal subnetwork under the Wasserstein gap is comprised of the $S$ weights with the largest variances $\sigma _ { d } ^ { 2 }$ . + +Finally, since a GGN-linearized neural network, as in Eq. (7), corresponds to a GLM with basis expansion $\phi _ { n } = J _ { W _ { M A P } } ( \pmb { x } _ { n } ) = \partial \pmb { f } ( \pmb { x } _ { n } , \pmb { W } _ { M A P } ) / \partial W _ { M A P }$ (see Step #3 in Section 3), Corollary 1.1 implies that under decorrelation, the optimal subnetwork comprises of the weights with the largest variances. In practice, even just computing the diagonal of the covariance matrix is challenging, so we use a diagonal Laplace approximation which instead computes the inverse of the diagonal of the GGN (see e.g. Ritter et al. (2018)). Finally, note that we only have to make the decorrelation assumption for the purposes of subnetwork selection – when doing posterior inference over the selected subnetwork, we estimate a full covariance matrix for maximal expressiveness, as described in Step #3 in Section 3. In our experiments in Section 5, we empirically show that making the decorrelation assumption for subnetwork selection but then using a full-covariance Gaussian for inference performs significantly better than directly making the decorrelation assumption for inference (e.g. mean-field variational inference, diagonal Laplace). + +# 5 EMPIRICAL ANALYSIS + +We empirically assess the effectiveness of subnetwork inference compared to point-estimated NNs and methods that do less expressive inference over the full network. We consider three tasks: 1) small-scale toy regression, 2) medium-scale tabular regression, and 3) large-scale image classification. + +![](images/348c77967767ca7b767128552cfde1a6eab11fff719d21ab1885ac59de9d2e18.jpg) +Figure 2: Predictive distributions (mean $\pm$ std) for 1D regression. The numbers in brackets denote the number of parameters over which inference was done (out of 2600 in total). Wasserstein-based subnetwork inference maintains richer predictive uncertainties at smaller parameter counts. + +# 5.1 HOW DOES SUBNETWORK INFERENCE RETAIN POSTERIOR PREDICTIVE UNCERTAINTY? + +We first assess how the predictive distribution of a full-covariance Gaussian posterior over a selected subnetwork qualitatively compares to that obtained from 1) a full-covariance Gaussian over the full network (Full Cov), 2) a factorised Gaussian posterior over the full network (Diag), 3) a fullcovariance Gaussian over only the (Final layer) of the network (Kristiadi et al., 2020), and 4) a point estimate $( M A P )$ . For subnetwork inference, we consider both Wasserstein (Wass) (as described in Section 4) and uniform random subnetwork selection (Rand) to obtain subnetworks that comprise of only $50 \%$ , $3 \%$ and $1 \%$ of the model parameters. Note that while for this toy example, we could in principle use the full covariance matrix for the purpose of subnetwork selection, we still just use its diagonal (as described in Section 4) for consistency. Our NN consists of 2 ReLU hidden layers with 50 hidden units each. We employ a homoscedastic Gaussian likelihood function where the noise variance is optimised with maximum likelihood. We use GGN Laplace inference over network weights (not biases) in combination with the linearized predictive distribution in Eq. (8). Thus, all approaches considered share their predictive mean, allowing us to better compare their uncertainty estimates. All approaches share a single prior precision of $\lambda = 3$ . + +We use a synthetic 1D regression task with two separated clusters of inputs (Antoran et al., 2020), ´ allowing us to probe for ‘in-between’ uncertainty (Foong et al., 2019b). Results are shown in Fig. 2. Subnetwork inference preserves more of the uncertainty of full network inference than diagonal Gaussian or final layer inference while doing inference over fewer weights. By capturing weight correlations, subnetwork inference retains uncertainty in between clusters of data. This is true for both random and Wasserstein subnetwork selection. However, the latter preserves more uncertainty with smaller subnetworks. Finally, the strong superiority to diagonal Laplace shows that making a diagonal assumption for subnetwork selection but then using a full-covariance Gaussian for inference (as we do) performs much signficantly better than making a diagonal assumption for the inferered posterior directly. These results suggest that expressive inference over a carefully selected subnetwork retains more predictive uncertainty than crude approximations over the full network. + +5.2 SUBNETWORK INFERENCE IN LARGE MODELS VS FULL INFERENCE OVER SMALL MODELS + +Secondly, we study the following natural question: “Why should one use subnetwork inference in a large model when one can just perform full network inference over a smaller model?” We explore this by considering 4 fully connected models of increasing size. These have numbers of hidden layers $h _ { d } = \{ 1 , 2 \}$ and hidden layer widths ${ w _ { d } } \mathrm { { = } } \{ 5 0 , 1 0 0 \}$ . For a dataset with input dimension $i _ { d }$ , the number of weights is given by $D { = } ( i _ { d } { + } 1 ) w _ { d } { + } ( h _ { d } { - } 1 ) w _ { d } ^ { 2 }$ . Our 2 hidden layer, 100 hidden unit models have a weight count of the order $1 0 ^ { 4 }$ . Full covariance inference in these models borders the limit of computational tractability on commercial hardware. We first obtain a MAP estimate of each model’s weights and our homoscedastic likelihood function’s noise variance. We then perform full network GGN Laplace inference for each model. We also use our proposed Wassertein rule to prune every network’s weight variances such that the number of variances that remain matches the size of every smaller network under consideration. In all cases, we employ the linearized predictive in Eq. (7). Consequently, networks with the same number of weights make the same mean predictions. Increasing the number of weight variances considered will thus only increase predictive uncertainty. + +![](images/66c302f1bb14b542bd24077fff89bad6dabe0c7a84ec71a09c89a80dccf32d61.jpg) +Figure 3: Mean test log-likelihood values obtained on UCI datasets across all splits. Different markers indicate models with different numbers of weights. The horizontal axis indicates the number of weights over which full covariance inference is performed. 0 corresponds to MAP parameter estimation, and the rightmost setting for each marker corresponds to full network inference. + +We employ 3 tabular datasets of increasing size (input dimensionality, n. points): wine (11, 1439), kin8nm (8, 7373) and protein (9, 41157). We consider their standard train-test splits (Hernandez- ´ Lobato & Adams, 2015) and their gap variants (Foong et al., 2019b), designed to test for out-ofdistribution uncertainty. Details are provided in Appendix C.4. For each split, we set aside $1 5 \%$ of the train data as a validation set. We use these for early stopping when finding MAP estimates and for selecting the weights’ prior precision. We keep other hyperparameters fixed across all models and datasets. Results are in Fig. 3. + +We present mean test log-likelihood (LL) values, as these take into account both accuracy and uncertainty. Larger models tend to perform better when doing MAP inference, with wine-gap and protein-gap being exceptions. We also find larger models improve over their respective MAP LLs more than small ones when performing approximate inference over the same numbers of weights. We conjecture this is due to an abundance of degenerate directions (weights) in the weight posterior of all models (Maddox et al., 2020). Full network inference in small models captures information about both useful and non-useful weights. In larger models, our subnetwork selection strategy allows us to dedicate a larger proportion of our resources to modelling informative weight variances and covariances. In 3 out of 6 datasets, we find abrupt increases in LL as we increase the number of weights over which we perform inference, followed by a plateau. Such plateaus might be explained by all of the most informative weight variances having already been accounted for. These results suggest that, given the same amount of compute, larger models benefit more from subnetwork inference than small ones. + +# 5.3 SCALING TO IMAGE CLASSIFICATION WITH DISTRIBUTION SHIFT + +We now assess the robustness of large convolutional neural networks with subnetwork inference to distribution shift on image classification tasks compared to the following baselines: point-estimated networks (MAP), Bayesian deep learning methods that do less expressive inference over the full network: MC Dropout (Gal & Ghahramani, 2016), diagonal Laplace, VOGN (Osawa et al., 2019) (all of which assume factorisation of the weight posterior), and SWAG (Maddox et al., 2019) (which assumes a diagonal plus low-rank posterior). We also benchmark deep ensembles (Lakshminarayanan et al., 2017). The latter is considered state-of-the-art for uncertainty quantification in deep learning (Ovadia et al., 2019; Ashukha et al., 2020a). We use ensembles of 5 DNNs, as suggested by (Ovadia et al., 2019), and 16 samples for MC Dropout, diagonal Laplace and SWAG. We use a Dropout probability of 0.1 and a prior precision of $\lambda = 4 0 , 0 0 0$ for diagonal Laplace, found via grid search. We apply all approaches to ResNet-18 (He et al., 2016), which is composed of an input convolutional block, 8 residual blocks and a linear layer, for a total of 11,168,000 parameters. For subnetwork inference, we compute the linearized predictive distribution in Eq. (8) via the closed-form approximation for integrals between Gaussians and multi-class cross-entropy likelihoods described in (Gibbs, 1998). We use Wasserstein subnetwork selection to retain only $0 . 3 8 \%$ of the weights, yielding a subnetwork with only 42,438 weights. This is the largest subnetwork for which we can tractably compute a full covariance matrix. Its size is 42 $, 4 3 8 ^ { 2 } \times \hat { 4 }$ Bytes $\approx 7 . 2$ GB. We use a prior precision of $\lambda = 5 0 0$ , found via grid search. Finally, to assess to importance of principled subnetwork selection, we also consider the baseline where we select the subnetwork uniformly at random (called Ours (Rand)). We perform the following two experiments, with results in Fig. 4. See Appendix B for additional results. + +![](images/966242861badeec22bbb4cab7022e457bc6a60cc58c9e8f8bac63f13ef1c8655.jpg) +Figure 4: Results on the rotated MNIST (left) and the corrupted CIFAR (right) benchmarks of Ovadia et al. (2019), showing the mean $\pm$ std of the error (top) and log-likelihood (bottom) across three different seeds. Subnetwork inference retains better uncertainty calibration and robustness to distribution shift than point estimated networks and other Bayesian deep learning approaches. + +Rotated MNIST: Following Ovadia et al. (2019); Antoran et al. (2020), we train all methods on ´ MNIST and evaluate their predictive distributions on increasingly rotated digits. While all methods perform well on the original MNIST test set, their accuracy degrades quickly for rotations larger than 30 degrees. In terms of LL, ensembles perform best out of our baselines. Subnetwork inference obtains significantly larger LL values than almost all baselines, including ensembles. The only exception is VOGN, which achieves slightly better performance. It was also observed in Ovadia et al. (2019) that mean-field variational inference (which VOGN also is an instance of) is very strong on MNIST, but its performance deteriorates on larger datasets. Subnetwork inference makes accurate predictions in-distribution while assigning higher uncertainty than the baselines to out-of-distribution points. Corrupted CIFAR: Again following Ovadia et al. (2019); Antoran et al. (2020), we train ´ on CIFAR10 and evaluate on data subject to 16 different corruptions with 5 levels of intensity each (Hendrycks & Dietterich, 2019). Our approach matches a MAP estimated network in terms of predictive error as local linearization makes their predictions the same. Ensembles and SWAG are the most accurate. Even so, subnetwork inference differentiates itself by being the least overconfident, outperforming all baselines in terms of log-likelihood at all corruption levels. Here, VOGN performs rather badly; while this might appear in stark contrast to its strong performance on the MNIST benchmark, the behaviour that mean-field VI performs well on MNIST but not on larger datasets was also observed in Ovadia et al. (2019). + +On both benchmarks, we furthermore find that randomly selecting the subnetwork performs substantially worse than using our more principled subnetwork selection strategy. This highlights the importance of the way subnetworks are selected. These results suggest that subnetwork inference results in better uncertainty calibration and robustness to distribution shift than other popular uncertainty quantification approaches. + +# 6 RELATED WORK + +Bayesian Deep Learning. There have significant efforts to characterise the posterior distribution over NN weights $p ( W | \bar { \mathcal { D } } )$ . Hamiltonian Monte Carlo (Neal, 1995) remains the golden standard for approximate inference in BNNs to this day. Although asymptotically unbiased, sampling based approaches are difficult to scale to the large datasets (Betancourt, 2015). As a result, approaches which find the best surrogate posterior among an approximating family (most often Gaussians) have gained popularity. The first of these was the Laplace approximation, introduced by MacKay (1992), who also proposed approximating the predictive posterior with that of the linearised model (Khan et al., 2019; Immer et al., 2020). The popularisation of larger NN models has made surrogate distributions that capture correlations between weights computationally intractable. Thus, most modern methods make use of the mean field assumption (Blundell et al., 2015; Hernandez-Lobato & ´ Adams, 2015; Gal & Ghahramani, 2016; Mishkin et al., 2018; Osawa et al., 2019). This comes at the cost of limited expressivity (Foong et al., 2019a) and empirical under-performance (Ovadia et al., 2019; Antoran et al., 2020) of uncertainty estimates. Our proposed approach recovers predictive ´ posterior expressivity while maintaining tractability by lowering the dimensionality of the weight space considered. This allows us to scale up approximations that do consider weight correlations (MacKay, 1992; Louizos & Welling, 2016; Maddox et al., 2019; Ritter et al., 2018). + +Neural Network Linearization. In the limit of infinite width, NNs converge to Gaussian process (GP) behaviour (Neal, 1995; Matthews, 2017; Garriga-Alonso et al., 2018). Recently, these results have been extended to finite width BNNs when the surrogate posterior is Gaussian (Khan et al., 2019). We draw upon these results to formulate a subnetwork selection strategy for BNNs. Neural linear methods perform inference over only the last layer of a NN, while keeping all other layers fixed (Snoek et al., 2015; Riquelme et al., 2018; Ovadia et al., 2019; Ober & Rasmussen, 2019; Pinsler et al., 2019; Kristiadi et al., 2020). These represent a different generalised linear model in which the basis functions are defined by the $l { - } 1$ first layers of a NN. They can also be viewed as a special case of subnetwork inference, in which the subnetwork is simply defined to be the last NN layer. + +Inference over Subspaces. The subfield of NN pruning aims to increase the computational efficiency of NNs by identifying the smallest subset of weights which are required to make accurate predictions. Approaches trade-off computational cost with compression efficiency, ranging from those that require multiple training runs (Frankle & Carbin, 2019) to those that prune before training (Wang et al., 2020). Our work differs in that it retains all NN weights but aims to find a small subset over which to perform probabilistic reasoning. More closely related work to ours is that of Izmailov et al. (2019), who propose to perform inference over a low-dimensional subspace of weights; e.g. one constructed from the principal components of the SGD trajectory. Moreover, several recent approaches use low-rank parameterizations of approximate posteriors in the context of variational inference (Rossi et al., 2019; Swiatkowski et al., 2020; Dusenberry et al., 2020). This could also be viewed as doing inference over an implicit subspace of weight space. In contrast, we propose a technique to find subsets of weights which are relevant to predictive uncertainty, i.e., we identify axis aligned subspaces. Finally, there have been recent works studying neural network sparsity / pruning from a Bayesian perspective (Ghosh & Doshi-Velez, 2017; Polson & Rockov ˇ a, 2018; Cui et al., 2020; Louizos et al., 2017; ´ Molchanov et al., 2017; Gomez et al., 2019; Lee et al., 2018). While these seem conceptually related at first glance, their goal is fundamentally different to ours: While those methods aim to perform model selection / sparsification by either explicitly or implicitly pruning unnecessary weights, our goal is to make inference more tractable. More precisely, while those sparse Bayesian deep learning methods prune individual weights, we instead just prune the variances of certain weights, which, importantly retains the full predictive power of the full network to retain high predictive accuracy. + +# 7 CONCLUSION + +In this paper, we develop a practical and scalable method for expressive yet tractable probabilistic inference in deep neural networks. We approximate the posterior over a subset of the weights while keeping all other weights deterministic. Computational cost is decoupled from network size, allowing us to scale expressive approximations, such as full-covariance Gaussian distributions, to real-world sized NNs. Our approach can be applied post-hoc to any pre-trained model, making it particularly attractive for practical use. Our empirical analysis suggests that subnetwork inference 1) is more expressive and retains more uncertainty than crude approximations over the full network, 2) allows us to employ larger NNs, which fit a broader range of functions, without sacrificing the quality of our uncertainty estimates, and 3) is competitive with state-of-the-art uncertainty quantification methods, like deep ensembles (Lakshminarayanan et al., 2017), on real-world scale problems. + +# REFERENCES + +Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mane.´ Concrete problems in ai safety. arXiv preprint arXiv:1606.06565, 2016. + +Javier Antoran, James Urquhart Allingham, and Jos ´ e Miguel Hern ´ andez-Lobato. Depth uncertainty ´ in neural networks, 2020. + +Arsenii Ashukha, Alexander Lyzhov, Dmitry Molchanov, and Dmitry Vetrov. Pitfalls of in-domain uncertainty estimation and ensembling in deep learning. 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Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. 2017. + +Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. In Richard C. Wilson, Edwin R. Hancock, and William A. P. Smith (eds.), Proceedings of the British Machine Vision Conference 2016, BMVC 2016, York, UK, September 19-22, 2016. BMVA Press, 2016. + +# A PROOFS FOR THE THEORETICAL RESULTS + +We now provide the proofs for the results in Section 4. + +# A.1 PROOF OF PROPOSITION 1 + +Proof. Note that the posterior distributions $\widetilde { p } ( \mathbf { w } | \mathbf { y } , \Phi )$ and $p _ { S } ( { \bf w } | { \pmb y } , { \pmb \Phi } )$ are both Gaussian. We thus econsider the squared 2-Wasserstein distance between two Gaussian distributions $\mathcal { N } ( \mu _ { 1 } , \Sigma _ { 1 } )$ and $\mathcal { N } ( \mu _ { 2 } , \Sigma _ { 2 } )$ , which has the following closed-form expression (Givens et al., $1 9 8 4 ) ^ { 4 }$ : + +$$ +\begin{array} { r } { W \left[ \mathcal { N } \left( \mu _ { 1 } , \Sigma _ { 1 } \right) \ | | \mathcal { N } \left( \mu _ { 2 } , \Sigma _ { 2 } \right) \right] = \| \mu _ { 1 } - \mu _ { 2 } \| _ { 2 } ^ { 2 } + \mathrm { t r a c e } \left( \Sigma _ { 1 } + \Sigma _ { 2 } - 2 \left( \Sigma _ { 1 } \Sigma _ { 2 } \right) ^ { 1 / 2 } \right) \ . } \end{array} +$$ + +Plugging in ${ \pmb { \mu } } _ { 1 } = { \pmb { \mu } } _ { 2 } = \mathbf { w } _ { M A P }$ , $\pmb { \Sigma } _ { 1 } = H ^ { - 1 }$ and $\Sigma _ { 2 } = M _ { S } \odot H ^ { - 1 }$ , we obtain + +$$ +\begin{array} { r l } & { W \left[ \widetilde { p } ( \mathbf { w } | y , X ) \mid | p _ { \mathrm { S } } ( \mathbf { w } | y , X ) \right] } \\ & { \quad = W \left[ \mathcal { N } ( \mathbf { w } _ { M A P } , H ^ { - 1 } ) \mid | \mathcal { N } ( \mathbf { w } _ { M A P } , M _ { S } \odot H ^ { - 1 } ) \right] } \\ & { \quad = \underbrace { \left. \mathbf { w } _ { M A P } \longleftrightarrow \widetilde { { \mathbf { w } } _ { M A P } } \right. _ { 2 } ^ { 2 } } _ { = \mathrm { t r a c e } } \left( H ^ { - 1 } + ( M _ { S } \odot H ^ { - 1 } ) - 2 \left( H ^ { - 1 } ( M _ { S } \odot H ^ { - 1 } ) \right) ^ { 1 / 2 } \right) } \\ & { \quad = \mathrm { t r a c e } \left( ( 1 + M _ { S } ) \odot H ^ { - 1 } \right) - \mathrm { t r a c e } \left( 2 \left( H ^ { - 1 } \left( M _ { S } \odot H ^ { - 1 } \right) \right) ^ { 1 / 2 } \right) } \\ & { \quad = \displaystyle \sum _ { d = 1 } ^ { D } ( 1 + m _ { d d } ) \sigma _ { d } ^ { 2 } - \mathrm { t r a c e } \left( 2 \left( H ^ { - 1 } \left( M _ { S } \odot H ^ { - 1 } \right) \right) ^ { 1 / 2 } \right) } \end{array} +$$ + +# A.2 PROOF OF COROLLARY 1.1 + +Proof. For $H ^ { - 1 } = \mathrm { d i a g } ( \sigma _ { 1 } ^ { 2 } , . . . , \sigma _ { D } ^ { 2 } )$ , the Wasserstein posterior gap in Eq. (12) simplifies to + +$$ +W \left[ \widetilde { p } ( { \mathbf { w } } | { \pmb y } , { \pmb \Phi } ) | | p _ { \mathrm { S } } ( { \mathbf { w } } | { \pmb y } , { \pmb \Phi } ) \right] = \sum _ { d = 1 } ^ { D } \left( \left( 1 + m _ { d d } \right) \sigma _ { d } ^ { 2 } - 2 m _ { d d } \sigma _ { d } ^ { 2 } \right) . +$$ + +The optimal subnetwork selection strategy amounts to choosing the binary vector $\pmb { m } = [ m _ { d d } ] _ { d = 1 } ^ { D }$ with $\textstyle \sum _ { d = 1 } ^ { D } m _ { d } = S$ (i.e., we select $S$ out of $D$ parameters) s.t. the posterior gap in Eq. (14) is minimized. Observing that the contribution of the $d$ -th parameter to the posterior gap is $( 1 + 1 ) \sigma _ { d } ^ { 2 } - 1 \times 2 \sigma _ { d } ^ { 2 } = 0$ if it is selected (i.e. if $m _ { d d } = 1$ ), and $( 1 + \bar { 0 } ) \sigma _ { d } ^ { 2 } - 0 \times 2 \sigma _ { d } ^ { 2 } \stackrel { . } { = } \sigma _ { d } ^ { 2 }$ if it is not selected (i.e. if $m _ { d d } = 0$ ), we see that the optimal subnetwork comprises of the $S$ weights with the largest variances $\sigma _ { d } ^ { 2 }$ . □ + +# B ADDITIONAL IMAGE CLASSIFICATION RESULTS + +Table 1: AUC-ROC scores for out-of-distribution detection, using CIFAR10 vs SVHN and MNIST vs FashionMNIST as in- (source) and out-of-distribution (target) datasets, respectively (Nalisnick et al., 2019). + +
SOURCETARGETOURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAG
CIFAR10SVHN0.85±0.030.86±0.020.85±0.010.86±0.020.91±0.000.86±0.020.83±0.00
MNISTFashion0.92±0.050.75±0.020.82±0.120.75±0.010.90±0.090.72±0.030.97±0.01
+ +Table 2: MNIST – no rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.07±0.01-0.01±0.00-0.01±0.00-0.04±0.03-0.01±0.00-0.01±0.00-0.01±0.00-0.14±nan
error0.01±0.000.00±0.000.00±0.000.01±0.010.00±0.000.00±0.000.00±0.000.01±nan
ECE0.05±0.010.00±0.000.00±0.000.00±0.000.00±0.000.00±0.000.00±0.000.10±nan
brier score0.02±0.000.01±0.000.01±0.000.02±0.010.01±0.000.01±0.000.01±0.000.04±nan
+ +4This also holds for our case of a degenerate Gaussian with singular covariance matrix (Givens et al., 1984). + +![](images/aad024300c6bf0b50f42234321d229a99d769b38ed3b300a3faab5c6a4f64a35.jpg) +Figure 5: Rotated MNIST (left) and Corrupted CIFAR10 (right) results for deep ensembles (Lakshminarayanan et al., 2017) with large numbers of ensemble members (i.e. up to 55). Horizontal axis denotes number of ensemble members, and vertical axis denotes performance in terms of loglikelihood. Straight horizontal lines correspond to the performance of our method, as a reference. Colors denote different levels of rotation (left) and corruption (right). It can clearly be observed that the performance of deep ensembles saturates after around 15 ensemble members, meaning that adding more members yields strongly diminishing returns. This is in agreement with recent works (Antoran´ et al., 2020; Ashukha et al., 2020a; Lobacheva et al., 2020). Our method significantly outperforms even very large deep ensembles, especially for high degrees of rotation/corruption. + +![](images/7ac4e36b330bb3a0ad67b1c0ff314010923ad8edfb3ea97ca805ca45b4ba2b70.jpg) +Figure 6: Rejection-classification plots. We simulate a realistic OOD rejection scenario (Filos et al., 2019b) by jointly evaluating our models on an in-distribution and an OOD test set. We allow our methods to reject increasing proportions of the data based on predictive entropy before classifying the rest. All predictions on OOD samples are treated as incorrect. Following (Nalisnick et al., 2019), we use CIFAR10 vs SVHN and MNIST vs FashionMNIST as in- and out-of-distribution datasets, respectively. Note that the SVHN test set is randomly sub-sampled down to a size of 10,000. + +![](images/c3a908807cb61e57ccfb703b4ff067fc82ee203ac08066346bfa9a6f26a046e7.jpg) +Figure 7: Full MNIST rotation and CIFAR10 corruption results, for ResNet-18, reporting predictive error, log-likelihood (LL), expected calibration error (ECE) and brier score, respectively (from top to bottom). + +![](images/84c92e68a29bb694ec93250940373bf76b1df5a4d1136c7e674ba7159aed3f4b.jpg) +Figure 8: MNIST rotation results for ResNet-50, reporting predictive error, log-likelihood (LL), expected calibration error (ECE) and brier score. We choose a subnetwork containing only $0 . 1 6 7 \%$ (39,190 / 23,466,560) of the parameters of the full network. We see that subnetwork inference still results in an improvement in the calibration of predictive uncertainty. As expected, however, for ResNet-50 the improvement over MAP is smaller than for ResNet-18 where we were able to choose a subnetwork containing $0 . 3 8 \%$ of the parameters. + +Table 3: MNIST – 15◦ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.14±0.02-0.05±0.00-0.05±0.00-0.11±0.08-0.04±0.00-0.05±0.00-0.04±0.00-0.19±nan
error0.02±0.000.02±0.000.01±0.000.03±0.020.01±0.000.02±0.000.01±0.000.02±nan
ECE0.08±0.010.00±0.000.00±0.000.01±0.010.00±0.000.00±0.000.00±0.000.12±nan
brier score0.05±0.010.03±0.000.02±0.000.05±0.030.02±0.000.02±0.000.02±0.000.07±nan
+ +Table 4: MNIST – $3 0 ^ { \circ }$ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.42±0.04-0.36±0.01-0.32±0.02-0.44±0.06-0.28±0.02-0.39±0.01-0.30±0.00-0.51±nan
error0.11±0.010.10±0.000.09±0.010.12±0.010.08±0.010.10±0.000.08±0.000.14±nan
ECE0.10±0.020.04±0.010.03±0.000.06±0.010.02±0.000.05±0.000.04±0.000.13±nan
brier score0.19±0.020.16±0.000.14±0.010.18±0.020.12±0.010.16±0.000.12±0.000.23±nan
+ +Table 5: MNIST – 45◦ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
midrule LL-1.09±0.03-1.60±0.05−1.44±0.11-1.68±0.20-1.36±0.07-1.75±0.06-1.35±0.02-1.15±nan
error0.36±0.010.35±0.010.33±0.010.35±0.030.31±0.010.35±0.010.29±0.000.40±nan
ECE0.03±0.010.22±0.010.19±0.020.22±0.020.17±0.010.23±0.010.18±0.000.01±nan
brier score0.49±0.020.55±0.020.52±0.020.55±0.040.48±0.020.56±0.020.46±0.010.53±nan
+ +Table 6: MNIST – $6 0 ^ { \circ }$ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-2.10±0.03-3.85±0.18-3.54±0.23-4.11±0.66-3.60±0.10-4.29±0.21-2.95±0.08-1.92±nan
error0.63±0.010.63±0.010.62±0.010.62±0.050.61±0.010.63±0.010.53±0.020.64±nan
ECE0.25±0.020.46±0.020.43±0.020.47±0.060.42±0.010.48±0.020.36±0.020.17±nan
brier score0.85±0.021.04±0.031.00±0.031.05±0.100.98±0.021.07±0.030.86±0.030.80±nan
+ +Table 7: MNIST – $7 5 ^ { \circ }$ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.02±0.07-5.93±0.28-5.49±0.38-6.92±0.32-5.74±0.15-6.63±0.33-4.46±0.18-2.54±nan
error0.80±0.020.79±0.010.79±0.010.81±0.000.78±0.010.79±0.010.72±0.020.77±nan
ECE0.41±0.040.62±0.030.59±0.010.65±0.010.58±0.010.64±0.030.51±0.020.26±nan
brier score1.08±0.041.34±0.041.30±0.021.39±0.011.29±0.021.37±0.041.17±0.040.95±nan
+ +Table 8: MNIST – 90◦ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.35±0.13-6.46±0.15-6.18±0.41-7.32±0.67-6.39±0.17-7.18±0.22-5.63±0.12-2.91±nan
error0.84±0.020.84±0.010.84±0.010.85±0.010.84±0.010.84±0.010.82±0.020.81±nan
ECE0.43±0.030.64±0.040.62±0.010.66±0.030.62±0.010.66±0.040.60±0.010.29±nan
brier score1.13±0.031.40±0.051.37±0.011.44±0.041.36±0.011.43±0.051.34±0.021.02±nan
+ +Table 9: MNIST – 105◦ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.59±0.05-7.06±0.45-6.70±0.52-7.69±0.99-7.01±0.17-7.87±0.53-6.28±0.19-3.10±nan
error0.85±0.020.84±0.020.84±0.010.85±0.010.84±0.010.84±0.020.81±0.000.81±nan
ECE0.47±0.040.67±0.050.63±0.010.67±0.030.64±0.010.68±0.040.61±0.010.34±nan
brier score1.17±0.051.44±0.071.38±0.021.44±0.041.40±0.011.46±0.071.34±0.021.07±nan
+ +Table 10: MNIST – 120◦ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.43±0.07-6.73±0.53-6.62±0.39-7.92±0.59-6.73±0.11-7.53±0.63-6.49±0.36-3.07±nan
error0.80±0.020.79±0.020.78±0.010.81±0.010.78±0.010.79±0.020.76±0.020.76±nan
ECE0.40±0.030.62±0.050.58±0.010.65±0.040.59±0.010.63±0.040.58±0.030.30±nan
brier score1.10±0.031.35±0.071.29±0.021.39±0.061.30±0.011.36±0.071.27±0.041.04±nan
+ +Table 11: MNIST – 135◦ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.24±0.06-6.43±0.38-6.46±0.28-7.05±0.88-6.57±0.10-7.24±0.48-6.40±0.37-2.89±nan
error0.71±0.020.71±0.020.70±0.010.71±0.010.70±0.010.71±0.020.70±0.020.67±nan
ECE0.32±0.010.55±0.030.52±0.010.56±0.020.52±0.010.56±0.030.53±0.020.25±nan
brier score0.99±0.021.21±0.051.17±0.021.22±0.041.17±0.011.23±0.051.18±0.040.94±nan
+ +Table 12: MNIST – 150◦ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.25±0.05-6.56±0.18-6.62±0.33-7.04±0.36-6.88±0.11-7.41±0.25-6.39±0.27-2.69±nan
error0.63±0.020.63±0.010.63±0.000.65±0.010.62±0.010.63±0.010.63±0.010.60±nan
ECE0.29±0.010.50±0.010.48±0.010.52±0.010.48±0.010.51±0.010.49±0.010.23±nan
brier score0.92±0.021.10±0.021.07±0.011.13±0.021.06±0.011.11±0.021.08±0.020.85±nan
+ +Table 13: MNIST – 165◦ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.42±0.12-7.01±0.15-7.08±0.39-7.80±0.12−7.51±0.11-7.91±0.18-6.63±0.24-2.67±nan
error0.58±0.010.58±0.010.58±0.010.58±0.000.57±0.010.58±0.010.59±0.000.56±nan
ECE0.32±0.020.49±0.010.48±0.010.49±0.010.48±0.000.51±0.010.48±0.000.25±nan
brier score0.90±0.021.05±0.011.04±0.011.05±0.011.03±0.011.07±0.021.03±0.010.82±nan
+ +Table 14: MNIST – 180◦ rotation. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.32±0.13-6.63±0.18-6.87±0.32-7.10±0.47-7.16±0.16-7.43±0.20-6.61±0.22-2.71±nan
error0.56±0.010.56±0.010.56±0.000.55±0.010.55±0.000.56±0.010.57±0.000.55±nan
ECE0.29±0.020.46±0.010.45±0.000.46±0.000.46±0.010.48±0.010.47±0.010.25±nan
brier score0.86±0.021.00±0.010.99±0.010.99±0.010.99±0.001.01±0.021.01±0.010.82±nan
+ +Table 15: CIFAR10 – no corruption. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.27±0.00-0.43±0.01-0.37±0.01-0.50±0.02-0.21±0.01-0.46±0.02-0.48±0.01-0.61±nan
error0.09±0.000.08±0.000.08±0.000.09±0.000.06±0.000.08±0.000.11±0.000.21±nan
ECE0.01±0.000.06±0.000.04±0.000.06±0.000.01±0.000.06±0.000.07±0.000.03±nan
brier score0.13±0.000.14±0.000.13±0.000.15±0.000.09±0.000.14±0.000.17±0.000.30±nan
+ +Table 16: CIFAR10 – level 1 corruption. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.51±0.01-0.91±0.01-0.80±0.02−1.03±0.02-0.50±0.02-0.96±0.02-0.89±0.02-0.99±nan
error0.17±0.010.16±0.000.16±0.000.17±0.000.13±0.000.16±0.000.17±0.000.32±nan
ECE0.03±0.000.11±0.000.10±0.000.13±0.000.04±0.000.12±0.010.11±0.000.03±nan
brier score0.24±0.000.27±0.000.25±0.000.29±0.000.19±0.000.27±0.010.29±0.000.44±nan
+ +Table 17: CIFAR10 – level 2 corruption. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.73±0.01-1.29±0.06-1.20±0.02−1.50±0.12-0.80±0.01-1.40±0.03-1.21±0.00-1.31±nan
error0.23±0.000.22±0.010.22±0.000.23±0.010.19±0.000.22±0.000.22±0.000.40±nan
ECE0.06±0.000.16±0.010.14±0.000.17±0.010.07±0.000.16±0.000.15±0.000.10±nan
brier score0.33±0.000.37±0.010.35±0.010.40±0.020.28±0.000.37±0.010.37±0.000.56±nan
+ +Table 18: CIFAR10 – level 3 corruption. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-1.06±0.02-2.06±0.12-1.85±0.07-2.13±0.17-1.28±0.03-2.18±0.08-1.63±0.03-1.83±nan
error0.32±0.010.31±0.010.31±0.010.31±0.010.28±0.000.31±0.010.28±0.000.51±nan
ECE0.11±0.010.24±0.010.21±0.010.24±0.010.12±0.000.24±0.010.20±0.000.19±nan
brier score0.46±0.010.54±0.020.50±0.020.54±0.030.42±0.000.54±0.020.47±0.010.72±nan
+ +Table 19: CIFAR10 – level 4 corruption. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-1.25±0.03-2.43±0.18-2.28±0.10-2.54±0.18-1.56±0.05-2.57±0.15-1.95±0.04-1.99±nan
error0.36±0.010.35±0.010.35±0.010.35±0.010.32±0.010.35±0.010.32±0.000.54±nan
ECE0.13±0.010.27±0.010.24±0.010.27±0.010.14±0.010.27±0.020.23±0.000.22±nan
brier score0.51±0.020.60±0.030.57±0.010.61±0.020.47±0.010.60±0.030.53±0.000.76±nan
+ +Table 20: CIFAR10 – level 5 corruption. + +
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL−1.47±0.03-2.82±0.11-2.71±0.13-3.20±0.13-1.88±0.05-3.03±0.10-2.31±0.09-2.00±nan
error0.41±0.000.40±0.010.40±0.010.41±0.010.37±0.010.40±0.000.36±0.010.54±nan
ECE0.16±0.010.31±0.010.28±0.010.33±0.020.17±0.010.31±0.010.27±0.010.19±nan
brier score0.58±0.000.69±0.010.65±0.010.72±0.030.55±0.010.69±0.010.61±0.010.75±nan
+ +# C EXPERIMENTAL SETUP + +# C.1 TOY EXPERIMENTS + +We train a single, 2 hidden layer network, with 50 hidden ReLU units per layer using MAP inference until convergence. Specifically, we use SGD with a learning rate of $1 \times 1 0 ^ { - 3 }$ , momentum of 0.9 and weight decay of $1 \times 1 0 ^ { - 4 }$ . We use a batch size of 512. The objective we optimise is the Gaussian log-likelihood of our data, where the mean is outputted by the network and the the variance is a hyperparameter learnt jointly with NN parameters by SGD. This variance parameters is shared among all datapoints. Once the network is trained, we perform post-hoc inference on it using different approaches. Since all of these involve the linearized approximation, the mean prediction is the same for all methods. Only their uncertainty estimates vary. + +Note that while for this toy example, we could in principle use the full covariance matrix for the purpose of subnetwork selection, we still just use its diagonal (as described in Section 4) for consistency. We use GGN Laplace inference over network weights (not biases) in combination with the linearized predictive distribution in Eq. (8). Thus, all approaches considered share their predictive mean, allowing us to better compare their uncertainty estimates. + +All approaches share a single prior precision of $\lambda = 3$ . We chose to select the prior precision such that the full covariance approach (optimistic baseline) presents reasonable results. We use the same value for all other methods. We first tried a precision of 1 and found the full covariance approach to produce excessively large errorbars (covering the whole plot). A value of 3 produces more reasonable results. + +Final layer inference is performed by computing the full Laplace covariance matrix and discarding all entries except those corresponding to the final layer of the NN. Results for random sub-network selection are obtained with a single sample from a scaled uniform distribution over weight choice. + +# C.2 UCI EXPERIMENTS + +In this experiment, our fully connected NNs have numbers of hidden layers $h _ { d } = \{ 1 , 2 \}$ and hidden layer widths ${ w _ { d } } \mathrm { { = } } \{ 5 0 , 1 0 0 \}$ . For a dataset with input dimension $i _ { d }$ , the number of weights is given by $D { = } ( i _ { d } { + } 1 ) w _ { d } { + } ( h _ { d } { - } 1 ) \bar { w } _ { d } ^ { 2 }$ . Our 2 hidden layer, 100 hidden unit models have a weight count of the order $1 0 ^ { 4 }$ . The non-linearity used is ReLU. + +We first obtain a MAP estimate of each model’s weights. Specifically, we use SGD with a learning rate of $1 \times 1 0 ^ { - 3 }$ , momentum of 0.9 and weight decay of $1 \bar { \times } 1 0 ^ { - 4 }$ . We use a batch size of 512. The objective we optimise is the Gaussian log-likelihood of our data, where the mean is outputted by the network and the the variance is a hyperparameter learnt jointly with NN parameters by SGD. + +For each dataset split, we set aside $1 5 \%$ of the train data as a validation set. We use these for early stopping training. Training runs for a maximum of 2000 epochs but early stops with a patience of 500 if validation performance does not increase. For the larger Protein dataset, these values are 500 and 125. The weight settings which provide best validation performance are kept. + +We then perform full network GGN Laplace inference for each model. We also use our proposed Wassertein rule together with the diagonal Hessian assumption to prune every network’s weight variances such that the number of variances that remain matches the size of every smaller network under consideration. The prior precision used for these steps is chosen such that the resulting predictor’s logliklihood performance on the validation set is maximised. Specifically, we employ a grid search over the values: $\lambda : [ 0 . 0 0 0 1 , 0 . 0 0 1 , 0 . 1 , 0 . 5 , 1 , 2 , 5 , 1 0 , 1 0 0 , 1 0 0 0 ]$ . In all cases, we employ the linearized predictive in Eq. (7). Consequently, networks with the same number of weights make the same mean predictions. Increasing the number of weight variances considered will thus only increase predictive uncertainty. + +# C.3 IMAGE EXPERIMENTS + +The results shown in Section 5.3 and Appendix B are obtained by training ResNet-18 (and ResNet-50) models using SGD with momentum. For each experiment repetition, we train 7 different models: The first is for: ‘MAP’, ‘Ours’, ‘Ours (Rand)’, ‘SWAG’, ‘Diag-Laplace’ and as the first element of ‘Ensemble’. We train 4 additional ‘Ensemble’ elements, 1 network with ‘Dropout’, and, finally 1 network for ‘VOGN’. The methods ‘Ours’, ‘Ours (Rand)’, ‘SWAG’, and ‘Diag-Laplace’ are applied post training. + +For all methods except ‘VOGN’ we use the following training procedure. The (initial) learning rate, momentum, and weight decay are 0.1, 0.9, and $1 \times \mathrm { i 0 ^ { - 4 } }$ , respectively. For ‘MAP’ we use 4 Nvidia P100 GPUs with a total batch size of 2048. For the calculation of the Jacobian in the subnetwork selection phase we use a single P100 GPU with a batch size of 4. For the calculation of the hessian we use a single P100 GPU with a batch size of 2. We train on 1 Nvidia P100 GPU with a batch size of 256 for all other methods. Each dataset is trained for a different number of epochs, shown in Table 21. We decay the learning rate by a factor of 10 at scheduled epochs, also shown in Table 21. Otherwise, all methods and datasets share hyperparameters. These hyperparameter settings are the defaults provided by PyTorch for training on ImageNet. We found them to perform well across the board. We report results obtained at the final training epoch. We do not use a separate validation set to determine the best epoch as we found ResNet-18 and ResNet-50 to not overfit with the chosen schedules. + +Table 21: Per-dataset training configuration for image experiments. + +
DATASETNo.EPOCHSLR SCHEDULE
MNIST9040,70
CIFAR10300150,225
+ +For ‘Dropout’, we add dropout to the standard ResNet-50 model (He et al., 2016) in between the 2 and 3 convolutions in the bottleneck blocks. This approach follows Zagoruyko & Komodakis (2016) and Ashukha et al. (2020b) who add dropout in-between the two convolutions of a WideResNet-50’s basic block. Following Antoran et al. (2020), we choose a dropout probability of 0.1, as they found ´ it to perform better than the value of 0.3 suggested by Ashukha et al. (2020b). We use $\mathrm { \dot { 1 } 6 \mathbf { M } C }$ samples for predictions. ‘Ensemble’ uses 5 elements for prediction. Ensemble elements differ from each other in their initialisation, which is sampled from the He initialisation distribution (He et al., 2015). We do not use adversarial training as, inline with Ashukha et al. (2020b), we do not find it to improve results. For ‘VOGN’ we use the same procedure and hyper-parameters as used by Osawa et al. (2019) in their CIFAR10 experiments, with the exception that we use a learning rate of $1 \times 1 0 ^ { - 3 }$ as we we found a value of $1 \times 1 0 ^ { - 4 }$ not to result in convergence. We train on a single Nvidia P100 GPU with a batch size of 256. See the authors’ GitHub for more details: github.com/team-approx-bayes/dl-with-bayes/blob/master/ distributed/classification/configs/cifar10/resnet18_vogn_bs256_ 8gpu.json. + +We modify the standard ResNet-50 and ResNet-18 architectures such that the first $7 \times 7$ convolution is replaced with a $3 \times 3$ convolution. Additionally, we remove the first max-pooling layer. Following Goyal et al. (2017), we zero-initialise the last batch normalisation layer in residual blocks so that they act as identity functions at the start of training. + +At test time, we tune the prior precision used for ‘Ours’, ‘Diag-Laplace’ and ‘SWAG’ approximation on a validation set for each approach individually, as in Ritter et al. (2018); Kristiadi et al. (2020). We use a grid search from $1 \times \bar { 1 } 0 ^ { - 4 }$ to $1 \times 1 0 ^ { 4 }$ in logarithmic steps, and then a second, finer-grained grid search between the two best performing values (again with logarithmic steps). + +# C.4 DATASETS + +The 1d toy dataset used in Section 5.1 was taken from Antoran et al. (2020). We obtained it from the ´ authors’ github repo: https://github.com/cambridge-mlg/DUN. Table 22 summarises the datasets used in Section 5.2. Wine and Protein are available from the UCI dataset repository Dua & Graff (2017). $\operatorname { K i n } 8 \mathrm { n m }$ is available from https://www.openml.org/d/189 Foong et al. (2019b). For the standard splits (Hernandez-Lobato & Adams, 2015)´ $90 \%$ of the data is used for training and $10 \%$ for validation. For the gap splits (Foong et al., 2019b) a split is obtained per input dimension by ordering points by their values across that dimension and removing the middle $33 \%$ of the points. These are used for validation. The datasets used for our image experiments are outlined in Table 23. + +Table 22: Datasets from tabular regression used in Section 5.2 + +
DatasetN TrainN Val (15% train)NTestSplitsOutput Dim Output TypeInput DimInput Type
Wine1223216160201Continous11Continous
Wine Gap906161532111Continous11Continous
Kin8nm62671106819201Continous8Continous
Kin8nm Gap4642820273081Continous8Continous
Protein349836174457351Continous9Continous
Protein Gap2591345731524491Continous9Continous
+ +Table 23: Summary of image datasets. The test and train set sizes are shown in brackets, e.g. (test & train). + +
NAMESIZEINPUT DIM.No. CLASSESNO. SPLITS
MNIST (LeCun et al., 1998)70,000 (60.000&10,000)784(28 × 28)102
Fashion-MNIST (Xiao et al.,2017)70,000 (60.000 &10.000)784(28 × 28)102
CIFAR10(Krizhevsky & Hinton,2009)60,000 (50.000&10.000)3072(32 ×32 ×3)102
SVHN (Netzer et al., 2011)99,289 (73,257& 26,032)3072(32 ×32 × 3)102
\ No newline at end of file diff --git a/parse/train/C4-QQ1EHNcI/C4-QQ1EHNcI_content_list.json b/parse/train/C4-QQ1EHNcI/C4-QQ1EHNcI_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..3af9102f62ea816afa73b72d815f0e1edb90f9cd --- /dev/null +++ b/parse/train/C4-QQ1EHNcI/C4-QQ1EHNcI_content_list.json @@ -0,0 +1,2598 @@ +[ + { + "type": "text", + "text": "EXPRESSIVE YET TRACTABLE BAYESIAN DEEP LEARNING VIA SUBNETWORK INFERENCE ", + "text_level": 1, + "bbox": [ + 176, + 98, + 745, + 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": "The Bayesian paradigm has the potential to solve some of the core issues in modern deep learning, such as poor calibration, data inefficiency, and catastrophic forgetting. However, scaling Bayesian inference to the high-dimensional parameter spaces of deep neural networks requires restrictive approximations. In this paper, we propose performing inference over only a small subset of the model parameters while keeping all others as point estimates. This enables us to use expressive posterior approximations that would otherwise be intractable for the full model. In particular, we develop a practical and scalable Bayesian deep learning method that first trains a point estimate, and then infers a full covariance Gaussian posterior approximation over a subnetwork. We propose a subnetwork selection procedure which aims to maximally preserve posterior uncertainty. We empirically demonstrate the effectiveness of our approach compared to point-estimated networks and methods that use less expressive posterior approximations over the full network. ", + "bbox": [ + 233, + 265, + 766, + 445 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 470, + 336, + 486 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural networks (DNNs) still suffer from critical shortcomings that make them unfit for important applications. For instance, DNNs tend to be poorly calibrated and overconfident in their predictions, especially when there is a shift in the train and test distributions (Nguyen et al., 2015; Guo et al., 2017). To reliably inform decision making, DNNs must be able to robustly quantify the uncertainty in their predictions, which is particularly important in safety-critical areas such as healthcare or autonomous driving (Amodei et al., 2016; Filos et al., 2019a; Fridman et al., 2019). ", + "bbox": [ + 174, + 501, + 825, + 585 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Bayesian modeling (Ghahramani, 2015; Gal, 2016) presents a principled way to capture predictive uncertainty via the posterior distribution over model parameters. Unfortunately, due to their nonlinearities, exact posterior inference is intractable in DNNs. Despite recent successes in the field of Bayesian deep learning (Blundell et al., 2015; Gal & Ghahramani, 2016; Osawa et al., 2019; Maddox et al., 2019; Dusenberry et al., 2020), existing methods are only made scalable to modern DNNs with large numbers of parameters by invoking unrealistic assumptions. This severely limits the expressiveness of the inferred posterior and thus deteriorates the quality of the induced uncertainty estimates (Ovadia et al., 2019; Fort et al., 2019; Foong et al., 2019a; Ashukha et al., 2020a). ", + "bbox": [ + 174, + 592, + 825, + 704 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Due to the heavy overparameterization of DNNs, their accuracy is well-preserved by a small subnetwork (Cheng et al., 2017). Additionally, recent work by Izmailov et al. (2019) has shown how performing inference over a low dimensional subspace of the weights can result in accurate uncertainty quantification. These observations prompt the following question for a DNN’s uncertainty: Can a full DNN’s model uncertainty be well-preserved by a small subnetwork’s model uncertainty? We answer this question in the affirmative. We show both theoretically and empirically that the full network posterior can be well represented by a subnetwork’s posterior. As a result, we can use more expensive but faithful posterior approximations over just that subnetwork. We show that this achieves better uncertainty quantification than if we use cheaper, but more crude, posterior approximations over the full network. ", + "bbox": [ + 174, + 710, + 825, + 849 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The contributions of this paper are as follows: ", + "bbox": [ + 176, + 857, + 473, + 871 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1. We propose a new Bayesian deep learning approach that performs Bayesian inference over only a small subset of the model weights and keeps all other weights deterministic. This allows us to use expressive posterior approximations that are typically intractable in DNNs. ", + "bbox": [ + 215, + 882, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/69ba0d7682cd0284507b2c8b65f0296f13b556dfc570af6f9e27b927c22c918d.jpg", + "image_caption": [ + "Figure 1: Schematic illustration of our proposed approach. (a) We train a neural network using standard techniques to obtain a point estimate of the weights. (b) We identify a small subset of the weights. (c) We estimate a posterior distribution over the selected subnetwork via Bayesian inference techniques. (d) We make predictions using the full network of mixed Bayesian/deterministic weights. " + ], + "image_footnote": [], + "bbox": [ + 183, + 104, + 816, + 237 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2. As a concrete instantiation of this framework, we develop a practical and scalable Bayesian deep learning method that uses the linearized Laplace approximation to infer a fullcovariance Gaussian posterior over a subnetwork within a point-estimated neural network. \n3. We formally characterize the discrepancy between the posterior distributions over a subnetwork and the full network (in terms of their Wasserstein distance) in the linearized model, and derive a theoretically motivated strategy to select a subnetwork that minimizes this discrepancy under certain assumptions. \n4. We empirically show, on various benchmarks, that our method compares favourably against point-estimated networks and other Bayesian deep learning methods, experimentally confirming that expressive subnetwork inference is superior to crude inference over full networks. ", + "bbox": [ + 210, + 333, + 826, + 483 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 SUBNETWORK POSTERIOR APPROXIMATION ", + "text_level": 1, + "bbox": [ + 173, + 506, + 575, + 522 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Bayesian neural networks (BNNs) aim to capture model uncertainty, i.e., uncertainty about the choice of weights $W$ which arises due to multiple plausible explanations of the training data $\\{ y , X \\}$ . Here, $\\textbf { { y } }$ is the dependent variable (e.g. classification label) and $\\boldsymbol { X }$ is the feature matrix. A prior distribution $p ( \\mathbf { W } )$ is specified over the BNN’s weights. We wish to infer their full posterior distribution ", + "bbox": [ + 173, + 537, + 826, + 593 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/49d86c485e154e77f1466b70cb3e98b045e2d1e38db86a79471ba12be996c0e4.jpg", + "text": "$$\n\\begin{array} { r l r } { p ( { \\bf W } | { \\bf y } , { \\pmb X } ) } & { \\propto } & { p ( { \\pmb y } | { \\pmb X } , { \\bf W } ) p ( { \\bf W } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 367, + 598, + 630, + 616 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To make predictions, we then estimate the posterior predictive distribution that averages the network’s predictions across all possible settings of the weights, weighted by their posterior probability, i.e. ", + "bbox": [ + 173, + 621, + 831, + 650 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/e42515f2fa3f51cd24be9f49825e0fd023d019b5260f20bf8c4257d81733408c.jpg", + "text": "$$\np ( { \\mathbf { y } ^ { * } } | X ^ { * } , y , X ) = \\int _ { \\mathbf { W } } p ( { \\mathbf { y } ^ { * } } | X ^ { * } , \\mathbf { W } ) p ( \\mathbf { W } | y , X ) d \\mathbf { W } .\n$$", + "text_format": "latex", + "bbox": [ + 312, + 654, + 686, + 688 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Unfortunately, due to the size of modern deep neural networks, it is not only intractable to infer the exact posterior distribution $p ( \\mathbf { W } | \\mathbf { \\boldsymbol { y } } , X )$ in Eq. (1), but it is even computationally challenging to properly approximate it. As a consequence, crude posterior approximations such as complete factorization are commonly employed (Blundell et al., 2015; Hernandez-Lobato & Adams, 2015; ´ Kingma et al., 2015; Khan et al., 2018; Osawa et al., 2019), i.e. $\\begin{array} { r } { p ( \\mathbf { W } | \\pmb { y } , \\pmb { X } ) \\approx \\prod _ { d = 1 } ^ { D } q ( \\mathbf { w } _ { d } ) } \\end{array}$ where $\\mathbf { W } _ { d }$ denotes the $d$ -th weight in the $D$ -dimensional neural network weight vector $\\mathbf { W _ { \\lambda } } \\in \\mathbb { R } ^ { D }$ (the concatenation and flattening of all layers’ weight matrices). Clearly, this is a very wishful assumption; In practise, it suffers from severe pathologies (Foong et al., 2019a;b). ", + "bbox": [ + 173, + 691, + 825, + 808 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we question the implicit assumption that a good posterior approximation needs to include all BNN parameters. Instead, we aim to perform inference only over a small subset of the weights. This approach is well-motivated for two reasons: ", + "bbox": [ + 174, + 814, + 823, + 856 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. Overparameterization: Maddox et al. (2020) have shown that, in the neighborhood of local optima, there are many directions that leave the NN’s predictions unchanged. Moreover, NNs can be heavily pruned without sacrificing test-set accuracy (Frankle & Carbin, 2019). Thus, the majority of a NN’s predictive power might be isolated to a small subnetwork. ", + "bbox": [ + 215, + 867, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2. Inference over submodels: Previous work1 has provided evidence that inference can be effective even when not done on the full parameter space. Izmailov et al. (2019) performed inference over a low-dimensional projection of the weights. Neural-linear models, which give a Bayesian treatment to only the last layer of a DNN, have shown to be competitive with full-network approaches (Riquelme et al., 2018; Ober & Rasmussen, 2019). ", + "bbox": [ + 212, + 102, + 825, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We thus combine these ideas, making the following two-step approximation of the posterior in Eq. (1): ", + "bbox": [ + 171, + 184, + 823, + 200 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8e94cca0ea87a069536e2aa115ec5b28ac77bb498de51531bd7817fb6c1bbd28.jpg", + "text": "$$\np ( \\mathbf { W } | \\pmb { y } , \\pmb { X } ) \\ \\approx \\ p ( \\mathbf { W } _ { S } | \\pmb { y } , \\pmb { X } ) \\prod _ { r } \\delta ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } ) \\ \\approx \\ q ( \\mathbf { W } _ { S } ) \\prod _ { r } \\delta ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } ) .\n$$", + "text_format": "latex", + "bbox": [ + 236, + 207, + 761, + 241 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The first approximation decomposes the full neural network posterior $p ( \\mathbf { W } | \\mathbf { \\boldsymbol { y } } , X )$ into a posterior $p ( \\mathbf { W } _ { S } | \\pmb { y } , \\pmb { X } )$ over the subnetwork $\\mathbf { W } _ { S }$ and delta functions $\\delta \\big ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } \\big )$ over all remaining weights $\\left\\{ \\mathbf { w } _ { r } \\right\\} _ { r }$ , keeping them at fixed values $\\mathbf { w } _ { r } ^ { * } \\in \\mathbb { R }$ . This can be viewed as pruning the variances of the weights $\\left\\{ \\mathbf { w } _ { r } \\right\\} _ { r }$ to zero, which is in contrast to ordinary weight pruning methods that set the weights $\\left\\{ \\mathbf { w } _ { r } \\right\\} _ { r }$ themselves to zero. The second approximation is a result of posterior inference over the subnetwork still being intractable. In turn, we introduce the approximate distribution $q ( \\mathbf { W } _ { S } )$ . Yet, as the subnetwork is much smaller than the full network, we can afford to make $q ( \\mathbf { W } _ { S } )$ expressive and able to capture rich dependencies across the weights within the subnetwork. ", + "bbox": [ + 173, + 246, + 825, + 359 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 SUBNETWORK INFERENCE VIA LAPLACE APPROXIMATION ", + "text_level": 1, + "bbox": [ + 176, + 378, + 692, + 395 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To obtain a method that is as practical as possible, we propose to use inference techniques that can estimate a posterior distribution post-hoc from a point-estimated network. The Laplace approximation (MacKay, 1992) is well-suited to this task as it derives the approximate posterior from the local optimization landscape. Other inference procedures, such as SWAG (Maddox et al., 2019), could also be used. Nevertheless, we focus on Laplace due to it being a well-studied, fundamental technique. ", + "bbox": [ + 173, + 410, + 825, + 482 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Step #1: Point Estimation. The first step of the procedure is to train a neural network to obtain a point estimate of the weights, denoted $W _ { M A P }$ . This estimate should respect the Bayesian model given in Eq. (1), and therefore we optimize the maximum $a$ -posteriori (MAP) objective: ", + "bbox": [ + 174, + 496, + 825, + 539 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e21589daff947dc49100ee8f2c8e4cfb7672c9f03346aca10d7080dc439468d1.jpg", + "text": "$$\n\\begin{array} { r } { { \\cal W } _ { M A P } = \\arg \\operatorname* { m a x } _ { W } \\left[ \\log p ( { \\bf y } | X , W ) + \\log p ( W ) \\right] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 546, + 679, + 564 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This can be done using standard stochastic gradient-based optimization methods commonly-used in modern deep learning (Goodfellow et al., 2016). This step is illustrated in Fig. 1 (a). ", + "bbox": [ + 174, + 570, + 825, + 599 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Step #2: Subnetwork Selection. The second step is to identify a small subnetwork $W _ { S }$ . Ideally, we would like to identify the subnetwork whose posterior is ‘closest’ to the full-network posterior. We formalize this argument in Section 4 and describe a principled strategy that, under certain conditions, minimizes the 2-Wasserstein distance between the sub- and full-network posteriors. All other weights not belonging to that subnetwork are then assigned fixed values: the MAP estimates obtained in Step #1. See Fig. 1 (b) for an illustration of this step. ", + "bbox": [ + 173, + 613, + 825, + 699 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Step #3: Bayesian Inference. Given the subnetwork point estimate $W _ { M A P } ^ { S }$ , we use the Laplace approximation to infer a full-covariance Gaussian posterior distribution over the subnetwork $\\mathbf { W } _ { S }$ : ", + "bbox": [ + 173, + 712, + 823, + 742 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/e5ae43e1c2a85753e76aab6440453d272df2c45247c3fbe1213293fb21a8796f.jpg", + "text": "$$\np ( \\mathbf { W } _ { S } | \\mathbf { \\boldsymbol { y } } , \\boldsymbol { X } ) \\approx \\ q ( \\mathbf { W } _ { S } ) \\ = \\ N \\left( \\mathbf { W } _ { S } ; \\mathbf { \\boldsymbol { W } } _ { M A P } ^ { S } , H ^ { - 1 } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 316, + 750, + 679, + 768 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the posterior covariance matrix $H ^ { - 1 } \\in \\mathbb { R } ^ { D \\times D }$ corresponds to the inverse of the average Hessian of the negative log posterior, i.e. $H = N \\mathbb { E } \\left[ - \\partial ^ { 2 } \\log \\hat { p } ( \\mathbf { y } | X , W ) / \\partial W ^ { 2 } \\right] + \\lambda I$ . Here, the expectation is w.r.t. the data generating distribution and $\\lambda$ is the precision of a zero-mean factorized Gaussian prior $p ( \\mathbf { W } ) = \\tilde { \\mathcal { N } } ( \\mathbf { W } ; \\mathbf { 0 } , \\lambda ^ { \\top } I )$ . In practice, we approximate the Hessian $H$ with the generalized Gauss-Newton (GGN) matrix $\\widetilde { H }$ (Schraudolph, 2002), i.e. ", + "bbox": [ + 173, + 775, + 826, + 851 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/fbd825b8cd18c03ac03c726c53875d2b7ce4542283a95d6a708c3dc3f3503d2a.jpg", + "text": "$$\n\\widetilde { H } = \\sum _ { n = 1 } ^ { N } J _ { n } ^ { \\top } H _ { n } J _ { n } + \\lambda I , \\mathrm { w i t h } J _ { n } = \\frac { \\partial f ( x _ { n } , W ) } { \\partial W } \\mathrm { a n d } H _ { n } = \\frac { \\partial ^ { 2 } L ( y _ { n } , f ( x _ { n } , W ) ) } { \\partial ^ { 2 } f ( x _ { n } , W ) }\n$$", + "text_format": "latex", + "bbox": [ + 199, + 856, + 779, + 900 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $J _ { n } \\in \\mathbb { R } ^ { O \\times D }$ is the Jacobian of the neural network features $\\pmb { f } ( \\pmb { x } _ { n } , \\pmb { W } ) \\in \\mathbb { R } ^ { O }$ w.r.t. the weights $W$ , and $\\pmb { H } _ { n } \\in \\mathbb { R } ^ { O \\times O }$ is the Hessian of the loss $L ( { \\pmb y } _ { n } , { \\pmb f } ( { \\pmb x } _ { n } , { \\pmb W } ) )$ w.r.t. the features $\\mathbf { { \\mathit { f } } } ( \\mathbf { { \\mathit { x } } } _ { n } , W )$ .The GGN $\\widetilde { H }$ has clear practical advantages over the Hessian $H$ ; see Martens $\\&$ Sutskever (2011) and Martens (2016). Using the Laplace approximation with the GGN Hessian can be viewed as an implicit local linearization of the underlying neural network $\\mathbf { \\Delta } f ( \\mathbf { \\Delta } x , W )$ at its MAP estimate $W _ { M A P }$ , ", + "bbox": [ + 173, + 102, + 825, + 178 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/fb8baf2073b26a7293798c22fa151b919456cae195a501b5e27e27f3117f5c62.jpg", + "text": "$$\nf _ { l i n } ^ { M A P } ( { \\pmb x } , { \\pmb W } ) = { \\pmb f } ( { \\pmb x } , { \\pmb W } _ { M A P } ) + J _ { { \\pmb W } _ { M A P } } ( { \\pmb x } ) ( { \\pmb W } - { \\pmb W } _ { M A P } )\n$$", + "text_format": "latex", + "bbox": [ + 289, + 183, + 709, + 200 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $J _ { W _ { M A P } } ( \\pmb { x } ) = \\partial \\pmb { f } ( \\pmb { x } , \\pmb { W } _ { M A P } ) / \\partial \\pmb { W } _ { M A P } \\in \\mathbb { R } ^ { O \\times D }$ (Immer et al., 2020). Note that the model in Eq. (7) is linear in $W$ , as only the term $J _ { W _ { M A P } } ( { \\pmb x } ) { \\pmb W }$ depends linearly on $W$ , while the other terms are constant w.r.t. $W$ and can thus be subsumed into an additive bias term (Khan et al., 2019). The GGN approximation thus locally turns the underlying probabilistic model from a Bayesian neural network into a (generalized) linear model, with basis function expansion $J _ { W _ { M A P } } ( { \\pmb x } )$ of covariate $_ { \\textbf { \\em x } }$ (Immer et al., 2020). Put differently, linearized Laplace in the neural network $\\mathbf { \\Delta } f ( \\mathbf { \\Delta } x , W )$ is equivalent to ordinary Laplace in the linear model $\\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { x } , \\pmb { W } )$ in Eq. (7), as the GGN $\\widetilde { H }$ corresponding to in Eq. (6) is equivalent to the Hessian 9). This is a useful property that will $H$ corresponding to ow us to derive a $\\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { x } , \\pmb { W } )$ in Eq. (7) (Khanetwork selection strategy in Section 4. This step is illustrated in Fig. 1 (c). We emphasize that this whole procedure (i.e. Steps #1-#3) is a perfectly valid mixed inference strategy, performing full Laplace inference over the selected subnetwork and MAP inference over all remaining weights. ", + "bbox": [ + 173, + 204, + 826, + 377 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Step #4: Prediction. Given the linearized Laplace approximation over the subnetwork $\\mathbf { W } _ { S }$ in Eqs. (5) and (6), i.e. $q ( \\mathbf { W } _ { S } ) = \\mathcal { N } ( \\mathbf { W } _ { S } ; { W _ { M A P } ^ { S } } , \\mathbf { \\widetilde { H } } ^ { - 1 } )$ , we can then compute the posterior predictive distribution. While, traditionally, one would compute the predictive distribution using the original Bayesian neural network likelihood, i.e. $p ( \\mathbf { y } | X , \\bar { \\mathbf { W } } ) = p ( \\bar { \\mathbf { y } } | f ( \\mathbf { x } , W ) )$ , Immer et al. (2020) recently suggested that, since inference was (implicitly) done in the GGpled to instead predict using the linearized likelihood Eq. (7), i.e. $p ( \\mathbf { y } \\vert X , \\mathbf { W } ) = p ( \\mathbf { y } \\vert \\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { \\dot { x } } , \\pmb { W } ) )$ -. This provides a formal justification for the empirical superiority of this approach observed previously (Lawrence, 2001; Foong et al., 2019b). We thus obtain the linearized predictive distribution ", + "bbox": [ + 173, + 390, + 826, + 506 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/03d743074fe2db716bb9b3a71d773f0e17ca75ff3ab8383c542e3e61d56ab0e0.jpg", + "text": "$$\np ( \\mathbf { y } ^ { * } | \\boldsymbol { X } ^ { * } , \\boldsymbol { y } , \\boldsymbol { X } ) \\approx \\int _ { \\mathbf { W } } p ( \\mathbf { y } ^ { * } | f _ { l i n } ^ { M A P } ( \\boldsymbol { X } ^ { * } , \\boldsymbol { W } ) ) \\mathcal { N } ( \\mathbf { W } _ { S } ; \\boldsymbol { W } _ { M A P } ^ { S } , \\tilde { H } ^ { - 1 } ) \\prod _ { r } \\delta ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } ) d \\mathbf { W } .\n$$", + "text_format": "latex", + "bbox": [ + 176, + 510, + 799, + 547 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "There are two ways to compute Eq. (8): Firstly, via a Monte Carlo approximation $p ( \\mathbf { y } ^ { * } | X ^ { * } , y , X ) \\simeq$ $\\begin{array} { r } { \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } p ( \\mathbf { y } ^ { * } | \\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { X } ^ { * } , \\widetilde { W } _ { m } ) ) } \\end{array}$ by sampling $\\widetilde { W } _ { m }$ from $\\mathcal { N } ( W _ { M A P } ^ { S } , \\widetilde { H } ^ { - 1 } )$ and $\\begin{array} { r } { \\prod _ { r } \\delta ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } ) } \\end{array}$ , M m=1 | lin f f N MAP e r − the latter of which is trivial. Secondly, due to linearity of p(y∗|f MAPlin (X∗, W )), there are closed- form expressions which are exact for Gaussian likelihoods (i.e. regression) and approximate for categorical ones (i.e. classification) (Bishop, 2006; Gibbs, 1998). This step is illustrated in Fig. 1 (d). ", + "bbox": [ + 173, + 551, + 826, + 628 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 PRINCIPLED SUBNETWORK SELECTION FOR LINEAR(IZED) MODELS ", + "text_level": 1, + "bbox": [ + 171, + 647, + 776, + 665 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We next analyze the subnetwork inference procedure described in Section 3 for the case of a generalized linear model (GLM) (Nelder & Baker, 1972), which models the expected response $y _ { n }$ given the basis function expansion of the covariates $\\phi _ { n } = \\phi ( { \\mathbf { { x } } } _ { n } )$ as ", + "bbox": [ + 173, + 678, + 826, + 722 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/62b1f43811f26b9e440302e041b0cf21d7014c77eb3f3538b7e3b14e7afd2065.jpg", + "text": "$$\n\\mathbb { E } [ y _ { n } | \\phi _ { n } ] = g ^ { - 1 } ( \\mathbf { w } ^ { T } \\phi _ { n } ) .\n$$", + "text_format": "latex", + "bbox": [ + 405, + 727, + 580, + 746 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here, $\\mathbf { w } \\in \\mathbb { R } ^ { D }$ is the vector of model parameters (which subsumes a scalar bias $\\beta _ { 0 }$ for notational convenience) and $g ^ { - 1 } ( \\cdot )$ denotes a link function such that $g ^ { - 1 } : \\mathbb { R } \\mapsto \\mu _ { y | \\phi }$ . In particular, we consider a Bayesian GLM, by specifying a prior distribution $p ( \\mathbf { w } )$ over model parameters and aiming to infer the posterior distribution $p ( \\mathbf { w } | \\mathbf { y } , \\pmb { \\Phi } ) \\propto p ( \\mathbf { y } | \\pmb { \\Phi } , \\mathbf { w } ) p ( \\mathbf { w } )$ , where $\\begin{array} { r } { \\pmb { \\Phi } = [ \\phi _ { 1 } , \\dots \\phi _ { N } ] ^ { T } } \\end{array}$ . ", + "bbox": [ + 173, + 750, + 825, + 809 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1. Point Estimation. Obtain the MAP estimate, $\\begin{array} { r } { \\mathbf { w } _ { M A P } = \\arg \\operatorname* { m a x } _ { \\mathbf { w } } \\log p ( \\pmb { y } | \\Phi , \\mathbf { w } ) + \\log p ( \\mathbf { w } ) } \\end{array}$ . For commonly-used link functions (e.g. the identity in case of a Gaussian likelihood for regression, or the sigmoid/softmax function in case of a categorical likelihood for classification) and commonly-used priors (e.g. a Gaussian), the log-posterior $\\propto \\log p ( \\pmb { y } | \\pmb { \\Phi } , \\mathbf { w } ) + \\log p ( \\mathbf { w } )$ is concave. This allows for simple gradient-based MAP optimisation. It also makes a full-covariance Gaussian, estimated via Laplace, a faithful approximation to the true, uni-modal posterior, i.e. ", + "bbox": [ + 178, + 819, + 826, + 904 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c7f7d852affd773f73ec3a7d922522179f5dddbf5d751a765886b3f8ccc5d231.jpg", + "text": "$$\np ( \\mathbf { w } | \\mathbf { y } , \\boldsymbol { \\Phi } ) \\approx \\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\boldsymbol { \\Phi } ) = \\mathcal { N } ( \\mathbf { w } ; \\mathbf { w } _ { M A P } , H ^ { - 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 348, + 907, + 673, + 925 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $H$ is the Hessian defined in Section 3. Note that for the GLM we consider, the Hessian $H$ is equivalent to the GGN $\\widetilde { H }$ defined in Eq. (6), meaning that an ordinary Laplace approximation is equivalent to a linearized Laplace approximation (Martens, 2016). For the case of an identity link function (i.e. a Gaussian likelihood with noise variance $\\sigma _ { 0 } ^ { 2 }$ ) and a Gaussian prior w $\\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { \\dot { N } } _ { 0 } ^ { - 1 } )$ , the MAP estimate even has a closed-form expression, $\\begin{array} { r } { { \\bf w } _ { M A P } = ( \\Phi ^ { T } \\Phi + \\sigma _ { 0 } ^ { 2 } \\Lambda _ { 0 } ) ^ { - 1 } \\Phi ^ { T } { \\pmb y } . } \\end{array}$ . Here, the Laplace approximation in Eq. (10) exactly corresponds to the true posterior, i.e. $\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi ) =$ $p ( \\mathbf { w } | \\mathbf { y } , \\Phi )$ . We will thus refer to the posterior $\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi )$ e in Eq. (10) as the full posterior. ", + "bbox": [ + 196, + 103, + 825, + 205 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2. Subnetwork Selection. Select a subset of $S$ model weights via a method of choice, yielding a binary vector $\\pmb { m } \\in \\mathbb { R } ^ { D }$ where $m _ { d } = 1$ if the $d$ -th weight is part of the subset, and $m _ { d } = 0$ otherwise. For convenience, we define the binary mask matrix $\\mathbf { M } _ { S } = \\pmb { m } \\pmb { m } ^ { \\top } \\in \\mathbb { R } ^ { D \\times D }$ which contains 1s in the rows/columns corresponding to the $S$ subnetwork weights2, and 0s otherwise. ", + "bbox": [ + 179, + 207, + 825, + 263 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3. Bayesian Inference. Compute the posterior over the subnetwork via a Laplace approximation: ", + "bbox": [ + 176, + 266, + 820, + 281 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4b4c8b457ab7b86b66c9494f10dcb2f3dc11133832e707f12a7d9c2c703beb4b.jpg", + "text": "$$\np _ { S } ( \\mathbf { w } | \\pmb { y } , \\pmb { \\Phi } ) = \\mathcal { N } ( \\mathbf { w } ; \\mathbf { w } _ { M A P } , M _ { S } \\odot H ^ { - 1 } ) \\ .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 281, + 656, + 299 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Firstly, note that the mean of the subnetwork posterior in Eq. (11) is the MAP estimate $\\mathbf { w } _ { M A P }$ and thus equal to the mean of the full posterior $\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi )$ in Eq. (10). Secondly, note that the ecovariance matrix of the subnetwork posterior in Eq. (11) is the element-wise product $M _ { S } \\odot H ^ { - 1 }$ , which masks the (co-)variances of all weights not belonging to the subnetwork to zero, effectively making them deterministic. More precisely, the subnetwork covariance matrix, $M _ { S } \\odot H ^ { - 1 }$ , is a $D \\times D$ matrix that is equal to the full posterior covariance matrix $H ^ { - 1 }$ in the rows/columns of the $S$ weights in the subnetwork, and zero in the rows/columns of all other $D - S$ weights. ", + "bbox": [ + 196, + 299, + 825, + 397 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We consider what we term the posterior gap—the Wasserstein distance3 (in particular the squared 2-Wasserstein distance) between the posterior distribution over the full network and the posterior distribution over the subnetwork. The proofs for all results below will be presented in Appendix A. ", + "bbox": [ + 176, + 406, + 820, + 449 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 1 (Posterior Gap). For a subnetwork of size $S < D$ , the Wasserstein gap between the full posterior $\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi )$ in Eq. (10) and the subnetwork posterior $p _ { \\mathrm { S } } ( { \\bf w } | { \\pmb y } , { \\pmb \\Phi } )$ in Eq. (11) is: ", + "bbox": [ + 174, + 450, + 823, + 479 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e5300a416acdb777f1addd70a6ab798618ea7ba016490d0f8e6fee38ec287c74.jpg", + "text": "$$\n\\begin{array} { r } { W \\left[ \\widetilde { p } ( \\mathbf { w } | y , \\Phi ) \\mid | p _ { \\mathrm { S } } ( \\mathbf { w } | y , \\Phi ) \\right] = \\sum _ { d = 1 } ^ { D } \\left( 1 + m _ { d d } \\right) \\sigma _ { d } ^ { 2 } - \\mathrm { t r a c e } ( 2 ( H ^ { - 1 } ( M _ { S } \\odot H ^ { - 1 } ) ) ^ { 1 / 2 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 479, + 781, + 501 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The optimal subnetwork should then minimize the posterior gap in Eq. (12). However, for full covariance matrices $H ^ { - 1 }$ and a large number of weights $D$ , this will generally be infeasible as Eq. (12) depends on all entries of the $D \\times D$ -matrix $H ^ { - 1 }$ , which is intractable to compute/store. To derive a practical subnetwork selection strategy, we assume the covariance matrix to be diagonal. ", + "bbox": [ + 174, + 507, + 825, + 564 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Corollary 1.1 (Optimality of Maximum Variance Subnetwork Selection under Decorrelation). For $a$ generalized linear model with posterior covariance matrix $H ^ { - 1 } = d i a g ( \\sigma _ { 1 } ^ { 2 } , . . . , \\sigma _ { D } ^ { 2 } )$ , the optimal subnetwork under the Wasserstein gap is comprised of the $S$ weights with the largest variances $\\sigma _ { d } ^ { 2 }$ . ", + "bbox": [ + 174, + 565, + 825, + 608 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Finally, since a GGN-linearized neural network, as in Eq. (7), corresponds to a GLM with basis expansion $\\phi _ { n } = J _ { W _ { M A P } } ( \\pmb { x } _ { n } ) = \\partial \\pmb { f } ( \\pmb { x } _ { n } , \\pmb { W } _ { M A P } ) / \\partial W _ { M A P }$ (see Step #3 in Section 3), Corollary 1.1 implies that under decorrelation, the optimal subnetwork comprises of the weights with the largest variances. In practice, even just computing the diagonal of the covariance matrix is challenging, so we use a diagonal Laplace approximation which instead computes the inverse of the diagonal of the GGN (see e.g. Ritter et al. (2018)). Finally, note that we only have to make the decorrelation assumption for the purposes of subnetwork selection – when doing posterior inference over the selected subnetwork, we estimate a full covariance matrix for maximal expressiveness, as described in Step #3 in Section 3. In our experiments in Section 5, we empirically show that making the decorrelation assumption for subnetwork selection but then using a full-covariance Gaussian for inference performs significantly better than directly making the decorrelation assumption for inference (e.g. mean-field variational inference, diagonal Laplace). ", + "bbox": [ + 173, + 616, + 826, + 784 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 EMPIRICAL ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 803, + 392, + 819 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We empirically assess the effectiveness of subnetwork inference compared to point-estimated NNs and methods that do less expressive inference over the full network. We consider three tasks: 1) small-scale toy regression, 2) medium-scale tabular regression, and 3) large-scale image classification. ", + "bbox": [ + 174, + 833, + 825, + 876 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/348c77967767ca7b767128552cfde1a6eab11fff719d21ab1885ac59de9d2e18.jpg", + "image_caption": [ + "Figure 2: Predictive distributions (mean $\\pm$ std) for 1D regression. The numbers in brackets denote the number of parameters over which inference was done (out of 2600 in total). Wasserstein-based subnetwork inference maintains richer predictive uncertainties at smaller parameter counts. " + ], + "image_footnote": [], + "bbox": [ + 176, + 97, + 823, + 315 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 HOW DOES SUBNETWORK INFERENCE RETAIN POSTERIOR PREDICTIVE UNCERTAINTY? ", + "text_level": 1, + "bbox": [ + 173, + 397, + 813, + 411 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We first assess how the predictive distribution of a full-covariance Gaussian posterior over a selected subnetwork qualitatively compares to that obtained from 1) a full-covariance Gaussian over the full network (Full Cov), 2) a factorised Gaussian posterior over the full network (Diag), 3) a fullcovariance Gaussian over only the (Final layer) of the network (Kristiadi et al., 2020), and 4) a point estimate $( M A P )$ . For subnetwork inference, we consider both Wasserstein (Wass) (as described in Section 4) and uniform random subnetwork selection (Rand) to obtain subnetworks that comprise of only $50 \\%$ , $3 \\%$ and $1 \\%$ of the model parameters. Note that while for this toy example, we could in principle use the full covariance matrix for the purpose of subnetwork selection, we still just use its diagonal (as described in Section 4) for consistency. Our NN consists of 2 ReLU hidden layers with 50 hidden units each. We employ a homoscedastic Gaussian likelihood function where the noise variance is optimised with maximum likelihood. We use GGN Laplace inference over network weights (not biases) in combination with the linearized predictive distribution in Eq. (8). Thus, all approaches considered share their predictive mean, allowing us to better compare their uncertainty estimates. All approaches share a single prior precision of $\\lambda = 3$ . ", + "bbox": [ + 173, + 424, + 825, + 618 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We use a synthetic 1D regression task with two separated clusters of inputs (Antoran et al., 2020), ´ allowing us to probe for ‘in-between’ uncertainty (Foong et al., 2019b). Results are shown in Fig. 2. Subnetwork inference preserves more of the uncertainty of full network inference than diagonal Gaussian or final layer inference while doing inference over fewer weights. By capturing weight correlations, subnetwork inference retains uncertainty in between clusters of data. This is true for both random and Wasserstein subnetwork selection. However, the latter preserves more uncertainty with smaller subnetworks. Finally, the strong superiority to diagonal Laplace shows that making a diagonal assumption for subnetwork selection but then using a full-covariance Gaussian for inference (as we do) performs much signficantly better than making a diagonal assumption for the inferered posterior directly. These results suggest that expressive inference over a carefully selected subnetwork retains more predictive uncertainty than crude approximations over the full network. ", + "bbox": [ + 174, + 625, + 825, + 779 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.2 SUBNETWORK INFERENCE IN LARGE MODELS VS FULL INFERENCE OVER SMALL MODELS ", + "bbox": [ + 174, + 800, + 820, + 813 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Secondly, we study the following natural question: “Why should one use subnetwork inference in a large model when one can just perform full network inference over a smaller model?” We explore this by considering 4 fully connected models of increasing size. These have numbers of hidden layers $h _ { d } = \\{ 1 , 2 \\}$ and hidden layer widths ${ w _ { d } } \\mathrm { { = } } \\{ 5 0 , 1 0 0 \\}$ . For a dataset with input dimension $i _ { d }$ , the number of weights is given by $D { = } ( i _ { d } { + } 1 ) w _ { d } { + } ( h _ { d } { - } 1 ) w _ { d } ^ { 2 }$ . Our 2 hidden layer, 100 hidden unit models have a weight count of the order $1 0 ^ { 4 }$ . Full covariance inference in these models borders the limit of computational tractability on commercial hardware. We first obtain a MAP estimate of each model’s weights and our homoscedastic likelihood function’s noise variance. We then perform full network GGN Laplace inference for each model. We also use our proposed Wassertein rule to prune every network’s weight variances such that the number of variances that remain matches the size of every smaller network under consideration. In all cases, we employ the linearized predictive in Eq. (7). Consequently, networks with the same number of weights make the same mean predictions. Increasing the number of weight variances considered will thus only increase predictive uncertainty. ", + "bbox": [ + 174, + 827, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 188 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/66c302f1bb14b542bd24077fff89bad6dabe0c7a84ec71a09c89a80dccf32d61.jpg", + "image_caption": [ + "Figure 3: Mean test log-likelihood values obtained on UCI datasets across all splits. Different markers indicate models with different numbers of weights. The horizontal axis indicates the number of weights over which full covariance inference is performed. 0 corresponds to MAP parameter estimation, and the rightmost setting for each marker corresponds to full network inference. " + ], + "image_footnote": [], + "bbox": [ + 173, + 199, + 825, + 330 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We employ 3 tabular datasets of increasing size (input dimensionality, n. points): wine (11, 1439), kin8nm (8, 7373) and protein (9, 41157). We consider their standard train-test splits (Hernandez- ´ Lobato & Adams, 2015) and their gap variants (Foong et al., 2019b), designed to test for out-ofdistribution uncertainty. Details are provided in Appendix C.4. For each split, we set aside $1 5 \\%$ of the train data as a validation set. We use these for early stopping when finding MAP estimates and for selecting the weights’ prior precision. We keep other hyperparameters fixed across all models and datasets. Results are in Fig. 3. ", + "bbox": [ + 174, + 415, + 825, + 512 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We present mean test log-likelihood (LL) values, as these take into account both accuracy and uncertainty. Larger models tend to perform better when doing MAP inference, with wine-gap and protein-gap being exceptions. We also find larger models improve over their respective MAP LLs more than small ones when performing approximate inference over the same numbers of weights. We conjecture this is due to an abundance of degenerate directions (weights) in the weight posterior of all models (Maddox et al., 2020). Full network inference in small models captures information about both useful and non-useful weights. In larger models, our subnetwork selection strategy allows us to dedicate a larger proportion of our resources to modelling informative weight variances and covariances. In 3 out of 6 datasets, we find abrupt increases in LL as we increase the number of weights over which we perform inference, followed by a plateau. Such plateaus might be explained by all of the most informative weight variances having already been accounted for. These results suggest that, given the same amount of compute, larger models benefit more from subnetwork inference than small ones. ", + "bbox": [ + 174, + 520, + 825, + 699 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.3 SCALING TO IMAGE CLASSIFICATION WITH DISTRIBUTION SHIFT ", + "text_level": 1, + "bbox": [ + 176, + 717, + 668, + 732 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We now assess the robustness of large convolutional neural networks with subnetwork inference to distribution shift on image classification tasks compared to the following baselines: point-estimated networks (MAP), Bayesian deep learning methods that do less expressive inference over the full network: MC Dropout (Gal & Ghahramani, 2016), diagonal Laplace, VOGN (Osawa et al., 2019) (all of which assume factorisation of the weight posterior), and SWAG (Maddox et al., 2019) (which assumes a diagonal plus low-rank posterior). We also benchmark deep ensembles (Lakshminarayanan et al., 2017). The latter is considered state-of-the-art for uncertainty quantification in deep learning (Ovadia et al., 2019; Ashukha et al., 2020a). We use ensembles of 5 DNNs, as suggested by (Ovadia et al., 2019), and 16 samples for MC Dropout, diagonal Laplace and SWAG. We use a Dropout probability of 0.1 and a prior precision of $\\lambda = 4 0 , 0 0 0$ for diagonal Laplace, found via grid search. We apply all approaches to ResNet-18 (He et al., 2016), which is composed of an input convolutional block, 8 residual blocks and a linear layer, for a total of 11,168,000 parameters. For subnetwork inference, we compute the linearized predictive distribution in Eq. (8) via the closed-form approximation for integrals between Gaussians and multi-class cross-entropy likelihoods described in (Gibbs, 1998). We use Wasserstein subnetwork selection to retain only $0 . 3 8 \\%$ of the weights, yielding a subnetwork with only 42,438 weights. This is the largest subnetwork for which we can tractably compute a full covariance matrix. Its size is 42 $, 4 3 8 ^ { 2 } \\times \\hat { 4 }$ Bytes $\\approx 7 . 2$ GB. We use a prior precision of $\\lambda = 5 0 0$ , found via grid search. Finally, to assess to importance of principled subnetwork selection, we also consider the baseline where we select the subnetwork uniformly at random (called Ours (Rand)). We perform the following two experiments, with results in Fig. 4. See Appendix B for additional results. ", + "bbox": [ + 173, + 743, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/966242861badeec22bbb4cab7022e457bc6a60cc58c9e8f8bac63f13ef1c8655.jpg", + "image_caption": [ + "Figure 4: Results on the rotated MNIST (left) and the corrupted CIFAR (right) benchmarks of Ovadia et al. (2019), showing the mean $\\pm$ std of the error (top) and log-likelihood (bottom) across three different seeds. Subnetwork inference retains better uncertainty calibration and robustness to distribution shift than point estimated networks and other Bayesian deep learning approaches. " + ], + "image_footnote": [], + "bbox": [ + 176, + 99, + 821, + 303 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 398, + 825, + 511 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Rotated MNIST: Following Ovadia et al. (2019); Antoran et al. (2020), we train all methods on ´ MNIST and evaluate their predictive distributions on increasingly rotated digits. While all methods perform well on the original MNIST test set, their accuracy degrades quickly for rotations larger than 30 degrees. In terms of LL, ensembles perform best out of our baselines. Subnetwork inference obtains significantly larger LL values than almost all baselines, including ensembles. The only exception is VOGN, which achieves slightly better performance. It was also observed in Ovadia et al. (2019) that mean-field variational inference (which VOGN also is an instance of) is very strong on MNIST, but its performance deteriorates on larger datasets. Subnetwork inference makes accurate predictions in-distribution while assigning higher uncertainty than the baselines to out-of-distribution points. Corrupted CIFAR: Again following Ovadia et al. (2019); Antoran et al. (2020), we train ´ on CIFAR10 and evaluate on data subject to 16 different corruptions with 5 levels of intensity each (Hendrycks & Dietterich, 2019). Our approach matches a MAP estimated network in terms of predictive error as local linearization makes their predictions the same. Ensembles and SWAG are the most accurate. Even so, subnetwork inference differentiates itself by being the least overconfident, outperforming all baselines in terms of log-likelihood at all corruption levels. Here, VOGN performs rather badly; while this might appear in stark contrast to its strong performance on the MNIST benchmark, the behaviour that mean-field VI performs well on MNIST but not on larger datasets was also observed in Ovadia et al. (2019). ", + "bbox": [ + 173, + 516, + 825, + 767 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "On both benchmarks, we furthermore find that randomly selecting the subnetwork performs substantially worse than using our more principled subnetwork selection strategy. This highlights the importance of the way subnetworks are selected. These results suggest that subnetwork inference results in better uncertainty calibration and robustness to distribution shift than other popular uncertainty quantification approaches. ", + "bbox": [ + 174, + 773, + 825, + 844 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 864, + 344, + 880 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Bayesian Deep Learning. There have significant efforts to characterise the posterior distribution over NN weights $p ( W | \\bar { \\mathcal { D } } )$ . Hamiltonian Monte Carlo (Neal, 1995) remains the golden standard for approximate inference in BNNs to this day. Although asymptotically unbiased, sampling based approaches are difficult to scale to the large datasets (Betancourt, 2015). As a result, approaches which find the best surrogate posterior among an approximating family (most often Gaussians) have gained popularity. The first of these was the Laplace approximation, introduced by MacKay (1992), who also proposed approximating the predictive posterior with that of the linearised model (Khan et al., 2019; Immer et al., 2020). The popularisation of larger NN models has made surrogate distributions that capture correlations between weights computationally intractable. Thus, most modern methods make use of the mean field assumption (Blundell et al., 2015; Hernandez-Lobato & ´ Adams, 2015; Gal & Ghahramani, 2016; Mishkin et al., 2018; Osawa et al., 2019). This comes at the cost of limited expressivity (Foong et al., 2019a) and empirical under-performance (Ovadia et al., 2019; Antoran et al., 2020) of uncertainty estimates. Our proposed approach recovers predictive ´ posterior expressivity while maintaining tractability by lowering the dimensionality of the weight space considered. This allows us to scale up approximations that do consider weight correlations (MacKay, 1992; Louizos & Welling, 2016; Maddox et al., 2019; Ritter et al., 2018). ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 297 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Neural Network Linearization. In the limit of infinite width, NNs converge to Gaussian process (GP) behaviour (Neal, 1995; Matthews, 2017; Garriga-Alonso et al., 2018). Recently, these results have been extended to finite width BNNs when the surrogate posterior is Gaussian (Khan et al., 2019). We draw upon these results to formulate a subnetwork selection strategy for BNNs. Neural linear methods perform inference over only the last layer of a NN, while keeping all other layers fixed (Snoek et al., 2015; Riquelme et al., 2018; Ovadia et al., 2019; Ober & Rasmussen, 2019; Pinsler et al., 2019; Kristiadi et al., 2020). These represent a different generalised linear model in which the basis functions are defined by the $l { - } 1$ first layers of a NN. They can also be viewed as a special case of subnetwork inference, in which the subnetwork is simply defined to be the last NN layer. ", + "bbox": [ + 173, + 305, + 825, + 430 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Inference over Subspaces. The subfield of NN pruning aims to increase the computational efficiency of NNs by identifying the smallest subset of weights which are required to make accurate predictions. Approaches trade-off computational cost with compression efficiency, ranging from those that require multiple training runs (Frankle & Carbin, 2019) to those that prune before training (Wang et al., 2020). Our work differs in that it retains all NN weights but aims to find a small subset over which to perform probabilistic reasoning. More closely related work to ours is that of Izmailov et al. (2019), who propose to perform inference over a low-dimensional subspace of weights; e.g. one constructed from the principal components of the SGD trajectory. Moreover, several recent approaches use low-rank parameterizations of approximate posteriors in the context of variational inference (Rossi et al., 2019; Swiatkowski et al., 2020; Dusenberry et al., 2020). This could also be viewed as doing inference over an implicit subspace of weight space. In contrast, we propose a technique to find subsets of weights which are relevant to predictive uncertainty, i.e., we identify axis aligned subspaces. Finally, there have been recent works studying neural network sparsity / pruning from a Bayesian perspective (Ghosh & Doshi-Velez, 2017; Polson & Rockov ˇ a, 2018; Cui et al., 2020; Louizos et al., 2017; ´ Molchanov et al., 2017; Gomez et al., 2019; Lee et al., 2018). While these seem conceptually related at first glance, their goal is fundamentally different to ours: While those methods aim to perform model selection / sparsification by either explicitly or implicitly pruning unnecessary weights, our goal is to make inference more tractable. More precisely, while those sparse Bayesian deep learning methods prune individual weights, we instead just prune the variances of certain weights, which, importantly retains the full predictive power of the full network to retain high predictive accuracy. ", + "bbox": [ + 174, + 435, + 825, + 714 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 746, + 318, + 762 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we develop a practical and scalable method for expressive yet tractable probabilistic inference in deep neural networks. We approximate the posterior over a subset of the weights while keeping all other weights deterministic. Computational cost is decoupled from network size, allowing us to scale expressive approximations, such as full-covariance Gaussian distributions, to real-world sized NNs. Our approach can be applied post-hoc to any pre-trained model, making it particularly attractive for practical use. Our empirical analysis suggests that subnetwork inference 1) is more expressive and retains more uncertainty than crude approximations over the full network, 2) allows us to employ larger NNs, which fit a broader range of functions, without sacrificing the quality of our uncertainty estimates, and 3) is competitive with state-of-the-art uncertainty quantification methods, like deep ensembles (Lakshminarayanan et al., 2017), on real-world scale problems. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mane.´ Concrete problems in ai safety. arXiv preprint arXiv:1606.06565, 2016. 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Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. 2017. ", + "bbox": [ + 173, + 844, + 821, + 875 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. In Richard C. Wilson, Edwin R. Hancock, and William A. P. Smith (eds.), Proceedings of the British Machine Vision Conference 2016, BMVC 2016, York, UK, September 19-22, 2016. BMVA Press, 2016. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A PROOFS FOR THE THEORETICAL RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 102, + 560, + 119 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We now provide the proofs for the results in Section 4. ", + "bbox": [ + 174, + 133, + 531, + 148 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 PROOF OF PROPOSITION 1 ", + "text_level": 1, + "bbox": [ + 176, + 165, + 400, + 179 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. Note that the posterior distributions $\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi )$ and $p _ { S } ( { \\bf w } | { \\pmb y } , { \\pmb \\Phi } )$ are both Gaussian. We thus econsider the squared 2-Wasserstein distance between two Gaussian distributions $\\mathcal { N } ( \\mu _ { 1 } , \\Sigma _ { 1 } )$ and $\\mathcal { N } ( \\mu _ { 2 } , \\Sigma _ { 2 } )$ , which has the following closed-form expression (Givens et al., $1 9 8 4 ) ^ { 4 }$ : ", + "bbox": [ + 173, + 190, + 828, + 234 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/565e271de9a646bb950170fa9ae62280bab507517efe8da2e700d1818886faac.jpg", + "text": "$$\n\\begin{array} { r } { W \\left[ \\mathcal { N } \\left( \\mu _ { 1 } , \\Sigma _ { 1 } \\right) \\ | | \\mathcal { N } \\left( \\mu _ { 2 } , \\Sigma _ { 2 } \\right) \\right] = \\| \\mu _ { 1 } - \\mu _ { 2 } \\| _ { 2 } ^ { 2 } + \\mathrm { t r a c e } \\left( \\Sigma _ { 1 } + \\Sigma _ { 2 } - 2 \\left( \\Sigma _ { 1 } \\Sigma _ { 2 } \\right) ^ { 1 / 2 } \\right) \\ . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 239, + 767, + 267 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Plugging in ${ \\pmb { \\mu } } _ { 1 } = { \\pmb { \\mu } } _ { 2 } = \\mathbf { w } _ { M A P }$ , $\\pmb { \\Sigma } _ { 1 } = H ^ { - 1 }$ and $\\Sigma _ { 2 } = M _ { S } \\odot H ^ { - 1 }$ , we obtain ", + "bbox": [ + 173, + 273, + 696, + 291 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/8c57835d8e7ad8bc5624ac4612ad29551b0b3ee6c35f289491865fd5e75199dc.jpg", + "text": "$$\n\\begin{array} { r l } & { W \\left[ \\widetilde { p } ( \\mathbf { w } | y , X ) \\mid | p _ { \\mathrm { S } } ( \\mathbf { w } | y , X ) \\right] } \\\\ & { \\quad = W \\left[ \\mathcal { N } ( \\mathbf { w } _ { M A P } , H ^ { - 1 } ) \\mid | \\mathcal { N } ( \\mathbf { w } _ { M A P } , M _ { S } \\odot H ^ { - 1 } ) \\right] } \\\\ & { \\quad = \\underbrace { \\left. \\mathbf { w } _ { M A P } \\longleftrightarrow \\widetilde { { \\mathbf { w } } _ { M A P } } \\right. _ { 2 } ^ { 2 } } _ { = \\mathrm { t r a c e } } \\left( H ^ { - 1 } + ( M _ { S } \\odot H ^ { - 1 } ) - 2 \\left( H ^ { - 1 } ( M _ { S } \\odot H ^ { - 1 } ) \\right) ^ { 1 / 2 } \\right) } \\\\ & { \\quad = \\mathrm { t r a c e } \\left( ( 1 + M _ { S } ) \\odot H ^ { - 1 } \\right) - \\mathrm { t r a c e } \\left( 2 \\left( H ^ { - 1 } \\left( M _ { S } \\odot H ^ { - 1 } \\right) \\right) ^ { 1 / 2 } \\right) } \\\\ & { \\quad = \\displaystyle \\sum _ { d = 1 } ^ { D } ( 1 + m _ { d d } ) \\sigma _ { d } ^ { 2 } - \\mathrm { t r a c e } \\left( 2 \\left( H ^ { - 1 } \\left( M _ { S } \\odot H ^ { - 1 } \\right) \\right) ^ { 1 / 2 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 296, + 795, + 435 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 PROOF OF COROLLARY 1.1 ", + "text_level": 1, + "bbox": [ + 176, + 449, + 406, + 463 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. For $H ^ { - 1 } = \\mathrm { d i a g } ( \\sigma _ { 1 } ^ { 2 } , . . . , \\sigma _ { D } ^ { 2 } )$ , the Wasserstein posterior gap in Eq. (12) simplifies to ", + "bbox": [ + 169, + 473, + 779, + 491 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/2b2c1e90cabf866a74cd694f16146c77320d69c556701bf45bffec3c0f629b50.jpg", + "text": "$$\nW \\left[ \\widetilde { p } ( { \\mathbf { w } } | { \\pmb y } , { \\pmb \\Phi } ) | | p _ { \\mathrm { S } } ( { \\mathbf { w } } | { \\pmb y } , { \\pmb \\Phi } ) \\right] = \\sum _ { d = 1 } ^ { D } \\left( \\left( 1 + m _ { d d } \\right) \\sigma _ { d } ^ { 2 } - 2 m _ { d d } \\sigma _ { d } ^ { 2 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 279, + 497, + 717, + 541 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The optimal subnetwork selection strategy amounts to choosing the binary vector $\\pmb { m } = [ m _ { d d } ] _ { d = 1 } ^ { D }$ with $\\textstyle \\sum _ { d = 1 } ^ { D } m _ { d } = S$ (i.e., we select $S$ out of $D$ parameters) s.t. the posterior gap in Eq. (14) is minimized. Observing that the contribution of the $d$ -th parameter to the posterior gap is $( 1 + 1 ) \\sigma _ { d } ^ { 2 } - 1 \\times 2 \\sigma _ { d } ^ { 2 } = 0$ if it is selected (i.e. if $m _ { d d } = 1$ ), and $( 1 + \\bar { 0 } ) \\sigma _ { d } ^ { 2 } - 0 \\times 2 \\sigma _ { d } ^ { 2 } \\stackrel { . } { = } \\sigma _ { d } ^ { 2 }$ if it is not selected (i.e. if $m _ { d d } = 0$ ), we see that the optimal subnetwork comprises of the $S$ weights with the largest variances $\\sigma _ { d } ^ { 2 }$ . □ ", + "bbox": [ + 173, + 547, + 826, + 625 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B ADDITIONAL IMAGE CLASSIFICATION RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 643, + 606, + 661 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/b89a0de286e4ed847a4ac59b4fed7fd4d7e14d9bdda5e5d83d5f6abc2739032b.jpg", + "table_caption": [ + "Table 1: AUC-ROC scores for out-of-distribution detection, using CIFAR10 vs SVHN and MNIST vs FashionMNIST as in- (source) and out-of-distribution (target) datasets, respectively (Nalisnick et al., 2019). " + ], + "table_footnote": [], + "table_body": "
SOURCETARGETOURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAG
CIFAR10SVHN0.85±0.030.86±0.020.85±0.010.86±0.020.91±0.000.86±0.020.83±0.00
MNISTFashion0.92±0.050.75±0.020.82±0.120.75±0.010.90±0.090.72±0.030.97±0.01
", + "bbox": [ + 173, + 732, + 823, + 779 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/edc82dd838072a16788123a53e481cf8eef5afecb96f34e74e9499e9b030396b.jpg", + "table_caption": [ + "Table 2: MNIST – no rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.07±0.01-0.01±0.00-0.01±0.00-0.04±0.03-0.01±0.00-0.01±0.00-0.01±0.00-0.14±nan
error0.01±0.000.00±0.000.00±0.000.01±0.010.00±0.000.00±0.000.00±0.000.01±nan
ECE0.05±0.010.00±0.000.00±0.000.00±0.000.00±0.000.00±0.000.00±0.000.10±nan
brier score0.02±0.000.01±0.000.01±0.000.02±0.010.01±0.000.01±0.000.01±0.000.04±nan
", + "bbox": [ + 174, + 824, + 825, + 887 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "4This also holds for our case of a degenerate Gaussian with singular covariance matrix (Givens et al., 1984). ", + "bbox": [ + 187, + 910, + 821, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/aad024300c6bf0b50f42234321d229a99d769b38ed3b300a3faab5c6a4f64a35.jpg", + "image_caption": [ + "Figure 5: Rotated MNIST (left) and Corrupted CIFAR10 (right) results for deep ensembles (Lakshminarayanan et al., 2017) with large numbers of ensemble members (i.e. up to 55). Horizontal axis denotes number of ensemble members, and vertical axis denotes performance in terms of loglikelihood. Straight horizontal lines correspond to the performance of our method, as a reference. Colors denote different levels of rotation (left) and corruption (right). It can clearly be observed that the performance of deep ensembles saturates after around 15 ensemble members, meaning that adding more members yields strongly diminishing returns. This is in agreement with recent works (Antoran´ et al., 2020; Ashukha et al., 2020a; Lobacheva et al., 2020). Our method significantly outperforms even very large deep ensembles, especially for high degrees of rotation/corruption. " + ], + "image_footnote": [], + "bbox": [ + 176, + 137, + 820, + 338 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/7ac4e36b330bb3a0ad67b1c0ff314010923ad8edfb3ea97ca805ca45b4ba2b70.jpg", + "image_caption": [ + "Figure 6: Rejection-classification plots. We simulate a realistic OOD rejection scenario (Filos et al., 2019b) by jointly evaluating our models on an in-distribution and an OOD test set. We allow our methods to reject increasing proportions of the data based on predictive entropy before classifying the rest. All predictions on OOD samples are treated as incorrect. Following (Nalisnick et al., 2019), we use CIFAR10 vs SVHN and MNIST vs FashionMNIST as in- and out-of-distribution datasets, respectively. Note that the SVHN test set is randomly sub-sampled down to a size of 10,000. " + ], + "image_footnote": [], + "bbox": [ + 220, + 561, + 777, + 784 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/c3a908807cb61e57ccfb703b4ff067fc82ee203ac08066346bfa9a6f26a046e7.jpg", + "image_caption": [ + "Figure 7: Full MNIST rotation and CIFAR10 corruption results, for ResNet-18, reporting predictive error, log-likelihood (LL), expected calibration error (ECE) and brier score, respectively (from top to bottom). " + ], + "image_footnote": [], + "bbox": [ + 173, + 92, + 821, + 875 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/84c92e68a29bb694ec93250940373bf76b1df5a4d1136c7e674ba7159aed3f4b.jpg", + "image_caption": [ + "Figure 8: MNIST rotation results for ResNet-50, reporting predictive error, log-likelihood (LL), expected calibration error (ECE) and brier score. We choose a subnetwork containing only $0 . 1 6 7 \\%$ (39,190 / 23,466,560) of the parameters of the full network. We see that subnetwork inference still results in an improvement in the calibration of predictive uncertainty. As expected, however, for ResNet-50 the improvement over MAP is smaller than for ResNet-18 where we were able to choose a subnetwork containing $0 . 3 8 \\%$ of the parameters. " + ], + "image_footnote": [], + "bbox": [ + 176, + 107, + 821, + 361 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/46c800635cb8f4f7e6542a4563100b8cae22e2b0a531f511afc93ccd8fa91c7c.jpg", + "table_caption": [ + "Table 3: MNIST – 15◦ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.14±0.02-0.05±0.00-0.05±0.00-0.11±0.08-0.04±0.00-0.05±0.00-0.04±0.00-0.19±nan
error0.02±0.000.02±0.000.01±0.000.03±0.020.01±0.000.02±0.000.01±0.000.02±nan
ECE0.08±0.010.00±0.000.00±0.000.01±0.010.00±0.000.00±0.000.00±0.000.12±nan
brier score0.05±0.010.03±0.000.02±0.000.05±0.030.02±0.000.02±0.000.02±0.000.07±nan
", + "bbox": [ + 174, + 508, + 825, + 571 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/96012ccc35b17bc2b191eda259776728a82e89192427812f600e8679ef193c57.jpg", + "table_caption": [ + "Table 4: MNIST – $3 0 ^ { \\circ }$ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.42±0.04-0.36±0.01-0.32±0.02-0.44±0.06-0.28±0.02-0.39±0.01-0.30±0.00-0.51±nan
error0.11±0.010.10±0.000.09±0.010.12±0.010.08±0.010.10±0.000.08±0.000.14±nan
ECE0.10±0.020.04±0.010.03±0.000.06±0.010.02±0.000.05±0.000.04±0.000.13±nan
brier score0.19±0.020.16±0.000.14±0.010.18±0.020.12±0.010.16±0.000.12±0.000.23±nan
", + "bbox": [ + 173, + 621, + 823, + 684 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/13a1ee3b10024f6d00e32082b7418b3f2308984e88d7389674bfb46e3707e564.jpg", + "table_caption": [ + "Table 5: MNIST – 45◦ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
midrule LL-1.09±0.03-1.60±0.05−1.44±0.11-1.68±0.20-1.36±0.07-1.75±0.06-1.35±0.02-1.15±nan
error0.36±0.010.35±0.010.33±0.010.35±0.030.31±0.010.35±0.010.29±0.000.40±nan
ECE0.03±0.010.22±0.010.19±0.020.22±0.020.17±0.010.23±0.010.18±0.000.01±nan
brier score0.49±0.020.55±0.020.52±0.020.55±0.040.48±0.020.56±0.020.46±0.010.53±nan
", + "bbox": [ + 174, + 733, + 823, + 800 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/56f61bfdcc68d00480b38d0a28029e44cddcd49436a3b02e71bdf3acb1a3f40f.jpg", + "table_caption": [ + "Table 6: MNIST – $6 0 ^ { \\circ }$ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-2.10±0.03-3.85±0.18-3.54±0.23-4.11±0.66-3.60±0.10-4.29±0.21-2.95±0.08-1.92±nan
error0.63±0.010.63±0.010.62±0.010.62±0.050.61±0.010.63±0.010.53±0.020.64±nan
ECE0.25±0.020.46±0.020.43±0.020.47±0.060.42±0.010.48±0.020.36±0.020.17±nan
brier score0.85±0.021.04±0.031.00±0.031.05±0.100.98±0.021.07±0.030.86±0.030.80±nan
", + "bbox": [ + 173, + 849, + 823, + 912 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/2793bbdf6d23cb3767f4e4f4ea80e32405d51405a579dfc100ed5e24e79a8136.jpg", + "table_caption": [ + "Table 7: MNIST – $7 5 ^ { \\circ }$ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.02±0.07-5.93±0.28-5.49±0.38-6.92±0.32-5.74±0.15-6.63±0.33-4.46±0.18-2.54±nan
error0.80±0.020.79±0.010.79±0.010.81±0.000.78±0.010.79±0.010.72±0.020.77±nan
ECE0.41±0.040.62±0.030.59±0.010.65±0.010.58±0.010.64±0.030.51±0.020.26±nan
brier score1.08±0.041.34±0.041.30±0.021.39±0.011.29±0.021.37±0.041.17±0.040.95±nan
", + "bbox": [ + 173, + 128, + 823, + 191 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/8ed8672fd96d4068d0c7251b168b43d468ab736c285134a2776bb5b95a7c23fd.jpg", + "table_caption": [ + "Table 8: MNIST – 90◦ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.35±0.13-6.46±0.15-6.18±0.41-7.32±0.67-6.39±0.17-7.18±0.22-5.63±0.12-2.91±nan
error0.84±0.020.84±0.010.84±0.010.85±0.010.84±0.010.84±0.010.82±0.020.81±nan
ECE0.43±0.030.64±0.040.62±0.010.66±0.030.62±0.010.66±0.040.60±0.010.29±nan
brier score1.13±0.031.40±0.051.37±0.011.44±0.041.36±0.011.43±0.051.34±0.021.02±nan
", + "bbox": [ + 173, + 232, + 823, + 295 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/866256812fa7664a066b10c63d51b1ccc50be1c3fdf9ad64efcb4f4670fdc335.jpg", + "table_caption": [ + "Table 9: MNIST – 105◦ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.59±0.05-7.06±0.45-6.70±0.52-7.69±0.99-7.01±0.17-7.87±0.53-6.28±0.19-3.10±nan
error0.85±0.020.84±0.020.84±0.010.85±0.010.84±0.010.84±0.020.81±0.000.81±nan
ECE0.47±0.040.67±0.050.63±0.010.67±0.030.64±0.010.68±0.040.61±0.010.34±nan
brier score1.17±0.051.44±0.071.38±0.021.44±0.041.40±0.011.46±0.071.34±0.021.07±nan
", + "bbox": [ + 174, + 337, + 823, + 398 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/08bf36e9d9729b2ae770026349db0bb2b26edd646fa8f9b9f3fbfe476a0d01a6.jpg", + "table_caption": [ + "Table 10: MNIST – 120◦ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.43±0.07-6.73±0.53-6.62±0.39-7.92±0.59-6.73±0.11-7.53±0.63-6.49±0.36-3.07±nan
error0.80±0.020.79±0.020.78±0.010.81±0.010.78±0.010.79±0.020.76±0.020.76±nan
ECE0.40±0.030.62±0.050.58±0.010.65±0.040.59±0.010.63±0.040.58±0.030.30±nan
brier score1.10±0.031.35±0.071.29±0.021.39±0.061.30±0.011.36±0.071.27±0.041.04±nan
", + "bbox": [ + 174, + 440, + 823, + 502 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/d45b8c8a70b592cb58b7f006bbdfe9171a482eaaf308c3e9640054147ce90c5e.jpg", + "table_caption": [ + "Table 11: MNIST – 135◦ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.24±0.06-6.43±0.38-6.46±0.28-7.05±0.88-6.57±0.10-7.24±0.48-6.40±0.37-2.89±nan
error0.71±0.020.71±0.020.70±0.010.71±0.010.70±0.010.71±0.020.70±0.020.67±nan
ECE0.32±0.010.55±0.030.52±0.010.56±0.020.52±0.010.56±0.030.53±0.020.25±nan
brier score0.99±0.021.21±0.051.17±0.021.22±0.041.17±0.011.23±0.051.18±0.040.94±nan
", + "bbox": [ + 173, + 544, + 823, + 606 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/adb12f830d247d4729c15d9fc269bee919724601879718f3c7791a9967a067df.jpg", + "table_caption": [ + "Table 12: MNIST – 150◦ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.25±0.05-6.56±0.18-6.62±0.33-7.04±0.36-6.88±0.11-7.41±0.25-6.39±0.27-2.69±nan
error0.63±0.020.63±0.010.63±0.000.65±0.010.62±0.010.63±0.010.63±0.010.60±nan
ECE0.29±0.010.50±0.010.48±0.010.52±0.010.48±0.010.51±0.010.49±0.010.23±nan
brier score0.92±0.021.10±0.021.07±0.011.13±0.021.06±0.011.11±0.021.08±0.020.85±nan
", + "bbox": [ + 173, + 647, + 823, + 708 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/bdab14d86f13513019a8dc084f7a8defc9cca45cfe190b6b5c9645ee94026cd7.jpg", + "table_caption": [ + "Table 13: MNIST – 165◦ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.42±0.12-7.01±0.15-7.08±0.39-7.80±0.12−7.51±0.11-7.91±0.18-6.63±0.24-2.67±nan
error0.58±0.010.58±0.010.58±0.010.58±0.000.57±0.010.58±0.010.59±0.000.56±nan
ECE0.32±0.020.49±0.010.48±0.010.49±0.010.48±0.000.51±0.010.48±0.000.25±nan
brier score0.90±0.021.05±0.011.04±0.011.05±0.011.03±0.011.07±0.021.03±0.010.82±nan
", + "bbox": [ + 173, + 751, + 823, + 813 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/07c036ced1e105451a5698062b4b8ec61999c89bd0792155bc113876efda83e3.jpg", + "table_caption": [ + "Table 14: MNIST – 180◦ rotation. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.32±0.13-6.63±0.18-6.87±0.32-7.10±0.47-7.16±0.16-7.43±0.20-6.61±0.22-2.71±nan
error0.56±0.010.56±0.010.56±0.000.55±0.010.55±0.000.56±0.010.57±0.000.55±nan
ECE0.29±0.020.46±0.010.45±0.000.46±0.000.46±0.010.48±0.010.47±0.010.25±nan
brier score0.86±0.021.00±0.010.99±0.010.99±0.010.99±0.001.01±0.021.01±0.010.82±nan
", + "bbox": [ + 173, + 854, + 823, + 916 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/148eb02bbf59236ed9ee98c856d4a6c3528daf9e7dd8a60f931c43d4b5f97868.jpg", + "table_caption": [ + "Table 15: CIFAR10 – no corruption. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.27±0.00-0.43±0.01-0.37±0.01-0.50±0.02-0.21±0.01-0.46±0.02-0.48±0.01-0.61±nan
error0.09±0.000.08±0.000.08±0.000.09±0.000.06±0.000.08±0.000.11±0.000.21±nan
ECE0.01±0.000.06±0.000.04±0.000.06±0.000.01±0.000.06±0.000.07±0.000.03±nan
brier score0.13±0.000.14±0.000.13±0.000.15±0.000.09±0.000.14±0.000.17±0.000.30±nan
", + "bbox": [ + 173, + 147, + 825, + 210 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/9eb1217956a82f29d268d54c8e8c2cb1e3dd292adbd11b853710db5ce0bf07d6.jpg", + "table_caption": [ + "Table 16: CIFAR10 – level 1 corruption. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.51±0.01-0.91±0.01-0.80±0.02−1.03±0.02-0.50±0.02-0.96±0.02-0.89±0.02-0.99±nan
error0.17±0.010.16±0.000.16±0.000.17±0.000.13±0.000.16±0.000.17±0.000.32±nan
ECE0.03±0.000.11±0.000.10±0.000.13±0.000.04±0.000.12±0.010.11±0.000.03±nan
brier score0.24±0.000.27±0.000.25±0.000.29±0.000.19±0.000.27±0.010.29±0.000.44±nan
", + "bbox": [ + 173, + 285, + 823, + 348 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/4fb663b90b5e77c278b96826beaa1a293735778800c420e7c2eeafcbf2efcc53.jpg", + "table_caption": [ + "Table 17: CIFAR10 – level 2 corruption. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.73±0.01-1.29±0.06-1.20±0.02−1.50±0.12-0.80±0.01-1.40±0.03-1.21±0.00-1.31±nan
error0.23±0.000.22±0.010.22±0.000.23±0.010.19±0.000.22±0.000.22±0.000.40±nan
ECE0.06±0.000.16±0.010.14±0.000.17±0.010.07±0.000.16±0.000.15±0.000.10±nan
brier score0.33±0.000.37±0.010.35±0.010.40±0.020.28±0.000.37±0.010.37±0.000.56±nan
", + "bbox": [ + 173, + 422, + 823, + 486 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/6aa924fbe0dedb316db42e4f51372755d961290b36fc676ec2d4fa71060203a9.jpg", + "table_caption": [ + "Table 18: CIFAR10 – level 3 corruption. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-1.06±0.02-2.06±0.12-1.85±0.07-2.13±0.17-1.28±0.03-2.18±0.08-1.63±0.03-1.83±nan
error0.32±0.010.31±0.010.31±0.010.31±0.010.28±0.000.31±0.010.28±0.000.51±nan
ECE0.11±0.010.24±0.010.21±0.010.24±0.010.12±0.000.24±0.010.20±0.000.19±nan
brier score0.46±0.010.54±0.020.50±0.020.54±0.030.42±0.000.54±0.020.47±0.010.72±nan
", + "bbox": [ + 173, + 560, + 823, + 625 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/52acece80b4112f1cd6df14e3284b0bfdef08491258797e3f98b3012ac1790f9.jpg", + "table_caption": [ + "Table 19: CIFAR10 – level 4 corruption. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-1.25±0.03-2.43±0.18-2.28±0.10-2.54±0.18-1.56±0.05-2.57±0.15-1.95±0.04-1.99±nan
error0.36±0.010.35±0.010.35±0.010.35±0.010.32±0.010.35±0.010.32±0.000.54±nan
ECE0.13±0.010.27±0.010.24±0.010.27±0.010.14±0.010.27±0.020.23±0.000.22±nan
brier score0.51±0.020.60±0.030.57±0.010.61±0.020.47±0.010.60±0.030.53±0.000.76±nan
", + "bbox": [ + 173, + 699, + 823, + 762 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/836c5447a448eba6dfda9330a6f64765a410e55c8bc33aeb581e8a36881420ec.jpg", + "table_caption": [ + "Table 20: CIFAR10 – level 5 corruption. " + ], + "table_footnote": [], + "table_body": "
OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL−1.47±0.03-2.82±0.11-2.71±0.13-3.20±0.13-1.88±0.05-3.03±0.10-2.31±0.09-2.00±nan
error0.41±0.000.40±0.010.40±0.010.41±0.010.37±0.010.40±0.000.36±0.010.54±nan
ECE0.16±0.010.31±0.010.28±0.010.33±0.020.17±0.010.31±0.010.27±0.010.19±nan
brier score0.58±0.000.69±0.010.65±0.010.72±0.030.55±0.010.69±0.010.61±0.010.75±nan
", + "bbox": [ + 174, + 837, + 823, + 900 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "C EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 174, + 102, + 401, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.1 TOY EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 135, + 352, + 150 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We train a single, 2 hidden layer network, with 50 hidden ReLU units per layer using MAP inference until convergence. Specifically, we use SGD with a learning rate of $1 \\times 1 0 ^ { - 3 }$ , momentum of 0.9 and weight decay of $1 \\times 1 0 ^ { - 4 }$ . We use a batch size of 512. The objective we optimise is the Gaussian log-likelihood of our data, where the mean is outputted by the network and the the variance is a hyperparameter learnt jointly with NN parameters by SGD. This variance parameters is shared among all datapoints. Once the network is trained, we perform post-hoc inference on it using different approaches. Since all of these involve the linearized approximation, the mean prediction is the same for all methods. Only their uncertainty estimates vary. ", + "bbox": [ + 174, + 161, + 825, + 273 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Note that while for this toy example, we could in principle use the full covariance matrix for the purpose of subnetwork selection, we still just use its diagonal (as described in Section 4) for consistency. We use GGN Laplace inference over network weights (not biases) in combination with the linearized predictive distribution in Eq. (8). Thus, all approaches considered share their predictive mean, allowing us to better compare their uncertainty estimates. ", + "bbox": [ + 174, + 280, + 825, + 349 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "All approaches share a single prior precision of $\\lambda = 3$ . We chose to select the prior precision such that the full covariance approach (optimistic baseline) presents reasonable results. We use the same value for all other methods. We first tried a precision of 1 and found the full covariance approach to produce excessively large errorbars (covering the whole plot). A value of 3 produces more reasonable results. ", + "bbox": [ + 174, + 356, + 825, + 426 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Final layer inference is performed by computing the full Laplace covariance matrix and discarding all entries except those corresponding to the final layer of the NN. Results for random sub-network selection are obtained with a single sample from a scaled uniform distribution over weight choice. ", + "bbox": [ + 176, + 433, + 825, + 476 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.2 UCI EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 494, + 352, + 508 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In this experiment, our fully connected NNs have numbers of hidden layers $h _ { d } = \\{ 1 , 2 \\}$ and hidden layer widths ${ w _ { d } } \\mathrm { { = } } \\{ 5 0 , 1 0 0 \\}$ . For a dataset with input dimension $i _ { d }$ , the number of weights is given by $D { = } ( i _ { d } { + } 1 ) w _ { d } { + } ( h _ { d } { - } 1 ) \\bar { w } _ { d } ^ { 2 }$ . Our 2 hidden layer, 100 hidden unit models have a weight count of the order $1 0 ^ { 4 }$ . The non-linearity used is ReLU. ", + "bbox": [ + 174, + 521, + 825, + 577 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We first obtain a MAP estimate of each model’s weights. Specifically, we use SGD with a learning rate of $1 \\times 1 0 ^ { - 3 }$ , momentum of 0.9 and weight decay of $1 \\bar { \\times } 1 0 ^ { - 4 }$ . We use a batch size of 512. The objective we optimise is the Gaussian log-likelihood of our data, where the mean is outputted by the network and the the variance is a hyperparameter learnt jointly with NN parameters by SGD. ", + "bbox": [ + 174, + 584, + 825, + 640 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "For each dataset split, we set aside $1 5 \\%$ of the train data as a validation set. We use these for early stopping training. Training runs for a maximum of 2000 epochs but early stops with a patience of 500 if validation performance does not increase. For the larger Protein dataset, these values are 500 and 125. The weight settings which provide best validation performance are kept. ", + "bbox": [ + 174, + 647, + 825, + 704 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We then perform full network GGN Laplace inference for each model. We also use our proposed Wassertein rule together with the diagonal Hessian assumption to prune every network’s weight variances such that the number of variances that remain matches the size of every smaller network under consideration. The prior precision used for these steps is chosen such that the resulting predictor’s logliklihood performance on the validation set is maximised. Specifically, we employ a grid search over the values: $\\lambda : [ 0 . 0 0 0 1 , 0 . 0 0 1 , 0 . 1 , 0 . 5 , 1 , 2 , 5 , 1 0 , 1 0 0 , 1 0 0 0 ]$ . In all cases, we employ the linearized predictive in Eq. (7). Consequently, networks with the same number of weights make the same mean predictions. Increasing the number of weight variances considered will thus only increase predictive uncertainty. ", + "bbox": [ + 174, + 710, + 825, + 835 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C.3 IMAGE EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 854, + 369, + 869 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The results shown in Section 5.3 and Appendix B are obtained by training ResNet-18 (and ResNet-50) models using SGD with momentum. For each experiment repetition, we train 7 different models: The first is for: ‘MAP’, ‘Ours’, ‘Ours (Rand)’, ‘SWAG’, ‘Diag-Laplace’ and as the first element of ‘Ensemble’. We train 4 additional ‘Ensemble’ elements, 1 network with ‘Dropout’, and, finally 1 network for ‘VOGN’. The methods ‘Ours’, ‘Ours (Rand)’, ‘SWAG’, and ‘Diag-Laplace’ are applied post training. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For all methods except ‘VOGN’ we use the following training procedure. The (initial) learning rate, momentum, and weight decay are 0.1, 0.9, and $1 \\times \\mathrm { i 0 ^ { - 4 } }$ , respectively. For ‘MAP’ we use 4 Nvidia P100 GPUs with a total batch size of 2048. For the calculation of the Jacobian in the subnetwork selection phase we use a single P100 GPU with a batch size of 4. For the calculation of the hessian we use a single P100 GPU with a batch size of 2. We train on 1 Nvidia P100 GPU with a batch size of 256 for all other methods. Each dataset is trained for a different number of epochs, shown in Table 21. We decay the learning rate by a factor of 10 at scheduled epochs, also shown in Table 21. Otherwise, all methods and datasets share hyperparameters. These hyperparameter settings are the defaults provided by PyTorch for training on ImageNet. We found them to perform well across the board. We report results obtained at the final training epoch. We do not use a separate validation set to determine the best epoch as we found ResNet-18 and ResNet-50 to not overfit with the chosen schedules. ", + "bbox": [ + 173, + 152, + 826, + 318 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/f413cddb64dfe36d9328f596a742ad4889010a00ee986534ca3d7836e63b776e.jpg", + "table_caption": [ + "Table 21: Per-dataset training configuration for image experiments. " + ], + "table_footnote": [], + "table_body": "
DATASETNo.EPOCHSLR SCHEDULE
MNIST9040,70
CIFAR10300150,225
", + "bbox": [ + 336, + 363, + 655, + 419 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For ‘Dropout’, we add dropout to the standard ResNet-50 model (He et al., 2016) in between the 2 and 3 convolutions in the bottleneck blocks. This approach follows Zagoruyko & Komodakis (2016) and Ashukha et al. (2020b) who add dropout in-between the two convolutions of a WideResNet-50’s basic block. Following Antoran et al. (2020), we choose a dropout probability of 0.1, as they found ´ it to perform better than the value of 0.3 suggested by Ashukha et al. (2020b). We use $\\mathrm { \\dot { 1 } 6 \\mathbf { M } C }$ samples for predictions. ‘Ensemble’ uses 5 elements for prediction. Ensemble elements differ from each other in their initialisation, which is sampled from the He initialisation distribution (He et al., 2015). We do not use adversarial training as, inline with Ashukha et al. (2020b), we do not find it to improve results. For ‘VOGN’ we use the same procedure and hyper-parameters as used by Osawa et al. (2019) in their CIFAR10 experiments, with the exception that we use a learning rate of $1 \\times 1 0 ^ { - 3 }$ as we we found a value of $1 \\times 1 0 ^ { - 4 }$ not to result in convergence. We train on a single Nvidia P100 GPU with a batch size of 256. See the authors’ GitHub for more details: github.com/team-approx-bayes/dl-with-bayes/blob/master/ distributed/classification/configs/cifar10/resnet18_vogn_bs256_ 8gpu.json. ", + "bbox": [ + 174, + 440, + 826, + 648 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We modify the standard ResNet-50 and ResNet-18 architectures such that the first $7 \\times 7$ convolution is replaced with a $3 \\times 3$ convolution. Additionally, we remove the first max-pooling layer. Following Goyal et al. (2017), we zero-initialise the last batch normalisation layer in residual blocks so that they act as identity functions at the start of training. ", + "bbox": [ + 174, + 656, + 825, + 712 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "At test time, we tune the prior precision used for ‘Ours’, ‘Diag-Laplace’ and ‘SWAG’ approximation on a validation set for each approach individually, as in Ritter et al. (2018); Kristiadi et al. (2020). We use a grid search from $1 \\times \\bar { 1 } 0 ^ { - 4 }$ to $1 \\times 1 0 ^ { 4 }$ in logarithmic steps, and then a second, finer-grained grid search between the two best performing values (again with logarithmic steps). ", + "bbox": [ + 174, + 718, + 825, + 775 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "C.4 DATASETS ", + "text_level": 1, + "bbox": [ + 174, + 797, + 289, + 811 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "The 1d toy dataset used in Section 5.1 was taken from Antoran et al. (2020). We obtained it from the ´ authors’ github repo: https://github.com/cambridge-mlg/DUN. Table 22 summarises the datasets used in Section 5.2. Wine and Protein are available from the UCI dataset repository Dua & Graff (2017). $\\operatorname { K i n } 8 \\mathrm { n m }$ is available from https://www.openml.org/d/189 Foong et al. (2019b). For the standard splits (Hernandez-Lobato & Adams, 2015)´ $90 \\%$ of the data is used for training and $10 \\%$ for validation. For the gap splits (Foong et al., 2019b) a split is obtained per input dimension by ordering points by their values across that dimension and removing the middle $33 \\%$ of the points. These are used for validation. The datasets used for our image experiments are outlined in Table 23. ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/9d48240bc943a4eedeb6f64a08cc446779eed498e0a1fdfa0eb3aa31a63c1730.jpg", + "table_caption": [ + "Table 22: Datasets from tabular regression used in Section 5.2 " + ], + "table_footnote": [], + "table_body": "
DatasetN TrainN Val (15% train)NTestSplitsOutput Dim Output TypeInput DimInput Type
Wine1223216160201Continous11Continous
Wine Gap906161532111Continous11Continous
Kin8nm62671106819201Continous8Continous
Kin8nm Gap4642820273081Continous8Continous
Protein349836174457351Continous9Continous
Protein Gap2591345731524491Continous9Continous
", + "bbox": [ + 173, + 127, + 964, + 239 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 265, + 823, + 292 + ], + "page_idx": 21 + }, + { + "type": "table", + "img_path": "images/20526b4e69c99fddd75d6474815a7f561493225ebfc32b7b6c15ea8ab3f95f0b.jpg", + "table_caption": [ + "Table 23: Summary of image datasets. The test and train set sizes are shown in brackets, e.g. (test & train). " + ], + "table_footnote": [], + "table_body": "
NAMESIZEINPUT DIM.No. CLASSESNO. SPLITS
MNIST (LeCun et al., 1998)70,000 (60.000&10,000)784(28 × 28)102
Fashion-MNIST (Xiao et al.,2017)70,000 (60.000 &10.000)784(28 × 28)102
CIFAR10(Krizhevsky & Hinton,2009)60,000 (50.000&10.000)3072(32 ×32 ×3)102
SVHN (Netzer et al., 2011)99,289 (73,257& 26,032)3072(32 ×32 × 3)102
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However, scaling Bayesian inference to the high-dimensional parameter", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 471, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 471, + 257 + ], + "score": 1.0, + "content": "spaces of deep neural networks requires restrictive approximations. In this paper,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 255, + 469, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 255, + 469, + 267 + ], + "score": 1.0, + "content": "we propose performing inference over only a small subset of the model parameters", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "spans": [ + { + "bbox": [ + 141, + 266, + 469, + 277 + ], + "score": 1.0, + "content": "while keeping all others as point estimates. This enables us to use expressive", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 277, + 469, + 288 + ], + "spans": [ + { + "bbox": [ + 141, + 277, + 469, + 288 + ], + "score": 1.0, + "content": "posterior approximations that would otherwise be intractable for the full model. In", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 288, + 469, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 288, + 469, + 299 + ], + "score": 1.0, + "content": "particular, we develop a practical and scalable Bayesian deep learning method that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 469, + 311 + ], + "score": 1.0, + "content": "first trains a point estimate, and then infers a full covariance Gaussian posterior", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 309, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 309, + 469, + 322 + ], + "score": 1.0, + "content": "approximation over a subnetwork. We propose a subnetwork selection procedure", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 320, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 320, + 470, + 333 + ], + "score": 1.0, + "content": "which aims to maximally preserve posterior uncertainty. We empirically demon-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 332, + 469, + 343 + ], + "spans": [ + { + "bbox": [ + 141, + 332, + 469, + 343 + ], + "score": 1.0, + "content": "strate the effectiveness of our approach compared to point-estimated networks and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 342, + 464, + 353 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 464, + 353 + ], + "score": 1.0, + "content": "methods that use less expressive posterior approximations over the full network.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 141, + 210, + 471, + 353 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 373, + 206, + 385 + ], + "lines": [ + { + "bbox": [ + 105, + 372, + 208, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 208, + 389 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 397, + 505, + 464 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 505, + 409 + ], + "score": 1.0, + "content": "Deep neural networks (DNNs) still suffer from critical shortcomings that make them unfit for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 409, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 506, + 421 + ], + "score": 1.0, + "content": "important applications. For instance, DNNs tend to be poorly calibrated and overconfident in their", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "predictions, especially when there is a shift in the train and test distributions (Nguyen et al., 2015;", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "Guo et al., 2017). To reliably inform decision making, DNNs must be able to robustly quantify", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 454 + ], + "score": 1.0, + "content": "the uncertainty in their predictions, which is particularly important in safety-critical areas such as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 452, + 496, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 496, + 465 + ], + "score": 1.0, + "content": "healthcare or autonomous driving (Amodei et al., 2016; Filos et al., 2019a; Fridman et al., 2019).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 398, + 506, + 465 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "score": 1.0, + "content": "Bayesian modeling (Ghahramani, 2015; Gal, 2016) presents a principled way to capture predictive", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 481, + 507, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 507, + 493 + ], + "score": 1.0, + "content": "uncertainty via the posterior distribution over model parameters. Unfortunately, due to their non-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 491, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 506, + 503 + ], + "score": 1.0, + "content": "linearities, exact posterior inference is intractable in DNNs. Despite recent successes in the field", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 502, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 506, + 515 + ], + "score": 1.0, + "content": "of Bayesian deep learning (Blundell et al., 2015; Gal & Ghahramani, 2016; Osawa et al., 2019;", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "Maddox et al., 2019; Dusenberry et al., 2020), existing methods are only made scalable to modern", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 537 + ], + "score": 1.0, + "content": "DNNs with large numbers of parameters by invoking unrealistic assumptions. This severely limits the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 548 + ], + "score": 1.0, + "content": "expressiveness of the inferred posterior and thus deteriorates the quality of the induced uncertainty", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 546, + 475, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 475, + 558 + ], + "score": 1.0, + "content": "estimates (Ovadia et al., 2019; Fort et al., 2019; Foong et al., 2019a; Ashukha et al., 2020a).", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 469, + 507, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 575 + ], + "score": 1.0, + "content": "Due to the heavy overparameterization of DNNs, their accuracy is well-preserved by a small sub-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "network (Cheng et al., 2017). Additionally, recent work by Izmailov et al. (2019) has shown how", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 586, + 507, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 507, + 597 + ], + "score": 1.0, + "content": "performing inference over a low dimensional subspace of the weights can result in accurate uncer-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 506, + 609 + ], + "score": 1.0, + "content": "tainty quantification. These observations prompt the following question for a DNN’s uncertainty:", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "Can a full DNN’s model uncertainty be well-preserved by a small subnetwork’s model uncertainty?", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "We answer this question in the affirmative. We show both theoretically and empirically that the full", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "network posterior can be well represented by a subnetwork’s posterior. As a result, we can use more", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "expensive but faithful posterior approximations over just that subnetwork. We show that this achieves", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "better uncertainty quantification than if we use cheaper, but more crude, posterior approximations", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 662, + 195, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 195, + 673 + ], + "score": 1.0, + "content": "over the full network.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 563, + 507, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 679, + 290, + 690 + ], + "lines": [ + { + "bbox": [ + 107, + 678, + 291, + 691 + ], + "spans": [ + { + "bbox": [ + 107, + 678, + 291, + 691 + ], + "score": 1.0, + "content": "The contributions of this paper are as follows:", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43, + "bbox_fs": [ + 107, + 678, + 291, + 691 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 129, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 129, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "1. We propose a new Bayesian deep learning approach that performs Bayesian inference over", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 142, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 142, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "only a small subset of the model weights and keeps all other weights deterministic. This", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 142, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 142, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "allows us to use expressive posterior approximations that are typically intractable in DNNs.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 129, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 83, + 500, + 188 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 83, + 500, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 83, + 500, + 188 + ], + "spans": [ + { + "bbox": [ + 112, + 83, + 500, + 188 + ], + "score": 0.966, + "type": "image", + "image_path": "69ba0d7682cd0284507b2c8b65f0296f13b556dfc570af6f9e27b927c22c918d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 83, + 500, + 118.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 118.0, + 500, + 153.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 153.0, + 500, + 188.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 200, + 506, + 245 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 200, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 506, + 214 + ], + "score": 1.0, + "content": "Figure 1: Schematic illustration of our proposed approach. (a) We train a neural network using", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 225 + ], + "score": 1.0, + "content": "standard techniques to obtain a point estimate of the weights. (b) We identify a small subset of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "weights. (c) We estimate a posterior distribution over the selected subnetwork via Bayesian inference", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 234, + 506, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 506, + 246 + ], + "score": 1.0, + "content": "techniques. (d) We make predictions using the full network of mixed Bayesian/deterministic weights.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 129, + 264, + 506, + 383 + ], + "lines": [ + { + "bbox": [ + 129, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 129, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "2. 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A prior distribution", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 459, + 476, + 471 + ], + "spans": [ + { + "bbox": [ + 107, + 459, + 132, + 471 + ], + "score": 0.91, + "content": "p ( \\mathbf { W } )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 459, + 476, + 471 + ], + "score": 1.0, + "content": "is specified over the BNN’s weights. We wish to infer their full posterior distribution", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 426, + 506, + 471 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 474, + 386, + 488 + ], + "lines": [ + { + "bbox": [ + 225, + 474, + 386, + 488 + ], + "spans": [ + { + "bbox": [ + 225, + 474, + 386, + 488 + ], + "score": 0.91, + "content": "\\begin{array} { r l r } { p ( { \\bf W } | { \\bf y } , { \\pmb X } ) } & { \\propto } & { p ( { \\pmb y } | { \\pmb X } , { \\bf W } ) p ( { \\bf W } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "49d86c485e154e77f1466b70cb3e98b045e2d1e38db86a79471ba12be996c0e4.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 225, + 474, + 386, + 488 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 492, + 509, + 515 + ], + "lines": [ + { + "bbox": [ + 106, + 492, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 505, + 505 + ], + "score": 1.0, + "content": "To make predictions, we then estimate the posterior predictive distribution that averages the network’s", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 503, + 495, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 495, + 516 + ], + "score": 1.0, + "content": "predictions across all possible settings of the weights, weighted by their posterior probability, i.e.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 492, + 505, + 516 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 191, + 518, + 420, + 545 + ], + "lines": [ + { + "bbox": [ + 191, + 518, + 420, + 545 + ], + "spans": [ + { + "bbox": [ + 191, + 518, + 420, + 545 + ], + "score": 0.93, + "content": "p ( { \\mathbf { y } ^ { * } } | X ^ { * } , y , X ) = \\int _ { \\mathbf { W } } p ( { \\mathbf { y } ^ { * } } | X ^ { * } , \\mathbf { W } ) p ( \\mathbf { W } | y , X ) d \\mathbf { W } .", + "type": "interline_equation", + "image_path": "e42515f2fa3f51cd24be9f49825e0fd023d019b5260f20bf8c4257d81733408c.jpg" + } + ] + } + ], + "index": 25.5, + "virtual_lines": [ + { + "bbox": [ + 191, + 518, + 420, + 531.5 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 191, + 531.5, + 420, + 545.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 548, + 505, + 640 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "Unfortunately, due to the size of modern deep neural networks, it is not only intractable to infer", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 558, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 234, + 573 + ], + "score": 1.0, + "content": "the exact posterior distribution", + "type": "text" + }, + { + "bbox": [ + 235, + 559, + 284, + 571 + ], + "score": 0.93, + "content": "p ( \\mathbf { W } | \\mathbf { \\boldsymbol { y } } , X )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 558, + 506, + 573 + ], + "score": 1.0, + "content": "in Eq. (1), but it is even computationally challenging", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 571, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 505, + 582 + ], + "score": 1.0, + "content": "to properly approximate it. As a consequence, crude posterior approximations such as complete", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 581, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 506, + 594 + ], + "score": 1.0, + "content": "factorization are commonly employed (Blundell et al., 2015; Hernandez-Lobato & Adams, 2015; ´", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 592, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 363, + 609 + ], + "score": 1.0, + "content": "Kingma et al., 2015; Khan et al., 2018; Osawa et al., 2019), i.e.", + "type": "text" + }, + { + "bbox": [ + 364, + 592, + 477, + 607 + ], + "score": 0.92, + "content": "\\begin{array} { r } { p ( \\mathbf { W } | \\pmb { y } , \\pmb { X } ) \\approx \\prod _ { d = 1 } ^ { D } q ( \\mathbf { w } _ { d } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 592, + 506, + 609 + ], + "score": 1.0, + "content": "where", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 605, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 119, + 618 + ], + "score": 0.85, + "content": "\\mathbf { W } _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 605, + 172, + 620 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 172, + 607, + 178, + 617 + ], + "score": 0.79, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 605, + 250, + 620 + ], + "score": 1.0, + "content": "-th weight in the", + "type": "text" + }, + { + "bbox": [ + 250, + 607, + 260, + 617 + ], + "score": 0.78, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 605, + 441, + 620 + ], + "score": 1.0, + "content": "-dimensional neural network weight vector", + "type": "text" + }, + { + "bbox": [ + 441, + 606, + 484, + 617 + ], + "score": 0.89, + "content": "\\mathbf { W _ { \\lambda } } \\in \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 605, + 506, + 620 + ], + "score": 1.0, + "content": "(the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 618, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 506, + 630 + ], + "score": 1.0, + "content": "concatenation and flattening of all layers’ weight matrices). Clearly, this is a very wishful assumption;", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 628, + 384, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 384, + 641 + ], + "score": 1.0, + "content": "In practise, it suffers from severe pathologies (Foong et al., 2019a;b).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 548, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 504, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "In this work, we question the implicit assumption that a good posterior approximation needs to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "include all BNN parameters. 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Inference over submodels: Previous work1 has provided evidence that inference can be", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 141, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "effective even when not done on the full parameter space. Izmailov et al. (2019) performed", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 141, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "inference over a low-dimensional projection of the weights. Neural-linear models, which", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "give a Bayesian treatment to only the last layer of a DNN, have shown to be competitive", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 127, + 466, + 138 + ], + "spans": [ + { + "bbox": [ + 141, + 127, + 466, + 138 + ], + "score": 1.0, + "content": "with full-network approaches (Riquelme et al., 2018; Ober & Rasmussen, 2019).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 105, + 146, + 504, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 507, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 507, + 162 + ], + "score": 1.0, + "content": "We thus combine these ideas, making the following two-step approximation of the posterior in Eq. (1):", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 164, + 466, + 191 + ], + "lines": [ + { + "bbox": [ + 145, + 164, + 466, + 191 + ], + "spans": [ + { + "bbox": [ + 145, + 164, + 466, + 191 + ], + "score": 0.92, + "content": "p ( \\mathbf { W } | \\pmb { y } , \\pmb { X } ) \\ \\approx \\ p ( \\mathbf { W } _ { S } | \\pmb { y } , \\pmb { X } ) \\prod _ { r } \\delta ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } ) \\ \\approx \\ q ( \\mathbf { W } _ { S } ) \\prod _ { r } \\delta ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } ) .", + "type": "interline_equation", + "image_path": "8e94cca0ea87a069536e2aa115ec5b28ac77bb498de51531bd7817fb6c1bbd28.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 145, + 164, + 466, + 173.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 145, + 173.0, + 466, + 182.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 145, + 182.0, + 466, + 191.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 195, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 104, + 195, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 104, + 195, + 390, + 209 + ], + "score": 1.0, + "content": "The first approximation decomposes the full neural network posterior", + "type": "text" + }, + { + "bbox": [ + 390, + 196, + 440, + 208 + ], + "score": 0.93, + "content": "p ( \\mathbf { W } | \\mathbf { \\boldsymbol { y } } , X )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 195, + 506, + 209 + ], + "score": 1.0, + "content": "into a posterior", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 162, + 219 + ], + "score": 0.91, + "content": "p ( \\mathbf { W } _ { S } | \\pmb { y } , \\pmb { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 207, + 246, + 221 + ], + "score": 1.0, + "content": "over the subnetwork", + "type": "text" + }, + { + "bbox": [ + 247, + 208, + 266, + 218 + ], + "score": 0.89, + "content": "\\mathbf { W } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 207, + 345, + 221 + ], + "score": 1.0, + "content": "and delta functions", + "type": "text" + }, + { + "bbox": [ + 346, + 208, + 394, + 219 + ], + "score": 0.92, + "content": "\\delta \\big ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "over all remaining weights", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 107, + 219, + 133, + 230 + ], + "score": 0.9, + "content": "\\left\\{ \\mathbf { w } _ { r } \\right\\} _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 218, + 256, + 231 + ], + "score": 1.0, + "content": ", keeping them at fixed values", + "type": "text" + }, + { + "bbox": [ + 257, + 218, + 289, + 230 + ], + "score": 0.93, + "content": "\\mathbf { w } _ { r } ^ { * } \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 218, + 505, + 231 + ], + "score": 1.0, + "content": ". This can be viewed as pruning the variances of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 229, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 140, + 241 + ], + "score": 1.0, + "content": "weights", + "type": "text" + }, + { + "bbox": [ + 140, + 230, + 167, + 241 + ], + "score": 0.93, + "content": "\\left\\{ \\mathbf { w } _ { r } \\right\\} _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 229, + 505, + 241 + ], + "score": 1.0, + "content": "to zero, which is in contrast to ordinary weight pruning methods that set the weights", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 107, + 240, + 133, + 252 + ], + "score": 0.92, + "content": "\\left\\{ \\mathbf { w } _ { r } \\right\\} _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "themselves to zero. The second approximation is a result of posterior inference over the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 441, + 263 + ], + "score": 1.0, + "content": "subnetwork still being intractable. In turn, we introduce the approximate distribution", + "type": "text" + }, + { + "bbox": [ + 441, + 251, + 471, + 263 + ], + "score": 0.92, + "content": "q ( \\mathbf { W } _ { S } )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 251, + 505, + 263 + ], + "score": 1.0, + "content": ". Yet, as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 412, + 275 + ], + "score": 1.0, + "content": "the subnetwork is much smaller than the full network, we can afford to make", + "type": "text" + }, + { + "bbox": [ + 412, + 262, + 443, + 274 + ], + "score": 0.91, + "content": "q ( \\mathbf { W } _ { S } )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "expressive and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 273, + 411, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 411, + 285 + ], + "score": 1.0, + "content": "able to capture rich dependencies across the weights within the subnetwork.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 300, + 424, + 313 + ], + "lines": [ + { + "bbox": [ + 104, + 299, + 426, + 315 + ], + "spans": [ + { + "bbox": [ + 104, + 299, + 426, + 315 + ], + "score": 1.0, + "content": "3 SUBNETWORK INFERENCE VIA LAPLACE APPROXIMATION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 325, + 505, + 382 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 339 + ], + "score": 1.0, + "content": "To obtain a method that is as practical as possible, we propose to use inference techniques that can", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "estimate a posterior distribution post-hoc from a point-estimated network. The Laplace approximation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "(MacKay, 1992) is well-suited to this task as it derives the approximate posterior from the local", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "optimization landscape. Other inference procedures, such as SWAG (Maddox et al., 2019), could also", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 369, + 500, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 500, + 383 + ], + "score": 1.0, + "content": "be used. Nevertheless, we focus on Laplace due to it being a well-studied, fundamental technique.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 505, + 427 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 506, + 406 + ], + "score": 1.0, + "content": "Step #1: Point Estimation. The first step of the procedure is to train a neural network to obtain a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 265, + 417 + ], + "score": 1.0, + "content": "point estimate of the weights, denoted", + "type": "text" + }, + { + "bbox": [ + 265, + 405, + 298, + 416 + ], + "score": 0.92, + "content": "W _ { M A P }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 405, + 505, + 417 + ], + "score": 1.0, + "content": ". This estimate should respect the Bayesian model", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 459, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 336, + 428 + ], + "score": 1.0, + "content": "given in Eq. (1), and therefore we optimize the maximum", + "type": "text" + }, + { + "bbox": [ + 336, + 418, + 342, + 426 + ], + "score": 0.37, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 416, + 459, + 428 + ], + "score": 1.0, + "content": "-posteriori (MAP) objective:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 433, + 416, + 447 + ], + "lines": [ + { + "bbox": [ + 195, + 433, + 416, + 447 + ], + "spans": [ + { + "bbox": [ + 195, + 433, + 416, + 447 + ], + "score": 0.89, + "content": "\\begin{array} { r } { { \\cal W } _ { M A P } = \\arg \\operatorname* { m a x } _ { W } \\left[ \\log p ( { \\bf y } | X , W ) + \\log p ( W ) \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "e21589daff947dc49100ee8f2c8e4cfb7672c9f03346aca10d7080dc439468d1.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 195, + 433, + 416, + 447 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "This can be done using standard stochastic gradient-based optimization methods commonly-used in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 463, + 444, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 444, + 476 + ], + "score": 1.0, + "content": "modern deep learning (Goodfellow et al., 2016). This step is illustrated in Fig. 1 (a).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 451, + 499 + ], + "score": 1.0, + "content": "Step #2: Subnetwork Selection. The second step is to identify a small subnetwork", + "type": "text" + }, + { + "bbox": [ + 452, + 487, + 469, + 498 + ], + "score": 0.89, + "content": "W _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 486, + 506, + 499 + ], + "score": 1.0, + "content": ". Ideally,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "we would like to identify the subnetwork whose posterior is ‘closest’ to the full-network posterior. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "formalize this argument in Section 4 and describe a principled strategy that, under certain conditions,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "minimizes the 2-Wasserstein distance between the sub- and full-network posteriors. All other weights", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 529, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 544 + ], + "score": 1.0, + "content": "not belonging to that subnetwork are then assigned fixed values: the MAP estimates obtained in Step", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 541, + 299, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 299, + 555 + ], + "score": 1.0, + "content": "#1. See Fig. 1 (b) for an illustration of this step.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 102, + 559, + 509, + 584 + ], + "spans": [ + { + "bbox": [ + 102, + 559, + 388, + 584 + ], + "score": 1.0, + "content": "Step #3: Bayesian Inference. 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Inference over submodels: Previous work1 has provided evidence that inference can be", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 141, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "effective even when not done on the full parameter space. Izmailov et al. (2019) performed", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 141, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 141, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "inference over a low-dimensional projection of the weights. Neural-linear models, which", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "give a Bayesian treatment to only the last layer of a DNN, have shown to be competitive", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 127, + 466, + 138 + ], + "spans": [ + { + "bbox": [ + 141, + 127, + 466, + 138 + ], + "score": 1.0, + "content": "with full-network approaches (Riquelme et al., 2018; Ober & Rasmussen, 2019).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 128, + 82, + 506, + 138 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 146, + 504, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 145, + 507, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 507, + 162 + ], + "score": 1.0, + "content": "We thus combine these ideas, making the following two-step approximation of the posterior in Eq. 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The second approximation is a result of posterior inference over the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 441, + 263 + ], + "score": 1.0, + "content": "subnetwork still being intractable. In turn, we introduce the approximate distribution", + "type": "text" + }, + { + "bbox": [ + 441, + 251, + 471, + 263 + ], + "score": 0.92, + "content": "q ( \\mathbf { W } _ { S } )", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 251, + 505, + 263 + ], + "score": 1.0, + "content": ". Yet, as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 412, + 275 + ], + "score": 1.0, + "content": "the subnetwork is much smaller than the full network, we can afford to make", + "type": "text" + }, + { + "bbox": [ + 412, + 262, + 443, + 274 + ], + "score": 0.91, + "content": "q ( \\mathbf { W } _ { S } )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "expressive and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 273, + 411, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 411, + 285 + ], + "score": 1.0, + "content": "able to capture rich dependencies across the weights within the subnetwork.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 195, + 506, + 285 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 300, + 424, + 313 + ], + "lines": [ + { + "bbox": [ + 104, + 299, + 426, + 315 + ], + "spans": [ + { + "bbox": [ + 104, + 299, + 426, + 315 + ], + "score": 1.0, + "content": "3 SUBNETWORK INFERENCE VIA LAPLACE APPROXIMATION", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 325, + 505, + 382 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 339 + ], + "score": 1.0, + "content": "To obtain a method that is as practical as possible, we propose to use inference techniques that can", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 336, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 350 + ], + "score": 1.0, + "content": "estimate a posterior distribution post-hoc from a point-estimated network. The Laplace approximation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "(MacKay, 1992) is well-suited to this task as it derives the approximate posterior from the local", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "optimization landscape. Other inference procedures, such as SWAG (Maddox et al., 2019), could also", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 369, + 500, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 500, + 383 + ], + "score": 1.0, + "content": "be used. Nevertheless, we focus on Laplace due to it being a well-studied, fundamental technique.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 325, + 506, + 383 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 393, + 505, + 427 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 506, + 406 + ], + "score": 1.0, + "content": "Step #1: Point Estimation. The first step of the procedure is to train a neural network to obtain a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 265, + 417 + ], + "score": 1.0, + "content": "point estimate of the weights, denoted", + "type": "text" + }, + { + "bbox": [ + 265, + 405, + 298, + 416 + ], + "score": 0.92, + "content": "W _ { M A P }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 405, + 505, + 417 + ], + "score": 1.0, + "content": ". This estimate should respect the Bayesian model", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 416, + 459, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 336, + 428 + ], + "score": 1.0, + "content": "given in Eq. (1), and therefore we optimize the maximum", + "type": "text" + }, + { + "bbox": [ + 336, + 418, + 342, + 426 + ], + "score": 0.37, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 416, + 459, + 428 + ], + "score": 1.0, + "content": "-posteriori (MAP) objective:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 394, + 506, + 428 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 433, + 416, + 447 + ], + "lines": [ + { + "bbox": [ + 195, + 433, + 416, + 447 + ], + "spans": [ + { + "bbox": [ + 195, + 433, + 416, + 447 + ], + "score": 0.89, + "content": "\\begin{array} { r } { { \\cal W } _ { M A P } = \\arg \\operatorname* { m a x } _ { W } \\left[ \\log p ( { \\bf y } | X , W ) + \\log p ( W ) \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "e21589daff947dc49100ee8f2c8e4cfb7672c9f03346aca10d7080dc439468d1.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 195, + 433, + 416, + 447 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "This can be done using standard stochastic gradient-based optimization methods commonly-used in", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 463, + 444, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 444, + 476 + ], + "score": 1.0, + "content": "modern deep learning (Goodfellow et al., 2016). This step is illustrated in Fig. 1 (a).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 106, + 452, + 505, + 476 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 451, + 499 + ], + "score": 1.0, + "content": "Step #2: Subnetwork Selection. The second step is to identify a small subnetwork", + "type": "text" + }, + { + "bbox": [ + 452, + 487, + 469, + 498 + ], + "score": 0.89, + "content": "W _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 486, + 506, + 499 + ], + "score": 1.0, + "content": ". Ideally,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "we would like to identify the subnetwork whose posterior is ‘closest’ to the full-network posterior. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "formalize this argument in Section 4 and describe a principled strategy that, under certain conditions,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "minimizes the 2-Wasserstein distance between the sub- and full-network posteriors. All other weights", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 529, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 544 + ], + "score": 1.0, + "content": "not belonging to that subnetwork are then assigned fixed values: the MAP estimates obtained in Step", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 541, + 299, + 555 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 299, + 555 + ], + "score": 1.0, + "content": "#1. See Fig. 1 (b) for an illustration of this step.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 486, + 506, + 555 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 102, + 559, + 509, + 584 + ], + "spans": [ + { + "bbox": [ + 102, + 559, + 388, + 584 + ], + "score": 1.0, + "content": "Step #3: Bayesian Inference. 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Using the Laplace approximation with the GGN Hessian can be viewed as an", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 127, + 507, + 143 + ], + "spans": [ + { + "bbox": [ + 104, + 127, + 348, + 143 + ], + "score": 1.0, + "content": "implicit local linearization of the underlying neural network", + "type": "text" + }, + { + "bbox": [ + 348, + 129, + 387, + 141 + ], + "score": 0.93, + "content": "\\mathbf { \\Delta } f ( \\mathbf { \\Delta } x , W )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 127, + 469, + 143 + ], + "score": 1.0, + "content": "at its MAP estimate", + "type": "text" + }, + { + "bbox": [ + 470, + 129, + 502, + 140 + ], + "score": 0.91, + "content": "W _ { M A P }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 127, + 507, + 143 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 145, + 434, + 159 + ], + "lines": [ + { + "bbox": [ + 177, + 145, + 434, + 159 + ], + "spans": [ + { + "bbox": [ + 177, + 145, + 434, + 159 + ], + "score": 0.86, + "content": "f _ { l i n } ^ { M A P } ( { \\pmb x } , { \\pmb W } ) = { \\pmb f } ( { \\pmb x } , { \\pmb W } _ { M A P } ) + J _ { { \\pmb W } _ { M A P } } ( { \\pmb x } ) ( { \\pmb W } - { \\pmb W } _ { M A P } )", + "type": "interline_equation", + "image_path": "fb8baf2073b26a7293798c22fa151b919456cae195a501b5e27e27f3117f5c62.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 177, + 145, + 434, + 159 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 506, + 299 + ], + "lines": [ + { + "bbox": [ + 104, + 160, + 507, + 176 + ], + "spans": [ + { + "bbox": [ + 104, + 160, + 133, + 176 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 162, + 340, + 175 + ], + "score": 0.84, + "content": "J _ { W _ { M A P } } ( \\pmb { x } ) = \\partial \\pmb { f } ( \\pmb { x } , \\pmb { W } _ { M A P } ) / \\partial \\pmb { W } _ { M A P } \\in \\mathbb { R } ^ { O \\times D }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 160, + 507, + 176 + ], + "score": 1.0, + "content": "(Immer et al., 2020). Note that the model", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 173, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 173, + 194, + 189 + ], + "score": 1.0, + "content": "in Eq. 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Put differently, linearized Laplace in the neural network", + "type": "text" + }, + { + "bbox": [ + 413, + 219, + 451, + 230 + ], + "score": 0.89, + "content": "\\mathbf { \\Delta } f ( \\mathbf { \\Delta } x , W )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 218, + 506, + 230 + ], + "score": 1.0, + "content": "is equivalent", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 101, + 225, + 509, + 251 + ], + "spans": [ + { + "bbox": [ + 101, + 225, + 267, + 251 + ], + "score": 1.0, + "content": "to ordinary Laplace in the linear model", + "type": "text" + }, + { + "bbox": [ + 268, + 230, + 327, + 244 + ], + "score": 0.93, + "content": "\\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { x } , \\pmb { W } )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 225, + 422, + 251 + ], + "score": 1.0, + "content": "in Eq. (7), as the GGN", + "type": "text" + }, + { + "bbox": [ + 423, + 230, + 433, + 241 + ], + "score": 0.82, + "content": "\\widetilde { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 225, + 509, + 251 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 146, + 237, + 509, + 267 + ], + "spans": [ + { + "bbox": [ + 146, + 237, + 297, + 267 + ], + "score": 1.0, + "content": "in Eq. (6) is equivalent to the Hessian 9). This is a useful property that will", + "type": "text" + }, + { + "bbox": [ + 297, + 244, + 307, + 253 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 237, + 377, + 267 + ], + "score": 1.0, + "content": "corresponding to ow us to derive a", + "type": "text" + }, + { + "bbox": [ + 377, + 243, + 437, + 256 + ], + "score": 0.93, + "content": "\\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { x } , \\pmb { W } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 237, + 509, + 267 + ], + "score": 1.0, + "content": "in Eq. (7) (Khanetwork selection", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "strategy in Section 4. This step is illustrated in Fig. 1 (c). We emphasize that this whole procedure", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "(i.e. Steps #1-#3) is a perfectly valid mixed inference strategy, performing full Laplace inference over", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 287, + 396, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 396, + 299 + ], + "score": 1.0, + "content": "the selected subnetwork and MAP inference over all remaining weights.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 309, + 506, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 474, + 324 + ], + "score": 1.0, + "content": "Step #4: Prediction. Given the linearized Laplace approximation over the subnetwork", + "type": "text" + }, + { + "bbox": [ + 474, + 311, + 493, + 322 + ], + "score": 0.88, + "content": "\\mathbf { W } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 102, + 318, + 509, + 341 + ], + "spans": [ + { + "bbox": [ + 102, + 318, + 186, + 341 + ], + "score": 1.0, + "content": "Eqs. (5) and (6), i.e.", + "type": "text" + }, + { + "bbox": [ + 186, + 321, + 326, + 336 + ], + "score": 0.91, + "content": "q ( \\mathbf { W } _ { S } ) = \\mathcal { N } ( \\mathbf { W } _ { S } ; { W _ { M A P } ^ { S } } , \\mathbf { \\widetilde { H } } ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 318, + 509, + 341 + ], + "score": 1.0, + "content": ", we can then compute the posterior predictive", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "distribution. While, traditionally, one would compute the predictive distribution using the original", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 266, + 358 + ], + "score": 1.0, + "content": "Bayesian neural network likelihood, i.e.", + "type": "text" + }, + { + "bbox": [ + 266, + 345, + 388, + 357 + ], + "score": 0.93, + "content": "p ( \\mathbf { y } | X , \\bar { \\mathbf { W } } ) = p ( \\bar { \\mathbf { y } } | f ( \\mathbf { x } , W ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 344, + 505, + 358 + ], + "score": 1.0, + "content": ", Immer et al. (2020) recently", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 102, + 356, + 507, + 387 + ], + "spans": [ + { + "bbox": [ + 102, + 356, + 360, + 387 + ], + "score": 1.0, + "content": "suggested that, since inference was (implicitly) done in the GGpled to instead predict using the linearized likelihood Eq. (7), i.e.", + "type": "text" + }, + { + "bbox": [ + 360, + 367, + 502, + 380 + ], + "score": 0.92, + "content": "p ( \\mathbf { y } \\vert X , \\mathbf { W } ) = p ( \\mathbf { y } \\vert \\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { \\dot { x } } , \\pmb { W } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 356, + 507, + 387 + ], + "score": 1.0, + "content": "-.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "This provides a formal justification for the empirical superiority of this approach observed previously", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 475, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 475, + 402 + ], + "score": 1.0, + "content": "(Lawrence, 2001; Foong et al., 2019b). We thus obtain the linearized predictive distribution", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 108, + 404, + 489, + 434 + ], + "lines": [ + { + "bbox": [ + 108, + 404, + 489, + 434 + ], + "spans": [ + { + "bbox": [ + 108, + 404, + 489, + 434 + ], + "score": 0.92, + "content": "p ( \\mathbf { y } ^ { * } | \\boldsymbol { X } ^ { * } , \\boldsymbol { y } , \\boldsymbol { X } ) \\approx \\int _ { \\mathbf { W } } p ( \\mathbf { y } ^ { * } | f _ { l i n } ^ { M A P } ( \\boldsymbol { X } ^ { * } , \\boldsymbol { W } ) ) \\mathcal { N } ( \\mathbf { W } _ { S } ; \\boldsymbol { W } _ { M A P } ^ { S } , \\tilde { H } ^ { - 1 } ) \\prod _ { r } \\delta ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } ) d \\mathbf { W } .", + "type": "interline_equation", + "image_path": "03d743074fe2db716bb9b3a71d773f0e17ca75ff3ab8383c542e3e61d56ab0e0.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 108, + 404, + 489, + 414.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 108, + 414.0, + 489, + 424.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 108, + 424.0, + 489, + 434.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 437, + 506, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 426, + 451 + ], + "score": 1.0, + "content": "There are two ways to compute Eq. (8): Firstly, via a Monte Carlo approximation", + "type": "text" + }, + { + "bbox": [ + 426, + 438, + 506, + 450 + ], + "score": 0.91, + "content": "p ( \\mathbf { y } ^ { * } | X ^ { * } , y , X ) \\simeq", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 446, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 107, + 449, + 249, + 465 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } p ( \\mathbf { y } ^ { * } | \\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { X } ^ { * } , \\widetilde { W } _ { m } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 446, + 302, + 469 + ], + "score": 1.0, + "content": "by sampling", + "type": "text" + }, + { + "bbox": [ + 302, + 449, + 322, + 463 + ], + "score": 0.91, + "content": "\\widetilde { W } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 446, + 345, + 469 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 345, + 449, + 419, + 464 + ], + "score": 0.92, + "content": "\\mathcal { N } ( W _ { M A P } ^ { S } , \\widetilde { H } ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 446, + 438, + 469 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 451, + 503, + 464 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\prod _ { r } \\delta ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 446, + 506, + 469 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 101, + 457, + 511, + 488 + ], + "spans": [ + { + "bbox": [ + 101, + 457, + 511, + 483 + ], + "score": 1.0, + "content": "M m=1 | lin f f N MAP e r − the latter of which is trivial. Secondly, due to linearity of p(y∗|f MAPlin (X∗, W )), there are closed-", + "type": "text" + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "form expressions which are exact for Gaussian likelihoods (i.e. regression) and approximate for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "categorical ones (i.e. classification) (Bishop, 2006; Gibbs, 1998). This step is illustrated in Fig. 1 (d).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 105, + 513, + 475, + 527 + ], + "lines": [ + { + "bbox": [ + 104, + 511, + 477, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 477, + 529 + ], + "score": 1.0, + "content": "4 PRINCIPLED SUBNETWORK SELECTION FOR LINEAR(IZED) MODELS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 506, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "We next analyze the subnetwork inference procedure described in Section 3 for the case of a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 547, + 504, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 492, + 564 + ], + "score": 1.0, + "content": "generalized linear model (GLM) (Nelder & Baker, 1972), which models the expected response", + "type": "text" + }, + { + "bbox": [ + 492, + 552, + 504, + 561 + ], + "score": 0.82, + "content": "y _ { n }", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 381, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 314, + 574 + ], + "score": 1.0, + "content": "given the basis function expansion of the covariates", + "type": "text" + }, + { + "bbox": [ + 315, + 560, + 368, + 572 + ], + "score": 0.93, + "content": "\\phi _ { n } = \\phi ( { \\mathbf { { x } } } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 560, + 381, + 574 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 576, + 355, + 591 + ], + "lines": [ + { + "bbox": [ + 248, + 576, + 355, + 591 + ], + "spans": [ + { + "bbox": [ + 248, + 576, + 355, + 591 + ], + "score": 0.91, + "content": "\\mathbb { E } [ y _ { n } | \\phi _ { n } ] = g ^ { - 1 } ( \\mathbf { w } ^ { T } \\phi _ { n } ) .", + "type": "interline_equation", + "image_path": "62b1f43811f26b9e440302e041b0cf21d7014c77eb3f3538b7e3b14e7afd2065.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 248, + 576, + 355, + 591 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 131, + 608 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 594, + 168, + 605 + ], + "score": 0.92, + "content": "\\mathbf { w } \\in \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 593, + 434, + 608 + ], + "score": 1.0, + "content": "is the vector of model parameters (which subsumes a scalar bias", + "type": "text" + }, + { + "bbox": [ + 435, + 595, + 446, + 606 + ], + "score": 0.88, + "content": "\\beta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 593, + 505, + 608 + ], + "score": 1.0, + "content": "for notational", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 177, + 620 + ], + "score": 1.0, + "content": "convenience) and", + "type": "text" + }, + { + "bbox": [ + 177, + 605, + 204, + 618 + ], + "score": 0.93, + "content": "g ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 604, + 333, + 620 + ], + "score": 1.0, + "content": "denotes a link function such that", + "type": "text" + }, + { + "bbox": [ + 333, + 605, + 399, + 619 + ], + "score": 0.92, + "content": "g ^ { - 1 } : \\mathbb { R } \\mapsto \\mu _ { y | \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 604, + 506, + 620 + ], + "score": 1.0, + "content": ". In particular, we consider", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 311, + 630 + ], + "score": 1.0, + "content": "a Bayesian GLM, by specifying a prior distribution", + "type": "text" + }, + { + "bbox": [ + 311, + 618, + 333, + 629 + ], + "score": 0.9, + "content": "p ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "over model parameters and aiming to infer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 628, + 440, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 207, + 641 + ], + "score": 1.0, + "content": "the posterior distribution", + "type": "text" + }, + { + "bbox": [ + 208, + 629, + 328, + 641 + ], + "score": 0.92, + "content": "p ( \\mathbf { w } | \\mathbf { y } , \\pmb { \\Phi } ) \\propto p ( \\mathbf { y } | \\pmb { \\Phi } , \\mathbf { w } ) p ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 628, + 359, + 641 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 360, + 628, + 436, + 641 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\pmb { \\Phi } = [ \\phi _ { 1 } , \\dots \\phi _ { N } ] ^ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 628, + 440, + 641 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 109, + 649, + 506, + 716 + ], + "lines": [ + { + "bbox": [ + 107, + 648, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 107, + 648, + 309, + 662 + ], + "score": 1.0, + "content": "1. Point Estimation. Obtain the MAP estimate,", + "type": "text" + }, + { + "bbox": [ + 309, + 649, + 503, + 661 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbf { w } _ { M A P } = \\arg \\operatorname* { m a x } _ { \\mathbf { w } } \\log p ( \\pmb { y } | \\Phi , \\mathbf { w } ) + \\log p ( \\mathbf { w } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 648, + 507, + 662 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 119, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 119, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "For commonly-used link functions (e.g. the identity in case of a Gaussian likelihood for re-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 119, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 119, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "gression, or the sigmoid/softmax function in case of a categorical likelihood for classification)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 120, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 120, + 682, + 377, + 695 + ], + "score": 1.0, + "content": "and commonly-used priors (e.g. a Gaussian), the log-posterior", + "type": "text" + }, + { + "bbox": [ + 378, + 682, + 494, + 694 + ], + "score": 0.92, + "content": "\\propto \\log p ( \\pmb { y } | \\pmb { \\Phi } , \\mathbf { w } ) + \\log p ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 120, + 693, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 120, + 693, + 505, + 705 + ], + "score": 1.0, + "content": "concave. This allows for simple gradient-based MAP optimisation. 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Using the Laplace approximation with the GGN Hessian can be viewed as an", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 127, + 507, + 143 + ], + "spans": [ + { + "bbox": [ + 104, + 127, + 348, + 143 + ], + "score": 1.0, + "content": "implicit local linearization of the underlying neural network", + "type": "text" + }, + { + "bbox": [ + 348, + 129, + 387, + 141 + ], + "score": 0.93, + "content": "\\mathbf { \\Delta } f ( \\mathbf { \\Delta } x , W )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 127, + 469, + 143 + ], + "score": 1.0, + "content": "at its MAP estimate", + "type": "text" + }, + { + "bbox": [ + 470, + 129, + 502, + 140 + ], + "score": 0.91, + "content": "W _ { M A P }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 127, + 507, + 143 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 104, + 80, + 507, + 143 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 145, + 434, + 159 + ], + "lines": [ + { + "bbox": [ + 177, + 145, + 434, + 159 + ], + "spans": [ + { + "bbox": [ + 177, + 145, + 434, + 159 + ], + "score": 0.86, + "content": "f _ { l i n } ^ { M A P } ( { \\pmb x } , { \\pmb W } ) = { \\pmb f } ( { \\pmb x } , { \\pmb W } _ { M A P } ) + J _ { { \\pmb W } _ { M A P } } ( { \\pmb x } ) ( { \\pmb W } - { \\pmb W } _ { M A P } )", + "type": "interline_equation", + "image_path": "fb8baf2073b26a7293798c22fa151b919456cae195a501b5e27e27f3117f5c62.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 177, + 145, + 434, + 159 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 506, + 299 + ], + "lines": [ + { + "bbox": [ + 104, + 160, + 507, + 176 + ], + "spans": [ + { + "bbox": [ + 104, + 160, + 133, + 176 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 162, + 340, + 175 + ], + "score": 0.84, + "content": "J _ { W _ { M A P } } ( \\pmb { x } ) = \\partial \\pmb { f } ( \\pmb { x } , \\pmb { W } _ { M A P } ) / \\partial \\pmb { W } _ { M A P } \\in \\mathbb { R } ^ { O \\times D }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 160, + 507, + 176 + ], + "score": 1.0, + "content": "(Immer et al., 2020). Note that the model", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 173, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 104, + 173, + 194, + 189 + ], + "score": 1.0, + "content": "in Eq. (7) is linear in", + "type": "text" + }, + { + "bbox": [ + 195, + 175, + 208, + 185 + ], + "score": 0.77, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 173, + 280, + 189 + ], + "score": 1.0, + "content": ", as only the term", + "type": "text" + }, + { + "bbox": [ + 280, + 174, + 341, + 186 + ], + "score": 0.91, + "content": "J _ { W _ { M A P } } ( { \\pmb x } ) { \\pmb W }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 173, + 424, + 189 + ], + "score": 1.0, + "content": "depends linearly on", + "type": "text" + }, + { + "bbox": [ + 425, + 174, + 438, + 185 + ], + "score": 0.78, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 173, + 507, + 189 + ], + "score": 1.0, + "content": ", while the other", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 185, + 505, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 185, + 203, + 197 + ], + "score": 1.0, + "content": "terms are constant w.r.t.", + "type": "text" + }, + { + "bbox": [ + 203, + 185, + 217, + 195 + ], + "score": 0.48, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 185, + 505, + 197 + ], + "score": 1.0, + "content": "and can thus be subsumed into an additive bias term (Khan et al., 2019).", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 196, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 506, + 208 + ], + "score": 1.0, + "content": "The GGN approximation thus locally turns the underlying probabilistic model from a Bayesian neural", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 206, + 504, + 221 + ], + "spans": [ + { + "bbox": [ + 104, + 206, + 397, + 221 + ], + "score": 1.0, + "content": "network into a (generalized) linear model, with basis function expansion", + "type": "text" + }, + { + "bbox": [ + 397, + 207, + 445, + 219 + ], + "score": 0.91, + "content": "J _ { W _ { M A P } } ( { \\pmb x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 206, + 496, + 221 + ], + "score": 1.0, + "content": "of covariate", + "type": "text" + }, + { + "bbox": [ + 496, + 209, + 504, + 217 + ], + "score": 0.68, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 412, + 230 + ], + "score": 1.0, + "content": "(Immer et al., 2020). Put differently, linearized Laplace in the neural network", + "type": "text" + }, + { + "bbox": [ + 413, + 219, + 451, + 230 + ], + "score": 0.89, + "content": "\\mathbf { \\Delta } f ( \\mathbf { \\Delta } x , W )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 218, + 506, + 230 + ], + "score": 1.0, + "content": "is equivalent", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 101, + 225, + 509, + 251 + ], + "spans": [ + { + "bbox": [ + 101, + 225, + 267, + 251 + ], + "score": 1.0, + "content": "to ordinary Laplace in the linear model", + "type": "text" + }, + { + "bbox": [ + 268, + 230, + 327, + 244 + ], + "score": 0.93, + "content": "\\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { x } , \\pmb { W } )", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 225, + 422, + 251 + ], + "score": 1.0, + "content": "in Eq. (7), as the GGN", + "type": "text" + }, + { + "bbox": [ + 423, + 230, + 433, + 241 + ], + "score": 0.82, + "content": "\\widetilde { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 225, + 509, + 251 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 146, + 237, + 509, + 267 + ], + "spans": [ + { + "bbox": [ + 146, + 237, + 297, + 267 + ], + "score": 1.0, + "content": "in Eq. (6) is equivalent to the Hessian 9). This is a useful property that will", + "type": "text" + }, + { + "bbox": [ + 297, + 244, + 307, + 253 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 237, + 377, + 267 + ], + "score": 1.0, + "content": "corresponding to ow us to derive a", + "type": "text" + }, + { + "bbox": [ + 377, + 243, + 437, + 256 + ], + "score": 0.93, + "content": "\\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { x } , \\pmb { W } )", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 237, + 509, + 267 + ], + "score": 1.0, + "content": "in Eq. (7) (Khanetwork selection", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "strategy in Section 4. This step is illustrated in Fig. 1 (c). We emphasize that this whole procedure", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 506, + 289 + ], + "score": 1.0, + "content": "(i.e. Steps #1-#3) is a perfectly valid mixed inference strategy, performing full Laplace inference over", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 287, + 396, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 396, + 299 + ], + "score": 1.0, + "content": "the selected subnetwork and MAP inference over all remaining weights.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11, + "bbox_fs": [ + 101, + 160, + 509, + 299 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 309, + 506, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 474, + 324 + ], + "score": 1.0, + "content": "Step #4: Prediction. Given the linearized Laplace approximation over the subnetwork", + "type": "text" + }, + { + "bbox": [ + 474, + 311, + 493, + 322 + ], + "score": 0.88, + "content": "\\mathbf { W } _ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 102, + 318, + 509, + 341 + ], + "spans": [ + { + "bbox": [ + 102, + 318, + 186, + 341 + ], + "score": 1.0, + "content": "Eqs. (5) and (6), i.e.", + "type": "text" + }, + { + "bbox": [ + 186, + 321, + 326, + 336 + ], + "score": 0.91, + "content": "q ( \\mathbf { W } _ { S } ) = \\mathcal { N } ( \\mathbf { W } _ { S } ; { W _ { M A P } ^ { S } } , \\mathbf { \\widetilde { H } } ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 318, + 509, + 341 + ], + "score": 1.0, + "content": ", we can then compute the posterior predictive", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "distribution. While, traditionally, one would compute the predictive distribution using the original", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 266, + 358 + ], + "score": 1.0, + "content": "Bayesian neural network likelihood, i.e.", + "type": "text" + }, + { + "bbox": [ + 266, + 345, + 388, + 357 + ], + "score": 0.93, + "content": "p ( \\mathbf { y } | X , \\bar { \\mathbf { W } } ) = p ( \\bar { \\mathbf { y } } | f ( \\mathbf { x } , W ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 344, + 505, + 358 + ], + "score": 1.0, + "content": ", Immer et al. (2020) recently", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 102, + 356, + 507, + 387 + ], + "spans": [ + { + "bbox": [ + 102, + 356, + 360, + 387 + ], + "score": 1.0, + "content": "suggested that, since inference was (implicitly) done in the GGpled to instead predict using the linearized likelihood Eq. (7), i.e.", + "type": "text" + }, + { + "bbox": [ + 360, + 367, + 502, + 380 + ], + "score": 0.92, + "content": "p ( \\mathbf { y } \\vert X , \\mathbf { W } ) = p ( \\mathbf { y } \\vert \\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { \\dot { x } } , \\pmb { W } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 356, + 507, + 387 + ], + "score": 1.0, + "content": "-.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "This provides a formal justification for the empirical superiority of this approach observed previously", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 475, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 475, + 402 + ], + "score": 1.0, + "content": "(Lawrence, 2001; Foong et al., 2019b). We thus obtain the linearized predictive distribution", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 102, + 310, + 509, + 402 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 108, + 404, + 489, + 434 + ], + "lines": [ + { + "bbox": [ + 108, + 404, + 489, + 434 + ], + "spans": [ + { + "bbox": [ + 108, + 404, + 489, + 434 + ], + "score": 0.92, + "content": "p ( \\mathbf { y } ^ { * } | \\boldsymbol { X } ^ { * } , \\boldsymbol { y } , \\boldsymbol { X } ) \\approx \\int _ { \\mathbf { W } } p ( \\mathbf { y } ^ { * } | f _ { l i n } ^ { M A P } ( \\boldsymbol { X } ^ { * } , \\boldsymbol { W } ) ) \\mathcal { N } ( \\mathbf { W } _ { S } ; \\boldsymbol { W } _ { M A P } ^ { S } , \\tilde { H } ^ { - 1 } ) \\prod _ { r } \\delta ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } ) d \\mathbf { W } .", + "type": "interline_equation", + "image_path": "03d743074fe2db716bb9b3a71d773f0e17ca75ff3ab8383c542e3e61d56ab0e0.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 108, + 404, + 489, + 414.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 108, + 414.0, + 489, + 424.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 108, + 424.0, + 489, + 434.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 437, + 506, + 498 + ], + "lines": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 426, + 451 + ], + "score": 1.0, + "content": "There are two ways to compute Eq. (8): Firstly, via a Monte Carlo approximation", + "type": "text" + }, + { + "bbox": [ + 426, + 438, + 506, + 450 + ], + "score": 0.91, + "content": "p ( \\mathbf { y } ^ { * } | X ^ { * } , y , X ) \\simeq", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 446, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 107, + 449, + 249, + 465 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } p ( \\mathbf { y } ^ { * } | \\pmb { f } _ { l i n } ^ { M A P } ( \\pmb { X } ^ { * } , \\widetilde { W } _ { m } ) ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 446, + 302, + 469 + ], + "score": 1.0, + "content": "by sampling", + "type": "text" + }, + { + "bbox": [ + 302, + 449, + 322, + 463 + ], + "score": 0.91, + "content": "\\widetilde { W } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 446, + 345, + 469 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 345, + 449, + 419, + 464 + ], + "score": 0.92, + "content": "\\mathcal { N } ( W _ { M A P } ^ { S } , \\widetilde { H } ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 446, + 438, + 469 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 451, + 503, + 464 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\prod _ { r } \\delta ( \\mathbf { w } _ { r } - \\mathbf { w } _ { r } ^ { * } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 446, + 506, + 469 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 101, + 457, + 511, + 488 + ], + "spans": [ + { + "bbox": [ + 101, + 457, + 511, + 483 + ], + "score": 1.0, + "content": "M m=1 | lin f f N MAP e r − the latter of which is trivial. Secondly, due to linearity of p(y∗|f MAPlin (X∗, W )), there are closed-", + "type": "text" + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "form expressions which are exact for Gaussian likelihoods (i.e. regression) and approximate for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "categorical ones (i.e. classification) (Bishop, 2006; Gibbs, 1998). This step is illustrated in Fig. 1 (d).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 101, + 438, + 511, + 499 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 513, + 475, + 527 + ], + "lines": [ + { + "bbox": [ + 104, + 511, + 477, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 477, + 529 + ], + "score": 1.0, + "content": "4 PRINCIPLED SUBNETWORK SELECTION FOR LINEAR(IZED) MODELS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 506, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "We next analyze the subnetwork inference procedure described in Section 3 for the case of a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 547, + 504, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 492, + 564 + ], + "score": 1.0, + "content": "generalized linear model (GLM) (Nelder & Baker, 1972), which models the expected response", + "type": "text" + }, + { + "bbox": [ + 492, + 552, + 504, + 561 + ], + "score": 0.82, + "content": "y _ { n }", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 381, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 314, + 574 + ], + "score": 1.0, + "content": "given the basis function expansion of the covariates", + "type": "text" + }, + { + "bbox": [ + 315, + 560, + 368, + 572 + ], + "score": 0.93, + "content": "\\phi _ { n } = \\phi ( { \\mathbf { { x } } } _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 560, + 381, + 574 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 538, + 505, + 574 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 576, + 355, + 591 + ], + "lines": [ + { + "bbox": [ + 248, + 576, + 355, + 591 + ], + "spans": [ + { + "bbox": [ + 248, + 576, + 355, + 591 + ], + "score": 0.91, + "content": "\\mathbb { E } [ y _ { n } | \\phi _ { n } ] = g ^ { - 1 } ( \\mathbf { w } ^ { T } \\phi _ { n } ) .", + "type": "interline_equation", + "image_path": "62b1f43811f26b9e440302e041b0cf21d7014c77eb3f3538b7e3b14e7afd2065.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 248, + 576, + 355, + 591 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 131, + 608 + ], + "score": 1.0, + "content": "Here,", + "type": "text" + }, + { + "bbox": [ + 131, + 594, + 168, + 605 + ], + "score": 0.92, + "content": "\\mathbf { w } \\in \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 593, + 434, + 608 + ], + "score": 1.0, + "content": "is the vector of model parameters (which subsumes a scalar bias", + "type": "text" + }, + { + "bbox": [ + 435, + 595, + 446, + 606 + ], + "score": 0.88, + "content": "\\beta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 593, + 505, + 608 + ], + "score": 1.0, + "content": "for notational", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 177, + 620 + ], + "score": 1.0, + "content": "convenience) and", + "type": "text" + }, + { + "bbox": [ + 177, + 605, + 204, + 618 + ], + "score": 0.93, + "content": "g ^ { - 1 } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 604, + 333, + 620 + ], + "score": 1.0, + "content": "denotes a link function such that", + "type": "text" + }, + { + "bbox": [ + 333, + 605, + 399, + 619 + ], + "score": 0.92, + "content": "g ^ { - 1 } : \\mathbb { R } \\mapsto \\mu _ { y | \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 604, + 506, + 620 + ], + "score": 1.0, + "content": ". In particular, we consider", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 618, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 311, + 630 + ], + "score": 1.0, + "content": "a Bayesian GLM, by specifying a prior distribution", + "type": "text" + }, + { + "bbox": [ + 311, + 618, + 333, + 629 + ], + "score": 0.9, + "content": "p ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 618, + 505, + 630 + ], + "score": 1.0, + "content": "over model parameters and aiming to infer", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 628, + 440, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 207, + 641 + ], + "score": 1.0, + "content": "the posterior distribution", + "type": "text" + }, + { + "bbox": [ + 208, + 629, + 328, + 641 + ], + "score": 0.92, + "content": "p ( \\mathbf { w } | \\mathbf { y } , \\pmb { \\Phi } ) \\propto p ( \\mathbf { y } | \\pmb { \\Phi } , \\mathbf { w } ) p ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 628, + 359, + 641 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 360, + 628, + 436, + 641 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\pmb { \\Phi } = [ \\phi _ { 1 } , \\dots \\phi _ { N } ] ^ { T } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 628, + 440, + 641 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 593, + 506, + 641 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 649, + 506, + 716 + ], + "lines": [ + { + "bbox": [ + 107, + 648, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 107, + 648, + 309, + 662 + ], + "score": 1.0, + "content": "1. Point Estimation. Obtain the MAP estimate,", + "type": "text" + }, + { + "bbox": [ + 309, + 649, + 503, + 661 + ], + "score": 0.87, + "content": "\\begin{array} { r } { \\mathbf { w } _ { M A P } = \\arg \\operatorname* { m a x } _ { \\mathbf { w } } \\log p ( \\pmb { y } | \\Phi , \\mathbf { w } ) + \\log p ( \\mathbf { w } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 648, + 507, + 662 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 119, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 119, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "For commonly-used link functions (e.g. the identity in case of a Gaussian likelihood for re-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 119, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 119, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "gression, or the sigmoid/softmax function in case of a categorical likelihood for classification)", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 120, + 682, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 120, + 682, + 377, + 695 + ], + "score": 1.0, + "content": "and commonly-used priors (e.g. a Gaussian), the log-posterior", + "type": "text" + }, + { + "bbox": [ + 378, + 682, + 494, + 694 + ], + "score": 0.92, + "content": "\\propto \\log p ( \\pmb { y } | \\pmb { \\Phi } , \\mathbf { w } ) + \\log p ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 682, + 505, + 695 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 120, + 693, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 120, + 693, + 505, + 705 + ], + "score": 1.0, + "content": "concave. This allows for simple gradient-based MAP optimisation. It also makes a full-covariance", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 120, + 703, + 497, + 717 + ], + "spans": [ + { + "bbox": [ + 120, + 703, + 497, + 717 + ], + "score": 1.0, + "content": "Gaussian, estimated via Laplace, a faithful approximation to the true, uni-modal posterior, i.e.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5, + "bbox_fs": [ + 107, + 648, + 507, + 717 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 719, + 412, + 733 + ], + "lines": [ + { + "bbox": [ + 213, + 719, + 412, + 733 + ], + "spans": [ + { + "bbox": [ + 213, + 719, + 412, + 733 + ], + "score": 0.89, + "content": "p ( \\mathbf { w } | \\mathbf { y } , \\boldsymbol { \\Phi } ) \\approx \\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\boldsymbol { \\Phi } ) = \\mathcal { N } ( \\mathbf { w } ; \\mathbf { w } _ { M A P } , H ^ { - 1 } )", + "type": "interline_equation", + "image_path": "c7f7d852affd773f73ec3a7d922522179f5dddbf5d751a765886b3f8ccc5d231.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 213, + 719, + 412, + 733 + ], + "spans": [], + "index": 46 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 120, + 82, + 505, + 163 + ], + "lines": [ + { + "bbox": [ + 120, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 120, + 82, + 147, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 147, + 83, + 158, + 92 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 82, + 493, + 95 + ], + "score": 1.0, + "content": "is the Hessian defined in Section 3. Note that for the GLM we consider, the Hessian", + "type": "text" + }, + { + "bbox": [ + 493, + 83, + 504, + 93 + ], + "score": 0.75, + "content": "H", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 119, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 119, + 94, + 219, + 107 + ], + "score": 1.0, + "content": "is equivalent to the GGN", + "type": "text" + }, + { + "bbox": [ + 219, + 93, + 230, + 105 + ], + "score": 0.85, + "content": "\\widetilde { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "defined in Eq. (6), meaning that an ordinary Laplace approximation is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 120, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 120, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "equivalent to a linearized Laplace approximation (Martens, 2016). For the case of an identity link", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 119, + 115, + 507, + 131 + ], + "spans": [ + { + "bbox": [ + 119, + 115, + 335, + 131 + ], + "score": 1.0, + "content": "function (i.e. a Gaussian likelihood with noise variance", + "type": "text" + }, + { + "bbox": [ + 336, + 116, + 347, + 128 + ], + "score": 0.88, + "content": "\\sigma _ { 0 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 115, + 443, + 131 + ], + "score": 1.0, + "content": ") and a Gaussian prior w", + "type": "text" + }, + { + "bbox": [ + 444, + 116, + 502, + 128 + ], + "score": 0.9, + "content": "\\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { \\dot { N } } _ { 0 } ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 115, + 507, + 131 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 118, + 127, + 507, + 142 + ], + "spans": [ + { + "bbox": [ + 118, + 127, + 337, + 142 + ], + "score": 1.0, + "content": "the MAP estimate even has a closed-form expression,", + "type": "text" + }, + { + "bbox": [ + 337, + 128, + 478, + 141 + ], + "score": 0.92, + "content": "\\begin{array} { r } { { \\bf w } _ { M A P } = ( \\Phi ^ { T } \\Phi + \\sigma _ { 0 } ^ { 2 } \\Lambda _ { 0 } ) ^ { - 1 } \\Phi ^ { T } { \\pmb y } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 127, + 507, + 142 + ], + "score": 1.0, + "content": ". Here,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 119, + 138, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 119, + 138, + 449, + 153 + ], + "score": 1.0, + "content": "the Laplace approximation in Eq. (10) exactly corresponds to the true posterior, i.e.", + "type": "text" + }, + { + "bbox": [ + 450, + 140, + 506, + 151 + ], + "score": 0.91, + "content": "\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi ) =", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 120, + 150, + 480, + 164 + ], + "spans": [ + { + "bbox": [ + 120, + 150, + 164, + 163 + ], + "score": 0.92, + "content": "p ( \\mathbf { w } | \\mathbf { y } , \\Phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 150, + 306, + 164 + ], + "score": 1.0, + "content": ". We will thus refer to the posterior", + "type": "text" + }, + { + "bbox": [ + 306, + 150, + 350, + 163 + ], + "score": 0.94, + "content": "\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 150, + 480, + 164 + ], + "score": 1.0, + "content": "e in Eq. (10) as the full posterior.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 110, + 164, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 163, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 296, + 177 + ], + "score": 1.0, + "content": "2. Subnetwork Selection. Select a subset of", + "type": "text" + }, + { + "bbox": [ + 297, + 165, + 304, + 174 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 163, + 506, + 177 + ], + "score": 1.0, + "content": "model weights via a method of choice, yielding", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 119, + 173, + 504, + 188 + ], + "spans": [ + { + "bbox": [ + 119, + 173, + 185, + 188 + ], + "score": 1.0, + "content": "a binary vector", + "type": "text" + }, + { + "bbox": [ + 185, + 175, + 225, + 186 + ], + "score": 0.92, + "content": "\\pmb { m } \\in \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 173, + 254, + 188 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 254, + 176, + 289, + 186 + ], + "score": 0.92, + "content": "m _ { d } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 173, + 315, + 188 + ], + "score": 1.0, + "content": "if the", + "type": "text" + }, + { + "bbox": [ + 316, + 176, + 322, + 185 + ], + "score": 0.82, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 173, + 469, + 188 + ], + "score": 1.0, + "content": "-th weight is part of the subset, and", + "type": "text" + }, + { + "bbox": [ + 469, + 176, + 504, + 186 + ], + "score": 0.88, + "content": "m _ { d } = 0", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 119, + 186, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 119, + 186, + 377, + 198 + ], + "score": 1.0, + "content": "otherwise. For convenience, we define the binary mask matrix", + "type": "text" + }, + { + "bbox": [ + 378, + 186, + 476, + 197 + ], + "score": 0.92, + "content": "\\mathbf { M } _ { S } = \\pmb { m } \\pmb { m } ^ { \\top } \\in \\mathbb { R } ^ { D \\times D }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 186, + 506, + 198 + ], + "score": 1.0, + "content": "which", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 119, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 119, + 197, + 335, + 210 + ], + "score": 1.0, + "content": "contains 1s in the rows/columns corresponding to the", + "type": "text" + }, + { + "bbox": [ + 335, + 198, + 343, + 207 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "subnetwork weights2, and 0s otherwise.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 108, + 211, + 502, + 223 + ], + "lines": [ + { + "bbox": [ + 107, + 210, + 501, + 224 + ], + "spans": [ + { + "bbox": [ + 107, + 210, + 501, + 224 + ], + "score": 1.0, + "content": "3. Bayesian Inference. Compute the posterior over the subnetwork via a Laplace approximation:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 223, + 402, + 237 + ], + "lines": [ + { + "bbox": [ + 221, + 223, + 402, + 237 + ], + "spans": [ + { + "bbox": [ + 221, + 223, + 402, + 237 + ], + "score": 0.84, + "content": "p _ { S } ( \\mathbf { w } | \\pmb { y } , \\pmb { \\Phi } ) = \\mathcal { N } ( \\mathbf { w } ; \\mathbf { w } _ { M A P } , M _ { S } \\odot H ^ { - 1 } ) \\ .", + "type": "interline_equation", + "image_path": "4b4c8b457ab7b86b66c9494f10dcb2f3dc11133832e707f12a7d9c2c703beb4b.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 221, + 223, + 402, + 237 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 120, + 237, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 118, + 235, + 504, + 252 + ], + "spans": [ + { + "bbox": [ + 118, + 235, + 473, + 252 + ], + "score": 1.0, + "content": "Firstly, note that the mean of the subnetwork posterior in Eq. (11) is the MAP estimate", + "type": "text" + }, + { + "bbox": [ + 473, + 240, + 504, + 249 + ], + "score": 0.78, + "content": "\\mathbf { w } _ { M A P }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 119, + 248, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 119, + 248, + 314, + 262 + ], + "score": 1.0, + "content": "and thus equal to the mean of the full posterior", + "type": "text" + }, + { + "bbox": [ + 315, + 249, + 358, + 260 + ], + "score": 0.92, + "content": "\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 248, + 506, + 262 + ], + "score": 1.0, + "content": "in Eq. (10). Secondly, note that the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 118, + 258, + 507, + 272 + ], + "spans": [ + { + "bbox": [ + 118, + 258, + 455, + 272 + ], + "score": 1.0, + "content": "ecovariance matrix of the subnetwork posterior in Eq. (11) is the element-wise product", + "type": "text" + }, + { + "bbox": [ + 455, + 259, + 502, + 271 + ], + "score": 0.92, + "content": "M _ { S } \\odot H ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 258, + 507, + 272 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 120, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 120, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "which masks the (co-)variances of all weights not belonging to the subnetwork to zero, effectively", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 119, + 280, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 119, + 280, + 435, + 294 + ], + "score": 1.0, + "content": "making them deterministic. More precisely, the subnetwork covariance matrix,", + "type": "text" + }, + { + "bbox": [ + 435, + 281, + 485, + 293 + ], + "score": 0.93, + "content": "M _ { S } \\odot H ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 280, + 507, + 294 + ], + "score": 1.0, + "content": ", is a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 121, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 121, + 293, + 151, + 303 + ], + "score": 0.9, + "content": "D \\times D", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 291, + 388, + 305 + ], + "score": 1.0, + "content": "matrix that is equal to the full posterior covariance matrix", + "type": "text" + }, + { + "bbox": [ + 388, + 292, + 408, + 302 + ], + "score": 0.93, + "content": "H ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "in the rows/columns of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 120, + 302, + 485, + 316 + ], + "spans": [ + { + "bbox": [ + 120, + 302, + 135, + 316 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 136, + 304, + 144, + 313 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 302, + 420, + 316 + ], + "score": 1.0, + "content": "weights in the subnetwork, and zero in the rows/columns of all other", + "type": "text" + }, + { + "bbox": [ + 420, + 303, + 449, + 313 + ], + "score": 0.91, + "content": "D - S", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 302, + 485, + 316 + ], + "score": 1.0, + "content": "weights.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 108, + 322, + 502, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "We consider what we term the posterior gap—the Wasserstein distance3 (in particular the squared", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 504, + 345 + ], + "score": 1.0, + "content": "2-Wasserstein distance) between the posterior distribution over the full network and the posterior", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 504, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 504, + 357 + ], + "score": 1.0, + "content": "distribution over the subnetwork. The proofs for all results below will be presented in Appendix A.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 504, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 337, + 370 + ], + "score": 1.0, + "content": "Proposition 1 (Posterior Gap). For a subnetwork of size", + "type": "text" + }, + { + "bbox": [ + 338, + 357, + 367, + 368 + ], + "score": 0.9, + "content": "S < D", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 357, + 505, + 370 + ], + "score": 1.0, + "content": ", the Wasserstein gap between the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 367, + 480, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 161, + 381 + ], + "score": 1.0, + "content": "full posterior", + "type": "text" + }, + { + "bbox": [ + 162, + 368, + 205, + 380 + ], + "score": 0.94, + "content": "\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 367, + 371, + 381 + ], + "score": 1.0, + "content": "in Eq. (10) and the subnetwork posterior", + "type": "text" + }, + { + "bbox": [ + 372, + 368, + 420, + 380 + ], + "score": 0.93, + "content": "p _ { \\mathrm { S } } ( { \\bf w } | { \\pmb y } , { \\pmb \\Phi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 367, + 480, + 381 + ], + "score": 1.0, + "content": "in Eq. (11) is:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 380, + 478, + 397 + ], + "lines": [ + { + "bbox": [ + 115, + 380, + 478, + 397 + ], + "spans": [ + { + "bbox": [ + 115, + 380, + 478, + 397 + ], + "score": 0.87, + "content": "\\begin{array} { r } { W \\left[ \\widetilde { p } ( \\mathbf { w } | y , \\Phi ) \\mid | p _ { \\mathrm { S } } ( \\mathbf { w } | y , \\Phi ) \\right] = \\sum _ { d = 1 } ^ { D } \\left( 1 + m _ { d d } \\right) \\sigma _ { d } ^ { 2 } - \\mathrm { t r a c e } ( 2 ( H ^ { - 1 } ( M _ { S } \\odot H ^ { - 1 } ) ) ^ { 1 / 2 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "e5300a416acdb777f1addd70a6ab798618ea7ba016490d0f8e6fee38ec287c74.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 115, + 380, + 478, + 397 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "The optimal subnetwork should then minimize the posterior gap in Eq. (12). However, for full", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 191, + 426 + ], + "score": 1.0, + "content": "covariance matrices", + "type": "text" + }, + { + "bbox": [ + 191, + 413, + 212, + 424 + ], + "score": 0.9, + "content": "H ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 413, + 345, + 426 + ], + "score": 1.0, + "content": "and a large number of weights", + "type": "text" + }, + { + "bbox": [ + 345, + 414, + 355, + 424 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 413, + 505, + 426 + ], + "score": 1.0, + "content": ", this will generally be infeasible as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 255, + 437 + ], + "score": 1.0, + "content": "Eq. (12) depends on all entries of the", + "type": "text" + }, + { + "bbox": [ + 255, + 425, + 286, + 435 + ], + "score": 0.91, + "content": "D \\times D", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 424, + 316, + 437 + ], + "score": 1.0, + "content": "-matrix", + "type": "text" + }, + { + "bbox": [ + 316, + 424, + 337, + 435 + ], + "score": 0.9, + "content": "H ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 424, + 505, + 437 + ], + "score": 1.0, + "content": ", which is intractable to compute/store. To", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 496, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 496, + 448 + ], + "score": 1.0, + "content": "derive a practical subnetwork selection strategy, we assume the covariance matrix to be diagonal.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 497, + 461 + ], + "score": 1.0, + "content": "Corollary 1.1 (Optimality of Maximum Variance Subnetwork Selection under Decorrelation). For", + "type": "text" + }, + { + "bbox": [ + 498, + 450, + 504, + 459 + ], + "score": 0.41, + "content": "a", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 347, + 473 + ], + "score": 1.0, + "content": "generalized linear model with posterior covariance matrix", + "type": "text" + }, + { + "bbox": [ + 347, + 459, + 452, + 471 + ], + "score": 0.92, + "content": "H ^ { - 1 } = d i a g ( \\sigma _ { 1 } ^ { 2 } , . . . , \\sigma _ { D } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 458, + 506, + 473 + ], + "score": 1.0, + "content": ", the optimal", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 467, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 467, + 340, + 486 + ], + "score": 1.0, + "content": "subnetwork under the Wasserstein gap is comprised of the", + "type": "text" + }, + { + "bbox": [ + 340, + 471, + 348, + 480 + ], + "score": 0.77, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 467, + 488, + 486 + ], + "score": 1.0, + "content": "weights with the largest variances", + "type": "text" + }, + { + "bbox": [ + 488, + 470, + 499, + 483 + ], + "score": 0.87, + "content": "\\sigma _ { d } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 467, + 505, + 486 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 506, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "Finally, since a GGN-linearized neural network, as in Eq. (7), corresponds to a GLM with basis ex-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 139, + 513 + ], + "score": 1.0, + "content": "pansion", + "type": "text" + }, + { + "bbox": [ + 140, + 500, + 344, + 513 + ], + "score": 0.89, + "content": "\\phi _ { n } = J _ { W _ { M A P } } ( \\pmb { x } _ { n } ) = \\partial \\pmb { f } ( \\pmb { x } _ { n } , \\pmb { W } _ { M A P } ) / \\partial W _ { M A P }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "(see Step #3 in Section 3), Corollary 1.1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "implies that under decorrelation, the optimal subnetwork comprises of the weights with the largest", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "variances. In practice, even just computing the diagonal of the covariance matrix is challenging, so we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "use a diagonal Laplace approximation which instead computes the inverse of the diagonal of the GGN", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "(see e.g. Ritter et al. (2018)). Finally, note that we only have to make the decorrelation assumption for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 567 + ], + "score": 1.0, + "content": "the purposes of subnetwork selection – when doing posterior inference over the selected subnetwork,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "we estimate a full covariance matrix for maximal expressiveness, as described in Step #3 in Section 3.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "In our experiments in Section 5, we empirically show that making the decorrelation assumption for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "subnetwork selection but then using a full-covariance Gaussian for inference performs significantly", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "better than directly making the decorrelation assumption for inference (e.g. mean-field variational", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 609, + 225, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 225, + 623 + ], + "score": 1.0, + "content": "inference, diagonal Laplace).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38.5 + }, + { + "type": "title", + "bbox": [ + 108, + 636, + 240, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 241, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 241, + 651 + ], + "score": 1.0, + "content": "5 EMPIRICAL ANALYSIS", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 661, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 505, + 673 + ], + "score": 1.0, + "content": "We empirically assess the effectiveness of subnetwork inference compared to point-estimated NNs", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 672, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 506, + 684 + ], + "score": 1.0, + "content": "and methods that do less expressive inference over the full network. We consider three tasks: 1)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 682, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 506, + 696 + ], + "score": 1.0, + "content": "small-scale toy regression, 2) medium-scale tabular regression, and 3) large-scale image classification.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 700, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 698, + 474, + 713 + ], + "spans": [ + { + "bbox": [ + 118, + 698, + 287, + 713 + ], + "score": 1.0, + "content": "2For consistency, we will keep referring to the", + "type": "text" + }, + { + "bbox": [ + 287, + 703, + 294, + 710 + ], + "score": 0.83, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 698, + 474, + 713 + ], + "score": 1.0, + "content": "selected linear model weights as a ”subnetwork”.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 118, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "3We use the Wasserstein distance instead of the more common Kullback–Leibler divergence because the", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 483, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 483, + 732 + ], + "score": 1.0, + "content": "Wasserstein is well-defined for degenerate distributions and is an actual distance metric (i.e. symmetric).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review 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": [ + 120, + 82, + 505, + 163 + ], + "lines": [ + { + "bbox": [ + 120, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 120, + 82, + 147, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 147, + 83, + 158, + 92 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 82, + 493, + 95 + ], + "score": 1.0, + "content": "is the Hessian defined in Section 3. Note that for the GLM we consider, the Hessian", + "type": "text" + }, + { + "bbox": [ + 493, + 83, + 504, + 93 + ], + "score": 0.75, + "content": "H", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 119, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 119, + 94, + 219, + 107 + ], + "score": 1.0, + "content": "is equivalent to the GGN", + "type": "text" + }, + { + "bbox": [ + 219, + 93, + 230, + 105 + ], + "score": 0.85, + "content": "\\widetilde { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 94, + 505, + 107 + ], + "score": 1.0, + "content": "defined in Eq. (6), meaning that an ordinary Laplace approximation is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 120, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 120, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "equivalent to a linearized Laplace approximation (Martens, 2016). For the case of an identity link", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 119, + 115, + 507, + 131 + ], + "spans": [ + { + "bbox": [ + 119, + 115, + 335, + 131 + ], + "score": 1.0, + "content": "function (i.e. a Gaussian likelihood with noise variance", + "type": "text" + }, + { + "bbox": [ + 336, + 116, + 347, + 128 + ], + "score": 0.88, + "content": "\\sigma _ { 0 } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 115, + 443, + 131 + ], + "score": 1.0, + "content": ") and a Gaussian prior w", + "type": "text" + }, + { + "bbox": [ + 444, + 116, + 502, + 128 + ], + "score": 0.9, + "content": "\\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { \\dot { N } } _ { 0 } ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 115, + 507, + 131 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 118, + 127, + 507, + 142 + ], + "spans": [ + { + "bbox": [ + 118, + 127, + 337, + 142 + ], + "score": 1.0, + "content": "the MAP estimate even has a closed-form expression,", + "type": "text" + }, + { + "bbox": [ + 337, + 128, + 478, + 141 + ], + "score": 0.92, + "content": "\\begin{array} { r } { { \\bf w } _ { M A P } = ( \\Phi ^ { T } \\Phi + \\sigma _ { 0 } ^ { 2 } \\Lambda _ { 0 } ) ^ { - 1 } \\Phi ^ { T } { \\pmb y } . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 127, + 507, + 142 + ], + "score": 1.0, + "content": ". Here,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 119, + 138, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 119, + 138, + 449, + 153 + ], + "score": 1.0, + "content": "the Laplace approximation in Eq. (10) exactly corresponds to the true posterior, i.e.", + "type": "text" + }, + { + "bbox": [ + 450, + 140, + 506, + 151 + ], + "score": 0.91, + "content": "\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi ) =", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 120, + 150, + 480, + 164 + ], + "spans": [ + { + "bbox": [ + 120, + 150, + 164, + 163 + ], + "score": 0.92, + "content": "p ( \\mathbf { w } | \\mathbf { y } , \\Phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 150, + 306, + 164 + ], + "score": 1.0, + "content": ". We will thus refer to the posterior", + "type": "text" + }, + { + "bbox": [ + 306, + 150, + 350, + 163 + ], + "score": 0.94, + "content": "\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 150, + 480, + 164 + ], + "score": 1.0, + "content": "e in Eq. (10) as the full posterior.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 118, + 82, + 507, + 164 + ] + }, + { + "type": "text", + "bbox": [ + 110, + 164, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 163, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 296, + 177 + ], + "score": 1.0, + "content": "2. Subnetwork Selection. Select a subset of", + "type": "text" + }, + { + "bbox": [ + 297, + 165, + 304, + 174 + ], + "score": 0.81, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 163, + 506, + 177 + ], + "score": 1.0, + "content": "model weights via a method of choice, yielding", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 119, + 173, + 504, + 188 + ], + "spans": [ + { + "bbox": [ + 119, + 173, + 185, + 188 + ], + "score": 1.0, + "content": "a binary vector", + "type": "text" + }, + { + "bbox": [ + 185, + 175, + 225, + 186 + ], + "score": 0.92, + "content": "\\pmb { m } \\in \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 173, + 254, + 188 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 254, + 176, + 289, + 186 + ], + "score": 0.92, + "content": "m _ { d } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 173, + 315, + 188 + ], + "score": 1.0, + "content": "if the", + "type": "text" + }, + { + "bbox": [ + 316, + 176, + 322, + 185 + ], + "score": 0.82, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 173, + 469, + 188 + ], + "score": 1.0, + "content": "-th weight is part of the subset, and", + "type": "text" + }, + { + "bbox": [ + 469, + 176, + 504, + 186 + ], + "score": 0.88, + "content": "m _ { d } = 0", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 119, + 186, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 119, + 186, + 377, + 198 + ], + "score": 1.0, + "content": "otherwise. For convenience, we define the binary mask matrix", + "type": "text" + }, + { + "bbox": [ + 378, + 186, + 476, + 197 + ], + "score": 0.92, + "content": "\\mathbf { M } _ { S } = \\pmb { m } \\pmb { m } ^ { \\top } \\in \\mathbb { R } ^ { D \\times D }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 186, + 506, + 198 + ], + "score": 1.0, + "content": "which", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 119, + 197, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 119, + 197, + 335, + 210 + ], + "score": 1.0, + "content": "contains 1s in the rows/columns corresponding to the", + "type": "text" + }, + { + "bbox": [ + 335, + 198, + 343, + 207 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 197, + 505, + 210 + ], + "score": 1.0, + "content": "subnetwork weights2, and 0s otherwise.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 106, + 163, + 506, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 211, + 502, + 223 + ], + "lines": [ + { + "bbox": [ + 107, + 210, + 501, + 224 + ], + "spans": [ + { + "bbox": [ + 107, + 210, + 501, + 224 + ], + "score": 1.0, + "content": "3. Bayesian Inference. Compute the posterior over the subnetwork via a Laplace approximation:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 107, + 210, + 501, + 224 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 223, + 402, + 237 + ], + "lines": [ + { + "bbox": [ + 221, + 223, + 402, + 237 + ], + "spans": [ + { + "bbox": [ + 221, + 223, + 402, + 237 + ], + "score": 0.84, + "content": "p _ { S } ( \\mathbf { w } | \\pmb { y } , \\pmb { \\Phi } ) = \\mathcal { N } ( \\mathbf { w } ; \\mathbf { w } _ { M A P } , M _ { S } \\odot H ^ { - 1 } ) \\ .", + "type": "interline_equation", + "image_path": "4b4c8b457ab7b86b66c9494f10dcb2f3dc11133832e707f12a7d9c2c703beb4b.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 221, + 223, + 402, + 237 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 120, + 237, + 505, + 315 + ], + "lines": [ + { + "bbox": [ + 118, + 235, + 504, + 252 + ], + "spans": [ + { + "bbox": [ + 118, + 235, + 473, + 252 + ], + "score": 1.0, + "content": "Firstly, note that the mean of the subnetwork posterior in Eq. (11) is the MAP estimate", + "type": "text" + }, + { + "bbox": [ + 473, + 240, + 504, + 249 + ], + "score": 0.78, + "content": "\\mathbf { w } _ { M A P }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 119, + 248, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 119, + 248, + 314, + 262 + ], + "score": 1.0, + "content": "and thus equal to the mean of the full posterior", + "type": "text" + }, + { + "bbox": [ + 315, + 249, + 358, + 260 + ], + "score": 0.92, + "content": "\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 248, + 506, + 262 + ], + "score": 1.0, + "content": "in Eq. (10). Secondly, note that the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 118, + 258, + 507, + 272 + ], + "spans": [ + { + "bbox": [ + 118, + 258, + 455, + 272 + ], + "score": 1.0, + "content": "ecovariance matrix of the subnetwork posterior in Eq. (11) is the element-wise product", + "type": "text" + }, + { + "bbox": [ + 455, + 259, + 502, + 271 + ], + "score": 0.92, + "content": "M _ { S } \\odot H ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 258, + 507, + 272 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 120, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 120, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "which masks the (co-)variances of all weights not belonging to the subnetwork to zero, effectively", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 119, + 280, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 119, + 280, + 435, + 294 + ], + "score": 1.0, + "content": "making them deterministic. More precisely, the subnetwork covariance matrix,", + "type": "text" + }, + { + "bbox": [ + 435, + 281, + 485, + 293 + ], + "score": 0.93, + "content": "M _ { S } \\odot H ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 280, + 507, + 294 + ], + "score": 1.0, + "content": ", is a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 121, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 121, + 293, + 151, + 303 + ], + "score": 0.9, + "content": "D \\times D", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 291, + 388, + 305 + ], + "score": 1.0, + "content": "matrix that is equal to the full posterior covariance matrix", + "type": "text" + }, + { + "bbox": [ + 388, + 292, + 408, + 302 + ], + "score": 0.93, + "content": "H ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "in the rows/columns of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 120, + 302, + 485, + 316 + ], + "spans": [ + { + "bbox": [ + 120, + 302, + 135, + 316 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 136, + 304, + 144, + 313 + ], + "score": 0.82, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 302, + 420, + 316 + ], + "score": 1.0, + "content": "weights in the subnetwork, and zero in the rows/columns of all other", + "type": "text" + }, + { + "bbox": [ + 420, + 303, + 449, + 313 + ], + "score": 0.91, + "content": "D - S", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 302, + 485, + 316 + ], + "score": 1.0, + "content": "weights.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 118, + 235, + 507, + 316 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 322, + 502, + 356 + ], + "lines": [ + { + "bbox": [ + 106, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "We consider what we term the posterior gap—the Wasserstein distance3 (in particular the squared", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 504, + 345 + ], + "score": 1.0, + "content": "2-Wasserstein distance) between the posterior distribution over the full network and the posterior", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 504, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 504, + 357 + ], + "score": 1.0, + "content": "distribution over the subnetwork. The proofs for all results below will be presented in Appendix A.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 322, + 505, + 357 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 504, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 337, + 370 + ], + "score": 1.0, + "content": "Proposition 1 (Posterior Gap). For a subnetwork of size", + "type": "text" + }, + { + "bbox": [ + 338, + 357, + 367, + 368 + ], + "score": 0.9, + "content": "S < D", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 357, + 505, + 370 + ], + "score": 1.0, + "content": ", the Wasserstein gap between the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 367, + 480, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 161, + 381 + ], + "score": 1.0, + "content": "full posterior", + "type": "text" + }, + { + "bbox": [ + 162, + 368, + 205, + 380 + ], + "score": 0.94, + "content": "\\widetilde { p } ( \\mathbf { w } | \\mathbf { y } , \\Phi )", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 367, + 371, + 381 + ], + "score": 1.0, + "content": "in Eq. (10) and the subnetwork posterior", + "type": "text" + }, + { + "bbox": [ + 372, + 368, + 420, + 380 + ], + "score": 0.93, + "content": "p _ { \\mathrm { S } } ( { \\bf w } | { \\pmb y } , { \\pmb \\Phi } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 367, + 480, + 381 + ], + "score": 1.0, + "content": "in Eq. (11) is:", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 357, + 505, + 381 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 115, + 380, + 478, + 397 + ], + "lines": [ + { + "bbox": [ + 115, + 380, + 478, + 397 + ], + "spans": [ + { + "bbox": [ + 115, + 380, + 478, + 397 + ], + "score": 0.87, + "content": "\\begin{array} { r } { W \\left[ \\widetilde { p } ( \\mathbf { w } | y , \\Phi ) \\mid | p _ { \\mathrm { S } } ( \\mathbf { w } | y , \\Phi ) \\right] = \\sum _ { d = 1 } ^ { D } \\left( 1 + m _ { d d } \\right) \\sigma _ { d } ^ { 2 } - \\mathrm { t r a c e } ( 2 ( H ^ { - 1 } ( M _ { S } \\odot H ^ { - 1 } ) ) ^ { 1 / 2 } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "e5300a416acdb777f1addd70a6ab798618ea7ba016490d0f8e6fee38ec287c74.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 115, + 380, + 478, + 397 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "The optimal subnetwork should then minimize the posterior gap in Eq. (12). However, for full", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 191, + 426 + ], + "score": 1.0, + "content": "covariance matrices", + "type": "text" + }, + { + "bbox": [ + 191, + 413, + 212, + 424 + ], + "score": 0.9, + "content": "H ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 413, + 345, + 426 + ], + "score": 1.0, + "content": "and a large number of weights", + "type": "text" + }, + { + "bbox": [ + 345, + 414, + 355, + 424 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 413, + 505, + 426 + ], + "score": 1.0, + "content": ", this will generally be infeasible as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 255, + 437 + ], + "score": 1.0, + "content": "Eq. (12) depends on all entries of the", + "type": "text" + }, + { + "bbox": [ + 255, + 425, + 286, + 435 + ], + "score": 0.91, + "content": "D \\times D", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 424, + 316, + 437 + ], + "score": 1.0, + "content": "-matrix", + "type": "text" + }, + { + "bbox": [ + 316, + 424, + 337, + 435 + ], + "score": 0.9, + "content": "H ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 424, + 505, + 437 + ], + "score": 1.0, + "content": ", which is intractable to compute/store. To", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 496, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 496, + 448 + ], + "score": 1.0, + "content": "derive a practical subnetwork selection strategy, we assume the covariance matrix to be diagonal.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 402, + 505, + 448 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 448, + 505, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 447, + 504, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 497, + 461 + ], + "score": 1.0, + "content": "Corollary 1.1 (Optimality of Maximum Variance Subnetwork Selection under Decorrelation). For", + "type": "text" + }, + { + "bbox": [ + 498, + 450, + 504, + 459 + ], + "score": 0.41, + "content": "a", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 347, + 473 + ], + "score": 1.0, + "content": "generalized linear model with posterior covariance matrix", + "type": "text" + }, + { + "bbox": [ + 347, + 459, + 452, + 471 + ], + "score": 0.92, + "content": "H ^ { - 1 } = d i a g ( \\sigma _ { 1 } ^ { 2 } , . . . , \\sigma _ { D } ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 458, + 506, + 473 + ], + "score": 1.0, + "content": ", the optimal", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 467, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 467, + 340, + 486 + ], + "score": 1.0, + "content": "subnetwork under the Wasserstein gap is comprised of the", + "type": "text" + }, + { + "bbox": [ + 340, + 471, + 348, + 480 + ], + "score": 0.77, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 467, + 488, + 486 + ], + "score": 1.0, + "content": "weights with the largest variances", + "type": "text" + }, + { + "bbox": [ + 488, + 470, + 499, + 483 + ], + "score": 0.87, + "content": "\\sigma _ { d } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 467, + 505, + 486 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 447, + 506, + 486 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 506, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "Finally, since a GGN-linearized neural network, as in Eq. (7), corresponds to a GLM with basis ex-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 139, + 513 + ], + "score": 1.0, + "content": "pansion", + "type": "text" + }, + { + "bbox": [ + 140, + 500, + 344, + 513 + ], + "score": 0.89, + "content": "\\phi _ { n } = J _ { W _ { M A P } } ( \\pmb { x } _ { n } ) = \\partial \\pmb { f } ( \\pmb { x } _ { n } , \\pmb { W } _ { M A P } ) / \\partial W _ { M A P }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "(see Step #3 in Section 3), Corollary 1.1", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 524 + ], + "score": 1.0, + "content": "implies that under decorrelation, the optimal subnetwork comprises of the weights with the largest", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 535 + ], + "score": 1.0, + "content": "variances. In practice, even just computing the diagonal of the covariance matrix is challenging, so we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "use a diagonal Laplace approximation which instead computes the inverse of the diagonal of the GGN", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "(see e.g. Ritter et al. (2018)). Finally, note that we only have to make the decorrelation assumption for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 556, + 506, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 567 + ], + "score": 1.0, + "content": "the purposes of subnetwork selection – when doing posterior inference over the selected subnetwork,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "we estimate a full covariance matrix for maximal expressiveness, as described in Step #3 in Section 3.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "In our experiments in Section 5, we empirically show that making the decorrelation assumption for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "subnetwork selection but then using a full-covariance Gaussian for inference performs significantly", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "better than directly making the decorrelation assumption for inference (e.g. mean-field variational", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 609, + 225, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 225, + 623 + ], + "score": 1.0, + "content": "inference, diagonal Laplace).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 489, + 506, + 623 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 636, + 240, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 241, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 241, + 651 + ], + "score": 1.0, + "content": "5 EMPIRICAL ANALYSIS", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 660, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 661, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 505, + 673 + ], + "score": 1.0, + "content": "We empirically assess the effectiveness of subnetwork inference compared to point-estimated NNs", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 672, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 672, + 506, + 684 + ], + "score": 1.0, + "content": "and methods that do less expressive inference over the full network. We consider three tasks: 1)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 682, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 506, + 696 + ], + "score": 1.0, + "content": "small-scale toy regression, 2) medium-scale tabular regression, and 3) large-scale image classification.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47, + "bbox_fs": [ + 106, + 661, + 506, + 696 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 77, + 504, + 250 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 77, + 504, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 77, + 504, + 250 + ], + "spans": [ + { + "bbox": [ + 108, + 77, + 504, + 250 + ], + "score": 0.973, + "type": "image", + "image_path": "348c77967767ca7b767128552cfde1a6eab11fff719d21ab1885ac59de9d2e18.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 77, + 504, + 134.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 134.66666666666666, + 504, + 192.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 192.33333333333331, + 504, + 249.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 258, + 506, + 292 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 269, + 271 + ], + "score": 1.0, + "content": "Figure 2: Predictive distributions (mean", + "type": "text" + }, + { + "bbox": [ + 269, + 259, + 280, + 269 + ], + "score": 0.59, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "std) for 1D regression. The numbers in brackets denote", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "the number of parameters over which inference was done (out of 2600 in total). Wasserstein-based", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 280, + 472, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 472, + 293 + ], + "score": 1.0, + "content": "subnetwork inference maintains richer predictive uncertainties at smaller parameter counts.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 106, + 315, + 498, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 500, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 500, + 327 + ], + "score": 1.0, + "content": "5.1 HOW DOES SUBNETWORK INFERENCE RETAIN POSTERIOR PREDICTIVE UNCERTAINTY?", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 336, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "We first assess how the predictive distribution of a full-covariance Gaussian posterior over a selected", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 348, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 359 + ], + "score": 1.0, + "content": "subnetwork qualitatively compares to that obtained from 1) a full-covariance Gaussian over the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "full network (Full Cov), 2) a factorised Gaussian posterior over the full network (Diag), 3) a full-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "covariance Gaussian over only the (Final layer) of the network (Kristiadi et al., 2020), and 4) a point", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 143, + 392 + ], + "score": 1.0, + "content": "estimate", + "type": "text" + }, + { + "bbox": [ + 144, + 380, + 171, + 391 + ], + "score": 0.54, + "content": "( M A P )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 380, + 505, + 392 + ], + "score": 1.0, + "content": ". For subnetwork inference, we consider both Wasserstein (Wass) (as described in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "Section 4) and uniform random subnetwork selection (Rand) to obtain subnetworks that comprise", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 138, + 415 + ], + "score": 1.0, + "content": "of only", + "type": "text" + }, + { + "bbox": [ + 138, + 402, + 158, + 413 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 402, + 161, + 415 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 161, + 402, + 176, + 413 + ], + "score": 0.82, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 402, + 195, + 415 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 195, + 402, + 209, + 412 + ], + "score": 0.87, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "of the model parameters. Note that while for this toy example, we could", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 413, + 504, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 504, + 426 + ], + "score": 1.0, + "content": "in principle use the full covariance matrix for the purpose of subnetwork selection, we still just use", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "its diagonal (as described in Section 4) for consistency. Our NN consists of 2 ReLU hidden layers", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "with 50 hidden units each. We employ a homoscedastic Gaussian likelihood function where the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "noise variance is optimised with maximum likelihood. We use GGN Laplace inference over network", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 457, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 469 + ], + "score": 1.0, + "content": "weights (not biases) in combination with the linearized predictive distribution in Eq. (8). Thus, all", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "approaches considered share their predictive mean, allowing us to better compare their uncertainty", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 479, + 370, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 340, + 492 + ], + "score": 1.0, + "content": "estimates. All approaches share a single prior precision of", + "type": "text" + }, + { + "bbox": [ + 340, + 479, + 365, + 489 + ], + "score": 0.9, + "content": "\\lambda = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 479, + 370, + 492 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "We use a synthetic 1D regression task with two separated clusters of inputs (Antoran et al., 2020), ´", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "allowing us to probe for ‘in-between’ uncertainty (Foong et al., 2019b). Results are shown in Fig. 2.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "Subnetwork inference preserves more of the uncertainty of full network inference than diagonal", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 528, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 542 + ], + "score": 1.0, + "content": "Gaussian or final layer inference while doing inference over fewer weights. By capturing weight", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "correlations, subnetwork inference retains uncertainty in between clusters of data. This is true for both", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "random and Wasserstein subnetwork selection. However, the latter preserves more uncertainty with", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "smaller subnetworks. Finally, the strong superiority to diagonal Laplace shows that making a diagonal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "assumption for subnetwork selection but then using a full-covariance Gaussian for inference (as we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "do) performs much signficantly better than making a diagonal assumption for the inferered posterior", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 594, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 608 + ], + "score": 1.0, + "content": "directly. These results suggest that expressive inference over a carefully selected subnetwork", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 606, + 470, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 470, + 618 + ], + "score": 1.0, + "content": "retains more predictive uncertainty than crude approximations over the full network.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 502, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "5.2 SUBNETWORK INFERENCE IN LARGE MODELS VS FULL INFERENCE OVER SMALL MODELS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "Secondly, we study the following natural question: “Why should one use subnetwork inference in a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "large model when one can just perform full network inference over a smaller model?” We explore", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "this by considering 4 fully connected models of increasing size. These have numbers of hidden", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 133, + 701 + ], + "score": 1.0, + "content": "layers", + "type": "text" + }, + { + "bbox": [ + 134, + 687, + 178, + 700 + ], + "score": 0.93, + "content": "h _ { d } = \\{ 1 , 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 687, + 281, + 701 + ], + "score": 1.0, + "content": "and hidden layer widths", + "type": "text" + }, + { + "bbox": [ + 281, + 687, + 341, + 699 + ], + "score": 0.91, + "content": "{ w _ { d } } \\mathrm { { = } } \\{ 5 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 687, + 493, + 701 + ], + "score": 1.0, + "content": ". For a dataset with input dimension", + "type": "text" + }, + { + "bbox": [ + 493, + 688, + 502, + 699 + ], + "score": 0.85, + "content": "i _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 687, + 506, + 701 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 246, + 712 + ], + "score": 1.0, + "content": "the number of weights is given by", + "type": "text" + }, + { + "bbox": [ + 246, + 699, + 354, + 711 + ], + "score": 0.88, + "content": "D { = } ( i _ { d } { + } 1 ) w _ { d } { + } ( h _ { d } { - } 1 ) w _ { d } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 698, + 506, + 712 + ], + "score": 1.0, + "content": ". Our 2 hidden layer, 100 hidden unit", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 270, + 721 + ], + "score": 1.0, + "content": "models have a weight count of the order", + "type": "text" + }, + { + "bbox": [ + 270, + 710, + 285, + 720 + ], + "score": 0.86, + "content": "1 0 ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 709, + 505, + 721 + ], + "score": 1.0, + "content": ". Full covariance inference in these models borders the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "limit of computational tractability on commercial hardware. We first obtain a MAP estimate of each", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 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 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 77, + 504, + 250 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 77, + 504, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 77, + 504, + 250 + ], + "spans": [ + { + "bbox": [ + 108, + 77, + 504, + 250 + ], + "score": 0.973, + "type": "image", + "image_path": "348c77967767ca7b767128552cfde1a6eab11fff719d21ab1885ac59de9d2e18.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 77, + 504, + 134.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 134.66666666666666, + 504, + 192.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 192.33333333333331, + 504, + 249.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 258, + 506, + 292 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 269, + 271 + ], + "score": 1.0, + "content": "Figure 2: Predictive distributions (mean", + "type": "text" + }, + { + "bbox": [ + 269, + 259, + 280, + 269 + ], + "score": 0.59, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "std) for 1D regression. The numbers in brackets denote", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 281 + ], + "score": 1.0, + "content": "the number of parameters over which inference was done (out of 2600 in total). Wasserstein-based", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 280, + 472, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 472, + 293 + ], + "score": 1.0, + "content": "subnetwork inference maintains richer predictive uncertainties at smaller parameter counts.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 106, + 315, + 498, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 500, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 500, + 327 + ], + "score": 1.0, + "content": "5.1 HOW DOES SUBNETWORK INFERENCE RETAIN POSTERIOR PREDICTIVE UNCERTAINTY?", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 336, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "We first assess how the predictive distribution of a full-covariance Gaussian posterior over a selected", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 348, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 359 + ], + "score": 1.0, + "content": "subnetwork qualitatively compares to that obtained from 1) a full-covariance Gaussian over the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "full network (Full Cov), 2) a factorised Gaussian posterior over the full network (Diag), 3) a full-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "covariance Gaussian over only the (Final layer) of the network (Kristiadi et al., 2020), and 4) a point", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 143, + 392 + ], + "score": 1.0, + "content": "estimate", + "type": "text" + }, + { + "bbox": [ + 144, + 380, + 171, + 391 + ], + "score": 0.54, + "content": "( M A P )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 380, + 505, + 392 + ], + "score": 1.0, + "content": ". For subnetwork inference, we consider both Wasserstein (Wass) (as described in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "Section 4) and uniform random subnetwork selection (Rand) to obtain subnetworks that comprise", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 138, + 415 + ], + "score": 1.0, + "content": "of only", + "type": "text" + }, + { + "bbox": [ + 138, + 402, + 158, + 413 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 402, + 161, + 415 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 161, + 402, + 176, + 413 + ], + "score": 0.82, + "content": "3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 402, + 195, + 415 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 195, + 402, + 209, + 412 + ], + "score": 0.87, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "of the model parameters. Note that while for this toy example, we could", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 413, + 504, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 504, + 426 + ], + "score": 1.0, + "content": "in principle use the full covariance matrix for the purpose of subnetwork selection, we still just use", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 437 + ], + "score": 1.0, + "content": "its diagonal (as described in Section 4) for consistency. Our NN consists of 2 ReLU hidden layers", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "with 50 hidden units each. We employ a homoscedastic Gaussian likelihood function where the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "noise variance is optimised with maximum likelihood. We use GGN Laplace inference over network", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 457, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 469 + ], + "score": 1.0, + "content": "weights (not biases) in combination with the linearized predictive distribution in Eq. (8). Thus, all", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "approaches considered share their predictive mean, allowing us to better compare their uncertainty", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 479, + 370, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 340, + 492 + ], + "score": 1.0, + "content": "estimates. All approaches share a single prior precision of", + "type": "text" + }, + { + "bbox": [ + 340, + 479, + 365, + 489 + ], + "score": 0.9, + "content": "\\lambda = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 479, + 370, + 492 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 336, + 506, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "We use a synthetic 1D regression task with two separated clusters of inputs (Antoran et al., 2020), ´", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "allowing us to probe for ‘in-between’ uncertainty (Foong et al., 2019b). Results are shown in Fig. 2.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "Subnetwork inference preserves more of the uncertainty of full network inference than diagonal", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 528, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 542 + ], + "score": 1.0, + "content": "Gaussian or final layer inference while doing inference over fewer weights. By capturing weight", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "correlations, subnetwork inference retains uncertainty in between clusters of data. This is true for both", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 563 + ], + "score": 1.0, + "content": "random and Wasserstein subnetwork selection. However, the latter preserves more uncertainty with", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "smaller subnetworks. Finally, the strong superiority to diagonal Laplace shows that making a diagonal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "assumption for subnetwork selection but then using a full-covariance Gaussian for inference (as we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 584, + 505, + 595 + ], + "score": 1.0, + "content": "do) performs much signficantly better than making a diagonal assumption for the inferered posterior", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 594, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 506, + 608 + ], + "score": 1.0, + "content": "directly. These results suggest that expressive inference over a carefully selected subnetwork", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 606, + 470, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 470, + 618 + ], + "score": 1.0, + "content": "retains more predictive uncertainty than crude approximations over the full network.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 496, + 506, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 634, + 502, + 644 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "5.2 SUBNETWORK INFERENCE IN LARGE MODELS VS FULL INFERENCE OVER SMALL MODELS", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 633, + 505, + 645 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "Secondly, we study the following natural question: “Why should one use subnetwork inference in a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "large model when one can just perform full network inference over a smaller model?” We explore", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "this by considering 4 fully connected models of increasing size. These have numbers of hidden", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 133, + 701 + ], + "score": 1.0, + "content": "layers", + "type": "text" + }, + { + "bbox": [ + 134, + 687, + 178, + 700 + ], + "score": 0.93, + "content": "h _ { d } = \\{ 1 , 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 687, + 281, + 701 + ], + "score": 1.0, + "content": "and hidden layer widths", + "type": "text" + }, + { + "bbox": [ + 281, + 687, + 341, + 699 + ], + "score": 0.91, + "content": "{ w _ { d } } \\mathrm { { = } } \\{ 5 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 687, + 493, + 701 + ], + "score": 1.0, + "content": ". For a dataset with input dimension", + "type": "text" + }, + { + "bbox": [ + 493, + 688, + 502, + 699 + ], + "score": 0.85, + "content": "i _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 687, + 506, + 701 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 246, + 712 + ], + "score": 1.0, + "content": "the number of weights is given by", + "type": "text" + }, + { + "bbox": [ + 246, + 699, + 354, + 711 + ], + "score": 0.88, + "content": "D { = } ( i _ { d } { + } 1 ) w _ { d } { + } ( h _ { d } { - } 1 ) w _ { d } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 698, + 506, + 712 + ], + "score": 1.0, + "content": ". Our 2 hidden layer, 100 hidden unit", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 270, + 721 + ], + "score": 1.0, + "content": "models have a weight count of the order", + "type": "text" + }, + { + "bbox": [ + 270, + 710, + 285, + 720 + ], + "score": 0.86, + "content": "1 0 ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 709, + 505, + 721 + ], + "score": 1.0, + "content": ". Full covariance inference in these models borders the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "limit of computational tractability on commercial hardware. We first obtain a MAP estimate of each", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "model’s weights and our homoscedastic likelihood function’s noise variance. We then perform full", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "network GGN Laplace inference for each model. We also use our proposed Wassertein rule to prune", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "every network’s weight variances such that the number of variances that remain matches the size", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "of every smaller network under consideration. In all cases, we employ the linearized predictive in", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 138 + ], + "score": 1.0, + "content": "Eq. (7). Consequently, networks with the same number of weights make the same mean predictions.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "Increasing the number of weight variances considered will thus only increase predictive uncertainty.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 655, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "model’s weights and our homoscedastic likelihood function’s noise variance. We then perform full", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "network GGN Laplace inference for each model. We also use our proposed Wassertein rule to prune", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "every network’s weight variances such that the number of variances that remain matches the size", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "of every smaller network under consideration. In all cases, we employ the linearized predictive in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 138 + ], + "score": 1.0, + "content": "Eq. (7). Consequently, networks with the same number of weights make the same mean predictions.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "Increasing the number of weight variances considered will thus only increase predictive uncertainty.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 106, + 158, + 505, + 262 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 158, + 505, + 262 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 158, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 262 + ], + "score": 0.967, + "type": "image", + "image_path": "66c302f1bb14b542bd24077fff89bad6dabe0c7a84ec71a09c89a80dccf32d61.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 106, + 158, + 505, + 192.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 106, + 192.66666666666666, + 505, + 227.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 106, + 227.33333333333331, + 505, + 262.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 273, + 505, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "Figure 3: Mean test log-likelihood values obtained on UCI datasets across all splits. Different", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "markers indicate models with different numbers of weights. The horizontal axis indicates the number", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "of weights over which full covariance inference is performed. 0 corresponds to MAP parameter", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 307, + 474, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 474, + 318 + ], + "score": 1.0, + "content": "estimation, and the rightmost setting for each marker corresponds to full network inference.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "We employ 3 tabular datasets of increasing size (input dimensionality, n. points): wine (11, 1439),", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "kin8nm (8, 7373) and protein (9, 41157). We consider their standard train-test splits (Hernandez- ´", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 351, + 507, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 507, + 364 + ], + "score": 1.0, + "content": "Lobato & Adams, 2015) and their gap variants (Foong et al., 2019b), designed to test for out-of-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 473, + 375 + ], + "score": 1.0, + "content": "distribution uncertainty. Details are provided in Appendix C.4. For each split, we set aside", + "type": "text" + }, + { + "bbox": [ + 473, + 362, + 493, + 373 + ], + "score": 0.87, + "content": "1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "the train data as a validation set. We use these for early stopping when finding MAP estimates and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "for selecting the weights’ prior precision. We keep other hyperparameters fixed across all models and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 395, + 229, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 229, + 408 + ], + "score": 1.0, + "content": "datasets. Results are in Fig. 3.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "We present mean test log-likelihood (LL) values, as these take into account both accuracy and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "uncertainty. Larger models tend to perform better when doing MAP inference, with wine-gap and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "protein-gap being exceptions. We also find larger models improve over their respective MAP LLs", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "more than small ones when performing approximate inference over the same numbers of weights.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "We conjecture this is due to an abundance of degenerate directions (weights) in the weight posterior", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 465, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 479 + ], + "score": 1.0, + "content": "of all models (Maddox et al., 2020). Full network inference in small models captures information", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "score": 1.0, + "content": "about both useful and non-useful weights. In larger models, our subnetwork selection strategy allows", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "us to dedicate a larger proportion of our resources to modelling informative weight variances and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "covariances. In 3 out of 6 datasets, we find abrupt increases in LL as we increase the number of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "weights over which we perform inference, followed by a plateau. Such plateaus might be explained", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "by all of the most informative weight variances having already been accounted for. These results", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 545 + ], + "score": 1.0, + "content": "suggest that, given the same amount of compute, larger models benefit more from subnetwork", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 543, + 219, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 219, + 556 + ], + "score": 1.0, + "content": "inference than small ones.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 108, + 568, + 409, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 411, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 411, + 581 + ], + "score": 1.0, + "content": "5.3 SCALING TO IMAGE CLASSIFICATION WITH DISTRIBUTION SHIFT", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "We now assess the robustness of large convolutional neural networks with subnetwork inference to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "distribution shift on image classification tasks compared to the following baselines: point-estimated", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "networks (MAP), Bayesian deep learning methods that do less expressive inference over the full", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "network: MC Dropout (Gal & Ghahramani, 2016), diagonal Laplace, VOGN (Osawa et al., 2019)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "(all of which assume factorisation of the weight posterior), and SWAG (Maddox et al., 2019) (which", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "assumes a diagonal plus low-rank posterior). We also benchmark deep ensembles (Lakshminarayanan", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 653, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 653, + 506, + 669 + ], + "score": 1.0, + "content": "et al., 2017). The latter is considered state-of-the-art for uncertainty quantification in deep learning", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "score": 1.0, + "content": "(Ovadia et al., 2019; Ashukha et al., 2020a). We use ensembles of 5 DNNs, as suggested by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "(Ovadia et al., 2019), and 16 samples for MC Dropout, diagonal Laplace and SWAG. We use a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 308, + 700 + ], + "score": 1.0, + "content": "Dropout probability of 0.1 and a prior precision of", + "type": "text" + }, + { + "bbox": [ + 309, + 688, + 358, + 699 + ], + "score": 0.79, + "content": "\\lambda = 4 0 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "for diagonal Laplace, found via grid", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "search. We apply all approaches to ResNet-18 (He et al., 2016), which is composed of an input", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "convolutional block, 8 residual blocks and a linear layer, for a total of 11,168,000 parameters. For", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "subnetwork inference, we compute the linearized predictive distribution in Eq. (8) via the closed-form", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 40 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 83, + 506, + 150 + ], + "lines_deleted": true + }, + { + "type": "image", + "bbox": [ + 106, + 158, + 505, + 262 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 158, + 505, + 262 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 158, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 158, + 505, + 262 + ], + "score": 0.967, + "type": "image", + "image_path": "66c302f1bb14b542bd24077fff89bad6dabe0c7a84ec71a09c89a80dccf32d61.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 106, + 158, + 505, + 192.66666666666666 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 106, + 192.66666666666666, + 505, + 227.33333333333331 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 106, + 227.33333333333331, + 505, + 262.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 273, + 505, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "Figure 3: Mean test log-likelihood values obtained on UCI datasets across all splits. Different", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 505, + 296 + ], + "score": 1.0, + "content": "markers indicate models with different numbers of weights. The horizontal axis indicates the number", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "of weights over which full covariance inference is performed. 0 corresponds to MAP parameter", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 307, + 474, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 474, + 318 + ], + "score": 1.0, + "content": "estimation, and the rightmost setting for each marker corresponds to full network inference.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + } + ], + "index": 8.75 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 506, + 342 + ], + "score": 1.0, + "content": "We employ 3 tabular datasets of increasing size (input dimensionality, n. points): wine (11, 1439),", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "kin8nm (8, 7373) and protein (9, 41157). We consider their standard train-test splits (Hernandez- ´", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 351, + 507, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 507, + 364 + ], + "score": 1.0, + "content": "Lobato & Adams, 2015) and their gap variants (Foong et al., 2019b), designed to test for out-of-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 473, + 375 + ], + "score": 1.0, + "content": "distribution uncertainty. Details are provided in Appendix C.4. For each split, we set aside", + "type": "text" + }, + { + "bbox": [ + 473, + 362, + 493, + 373 + ], + "score": 0.87, + "content": "1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 361, + 506, + 375 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "the train data as a validation set. We use these for early stopping when finding MAP estimates and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "for selecting the weights’ prior precision. We keep other hyperparameters fixed across all models and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 395, + 229, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 229, + 408 + ], + "score": 1.0, + "content": "datasets. Results are in Fig. 3.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 329, + 507, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "We present mean test log-likelihood (LL) values, as these take into account both accuracy and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "uncertainty. Larger models tend to perform better when doing MAP inference, with wine-gap and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "protein-gap being exceptions. We also find larger models improve over their respective MAP LLs", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "more than small ones when performing approximate inference over the same numbers of weights.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "We conjecture this is due to an abundance of degenerate directions (weights) in the weight posterior", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 465, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 505, + 479 + ], + "score": 1.0, + "content": "of all models (Maddox et al., 2020). Full network inference in small models captures information", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "score": 1.0, + "content": "about both useful and non-useful weights. In larger models, our subnetwork selection strategy allows", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "us to dedicate a larger proportion of our resources to modelling informative weight variances and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 512 + ], + "score": 1.0, + "content": "covariances. In 3 out of 6 datasets, we find abrupt increases in LL as we increase the number of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "weights over which we perform inference, followed by a plateau. Such plateaus might be explained", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "by all of the most informative weight variances having already been accounted for. These results", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 506, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 545 + ], + "score": 1.0, + "content": "suggest that, given the same amount of compute, larger models benefit more from subnetwork", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 543, + 219, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 219, + 556 + ], + "score": 1.0, + "content": "inference than small ones.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 412, + 506, + 556 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 568, + 409, + 580 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 411, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 411, + 581 + ], + "score": 1.0, + "content": "5.3 SCALING TO IMAGE CLASSIFICATION WITH DISTRIBUTION SHIFT", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "We now assess the robustness of large convolutional neural networks with subnetwork inference to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "distribution shift on image classification tasks compared to the following baselines: point-estimated", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 623 + ], + "score": 1.0, + "content": "networks (MAP), Bayesian deep learning methods that do less expressive inference over the full", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "score": 1.0, + "content": "network: MC Dropout (Gal & Ghahramani, 2016), diagonal Laplace, VOGN (Osawa et al., 2019)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "(all of which assume factorisation of the weight posterior), and SWAG (Maddox et al., 2019) (which", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "assumes a diagonal plus low-rank posterior). We also benchmark deep ensembles (Lakshminarayanan", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 653, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 104, + 653, + 506, + 669 + ], + "score": 1.0, + "content": "et al., 2017). The latter is considered state-of-the-art for uncertainty quantification in deep learning", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "score": 1.0, + "content": "(Ovadia et al., 2019; Ashukha et al., 2020a). We use ensembles of 5 DNNs, as suggested by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "(Ovadia et al., 2019), and 16 samples for MC Dropout, diagonal Laplace and SWAG. We use a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 308, + 700 + ], + "score": 1.0, + "content": "Dropout probability of 0.1 and a prior precision of", + "type": "text" + }, + { + "bbox": [ + 309, + 688, + 358, + 699 + ], + "score": 0.79, + "content": "\\lambda = 4 0 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "for diagonal Laplace, found via grid", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "search. We apply all approaches to ResNet-18 (He et al., 2016), which is composed of an input", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "convolutional block, 8 residual blocks and a linear layer, for a total of 11,168,000 parameters. For", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "subnetwork inference, we compute the linearized predictive distribution in Eq. (8) via the closed-form", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "approximation for integrals between Gaussians and multi-class cross-entropy likelihoods described", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 326, + 507, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 413, + 340 + ], + "score": 1.0, + "content": "in (Gibbs, 1998). We use Wasserstein subnetwork selection to retain only", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 414, + 327, + 441, + 339 + ], + "score": 0.89, + "content": "0 . 3 8 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 442, + 326, + 507, + 340 + ], + "score": 1.0, + "content": "of the weights,", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "yielding a subnetwork with only 42,438 weights. This is the largest subnetwork for which we can", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 332, + 362 + ], + "score": 1.0, + "content": "tractably compute a full covariance matrix. Its size is 42", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 333, + 349, + 373, + 360 + ], + "score": 0.86, + "content": ", 4 3 8 ^ { 2 } \\times \\hat { 4 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 374, + 349, + 399, + 362 + ], + "score": 1.0, + "content": "Bytes", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 399, + 349, + 425, + 360 + ], + "score": 0.8, + "content": "\\approx 7 . 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 425, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "GB. We use a prior", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 155, + 373 + ], + "score": 1.0, + "content": "precision of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 155, + 361, + 190, + 371 + ], + "score": 0.9, + "content": "\\lambda = 5 0 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 190, + 360, + 506, + 373 + ], + "score": 1.0, + "content": ", found via grid search. Finally, to assess to importance of principled subnetwork", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "selection, we also consider the baseline where we select the subnetwork uniformly at random (called", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "score": 1.0, + "content": "Ours (Rand)). We perform the following two experiments, with results in Fig. 4. See Appendix B for", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 393, + 180, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 180, + 405 + ], + "score": 1.0, + "content": "additional results.", + "type": "text", + "cross_page": true + } + ], + "index": 14 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 589, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 79, + 503, + 240 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 79, + 503, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 79, + 503, + 240 + ], + "spans": [ + { + "bbox": [ + 108, + 79, + 503, + 240 + ], + "score": 0.973, + "type": "image", + "image_path": "966242861badeec22bbb4cab7022e457bc6a60cc58c9e8f8bac63f13ef1c8655.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 79, + 503, + 132.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 132.66666666666666, + 503, + 186.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 186.33333333333331, + 503, + 239.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 252, + 505, + 296 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "Figure 4: Results on the rotated MNIST (left) and the corrupted CIFAR (right) benchmarks of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 262, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 267, + 276 + ], + "score": 1.0, + "content": "Ovadia et al. (2019), showing the mean", + "type": "text" + }, + { + "bbox": [ + 267, + 264, + 278, + 274 + ], + "score": 0.73, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 262, + 506, + 276 + ], + "score": 1.0, + "content": "std of the error (top) and log-likelihood (bottom) across", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "three different seeds. Subnetwork inference retains better uncertainty calibration and robustness to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 284, + 482, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 482, + 298 + ], + "score": 1.0, + "content": "distribution shift than point estimated networks and other Bayesian deep learning approaches.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 506, + 329 + ], + "score": 1.0, + "content": "approximation for integrals between Gaussians and multi-class cross-entropy likelihoods described", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 326, + 507, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 413, + 340 + ], + "score": 1.0, + "content": "in (Gibbs, 1998). We use Wasserstein subnetwork selection to retain only", + "type": "text" + }, + { + "bbox": [ + 414, + 327, + 441, + 339 + ], + "score": 0.89, + "content": "0 . 3 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 326, + 507, + 340 + ], + "score": 1.0, + "content": "of the weights,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "yielding a subnetwork with only 42,438 weights. This is the largest subnetwork for which we can", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 332, + 362 + ], + "score": 1.0, + "content": "tractably compute a full covariance matrix. Its size is 42", + "type": "text" + }, + { + "bbox": [ + 333, + 349, + 373, + 360 + ], + "score": 0.86, + "content": ", 4 3 8 ^ { 2 } \\times \\hat { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 349, + 399, + 362 + ], + "score": 1.0, + "content": "Bytes", + "type": "text" + }, + { + "bbox": [ + 399, + 349, + 425, + 360 + ], + "score": 0.8, + "content": "\\approx 7 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "GB. We use a prior", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 155, + 373 + ], + "score": 1.0, + "content": "precision of", + "type": "text" + }, + { + "bbox": [ + 155, + 361, + 190, + 371 + ], + "score": 0.9, + "content": "\\lambda = 5 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 360, + 506, + 373 + ], + "score": 1.0, + "content": ", found via grid search. Finally, to assess to importance of principled subnetwork", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 506, + 384 + ], + "score": 1.0, + "content": "selection, we also consider the baseline where we select the subnetwork uniformly at random (called", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "score": 1.0, + "content": "Ours (Rand)). We perform the following two experiments, with results in Fig. 4. See Appendix B for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 393, + 180, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 180, + 405 + ], + "score": 1.0, + "content": "additional results.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 409, + 505, + 608 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "Rotated MNIST: Following Ovadia et al. (2019); Antoran et al. (2020), we train all methods on ´", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 422, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 433 + ], + "score": 1.0, + "content": "MNIST and evaluate their predictive distributions on increasingly rotated digits. While all methods", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 431, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 446 + ], + "score": 1.0, + "content": "perform well on the original MNIST test set, their accuracy degrades quickly for rotations larger", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "than 30 degrees. In terms of LL, ensembles perform best out of our baselines. Subnetwork inference", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "obtains significantly larger LL values than almost all baselines, including ensembles. The only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "exception is VOGN, which achieves slightly better performance. It was also observed in Ovadia et al.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "(2019) that mean-field variational inference (which VOGN also is an instance of) is very strong on", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "MNIST, but its performance deteriorates on larger datasets. Subnetwork inference makes accurate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "predictions in-distribution while assigning higher uncertainty than the baselines to out-of-distribution", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 522 + ], + "score": 1.0, + "content": "points. Corrupted CIFAR: Again following Ovadia et al. (2019); Antoran et al. (2020), we train ´", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "on CIFAR10 and evaluate on data subject to 16 different corruptions with 5 levels of intensity each", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 506, + 542 + ], + "score": 1.0, + "content": "(Hendrycks & Dietterich, 2019). Our approach matches a MAP estimated network in terms of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 506, + 554 + ], + "score": 1.0, + "content": "predictive error as local linearization makes their predictions the same. Ensembles and SWAG are the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "most accurate. Even so, subnetwork inference differentiates itself by being the least overconfident,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 564, + 505, + 576 + ], + "score": 1.0, + "content": "outperforming all baselines in terms of log-likelihood at all corruption levels. Here, VOGN performs", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "rather badly; while this might appear in stark contrast to its strong performance on the MNIST", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 506, + 599 + ], + "score": 1.0, + "content": "benchmark, the behaviour that mean-field VI performs well on MNIST but not on larger datasets was", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 597, + 257, + 609 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 257, + 609 + ], + "score": 1.0, + "content": "also observed in Ovadia et al. 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(2020), we train all methods on ´", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 422, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 433 + ], + "score": 1.0, + "content": "MNIST and evaluate their predictive distributions on increasingly rotated digits. While all methods", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 431, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 506, + 446 + ], + "score": 1.0, + "content": "perform well on the original MNIST test set, their accuracy degrades quickly for rotations larger", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "than 30 degrees. In terms of LL, ensembles perform best out of our baselines. Subnetwork inference", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "obtains significantly larger LL values than almost all baselines, including ensembles. The only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "exception is VOGN, which achieves slightly better performance. 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Although asymptotically unbiased, sampling based", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "approaches are difficult to scale to the large datasets (Betancourt, 2015). As a result, approaches", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 507, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 507, + 118 + ], + "score": 1.0, + "content": "which find the best surrogate posterior among an approximating family (most often Gaussians)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "have gained popularity. 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Thus, most", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 506, + 171 + ], + "score": 1.0, + "content": "modern methods make use of the mean field assumption (Blundell et al., 2015; Hernandez-Lobato & ´", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 182 + ], + "score": 1.0, + "content": "Adams, 2015; Gal & Ghahramani, 2016; Mishkin et al., 2018; Osawa et al., 2019). This comes at the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 193 + ], + "score": 1.0, + "content": "cost of limited expressivity (Foong et al., 2019a) and empirical under-performance (Ovadia et al.,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "2019; Antoran et al., 2020) of uncertainty estimates. Our proposed approach recovers predictive ´", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 104, + 203, + 506, + 216 + ], + "score": 1.0, + "content": "posterior expressivity while maintaining tractability by lowering the dimensionality of the weight", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "space considered. This allows us to scale up approximations that do consider weight correlations", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 443, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 443, + 238 + ], + "score": 1.0, + "content": "(MacKay, 1992; Louizos & Welling, 2016; Maddox et al., 2019; Ritter et al., 2018).", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Neural Network Linearization. In the limit of infinite width, NNs converge to Gaussian process", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "(GP) behaviour (Neal, 1995; Matthews, 2017; Garriga-Alonso et al., 2018). Recently, these results", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "have been extended to finite width BNNs when the surrogate posterior is Gaussian (Khan et al., 2019).", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "We draw upon these results to formulate a subnetwork selection strategy for BNNs. Neural linear", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "methods perform inference over only the last layer of a NN, while keeping all other layers fixed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "(Snoek et al., 2015; Riquelme et al., 2018; Ovadia et al., 2019; Ober & Rasmussen, 2019; Pinsler", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "et al., 2019; Kristiadi et al., 2020). These represent a different generalised linear model in which the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 240, + 331 + ], + "score": 1.0, + "content": "basis functions are defined by the", + "type": "text" + }, + { + "bbox": [ + 240, + 319, + 258, + 329 + ], + "score": 0.76, + "content": "l { - } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "first layers of a NN. They can also be viewed as a special case", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 328, + 473, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 473, + 342 + ], + "score": 1.0, + "content": "of subnetwork inference, in which the subnetwork is simply defined to be the last NN layer.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "Inference over Subspaces. The subfield of NN pruning aims to increase the computational efficiency", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "of NNs by identifying the smallest subset of weights which are required to make accurate predictions.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "Approaches trade-off computational cost with compression efficiency, ranging from those that require", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 380, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 506, + 392 + ], + "score": 1.0, + "content": "multiple training runs (Frankle & Carbin, 2019) to those that prune before training (Wang et al., 2020).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "Our work differs in that it retains all NN weights but aims to find a small subset over which to perform", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "probabilistic reasoning. More closely related work to ours is that of Izmailov et al. (2019), who", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "propose to perform inference over a low-dimensional subspace of weights; e.g. one constructed from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 424, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 506, + 435 + ], + "score": 1.0, + "content": "the principal components of the SGD trajectory. Moreover, several recent approaches use low-rank", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 435, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 446 + ], + "score": 1.0, + "content": "parameterizations of approximate posteriors in the context of variational inference (Rossi et al., 2019;", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 445, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 506, + 457 + ], + "score": 1.0, + "content": "Swiatkowski et al., 2020; Dusenberry et al., 2020). This could also be viewed as doing inference", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "over an implicit subspace of weight space. In contrast, we propose a technique to find subsets of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "weights which are relevant to predictive uncertainty, i.e., we identify axis aligned subspaces. Finally,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "there have been recent works studying neural network sparsity / pruning from a Bayesian perspective", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "(Ghosh & Doshi-Velez, 2017; Polson & Rockov ˇ a, 2018; Cui et al., 2020; Louizos et al., 2017; ´", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Molchanov et al., 2017; Gomez et al., 2019; Lee et al., 2018). While these seem conceptually related", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "at first glance, their goal is fundamentally different to ours: While those methods aim to perform", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "model selection / sparsification by either explicitly or implicitly pruning unnecessary weights, our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "goal is to make inference more tractable. More precisely, while those sparse Bayesian deep learning", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "methods prune individual weights, we instead just prune the variances of certain weights, which,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 554, + 499, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 499, + 569 + ], + "score": 1.0, + "content": "importantly retains the full predictive power of the full network to retain high predictive accuracy.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 108, + 591, + 195, + 604 + ], + "lines": [ + { + "bbox": [ + 104, + 589, + 197, + 607 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 197, + 607 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "In this paper, we develop a practical and scalable method for expressive yet tractable probabilistic", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "inference in deep neural networks. We approximate the posterior over a subset of the weights while", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 643, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 658 + ], + "score": 1.0, + "content": "keeping all other weights deterministic. Computational cost is decoupled from network size, allowing", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "us to scale expressive approximations, such as full-covariance Gaussian distributions, to real-world", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "sized NNs. Our approach can be applied post-hoc to any pre-trained model, making it particularly", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "attractive for practical use. Our empirical analysis suggests that subnetwork inference 1) is more", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "expressive and retains more uncertainty than crude approximations over the full network, 2) allows", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "us to employ larger NNs, which fit a broader range of functions, without sacrificing the quality of our", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "uncertainty estimates, and 3) is competitive with state-of-the-art uncertainty quantification methods,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 444, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 444, + 733 + ], + "score": 1.0, + "content": "like deep ensembles (Lakshminarayanan et al., 2017), on real-world scale problems.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 48.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [], + "index": 6.5, + "bbox_fs": [ + 104, + 82, + 507, + 238 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "Neural Network Linearization. In the limit of infinite width, NNs converge to Gaussian process", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "(GP) behaviour (Neal, 1995; Matthews, 2017; Garriga-Alonso et al., 2018). Recently, these results", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "have been extended to finite width BNNs when the surrogate posterior is Gaussian (Khan et al., 2019).", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "We draw upon these results to formulate a subnetwork selection strategy for BNNs. Neural linear", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "methods perform inference over only the last layer of a NN, while keeping all other layers fixed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "(Snoek et al., 2015; Riquelme et al., 2018; Ovadia et al., 2019; Ober & Rasmussen, 2019; Pinsler", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "et al., 2019; Kristiadi et al., 2020). These represent a different generalised linear model in which the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 240, + 331 + ], + "score": 1.0, + "content": "basis functions are defined by the", + "type": "text" + }, + { + "bbox": [ + 240, + 319, + 258, + 329 + ], + "score": 0.76, + "content": "l { - } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "first layers of a NN. They can also be viewed as a special case", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 328, + 473, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 473, + 342 + ], + "score": 1.0, + "content": "of subnetwork inference, in which the subnetwork is simply defined to be the last NN layer.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 241, + 506, + 342 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 360 + ], + "score": 1.0, + "content": "Inference over Subspaces. The subfield of NN pruning aims to increase the computational efficiency", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 506, + 370 + ], + "score": 1.0, + "content": "of NNs by identifying the smallest subset of weights which are required to make accurate predictions.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "Approaches trade-off computational cost with compression efficiency, ranging from those that require", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 380, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 506, + 392 + ], + "score": 1.0, + "content": "multiple training runs (Frankle & Carbin, 2019) to those that prune before training (Wang et al., 2020).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "Our work differs in that it retains all NN weights but aims to find a small subset over which to perform", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "probabilistic reasoning. More closely related work to ours is that of Izmailov et al. (2019), who", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "propose to perform inference over a low-dimensional subspace of weights; e.g. one constructed from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 424, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 506, + 435 + ], + "score": 1.0, + "content": "the principal components of the SGD trajectory. Moreover, several recent approaches use low-rank", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 435, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 446 + ], + "score": 1.0, + "content": "parameterizations of approximate posteriors in the context of variational inference (Rossi et al., 2019;", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 445, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 506, + 457 + ], + "score": 1.0, + "content": "Swiatkowski et al., 2020; Dusenberry et al., 2020). This could also be viewed as doing inference", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "over an implicit subspace of weight space. In contrast, we propose a technique to find subsets of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "weights which are relevant to predictive uncertainty, i.e., we identify axis aligned subspaces. Finally,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "there have been recent works studying neural network sparsity / pruning from a Bayesian perspective", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 506, + 501 + ], + "score": 1.0, + "content": "(Ghosh & Doshi-Velez, 2017; Polson & Rockov ˇ a, 2018; Cui et al., 2020; Louizos et al., 2017; ´", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "Molchanov et al., 2017; Gomez et al., 2019; Lee et al., 2018). While these seem conceptually related", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "at first glance, their goal is fundamentally different to ours: While those methods aim to perform", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 535 + ], + "score": 1.0, + "content": "model selection / sparsification by either explicitly or implicitly pruning unnecessary weights, our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "goal is to make inference more tractable. More precisely, while those sparse Bayesian deep learning", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 544, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 506, + 556 + ], + "score": 1.0, + "content": "methods prune individual weights, we instead just prune the variances of certain weights, which,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 554, + 499, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 499, + 569 + ], + "score": 1.0, + "content": "importantly retains the full predictive power of the full network to retain high predictive accuracy.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 345, + 506, + 569 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 591, + 195, + 604 + ], + "lines": [ + { + "bbox": [ + 104, + 589, + 197, + 607 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 197, + 607 + ], + "score": 1.0, + "content": "7 CONCLUSION", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "In this paper, we develop a practical and scalable method for expressive yet tractable probabilistic", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "inference in deep neural networks. We approximate the posterior over a subset of the weights while", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 643, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 658 + ], + "score": 1.0, + "content": "keeping all other weights deterministic. Computational cost is decoupled from network size, allowing", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 667 + ], + "score": 1.0, + "content": "us to scale expressive approximations, such as full-covariance Gaussian distributions, to real-world", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "sized NNs. Our approach can be applied post-hoc to any pre-trained model, making it particularly", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "attractive for practical use. Our empirical analysis suggests that subnetwork inference 1) is more", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "expressive and retains more uncertainty than crude approximations over the full network, 2) allows", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "us to employ larger NNs, which fit a broader range of functions, without sacrificing the quality of our", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "uncertainty estimates, and 3) is competitive with state-of-the-art uncertainty quantification methods,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 444, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 444, + 733 + ], + "score": 1.0, + "content": "like deep ensembles (Lakshminarayanan et al., 2017), on real-world scale problems.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 621, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 176, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 100, + 504, + 123 + ], + "lines": [ + { + "bbox": [ + 106, + 100, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 506, + 113 + ], + "score": 1.0, + "content": "Dario Amodei, Chris Olah, Jacob Steinhardt, Paul Christiano, John Schulman, and Dan Mane.´", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 116, + 111, + 406, + 123 + ], + "spans": [ + { + "bbox": [ + 116, + 111, + 406, + 123 + ], + "score": 1.0, + "content": "Concrete problems in ai safety. arXiv preprint arXiv:1606.06565, 2016.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 129, + 503, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 127, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 143 + ], + "score": 1.0, + "content": "Javier Antoran, James Urquhart Allingham, and Jos ´ e Miguel Hern ´ andez-Lobato. 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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.14±0.02-0.05±0.00-0.05±0.00-0.11±0.08-0.04±0.00-0.05±0.00-0.04±0.00-0.19±nan
error0.02±0.000.02±0.000.01±0.000.03±0.020.01±0.000.02±0.000.01±0.000.02±nan
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midrule LL-1.09±0.03-1.60±0.05−1.44±0.11-1.68±0.20-1.36±0.07-1.75±0.06-1.35±0.02-1.15±nan
error0.36±0.010.35±0.010.33±0.010.35±0.030.31±0.010.35±0.010.29±0.000.40±nan
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brier score0.49±0.020.55±0.020.52±0.020.55±0.040.48±0.020.56±0.020.46±0.010.53±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-2.10±0.03-3.85±0.18-3.54±0.23-4.11±0.66-3.60±0.10-4.29±0.21-2.95±0.08-1.92±nan
error0.63±0.010.63±0.010.62±0.010.62±0.050.61±0.010.63±0.010.53±0.020.64±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.14±0.02-0.05±0.00-0.05±0.00-0.11±0.08-0.04±0.00-0.05±0.00-0.04±0.00-0.19±nan
error0.02±0.000.02±0.000.01±0.000.03±0.020.01±0.000.02±0.000.01±0.000.02±nan
ECE0.08±0.010.00±0.000.00±0.000.01±0.010.00±0.000.00±0.000.00±0.000.12±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.42±0.04-0.36±0.01-0.32±0.02-0.44±0.06-0.28±0.02-0.39±0.01-0.30±0.00-0.51±nan
error0.11±0.010.10±0.000.09±0.010.12±0.010.08±0.010.10±0.000.08±0.000.14±nan
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brier score0.19±0.020.16±0.000.14±0.010.18±0.020.12±0.010.16±0.000.12±0.000.23±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
midrule LL-1.09±0.03-1.60±0.05−1.44±0.11-1.68±0.20-1.36±0.07-1.75±0.06-1.35±0.02-1.15±nan
error0.36±0.010.35±0.010.33±0.010.35±0.030.31±0.010.35±0.010.29±0.000.40±nan
ECE0.03±0.010.22±0.010.19±0.020.22±0.020.17±0.010.23±0.010.18±0.000.01±nan
brier score0.49±0.020.55±0.020.52±0.020.55±0.040.48±0.020.56±0.020.46±0.010.53±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-2.10±0.03-3.85±0.18-3.54±0.23-4.11±0.66-3.60±0.10-4.29±0.21-2.95±0.08-1.92±nan
error0.63±0.010.63±0.010.62±0.010.62±0.050.61±0.010.63±0.010.53±0.020.64±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.02±0.07-5.93±0.28-5.49±0.38-6.92±0.32-5.74±0.15-6.63±0.33-4.46±0.18-2.54±nan
error0.80±0.020.79±0.010.79±0.010.81±0.000.78±0.010.79±0.010.72±0.020.77±nan
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LL−1.47±0.03-2.82±0.11-2.71±0.13-3.20±0.13-1.88±0.05-3.03±0.10-2.31±0.09-2.00±nan
error0.41±0.000.40±0.010.40±0.010.41±0.010.37±0.010.40±0.000.36±0.010.54±nan
ECE0.16±0.010.31±0.010.28±0.010.33±0.020.17±0.010.31±0.010.27±0.010.19±nan
brier score0.58±0.000.69±0.010.65±0.010.72±0.030.55±0.010.69±0.010.61±0.010.75±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.27±0.00-0.43±0.01-0.37±0.01-0.50±0.02-0.21±0.01-0.46±0.02-0.48±0.01-0.61±nan
error0.09±0.000.08±0.000.08±0.000.09±0.000.06±0.000.08±0.000.11±0.000.21±nan
ECE0.01±0.000.06±0.000.04±0.000.06±0.000.01±0.000.06±0.000.07±0.000.03±nan
brier score0.13±0.000.14±0.000.13±0.000.15±0.000.09±0.000.14±0.000.17±0.000.30±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.51±0.01-0.91±0.01-0.80±0.02−1.03±0.02-0.50±0.02-0.96±0.02-0.89±0.02-0.99±nan
error0.17±0.010.16±0.000.16±0.000.17±0.000.13±0.000.16±0.000.17±0.000.32±nan
ECE0.03±0.000.11±0.000.10±0.000.13±0.000.04±0.000.12±0.010.11±0.000.03±nan
brier score0.24±0.000.27±0.000.25±0.000.29±0.000.19±0.000.27±0.010.29±0.000.44±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.73±0.01-1.29±0.06-1.20±0.02−1.50±0.12-0.80±0.01-1.40±0.03-1.21±0.00-1.31±nan
error0.23±0.000.22±0.010.22±0.000.23±0.010.19±0.000.22±0.000.22±0.000.40±nan
ECE0.06±0.000.16±0.010.14±0.000.17±0.010.07±0.000.16±0.000.15±0.000.10±nan
brier score0.33±0.000.37±0.010.35±0.010.40±0.020.28±0.000.37±0.010.37±0.000.56±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-1.06±0.02-2.06±0.12-1.85±0.07-2.13±0.17-1.28±0.03-2.18±0.08-1.63±0.03-1.83±nan
error0.32±0.010.31±0.010.31±0.010.31±0.010.28±0.000.31±0.010.28±0.000.51±nan
ECE0.11±0.010.24±0.010.21±0.010.24±0.010.12±0.000.24±0.010.20±0.000.19±nan
brier score0.46±0.010.54±0.020.50±0.020.54±0.030.42±0.000.54±0.020.47±0.010.72±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-1.25±0.03-2.43±0.18-2.28±0.10-2.54±0.18-1.56±0.05-2.57±0.15-1.95±0.04-1.99±nan
error0.36±0.010.35±0.010.35±0.010.35±0.010.32±0.010.35±0.010.32±0.000.54±nan
ECE0.13±0.010.27±0.010.24±0.010.27±0.010.14±0.010.27±0.020.23±0.000.22±nan
brier score0.51±0.020.60±0.030.57±0.010.61±0.020.47±0.010.60±0.030.53±0.000.76±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL−1.47±0.03-2.82±0.11-2.71±0.13-3.20±0.13-1.88±0.05-3.03±0.10-2.31±0.09-2.00±nan
error0.41±0.000.40±0.010.40±0.010.41±0.010.37±0.010.40±0.000.36±0.010.54±nan
ECE0.16±0.010.31±0.010.28±0.010.33±0.020.17±0.010.31±0.010.27±0.010.19±nan
brier score0.58±0.000.69±0.010.65±0.010.72±0.030.55±0.010.69±0.010.61±0.010.75±nan
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Specifically, we use SGD with a learning rate of", + "type": "text" + }, + { + "bbox": [ + 373, + 139, + 411, + 150 + ], + "score": 0.92, + "content": "1 \\times 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 138, + 506, + 151 + ], + "score": 1.0, + "content": ", momentum of 0.9 and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 174, + 163 + ], + "score": 1.0, + "content": "weight decay of", + "type": "text" + }, + { + "bbox": [ + 174, + 150, + 212, + 161 + ], + "score": 0.92, + "content": "1 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 150, + 505, + 163 + ], + "score": 1.0, + "content": ". We use a batch size of 512. The objective we optimise is the Gaussian", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 174 + ], + "score": 1.0, + "content": "log-likelihood of our data, where the mean is outputted by the network and the the variance is a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "score": 1.0, + "content": "hyperparameter learnt jointly with NN parameters by SGD. This variance parameters is shared among", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "all datapoints. Once the network is trained, we perform post-hoc inference on it using different", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "approaches. Since all of these involve the linearized approximation, the mean prediction is the same", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 325, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 325, + 219 + ], + "score": 1.0, + "content": "for all methods. Only their uncertainty estimates vary.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 222, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "Note that while for this toy example, we could in principle use the full covariance matrix for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 234, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 505, + 245 + ], + "score": 1.0, + "content": "the purpose of subnetwork selection, we still just use its diagonal (as described in Section 4) for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 244, + 504, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 504, + 256 + ], + "score": 1.0, + "content": "consistency. We use GGN Laplace inference over network weights (not biases) in combination with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "the linearized predictive distribution in Eq. (8). Thus, all approaches considered share their predictive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 267, + 364, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 364, + 279 + ], + "score": 1.0, + "content": "mean, allowing us to better compare their uncertainty estimates.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 300, + 295 + ], + "score": 1.0, + "content": "All approaches share a single prior precision of", + "type": "text" + }, + { + "bbox": [ + 300, + 283, + 325, + 293 + ], + "score": 0.89, + "content": "\\lambda = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 283, + 505, + 295 + ], + "score": 1.0, + "content": ". We chose to select the prior precision such", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "score": 1.0, + "content": "that the full covariance approach (optimistic baseline) presents reasonable results. We use the same", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 318 + ], + "score": 1.0, + "content": "value for all other methods. We first tried a precision of 1 and found the full covariance approach to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "produce excessively large errorbars (covering the whole plot). A value of 3 produces more reasonable", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 138, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 138, + 339 + ], + "score": 1.0, + "content": "results.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 108, + 343, + 505, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "Final layer inference is performed by computing the full Laplace covariance matrix and discarding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "all entries except those corresponding to the final layer of the NN. Results for random sub-network", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 500, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 500, + 378 + ], + "score": 1.0, + "content": "selection are obtained with a single sample from a scaled uniform distribution over weight choice.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 392, + 216, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 217, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 217, + 405 + ], + "score": 1.0, + "content": "C.2 UCI EXPERIMENTS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 413, + 426 + ], + "score": 1.0, + "content": "In this experiment, our fully connected NNs have numbers of hidden layers", + "type": "text" + }, + { + "bbox": [ + 413, + 414, + 457, + 425 + ], + "score": 0.92, + "content": "h _ { d } = \\{ 1 , 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "and hidden", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 158, + 437 + ], + "score": 1.0, + "content": "layer widths", + "type": "text" + }, + { + "bbox": [ + 159, + 424, + 218, + 436 + ], + "score": 0.9, + "content": "{ w _ { d } } \\mathrm { { = } } \\{ 5 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 424, + 365, + 437 + ], + "score": 1.0, + "content": ". For a dataset with input dimension", + "type": "text" + }, + { + "bbox": [ + 365, + 425, + 374, + 435 + ], + "score": 0.85, + "content": "i _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 424, + 505, + 437 + ], + "score": 1.0, + "content": ", the number of weights is given", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 119, + 448 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 435, + 227, + 447 + ], + "score": 0.88, + "content": "D { = } ( i _ { d } { + } 1 ) w _ { d } { + } ( h _ { d } { - } 1 ) \\bar { w } _ { d } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 435, + 506, + 448 + ], + "score": 1.0, + "content": ". Our 2 hidden layer, 100 hidden unit models have a weight count of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 296, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 145, + 458 + ], + "score": 1.0, + "content": "the order", + "type": "text" + }, + { + "bbox": [ + 145, + 447, + 160, + 457 + ], + "score": 0.85, + "content": "1 0 ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 446, + 296, + 458 + ], + "score": 1.0, + "content": ". The non-linearity used is ReLU.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "We first obtain a MAP estimate of each model’s weights. Specifically, we use SGD with a learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 472, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 135, + 487 + ], + "score": 1.0, + "content": "rate of", + "type": "text" + }, + { + "bbox": [ + 135, + 474, + 173, + 485 + ], + "score": 0.91, + "content": "1 \\times 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 472, + 333, + 487 + ], + "score": 1.0, + "content": ", momentum of 0.9 and weight decay of", + "type": "text" + }, + { + "bbox": [ + 333, + 474, + 372, + 485 + ], + "score": 0.92, + "content": "1 \\bar { \\times } 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 472, + 505, + 487 + ], + "score": 1.0, + "content": ". We use a batch size of 512. The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "objective we optimise is the Gaussian log-likelihood of our data, where the mean is outputted by the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 497, + 478, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 478, + 508 + ], + "score": 1.0, + "content": "network and the the variance is a hyperparameter learnt jointly with NN parameters by SGD.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 248, + 525 + ], + "score": 1.0, + "content": "For each dataset split, we set aside", + "type": "text" + }, + { + "bbox": [ + 249, + 513, + 268, + 524 + ], + "score": 0.86, + "content": "1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "of the train data as a validation set. We use these for early", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "stopping training. Training runs for a maximum of 2000 epochs but early stops with a patience of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "500 if validation performance does not increase. For the larger Protein dataset, these values are 500", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 546, + 434, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 434, + 559 + ], + "score": 1.0, + "content": "and 125. The weight settings which provide best validation performance are kept.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 504, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 504, + 575 + ], + "score": 1.0, + "content": "We then perform full network GGN Laplace inference for each model. We also use our proposed", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "Wassertein rule together with the diagonal Hessian assumption to prune every network’s weight", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "variances such that the number of variances that remain matches the size of every smaller network", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 594, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 609 + ], + "score": 1.0, + "content": "under consideration. The prior precision used for these steps is chosen such that the resulting", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "predictor’s logliklihood performance on the validation set is maximised. Specifically, we employ", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 617, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 232, + 632 + ], + "score": 1.0, + "content": "a grid search over the values:", + "type": "text" + }, + { + "bbox": [ + 232, + 618, + 431, + 630 + ], + "score": 0.68, + "content": "\\lambda : [ 0 . 0 0 0 1 , 0 . 0 0 1 , 0 . 1 , 0 . 5 , 1 , 2 , 5 , 1 0 , 1 0 0 , 1 0 0 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 617, + 506, + 632 + ], + "score": 1.0, + "content": ". In all cases, we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "employ the linearized predictive in Eq. (7). Consequently, networks with the same number of weights", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "make the same mean predictions. Increasing the number of weight variances considered will thus", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 650, + 253, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 253, + 664 + ], + "score": 1.0, + "content": "only increase predictive uncertainty.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + }, + { + "type": "title", + "bbox": [ + 108, + 677, + 226, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 228, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 228, + 690 + ], + "score": 1.0, + "content": "C.3 IMAGE EXPERIMENTS", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "The results shown in Section 5.3 and Appendix B are obtained by training ResNet-18 (and ResNet-50)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "models using SGD with momentum. For each experiment repetition, we train 7 different models:", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "The first is for: ‘MAP’, ‘Ours’, ‘Ours (Rand)’, ‘SWAG’, ‘Diag-Laplace’ and as the first element of", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review 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": "title", + "bbox": [ + 107, + 81, + 246, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 248, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 248, + 96 + ], + "score": 1.0, + "content": "C EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 108, + 107, + 216, + 119 + ], + "lines": [ + { + "bbox": [ + 106, + 107, + 217, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 107, + 217, + 119 + ], + "score": 1.0, + "content": "C.1 TOY EXPERIMENTS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 128, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 129, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "We train a single, 2 hidden layer network, with 50 hidden ReLU units per layer using MAP inference", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 138, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 372, + 151 + ], + "score": 1.0, + "content": "until convergence. Specifically, we use SGD with a learning rate of", + "type": "text" + }, + { + "bbox": [ + 373, + 139, + 411, + 150 + ], + "score": 0.92, + "content": "1 \\times 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 138, + 506, + 151 + ], + "score": 1.0, + "content": ", momentum of 0.9 and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 150, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 174, + 163 + ], + "score": 1.0, + "content": "weight decay of", + "type": "text" + }, + { + "bbox": [ + 174, + 150, + 212, + 161 + ], + "score": 0.92, + "content": "1 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 150, + 505, + 163 + ], + "score": 1.0, + "content": ". We use a batch size of 512. The objective we optimise is the Gaussian", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 506, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 506, + 174 + ], + "score": 1.0, + "content": "log-likelihood of our data, where the mean is outputted by the network and the the variance is a", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 186 + ], + "score": 1.0, + "content": "hyperparameter learnt jointly with NN parameters by SGD. This variance parameters is shared among", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "score": 1.0, + "content": "all datapoints. Once the network is trained, we perform post-hoc inference on it using different", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 207 + ], + "score": 1.0, + "content": "approaches. Since all of these involve the linearized approximation, the mean prediction is the same", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 205, + 325, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 325, + 219 + ], + "score": 1.0, + "content": "for all methods. Only their uncertainty estimates vary.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 129, + 506, + 219 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 222, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 234 + ], + "score": 1.0, + "content": "Note that while for this toy example, we could in principle use the full covariance matrix for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 234, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 505, + 245 + ], + "score": 1.0, + "content": "the purpose of subnetwork selection, we still just use its diagonal (as described in Section 4) for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 244, + 504, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 504, + 256 + ], + "score": 1.0, + "content": "consistency. We use GGN Laplace inference over network weights (not biases) in combination with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 255, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "the linearized predictive distribution in Eq. (8). Thus, all approaches considered share their predictive", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 267, + 364, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 364, + 279 + ], + "score": 1.0, + "content": "mean, allowing us to better compare their uncertainty estimates.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 222, + 505, + 279 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 300, + 295 + ], + "score": 1.0, + "content": "All approaches share a single prior precision of", + "type": "text" + }, + { + "bbox": [ + 300, + 283, + 325, + 293 + ], + "score": 0.89, + "content": "\\lambda = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 283, + 505, + 295 + ], + "score": 1.0, + "content": ". We chose to select the prior precision such", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 505, + 307 + ], + "score": 1.0, + "content": "that the full covariance approach (optimistic baseline) presents reasonable results. We use the same", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 318 + ], + "score": 1.0, + "content": "value for all other methods. We first tried a precision of 1 and found the full covariance approach to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 329 + ], + "score": 1.0, + "content": "produce excessively large errorbars (covering the whole plot). A value of 3 produces more reasonable", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 138, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 138, + 339 + ], + "score": 1.0, + "content": "results.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 283, + 506, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 343, + 505, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 356 + ], + "score": 1.0, + "content": "Final layer inference is performed by computing the full Laplace covariance matrix and discarding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "all entries except those corresponding to the final layer of the NN. Results for random sub-network", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 500, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 500, + 378 + ], + "score": 1.0, + "content": "selection are obtained with a single sample from a scaled uniform distribution over weight choice.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 342, + 505, + 378 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 392, + 216, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 217, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 217, + 405 + ], + "score": 1.0, + "content": "C.2 UCI EXPERIMENTS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 413, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 413, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 413, + 426 + ], + "score": 1.0, + "content": "In this experiment, our fully connected NNs have numbers of hidden layers", + "type": "text" + }, + { + "bbox": [ + 413, + 414, + 457, + 425 + ], + "score": 0.92, + "content": "h _ { d } = \\{ 1 , 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 413, + 505, + 426 + ], + "score": 1.0, + "content": "and hidden", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 158, + 437 + ], + "score": 1.0, + "content": "layer widths", + "type": "text" + }, + { + "bbox": [ + 159, + 424, + 218, + 436 + ], + "score": 0.9, + "content": "{ w _ { d } } \\mathrm { { = } } \\{ 5 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 424, + 365, + 437 + ], + "score": 1.0, + "content": ". For a dataset with input dimension", + "type": "text" + }, + { + "bbox": [ + 365, + 425, + 374, + 435 + ], + "score": 0.85, + "content": "i _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 424, + 505, + 437 + ], + "score": 1.0, + "content": ", the number of weights is given", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 119, + 448 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 119, + 435, + 227, + 447 + ], + "score": 0.88, + "content": "D { = } ( i _ { d } { + } 1 ) w _ { d } { + } ( h _ { d } { - } 1 ) \\bar { w } _ { d } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 435, + 506, + 448 + ], + "score": 1.0, + "content": ". Our 2 hidden layer, 100 hidden unit models have a weight count of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 446, + 296, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 145, + 458 + ], + "score": 1.0, + "content": "the order", + "type": "text" + }, + { + "bbox": [ + 145, + 447, + 160, + 457 + ], + "score": 0.85, + "content": "1 0 ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 446, + 296, + 458 + ], + "score": 1.0, + "content": ". The non-linearity used is ReLU.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 413, + 506, + 458 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "We first obtain a MAP estimate of each model’s weights. Specifically, we use SGD with a learning", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 472, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 135, + 487 + ], + "score": 1.0, + "content": "rate of", + "type": "text" + }, + { + "bbox": [ + 135, + 474, + 173, + 485 + ], + "score": 0.91, + "content": "1 \\times 1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 472, + 333, + 487 + ], + "score": 1.0, + "content": ", momentum of 0.9 and weight decay of", + "type": "text" + }, + { + "bbox": [ + 333, + 474, + 372, + 485 + ], + "score": 0.92, + "content": "1 \\bar { \\times } 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 472, + 505, + 487 + ], + "score": 1.0, + "content": ". We use a batch size of 512. The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "objective we optimise is the Gaussian log-likelihood of our data, where the mean is outputted by the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 497, + 478, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 478, + 508 + ], + "score": 1.0, + "content": "network and the the variance is a hyperparameter learnt jointly with NN parameters by SGD.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 462, + 505, + 508 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 248, + 525 + ], + "score": 1.0, + "content": "For each dataset split, we set aside", + "type": "text" + }, + { + "bbox": [ + 249, + 513, + 268, + 524 + ], + "score": 0.86, + "content": "1 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "of the train data as a validation set. We use these for early", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 536 + ], + "score": 1.0, + "content": "stopping training. Training runs for a maximum of 2000 epochs but early stops with a patience of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 548 + ], + "score": 1.0, + "content": "500 if validation performance does not increase. For the larger Protein dataset, these values are 500", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 546, + 434, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 434, + 559 + ], + "score": 1.0, + "content": "and 125. The weight settings which provide best validation performance are kept.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 513, + 506, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 563, + 504, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 504, + 575 + ], + "score": 1.0, + "content": "We then perform full network GGN Laplace inference for each model. We also use our proposed", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 505, + 586 + ], + "score": 1.0, + "content": "Wassertein rule together with the diagonal Hessian assumption to prune every network’s weight", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 505, + 597 + ], + "score": 1.0, + "content": "variances such that the number of variances that remain matches the size of every smaller network", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 594, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 609 + ], + "score": 1.0, + "content": "under consideration. The prior precision used for these steps is chosen such that the resulting", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "predictor’s logliklihood performance on the validation set is maximised. Specifically, we employ", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 617, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 232, + 632 + ], + "score": 1.0, + "content": "a grid search over the values:", + "type": "text" + }, + { + "bbox": [ + 232, + 618, + 431, + 630 + ], + "score": 0.68, + "content": "\\lambda : [ 0 . 0 0 0 1 , 0 . 0 0 1 , 0 . 1 , 0 . 5 , 1 , 2 , 5 , 1 0 , 1 0 0 , 1 0 0 0 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 617, + 506, + 632 + ], + "score": 1.0, + "content": ". In all cases, we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 505, + 641 + ], + "score": 1.0, + "content": "employ the linearized predictive in Eq. (7). Consequently, networks with the same number of weights", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "make the same mean predictions. Increasing the number of weight variances considered will thus", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 650, + 253, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 253, + 664 + ], + "score": 1.0, + "content": "only increase predictive uncertainty.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 563, + 506, + 664 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 677, + 226, + 689 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 228, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 228, + 690 + ], + "score": 1.0, + "content": "C.3 IMAGE EXPERIMENTS", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "The results shown in Section 5.3 and Appendix B are obtained by training ResNet-18 (and ResNet-50)", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 721 + ], + "score": 1.0, + "content": "models using SGD with momentum. For each experiment repetition, we train 7 different models:", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "The first is for: ‘MAP’, ‘Ours’, ‘Ours (Rand)’, ‘SWAG’, ‘Diag-Laplace’ and as the first element of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "‘Ensemble’. We train 4 additional ‘Ensemble’ elements, 1 network with ‘Dropout’, and, finally 1", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "network for ‘VOGN’. The methods ‘Ours’, ‘Ours (Rand)’, ‘SWAG’, and ‘Diag-Laplace’ are applied", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 162, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 162, + 118 + ], + "score": 1.0, + "content": "post training.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 699, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "‘Ensemble’. We train 4 additional ‘Ensemble’ elements, 1 network with ‘Dropout’, and, finally 1", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "network for ‘VOGN’. The methods ‘Ours’, ‘Ours (Rand)’, ‘SWAG’, and ‘Diag-Laplace’ are applied", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 162, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 162, + 118 + ], + "score": 1.0, + "content": "post training.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 506, + 252 + ], + "lines": [ + { + "bbox": [ + 105, + 120, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 506, + 135 + ], + "score": 1.0, + "content": "For all methods except ‘VOGN’ we use the following training procedure. The (initial) learning rate,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 295, + 144 + ], + "score": 1.0, + "content": "momentum, and weight decay are 0.1, 0.9, and", + "type": "text" + }, + { + "bbox": [ + 296, + 132, + 334, + 143 + ], + "score": 0.92, + "content": "1 \\times \\mathrm { i 0 ^ { - 4 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 132, + 506, + 144 + ], + "score": 1.0, + "content": ", respectively. For ‘MAP’ we use 4 Nvidia", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "P100 GPUs with a total batch size of 2048. For the calculation of the Jacobian in the subnetwork", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 166 + ], + "score": 1.0, + "content": "selection phase we use a single P100 GPU with a batch size of 4. For the calculation of the hessian", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "we use a single P100 GPU with a batch size of 2. We train on 1 Nvidia P100 GPU with a batch", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 187 + ], + "score": 1.0, + "content": "size of 256 for all other methods. Each dataset is trained for a different number of epochs, shown in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "Table 21. We decay the learning rate by a factor of 10 at scheduled epochs, also shown in Table 21.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "Otherwise, all methods and datasets share hyperparameters. These hyperparameter settings are the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "defaults provided by PyTorch for training on ImageNet. We found them to perform well across the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 231 + ], + "score": 1.0, + "content": "board. We report results obtained at the final training epoch. We do not use a separate validation set", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "to determine the best epoch as we found ResNet-18 and ResNet-50 to not overfit with the chosen", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 150, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 150, + 253 + ], + "score": 1.0, + "content": "schedules.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 8.5 + }, + { + "type": "table", + "bbox": [ + 206, + 288, + 401, + 332 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 170, + 275, + 439, + 287 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 169, + 273, + 441, + 289 + ], + "spans": [ + { + "bbox": [ + 169, + 273, + 441, + 289 + ], + "score": 1.0, + "content": "Table 21: Per-dataset training configuration for image experiments.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "table_body", + "bbox": [ + 206, + 288, + 401, + 332 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 206, + 288, + 401, + 332 + ], + "spans": [ + { + "bbox": [ + 206, + 288, + 401, + 332 + ], + "score": 0.968, + "html": "
DATASETNo.EPOCHSLR SCHEDULE
MNIST9040,70
CIFAR10300150,225
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This approach follows Zagoruyko & Komodakis (2016)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 371, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 506, + 383 + ], + "score": 1.0, + "content": "and Ashukha et al. (2020b) who add dropout in-between the two convolutions of a WideResNet-50’s", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 382, + 506, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 506, + 395 + ], + "score": 1.0, + "content": "basic block. Following Antoran et al. (2020), we choose a dropout probability of 0.1, as they found ´", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 393, + 504, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 473, + 405 + ], + "score": 1.0, + "content": "it to perform better than the value of 0.3 suggested by Ashukha et al. (2020b). 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Each dataset is trained for a different number of epochs, shown in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 199 + ], + "score": 1.0, + "content": "Table 21. We decay the learning rate by a factor of 10 at scheduled epochs, also shown in Table 21.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "Otherwise, all methods and datasets share hyperparameters. These hyperparameter settings are the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "defaults provided by PyTorch for training on ImageNet. We found them to perform well across the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 231 + ], + "score": 1.0, + "content": "board. We report results obtained at the final training epoch. 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We use", + "type": "text" + }, + { + "bbox": [ + 474, + 393, + 504, + 404 + ], + "score": 0.34, + "content": "\\mathrm { \\dot { 1 } 6 \\mathbf { M } C }", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 404, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 506, + 417 + ], + "score": 1.0, + "content": "samples for predictions. ‘Ensemble’ uses 5 elements for prediction. Ensemble elements differ", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "from each other in their initialisation, which is sampled from the He initialisation distribution", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 439 + ], + "score": 1.0, + "content": "(He et al., 2015). We do not use adversarial training as, inline with Ashukha et al. (2020b), we", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "do not find it to improve results. For ‘VOGN’ we use the same procedure and hyper-parameters", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 448, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 460 + ], + "score": 1.0, + "content": "as used by Osawa et al. 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See the authors’ GitHub for", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 494 + ], + "score": 1.0, + "content": "more details: github.com/team-approx-bayes/dl-with-bayes/blob/master/", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 487, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 487, + 505 + ], + "score": 1.0, + "content": "distributed/classification/configs/cifar10/resnet18_vogn_bs256_", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 504, + 166, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 166, + 516 + ], + "score": 1.0, + "content": "8gpu.json.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 349, + 507, + 516 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 431, + 531 + ], + "score": 1.0, + "content": "We modify the standard ResNet-50 and ResNet-18 architectures such that the first", + "type": "text" + }, + { + "bbox": [ + 432, + 520, + 455, + 531 + ], + "score": 0.88, + "content": "7 \\times 7", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 520, + 505, + 531 + ], + "score": 1.0, + "content": "convolution", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 530, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 177, + 544 + ], + "score": 1.0, + "content": "is replaced with a", + "type": "text" + }, + { + "bbox": [ + 178, + 531, + 201, + 541 + ], + "score": 0.9, + "content": "3 \\times 3", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 530, + 506, + 544 + ], + "score": 1.0, + "content": "convolution. 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(2017), we zero-initialise the last batch normalisation layer in residual blocks so that they", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 552, + 294, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 294, + 566 + ], + "score": 1.0, + "content": "act as identity functions at the start of training.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 520, + 506, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 569, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "At test time, we tune the prior precision used for ‘Ours’, ‘Diag-Laplace’ and ‘SWAG’ approximation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 581, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 592 + ], + "score": 1.0, + "content": "on a validation set for each approach individually, as in Ritter et al. (2018); Kristiadi et al. (2020).", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 210, + 604 + ], + "score": 1.0, + "content": "We use a grid search from", + "type": "text" + }, + { + "bbox": [ + 211, + 591, + 250, + 602 + ], + "score": 0.91, + "content": "1 \\times \\bar { 1 } 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 591, + 261, + 604 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 261, + 591, + 294, + 602 + ], + "score": 0.92, + "content": "1 \\times 1 0 ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "in logarithmic steps, and then a second, finer-grained", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 603, + 438, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 438, + 614 + ], + "score": 1.0, + "content": "grid search between the two best performing values (again with logarithmic steps).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 569, + 506, + 614 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 632, + 177, + 643 + ], + "lines": [ + { + "bbox": [ + 106, + 631, + 179, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 179, + 644 + ], + "score": 1.0, + "content": "C.4 DATASETS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "The 1d toy dataset used in Section 5.1 was taken from Antoran et al. (2020). We obtained it from the ´", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 666, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 677 + ], + "score": 1.0, + "content": "authors’ github repo: https://github.com/cambridge-mlg/DUN. Table 22 summarises", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "the datasets used in Section 5.2. Wine and Protein are available from the UCI dataset repository Dua", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 176, + 700 + ], + "score": 1.0, + "content": "& Graff (2017).", + "type": "text" + }, + { + "bbox": [ + 176, + 688, + 211, + 699 + ], + "score": 0.34, + "content": "\\operatorname { K i n } 8 \\mathrm { n m }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "is available from https://www.openml.org/d/189 Foong et al.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 392, + 711 + ], + "score": 1.0, + "content": "(2019b). 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DatasetN TrainN Val (15% train)NTestSplitsOutput Dim Output TypeInput DimInput Type
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Wine Gap906161532111Continous11Continous
Kin8nm62671106819201Continous8Continous
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NAMESIZEINPUT DIM.No. CLASSESNO. SPLITS
MNIST (LeCun et al., 1998)70,000 (60.000&10,000)784(28 × 28)102
Fashion-MNIST (Xiao et al.,2017)70,000 (60.000 &10.000)784(28 × 28)102
CIFAR10(Krizhevsky & Hinton,2009)60,000 (50.000&10.000)3072(32 ×32 ×3)102
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DatasetN TrainN Val (15% train)NTestSplitsOutput Dim Output TypeInput DimInput Type
Wine1223216160201Continous11Continous
Wine Gap906161532111Continous11Continous
Kin8nm62671106819201Continous8Continous
Kin8nm Gap4642820273081Continous8Continous
Protein349836174457351Continous9Continous
Protein Gap2591345731524491Continous9Continous
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NAMESIZEINPUT DIM.No. CLASSESNO. SPLITS
MNIST (LeCun et al., 1998)70,000 (60.000&10,000)784(28 × 28)102
Fashion-MNIST (Xiao et al.,2017)70,000 (60.000 &10.000)784(28 × 28)102
CIFAR10(Krizhevsky & Hinton,2009)60,000 (50.000&10.000)3072(32 ×32 ×3)102
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-2.10±0.03-3.85±0.18-3.54±0.23-4.11±0.66-3.60±0.10-4.29±0.21-2.95±0.08-1.92±nan
error0.63±0.010.63±0.010.62±0.010.62±0.050.61±0.010.63±0.010.53±0.020.64±nan
ECE0.25±0.020.46±0.020.43±0.020.47±0.060.42±0.010.48±0.020.36±0.020.17±nan
brier score0.85±0.021.04±0.031.00±0.031.05±0.100.98±0.021.07±0.030.86±0.030.80±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-0.42±0.04-0.36±0.01-0.32±0.02-0.44±0.06-0.28±0.02-0.39±0.01-0.30±0.00-0.51±nan
error0.11±0.010.10±0.000.09±0.010.12±0.010.08±0.010.10±0.000.08±0.000.14±nan
ECE0.10±0.020.04±0.010.03±0.000.06±0.010.02±0.000.05±0.000.04±0.000.13±nan
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OURSOURS (RAND)DROPOUTDIAG-LAPENSEMBLEMAPSWAGVOGN
LL-3.42±0.12-7.01±0.15-7.08±0.39-7.80±0.12−7.51±0.11-7.91±0.18-6.63±0.24-2.67±nan
error0.58±0.010.58±0.010.58±0.010.58±0.000.57±0.010.58±0.010.59±0.000.56±nan
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brier score0.90±0.021.05±0.011.04±0.011.05±0.011.03±0.011.07±0.021.03±0.010.82±nan
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LL-1.06±0.02-2.06±0.12-1.85±0.07-2.13±0.17-1.28±0.03-2.18±0.08-1.63±0.03-1.83±nan
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DatasetN TrainN Val (15% train)NTestSplitsOutput Dim Output TypeInput DimInput Type
Wine1223216160201Continous11Continous
Wine Gap906161532111Continous11Continous
Kin8nm62671106819201Continous8Continous
Kin8nm Gap4642820273081Continous8Continous
Protein349836174457351Continous9Continous
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NAMESIZEINPUT DIM.No. CLASSESNO. SPLITS
MNIST (LeCun et al., 1998)70,000 (60.000&10,000)784(28 × 28)102
Fashion-MNIST (Xiao et al.,2017)70,000 (60.000 &10.000)784(28 × 28)102
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SVHN (Netzer et al., 2011)99,289 (73,257& 26,032)3072(32 ×32 × 3)102
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This may result in the whole training process (initialization/pre-training $^ +$ fine-tuning) to be very timeconsuming. In this paper, we combine the ideas of ensemble learning and deep learning, and present a novel deep learning framework called deep error-correcting output codes (DeepECOC). DeepECOC are composed of multiple layers of the ECOC module, which combines multiple binary classifiers for feature learning. Here, the weights learned for the binary classifiers can be considered as weights between two successive layers, while the outputs of the combined binary classifiers as the outputs of a hidden layer. On the one hand, the ECOC modules can be learned using given supervisory information, and on the other hand, based on the ternary coding design, the weights can be learned only using part of the training data. Hence, the supervised pre-training of DeepECOC is in general very effective and efficient. We have conducted extensive experiments to compare DeepECOC with traditional ECOC, feature learning and deep learning algorithms on several benchmark data sets. The results demonstrate that DeepECOC perform not only better than traditional ECOC and feature learning algorithms, but also state-ofthe-art deep learning models in most cases. + +# 1 INTRODUCTION + +Error correcting output codes (ECOC) are an ensemble learning framework to address multi-class classification problems (Dietterich & Bakiri, 1995). The work by (Zhong & Liu, 2013) shows that the ECOC methods can also be used for feature learning, in either a linear or a nonlinear manner. However, although sophisticated coding and decoding strategies are applied (Escalera et al., 2010; Zhong et al., 2012; Zhong & Cheriet, 2013), the learnability of ECOC is limited by its singlelayer structure. Therefore, to exploit the advantages of the ECOC framework, such as supervised ensemble learning and effective coding design, it’s necessary to combine its ideas with that of deep learning. + +In recent years, many deep learning models have been proposed to handle various challenging problems. Meantime, desirable performances in many domains have been achieved, such as image classification and detection, document analysis and recognition, natural language processing, and video analysis (Hinton & Salakhutdinov, 2006; Krizhevsky et al., 2012; Szegedy et al., 2014; Simonyan & Zisserman, 2014; Zhang et al., 2015; Wang & Ji, 2015; Hong et al., 2015). Among others, (Hinton & + +Salakhutdinov, 2006) presents the ground-breaking deep autoencoder that learns the weight matrices by pre-training the stacked restricted Boltzmann machines (RBMs) and fine-tuning the weights using gradient descent. It delivers much better representations of data than shallow feature learning algorithms, such as principal components analysis (PCA) (Jolliffe, 1986) and latent semantic analysis (LSA) (Deerwester et al., 1990). In order to boost the traditional autoencoder and prevent the “overfitting” problem, (Vincent et al., 2008) introduces the denosing autoencoder that corrupted the data with a random noise. Recently, most of the research focuses on deep convolutional neural networks (CNNs) and recurrent neural networks (RNNs), which greatly improves the state-of-the-art in the areas of object recognition, unsegmented handwriting recognition and speech recognition (Krizhevsky et al., 2012; Graves et al., 2009; Sak et al., 2014). However, existing deep networks are generally initialized with unsupervised methods, such as random assignments and greedy layerwise pre-training. In the case of random initialization, to obtain good results, many training data and a long training time are generally used; while in the case of greedy layerwise pre-training, as the whole training data set needs to be used, the pre-training process is very time-consuming and difficult to find a stable solution. + +To overcome the limitations of both traditional ECOC methods and deep learning models, and meanwhile, take advantages of both of them, in this paper, we propose a novel deep learning model called deep error-correcting output codes (DeepECOC). DeepECOC are composed of multiple stacked ECOC modules, each of which combines multiple binary classifiers for feature learning. Here, the weights learned for the binary classifiers can be considered as weights between two successive layers, while the probabilistic outputs of the combined binary classifiers as the outputs of a hidden layer or new representations of data. On the one hand, the ECOC modules can be learned layer by layer using the given supervisory information, and on the other hand, based on the ternary coding design, some classes of data are automatically neglected when training the binary classifiers, such that the weights are learned only using part of the training data. Hence, the supervised pre-training of DeepECOC is in general very effective and efficient. We have compared DeepECOC with traditional ECOC, feature learning and deep learning algorithms to demonstrate the effectiveness and superiority of DeepECOC. The results are reported in Section 4. + +The rest of this paper is organized as follows: In Section 2, we give a brief overview to related work. In Section 3, we present the proposed model, DeepECOC, in detail. The experimental results are reported in Section 4, while Section 5 concludes this paper with remarks and future work. + +# 2 RELATED WORK + +Traditional ECOC framework has two steps: coding and decoding. In the coding step, an ECOC matrix is defined or learned from data, and the binary classifiers are trained based on the ECOC coding; in the decoding step, the class label is given to a test sample based on a similarity measure between codewords and outputs of the binary classifiers. The widely used coding strategies include one-versus-all (OneVsAll) (Nilsson, 1965), one-versus-one (OneVsOne) (Hastie et al., 1998), discriminant ECOC (DECOC) (Pujol et al., 2006), ECOC optimizing node embedding (ECOCONE) (Escalera et al., 2006), dense and sparse coding (Escalera et al., 2009; Allwein et al., 2001), and so on. Among them, the OneVsAll, OneVsOne, dense and sparse coding strategies are problem-independent, whilst the DECOC and ECOCONE are problem-dependent. Generally, the length of the codeword by the OneVsAll, OneVsOne, DECOC and ECOCONE coding designs is related to the number of classes, but that by the dense and sparse coding design is relatively flexible. In this work, we design the structure of DeepECOC based on the properties of each coding strategy. The commonly used binary ECOC decoding strategies are the Hamming decoding (Nilsson, 1965) and Euclidean decoding (Hastie et al., 1998). For ternary ECOC decoding strategies, the attenuated Euclidean decoding (Pujol et al., 2008), loss-based decoding (Allwein et al., 2001), and probabilistic-based decoding (Passerini et al., 2004) are widely used. Currently, the state-of-the-art ternary ECOC decoding strategies are the discrete pessimistic beta density distribution decoding and loss-weighted decoding (Escalera et al., 2010). In this work, for the simplicity of back propagation, we directly add a Softmax layer at the top of DeepECOC for the decoding. Note that, although many sophisticated coding and decoding strategies have been proposed in recent years (Escalera et al., 2010; Zhong et al., 2012; Zhong & Cheriet, 2013), the learnability of ECOC is limited by its single-layer structure. To further exploit the advantages of ECOC, such as supervised ensemble learning and effective coding design, it’s necessary to combine its ideas with that of deep learning. + +![](images/86fe1fb2b334c231e08ad95e358eeac2a15ba70ca951e0e93d5b04445c7aad57.jpg) +Figure 1: Two coding matrices encoded with the one-versus-all (binary case) and one-versus-one (ternary case) coding strategies. + +In the literature of deep learning, there is some work that attempts to construct a deep architecture with multiple feature learning methods (Hinton & Salakhutdinov, 2006; Trigeorgis et al., 2014; Yuan et al., 2015; Zheng et al., 2015; 2014). For instance, deep autoencoder is built up by RBMs (Hinton & Salakhutdinov, 2006), and deep semi-NMF combines multiple steps of matrix factorization (Trigeorgis et al., 2014). Similarly, deep CNNs and RNNs can also be considered as deep models that learn the new representations of data layer by layer (Krizhevsky et al., 2012; Graves et al., 2009; Sak et al., 2014). The success of these existing models demonstrate that deep networks are beneficial to the representation learning tasks, especially for the large scale applications. However, as discussed in the previous section, existing deep learning models are generally initialized with unsupervised methods, such as random assignments and greedy layerwise pre-training, which result in a long training time of the deep models. In this work, we propose the DeepECOC model, which is based on the stacked ECOC modules. When pre-training DeepECOC, the ECOC modules can be learned with the available supervisory information. Intuitively, as this manner of supervised pre-training has deterministic objective, the learned value of the parameters will be very close to the best local minimum on the solution manifold. Experimental results shown in Section 4 also demonstrate this fact. + +# 3 DEEP ERROR-CORRECTING OUTPUT CODES (DEEPECOC) + +In this section, we first introduce the traditional ECOC framework, which is the important building block of DeepECOC. Then we present the learning procedures of DeepECOC in detail. + +# 3.1 THE ECOC FRAMEWORK + +Error correcting output codes (ECOC), which combine multiple binary classifiers to solve multiclass classification problems, are an ensemble learning framework. The ECOC methods in general consist of two steps: coding and decoding. In the coding step, the ECOC coding matrix $\mathbf { M } \in \{ - 1 , 1 \} ^ { C \times L }$ (binary case) or $\mathbf { M } \in \{ - 1 , \mathbf { \bar { 0 } } , 1 \} ^ { C \times L }$ (ternary case) is first defined or learned from the training data, where each row of $\mathbf { M }$ is the codeword of a class, each column corresponds to a dichotomizer (binary classifier), $L$ is the length of the codewords (the number of binary classifiers), $C$ is the number of classes, symbol ‘1’ indicates positive class, $^ { \bullet } - _ { 1 } \bullet$ indicates negative class, and $\cdot _ { 0 } \cdot \mathrm { ~ }$ indicates that a particular class is not considered by a given classifier. Then, the binary classifiers (dichotomizers) are trained according to the partition of the classes in the columns of M. Fig. 1 shows two coding matrices encoded with the one-versus-all (binary case) and one-versus-one (ternary case) coding strategies. The matrix is coded using several dichotomizers for a 4-class problem with respective codewords $\{ y _ { 1 } , \ldots , y _ { 4 } \}$ . The white girds are coded by 1 (considered as positive class by the respective dichotomizer $h _ { j }$ ), the dark girds by $^ { - 1 }$ (considered as the negative class), and the gray girds by 0 (classes that are not considered by the respective dichotomizer $h _ { j }$ ). In the decoding step, the test data are predicted based on an adopted decoding strategy and the outputs of the binary classifiers. + +In order to take the probabilistic outputs of the base classifiers as new representations of data, we adopt linear support vector machines (linear SVMs) as the binary classifiers (dichotomizers), which solve a quadratic programming problem + +$$ +\begin{array} { c l l } { \displaystyle \operatorname* { m i n } _ { \mathbf { w } , b , \xi _ { i } } J ( \mathbf { w } ) = \frac { 1 } { 2 } \| \mathbf { w } \| ^ { 2 } + C \displaystyle \sum _ { i = 1 } ^ { N } \xi _ { i } } \\ { s . t . } & { \ y _ { i } f ( \mathbf { x } _ { i } ) \geq 1 - \xi _ { i } , \xi _ { i } \geq 0 , i = 1 , \ldots , N } \end{array} +$$ + +where w and $b$ are the coefficients and bias of the binary classifier, $y _ { i } \in \{ + 1 , - 1 \}$ , $\xi _ { i }$ ’s are the slack variables, and $N$ is the number of the training data. The discriminant function can be expressed as + +$$ +f ( \mathbf { x } ) = \mathbf { w } ^ { T } \mathbf { x } + b . +$$ + +This problem can be solved as, + +$$ +\mathbf { w } = \sum _ { i = 1 } ^ { N } \alpha _ { i } y _ { i } x _ { i } , +$$ + +$$ +b = \frac { 1 } { N _ { S \mathcal { V } } } \sum _ { { \bf x } _ { i } \in S \mathcal { V } , i = 1 } ^ { N _ { S \mathcal { V } } } ( y _ { i } - { \bf w } ^ { T } { \bf x } _ { i } ) , +$$ + +where $\alpha _ { i }$ ’s are the non-negative Lagrange multipliers, $N _ { s v }$ is the number of support vectors and $\mathit { s v }$ is the set of support vectors. The dual form of Problem (1) can be written as + +$$ +\begin{array} { r l } { \displaystyle \operatorname* { m a x } _ { \alpha } } & { \displaystyle \sum _ { i = 1 } ^ { N } \alpha _ { i } - \frac { 1 } { 2 } \sum _ { i , j = 1 } ^ { N } \alpha _ { i } \alpha _ { j } y _ { i } y _ { j } \mathbf { x } _ { i } ^ { T } \mathbf { x } _ { j } } \\ { = } & { \displaystyle \sum _ { i = 1 } ^ { N } \alpha _ { i } - \frac { 1 } { 2 } \sum _ { i , j = 1 } ^ { N } \alpha _ { i } \alpha _ { j } y _ { i } y _ { j } k ( \mathbf { x } _ { i } , \mathbf { x } _ { j } ) } \\ { s . t . } & { 0 \le \alpha _ { i } \le \lambda , i = 1 , \dots , N , } \\ { \displaystyle } & { \displaystyle \sum _ { i = 1 } ^ { N } \alpha _ { i } y _ { i } = 0 , } \end{array} +$$ + +where $k ( \mathbf { x } _ { i } , \mathbf { x } _ { j } ) ~ = ~ \mathbf { x } _ { i } ^ { T } \mathbf { x } _ { j }$ is the linear kernel function, $\lambda$ is a constant number, and $\alpha =$ $\{ \alpha _ { 1 } , \dotsc , \alpha _ { N } \}$ is the vector of Lagrange multipliers. Replacing the linear kernel function with a nonlinear kernel, such as the Gaussian kernel + +$$ +k ( \mathbf { x } _ { i } , \mathbf { x } _ { j } ) = \exp ( - \sigma ^ { - 1 } \| \mathbf { x } _ { i } - \mathbf { x } _ { j } \| ^ { 2 } ) , +$$ + +we can learn a nonlinear SVM, where $\sigma$ is the parameter for the Gaussian kernel function. The discriminant function of SVMs with a nonlinear kernel can be written as + +$$ +f ( \mathbf { x } ) = \sum _ { i = 1 } ^ { N } \alpha _ { i } y _ { i } k ( \mathbf { x } _ { i } , \mathbf { x } ) + b . +$$ + +Applying a decoding strategy on the outputs of the binary classifiers, the ECOC framework can be used for multi-class learning, while applying the sigmoid function on the values of the discriminant function, ECOC can be used for feature learning (Zhong & Liu, 2013). This is also the foundation of the DeepECOC model. + +# 3.2 DEEPECOC + +To combine the advantages of ECOC and deep learning algorithms, we build the DeepECOC architecture as follows + +$$ +\mathbf { x } \ { \overset { q _ { D } } { \longrightarrow } } \ { \tilde { \mathbf { x } } } \ { \overset { \mathbf { W } _ { 1 } } { \underset { \mathbf { b } _ { 1 } } { \longrightarrow } } } \ \mathbf { h } _ { 1 } \ { \overset { \mathbf { W } _ { 2 } } { \underset { \mathbf { b } _ { 2 } } { \longrightarrow } } } \ \cdot \cdot \cdot \ { \overset { \mathbf { W } _ { n - 1 } } { \underset { \mathbf { b } _ { n - 1 } } { \longrightarrow } } } \ \mathbf { h } _ { n - 1 } \ { \overset { s o f t m a x } { \longrightarrow } } \ \mathbf { y } , +$$ + +where the first step makes the clean input $\mathbf { x } \in [ 0 , 1 ] ^ { d }$ partially destroyed by means of a stochastic mapping $\tilde { \textbf { x } } \sim q _ { D } ( \tilde { \textbf { x } } | \textbf { x } )$ . In the corrupting process, we set a parameter called denoising rate $\nu$ . For each input $\mathbf { x }$ , a fixed number $\nu d$ of components are chosen at random, and their value is forced to 0, while the others are left untouched. This operation makes the model more robust and prevent the overfitting problem in most cases (Vincent et al., 2008). Subsequently, the “corrupted” data are taken as inputs for the DeepECOC model. $\mathbf { W } _ { 1 }$ and $\mathbf { b } _ { 1 }$ are the weight matrix and bias vector learned from the first ECOC module. The output of the first hidden layer is denoted as + +$$ +\mathbf { h } _ { 1 } = s ( \mathbf { W } _ { 1 } ^ { T } \mathbf { x } + \mathbf { b } _ { 1 } ) , +$$ + +where $s ( \cdot )$ is the sigmoid activation function $\begin{array} { r } { s ( x ) = \frac { 1 } { 1 + e ^ { - x } } } \end{array}$ . From the second layer to the $( n - 1 )$ - th layer, we use the stacked ECOC modules to learn the weight matrices and biases, which can be considered as weights between two successive layers of a deep network. Similarly, we use the output of the $\left( k - 1 \right)$ -th layer as the input of the $k$ -th layer, + +$$ +\mathbf { h } _ { k } = s ( \mathbf { W } _ { k } ^ { T } \mathbf { h } _ { k - 1 } + \mathbf { b } _ { k } ) . +$$ + +Here, $\mathbf { h } _ { k }$ can be viewed as an activation output and a new representation of the input datum $\mathbf { x }$ . + +For example, if we adopt the OneVsAll coding strategy for one layer of the ECOC module, we first define the coding matrix $\mathbf { M } ^ { C \times C }$ , where $C$ is the number of classes. Then, we can train $C$ SVM classifiers to obtain the weight matrix $\mathbf { W } = \left\{ \mathbf { w } _ { 1 } , \dots , \mathbf { w } _ { i } , \dots , \mathbf { w } _ { C } \right\}$ and the bias $\mathbf { b } = \{ b _ { 1 } , \dots , b _ { i } , \dots , b _ { C } \}$ . Next, we calculate the output of the first layer by using Eq. (9). Subsequently, we repeat this process layer by layer to build the DeepECOC model. It’s obvious that, if we adopt different coding strategies, we can get different kinds of DeepECOC architectures. + +For the last layer of DeepECOC, we employ the softmax regression for the multi-class learning. Its cost function is defined as + +$$ +J ( \mathbf { w } ) = - \frac { 1 } { N } ( \sum _ { i = 1 } ^ { N } \sum _ { j = 1 } ^ { K } \mathbf { I } ( y _ { i } = j ) \log \frac { \exp ( \mathbf { w } _ { j } ^ { T } \mathbf { h } _ { i } ^ { n - 1 } ) } { \sum _ { l = 1 } ^ { K } \exp ( \mathbf { w } _ { l } ^ { T } \mathbf { h } _ { i } ^ { n - 1 } ) } ) , +$$ + +where $\mathbf { I } ( x )$ is the indicator function, $\mathbf { I } ( x ) = 1$ if $x$ is true, else $\mathbf { I } ( x ) = 0$ . $y _ { i }$ is the label corresponding to $\mathbf { x } _ { i }$ . It’s easy to compute the probability that $\mathbf { x } _ { i }$ is classified to class $j$ , + +$$ +p ( y _ { i } = j | \mathbf { x } _ { i } , \mathbf { w } ) = \frac { \exp ( \mathbf { w } _ { j } ^ { T } \mathbf { h } _ { i } ^ { n - 1 } ) } { \sum _ { l = 1 } ^ { K } \exp ( \mathbf { w } _ { l } ^ { T } \mathbf { h } _ { i } ^ { n - 1 } ) } . +$$ + +Taking derivatives, one can show that the gradient of $J ( \mathbf { w } )$ with respect to w is, + +$$ +\nabla J ( \mathbf { w } ) = - \frac { 1 } { N } \sum _ { i = 1 } ^ { N } [ \mathbf { x } _ { i } ( \mathbf { I } ( y _ { i } = j ) - p ( y _ { i } = j | \mathbf { x } _ { i } , \mathbf { w } ) ) ] . +$$ + +After the pre-training step, we use back propagation (Rumelhart et al., 1988) to fine tune the whole architecture. Moreover, we also employ a technique called “dropout” for regularization (Hinton et al., 2012). When a large feedforward neural network is trained on a small training set, dropout generally performs well on the test set. The basic idea of dropout is that each hidden node is randomly omitted from the network with a probability of $\beta$ . In another view, dropout is a very efficient way to perform model averaging with neural networks. Through these processes, we finally obtain the DeepECOC model, which is robust and easy to be applied to multi-class classification tasks. + +Note that, compared to existing deep learning algorithms, DeepECOC have some important advantages. Firstly, unlike previous deep learning algorithms, DeepECOC are built with the ECOC modules and pre-trained in a supervised learning fashion. Secondly, if we adopt ternary coding strategies, due to the natural merit of ECOC, the weights can be learned using only part of the training data. Thirdly, in contrast to the learning of the weight matrices in previous deep learning models, the binary classifiers in each ECOC module can be learned in parallel, which may greatly speed up the learning of DeepECOC. + +# 4 EXPERIMENTS + +To evaluate the effectiveness of the proposed method, DeepECOC, we conducted 4 parts of experiments. In the first part, we compared DeepECOC with some deep learning models and single-layer ECOC approaches on 16 data sets from the UCI machine learning repository 1. In the second part, we compared DeepECOC with traditional feature learning models, some deep learning models and single-layer ECOC approaches on the USPS handwritten digits 2, and tested DeepECOC with different number of hidden layers. In the third part, we used the MNIST handwritten digits 3 to further demonstrate the effectiveness of DeepECOC for handwritten digits recognition. Finally, the CIFAR10 data set 4 was used to demonstrate the effectiveness of DeepECOC on image classification tasks. For all the data sets, the features were normalized within $[ 0 , 1 ]$ . In the following, we report the experimental results in detail. + +Table 1: Details of the UCI data sets (T: training samples; A: attributes; C: classes). + +
Problem#ofT#ofA#ofCProblem#ofT#ofAofC
Dermatology366346Yeast1484810
Iris15043Satimage6435367
Ecoli33688Letter200001626
Wine178133Pendigits109921610
Glass21497Segmentation2310197
Thyroid21553Optdigits56206410
Vowel9901011Shuttle1450097
Balance62543Vehicle846184
+ +Table 2: Classification accuracy and standard deviation obtained by DeepECOC and the compared approaches on 16 UCI data sets. Here, DeepECOC(1) $\sim$ DeepECOC(3) are 3 variant of DeepECOC with the ECOCONE coding design initialized by one-versus-one, one-versus-all and DECOC respectively. The best results are highlighted in boldface. + +
ProblemSingleAEDAEDeepECOC (1)DeepECOC (2)DeepECOC (3）
Dermatology0.95130.9429 ±0.06710.9674 ±0.03120.9702 ±0.03540.9779 ±0.02080.9747 ±0.0318
Iris0.96000.9600 ±0.05620.9333 ±0.08890.9600 ±0.05350.9267 ±0.11090.9533 ±0.0383
Ecoli0.81470.7725 ±0.06080.8000 ±0.03620.8529 ±0.04030.8824 ±0.06260.9118 ±0.0636
Wine0.96050.9765 ±0.02640.9563 ±0.04220.9875 ±0.02640.9813 ±0.03020.9688 ±0.0329
Glass0.67620.6669 ±0.10320.6669 ±0.07150.7895 ±0.07880.7368 ±0.11400.7562 ±0.0879
Thyroid0.92100.9513 ±0.06140.9599 ±0.05670.9656 ±0.05130.9703 ±0.05400.9608 ±0.0518
Vowel0.71770.6985 ±0.07450.7101 ±0.07560.7475 ±0.09010.6010 ±0.06270.6863 ±0.0788
Balance0.82220.8036 ±0.03200.8268 ±0.05480.9137 ±0.04120.8333 ±0.03180.9167 ±0.0312
Yeast0.52170.5641 ±0.03460.5891 ±0.02720.5959 ±0.05990.5494 ±0.04340.5697 ±0.0462
Satimage0.85370.8675 ±0.0528 0.92340.8897 ±0.03040.8961 ±0.04800.8360 ±0.03900.9077 ±0.0555
Letter0.9192±0.0547 0.98310.9381 ±0.0641 0.98860.9532 ±0.03410.9247 ±0.03520.9501 ±0.0563
Pendigits0.9801±0.0123 0.9584±0.0034 0.95960.9908 ±0.0031 0.97110.9866 ±0.0107 0.95840.9899 ±0.0075 0.9711
Segmentation0.9701±0.0317 0.9785±0.0211 0.9856±0.0286 0.9867±0.0163 0.9848±0.0233 0.9911
Optdigits0.9982±0.0101 0.9953±0.0088 0.9976±0.0096 0.9988±0.0123 0.9983±0.0091 0.9993
Shuttle0.9988±0.0012 0.6987±0.0014 0.7348±0.0021 0.7561±0.0018 0.6908±0.0010 0.7195
Vehicle Mean rank0.7315 4.0938±0.0521 4.8750±0.0454 3.9375±0.0480 1.7500±0.04321 3.9375±0.0148 2.4063
+ +# 4.1 CLASSIFICATION ON 16 UCI MACHINE LEARNING REPOSITORY DATA SETS + +The detail of the UCI data sets are shown in Table 1. In these experiments, we compared DeepECOC with autoencoder (AE) (Hinton & Salakhutdinov, 2006), denoising autoencoder (DAE) (Vincent et al., 2008) and single-layer ECOC approaches (Single) (Escalera et al., 2010). We built DeepECOC with the ECOC optimizing node embedding (ECOCONE) coding method (Escalera et al., 2006). Here, since we initialized ECOCONE with 3 different coding methods, i.e. one-versus-one, oneversus-all and DECOC, DeepECOC had 3 variants. In addition, the state-of-the-art linear lossweighted (LLW) decoding strategy was used for ECOCONE. Finally, a structure with 3 hidden layers was adopted for DeepECOC, which had 0.1 denoising rate and 0.1 dropout rate: + +$$ +\begin{array} { r } { \textbf { x } \overset { q _ { D } } { \longrightarrow } \tilde { \textbf { x } } \xrightarrow { \textbf { W } _ { 1 } } \textbf { h } _ { 1 } \xrightarrow { \textbf { W } _ { 2 } } \textbf { h } _ { 2 } \xrightarrow { \textbf { W } _ { 3 } } \textbf { h } _ { 3 } \xrightarrow { s o f t m a x } \textbf { y } . } \end{array} +$$ + +For the fine-tuning process, we used the stochastic gradient descent algorithm. The learning rate and epoches from different data sets are described in Table 3. The autoencoder and denoising autoencoder’s architectures are as same as DeepECOC with ECOCONE initialized by one-versus-one. For single-layer ECOC approaches, we chose the best results shown in (Escalera et al., 2010) as our compared results. For all DeepECOC models, we used support vector machines (SVMs) with RBF kernel function as base classifiers. The parameters of SVMs were set to default (Chang & Lin, 2011). + +Table 2 shows the average classification accuracy and standard deviation on 16 UCI data sets. Except on the OptDigits data set, DeepECOC achieved the best results compared with autoencoder, denoising autoencoder and single-layer ECOC approaches. In fact, on the OptDigits data set, DeepECOC achieved comparative result with single-layer ECOC approaches. Among others, DeepECOC with ECOCONE (initialized by one-versus-one) coding strategy obtained the best results on 9 data sets, while DeepECOC with ECOCONE (initialized by DECOC) coding strategy obtained the best results on 5 data sets. From the mean rank values, we can see that DeepECOC with ECOCONE (initialized by one-versus-one and DECOC) strategy far surpass other compared methods. + +Table 3: Details of the learning rate $\eta$ and epoch on the UCI data sets. + +
ProblemmEpochProblemnEpoch
Dermatology0.12000Yeast0.014000
Iris0.1400Satimage0.014000
Ecoli0.12000Letter0.018000
Wine0.12000Pendigits0.012000
Glass0.014000Segmentation0.018000
Thyroid0.1800Optdigits0.012000
Vowel0.14000Shuttle0.12000
Balance0.14000Vehicle0.14000
+ +# 4.2 CLASSIFICATION ON THE USPS DATA SET + +The USPS handwritten digits data set includes 7291 training samples and 2007 test samples from 10 classes. The size of the images is $1 6 \times 1 6 = 2 5 6$ . Our experiments on this data set were divided into 2 parts. Firstly, we compared DeepECOC with two traditional feature learning models (principal components analysis (PCA) (Jolliffe, 2002) and marginal Fisher analysis (MFA) (Yan et al., 2007)), autoencoder (AE), denoising autoencoder (DAE), LeNet (LeCun et al., 1998), PCANet (Chan et al., 2015) and single-layer ECOC approaches. Here, PCA is an unsupervised method, MFA is a supervised method. For MFA, the number of nearest neighbors for constructing the intrinsic graph was set to 5, while that for constructing the penalty graph was set to 15. For DeepECOC, we also used 3 coding design methods in this experiment. We used batch gradient descent for the fine-tuning process, the batch size was set to 100, the learning rate was set to 1, the number of epoch was set to 40000, the denoising rate, and dropout rate were set to 0.1. We also used SVMs with RBF kernel and default parameters as base classifiers. For single-layer ECOC approaches, we adopted ECOCONE (initialized by one-versus-one) as coding design method and linear loss-weighted (LLW) decoding strategy. For the LeNet model, we used 2 convolutional layers, two pooling layers and two fully connected layers. The kernel size of the convolutional layers and pooling layers was set to $2 \times 2$ , the stride was set to 1, the number of nodes of the first layer was set to 200, the epoch was set to 8000, the initial learning rate was set to 0.001, learning rate policy was set to “inv”, and the momentum was set to 0.9. For the PCANet model, we used two PCA-filter stages, one binary hashing stage and one blockwise histograms. The filter size, the number of filters, and the block size were set to $k _ { 1 } = k _ { 2 } = 3$ , $L _ { 1 } = L _ { 2 } = 4$ , and $7 \times 7$ , respectively. The experimental results are shown in Fig. 2(a). + +From Fig. 2(a), we can see that DeepECOC with ECOCONE (initialized by one-versus-one) coding strategy achieved the best result than other methods include traditional feature learning models, existing deep learning methods and single-layer ECOC approaches. + +In the second part, we evaluated DeepECOC with different number of hidden layers. We used 2 to 6 hidden layers in our experiments. The parameter settings were as same as the first part. Fig. 2(b) + +![](images/69601db0f9abe1f64fba996cd93599b01ce074f8d2c7529a048c62226e3571e4.jpg) +Figure 2: (a) Classification accuracy obtained on the USPS data set. Here, DeepE ${ \mathrm { C O C } } ( 1 ) \sim$ DeepECOC(3) are 3 variant of DeepECOC with the ECOCONE coding design initialized by one-versusone, one-versus-all and DECOC respectively. (b) Classification accuracy with different numbers of hidden layers on the USPS data set. + +![](images/04d6c3cab8ccbbf66a6ef3819036db6e0b5b4cd06a21498a052a3052377c8fef.jpg) +Figure 3: Classification accuracy obtained on the MNIST data set for two architectures. + +shows the experimental results. We can see that DeepECOC obtained the best result when using 3 hidden layers. When the number of hidden layers is less than 3, the effectiveness of DeepECOC increases with the increasing of the number of hidden layers. Along with the number of hidden layers continues to grow, the effectiveness of DeepECOC decreases. + +# 4.3 CLASSIFICATION ON THE MNIST DATA SET + +MNIST handwritten digits data set has a training set of 60,000 examples, and a test set of 10,000 examples with 784 dimensional features. We designed 2 architectures for autoencoder, denoising autoencoder and DeepECOC. The first architecture was $7 8 4 - Z _ { 1 } - Z _ { 2 } - Z _ { 3 } - 1 0$ , where $Z _ { i }$ was the number of hidden neurons designed based on some ECOC coding strategies. We designed this architecture because we wanted to make autoencoder and denoising autoencoder had the same structure with DeepECOC. The second architecture is $7 8 4 - 5 0 0 - 5 0 0 - 2 0 0 0 - 1 0$ . This architecture was used in (Hinton & Salakhutdinov, 2006). In order to make DeepECOC adapt to this structure, we used the dense and sparse coding design methods that can control the codeword length. Note that, the dense and sparse coding design methods are totally random and data-independent. The denoising rate and dropout rate were set to 0.1, the batch size was set to 100, the learning rate was set to 0.01, and the number of epoch was set to 80000. For LeNet model, we adopted the parameters as same as (LeCun et al., 1998). For PCANet model, we used two PCA-filter stages, one binary hashing stage and one blockwise histograms. In the PCANet, the filter size, the number of filters, and the block size were set to $k _ { 1 } = k _ { 2 } = 8$ , $L _ { 1 } = L _ { 2 } = 7$ , and $7 \times 7$ , respectively. + +Fig. 3(a) and Fig. 3(b) show the experimental results on 2 architectures. We can see that DeepECOC are comparative with existing deep learning methods on the second architecture and outperform + +Table 4: Classification accuracy obtained on the LBP-CIFAR10 data set. The best result for each scenario is highlighted in bold face. + +
ProblemAEDAELeNetPCANetDeepECOC(1)DeepECOC(2)DeepECOC(3)
LBP-CIFAR10 (36)0.35010.36780.32560.25690.50890.45170.4752
LBP-CIFAR10 (256)0.43520.45870.32210.25690.55880.45890.5224
+ +them on the first architecture. In addition, DeepECOC with both two architectures outperform the single-layer ECOC approaches. + +# 4.4 CLASSIFICATION ON THE LBP-CIFAR10 DATA SET + +The CIFAR-10 dataset is a relative large scale data set which consists of $6 0 0 0 0 3 2 \times 3 2$ colour images in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images. For the purpose of reducing computational cost, we attempted to extract features of the data using an efficient local binary patterns algorithm. As a result, the representations with dimensionality 36 and 256 were adopted and the data were normalized to [0, 1] as well, called LBP-CIFAR10 (36) and LBP-CIFAR10 (256). We also used 3 hidden layers for all deep learning methods. The learning rate was set to 0.1, and the epoch was set to 4000. For the LeNet model, we used 2 convolutional layers and two fully connected layers without pooling layers. The kernel size was set to $2 \times 2$ , the stride was set to 1, the number of node of the first fully connected layer was set to 64, the epoch was set to 4000, the initial learning rate was set to 0.01, learning rate policy was set to “inv”, and the momentum was set to 0.9. For the PCANet model, we used two PCA-filter stages, one binary hashing stage and one blockwise histograms. In the PCANet, the filter size, the number of filters, and the block size were set to $k _ { 1 } = k _ { 2 } = 3$ , $L _ { 1 } = L _ { 2 } = 4$ , and $7 \times 7$ , respectively. The classification accuracy are reported in Table 4. + +From Table 4, we can easy to see that DeepECOC achieved the best results. Moreover, DeepECOC with ECOCONE (initialized by one-versus-one) coding strategy achieved the better results than autoencoder and denoising autoencoder, LeNet and PCANet. Hence, we can conclude that, DeepECOC are a general model to handle different real world applications and achieves desirable results in most cases. + +# 5 CONCLUSION + +In this paper, we propose a novel deep learning model, called deep error correcting output codes (DeepECOC). DeepECOC extend traditional ECOC algorithms to a deep architecture fashion, and meanwhile, brings new elements to the deep learning area, such as supervised initialization, and automatic neglecting of part of the data during network training. Extensive experiments on 16 data sets from the UCI machine learning repository, the USPS and MNIST handwritten digits and the CIFAR-10 data set demonstrate the superiority of DeepECOC over traditional ECOC, feature learning and deep learning methods. In future work, we will further exploit the learnability of DeepECOC on large scale applications. + +# REFERENCES + +E. L. Allwein, R. E. Schapire, and Y. Singer. 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Neural Computing and Applications, 21(4):715–724, 2012. \ No newline at end of file diff --git a/parse/train/Hkz6aNqle/Hkz6aNqle_content_list.json b/parse/train/Hkz6aNqle/Hkz6aNqle_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..faf9a3bc4475fe1252bde3966e671dab77c0d0ba --- /dev/null +++ b/parse/train/Hkz6aNqle/Hkz6aNqle_content_list.json @@ -0,0 +1,1013 @@ +[ + { + "type": "text", + "text": "DEEP ERROR-CORRECTING OUTPUT CODES ", + "text_level": 1, + "bbox": [ + 174, + 99, + 704, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Guoqiang Zhong \nDepartment of Computer Science and Technology \nOcean University of China \ngqzhong@ouc.edu.cn \nYuchen Zheng \nDepartment of Computer Science and Technology \nOcean University of China \nouczyc@outlook.com \nPeng Zhang \nDepartment of Computer Science and Technology \nOcean University of China \nsdrzbruce@163.com \nMengqi Li \nDepartment of International Trade and Economy \nOcean University of China \nenri9615@outlook.com \nJunyu Dong \nDepartment of Computer Science and Technology \nOcean University of China \ndongjunyu@ouc.edu.cn ", + "bbox": [ + 183, + 145, + 511, + 202 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 529, + 145, + 857, + 202 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 183, + 222, + 511, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 529, + 222, + 849, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 183, + 299, + 511, + 354 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 392, + 544, + 406 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Existing deep networks are generally initialized with unsupervised methods, such as random assignments and greedy layerwise pre-training. This may result in the whole training process (initialization/pre-training $^ +$ fine-tuning) to be very timeconsuming. In this paper, we combine the ideas of ensemble learning and deep learning, and present a novel deep learning framework called deep error-correcting output codes (DeepECOC). DeepECOC are composed of multiple layers of the ECOC module, which combines multiple binary classifiers for feature learning. Here, the weights learned for the binary classifiers can be considered as weights between two successive layers, while the outputs of the combined binary classifiers as the outputs of a hidden layer. On the one hand, the ECOC modules can be learned using given supervisory information, and on the other hand, based on the ternary coding design, the weights can be learned only using part of the training data. Hence, the supervised pre-training of DeepECOC is in general very effective and efficient. We have conducted extensive experiments to compare DeepECOC with traditional ECOC, feature learning and deep learning algorithms on several benchmark data sets. The results demonstrate that DeepECOC perform not only better than traditional ECOC and feature learning algorithms, but also state-ofthe-art deep learning models in most cases. ", + "bbox": [ + 233, + 425, + 764, + 674 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 703, + 336, + 719 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Error correcting output codes (ECOC) are an ensemble learning framework to address multi-class classification problems (Dietterich & Bakiri, 1995). The work by (Zhong & Liu, 2013) shows that the ECOC methods can also be used for feature learning, in either a linear or a nonlinear manner. However, although sophisticated coding and decoding strategies are applied (Escalera et al., 2010; Zhong et al., 2012; Zhong & Cheriet, 2013), the learnability of ECOC is limited by its singlelayer structure. Therefore, to exploit the advantages of the ECOC framework, such as supervised ensemble learning and effective coding design, it’s necessary to combine its ideas with that of deep learning. ", + "bbox": [ + 174, + 734, + 825, + 847 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In recent years, many deep learning models have been proposed to handle various challenging problems. Meantime, desirable performances in many domains have been achieved, such as image classification and detection, document analysis and recognition, natural language processing, and video analysis (Hinton & Salakhutdinov, 2006; Krizhevsky et al., 2012; Szegedy et al., 2014; Simonyan & Zisserman, 2014; Zhang et al., 2015; Wang & Ji, 2015; Hong et al., 2015). Among others, (Hinton & ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Salakhutdinov, 2006) presents the ground-breaking deep autoencoder that learns the weight matrices by pre-training the stacked restricted Boltzmann machines (RBMs) and fine-tuning the weights using gradient descent. It delivers much better representations of data than shallow feature learning algorithms, such as principal components analysis (PCA) (Jolliffe, 1986) and latent semantic analysis (LSA) (Deerwester et al., 1990). In order to boost the traditional autoencoder and prevent the “overfitting” problem, (Vincent et al., 2008) introduces the denosing autoencoder that corrupted the data with a random noise. Recently, most of the research focuses on deep convolutional neural networks (CNNs) and recurrent neural networks (RNNs), which greatly improves the state-of-the-art in the areas of object recognition, unsegmented handwriting recognition and speech recognition (Krizhevsky et al., 2012; Graves et al., 2009; Sak et al., 2014). However, existing deep networks are generally initialized with unsupervised methods, such as random assignments and greedy layerwise pre-training. In the case of random initialization, to obtain good results, many training data and a long training time are generally used; while in the case of greedy layerwise pre-training, as the whole training data set needs to be used, the pre-training process is very time-consuming and difficult to find a stable solution. ", + "bbox": [ + 173, + 104, + 825, + 310 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To overcome the limitations of both traditional ECOC methods and deep learning models, and meanwhile, take advantages of both of them, in this paper, we propose a novel deep learning model called deep error-correcting output codes (DeepECOC). DeepECOC are composed of multiple stacked ECOC modules, each of which combines multiple binary classifiers for feature learning. Here, the weights learned for the binary classifiers can be considered as weights between two successive layers, while the probabilistic outputs of the combined binary classifiers as the outputs of a hidden layer or new representations of data. On the one hand, the ECOC modules can be learned layer by layer using the given supervisory information, and on the other hand, based on the ternary coding design, some classes of data are automatically neglected when training the binary classifiers, such that the weights are learned only using part of the training data. Hence, the supervised pre-training of DeepECOC is in general very effective and efficient. We have compared DeepECOC with traditional ECOC, feature learning and deep learning algorithms to demonstrate the effectiveness and superiority of DeepECOC. The results are reported in Section 4. ", + "bbox": [ + 174, + 319, + 825, + 500 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The rest of this paper is organized as follows: In Section 2, we give a brief overview to related work. In Section 3, we present the proposed model, DeepECOC, in detail. The experimental results are reported in Section 4, while Section 5 concludes this paper with remarks and future work. ", + "bbox": [ + 176, + 506, + 825, + 549 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 571, + 343, + 587 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Traditional ECOC framework has two steps: coding and decoding. In the coding step, an ECOC matrix is defined or learned from data, and the binary classifiers are trained based on the ECOC coding; in the decoding step, the class label is given to a test sample based on a similarity measure between codewords and outputs of the binary classifiers. The widely used coding strategies include one-versus-all (OneVsAll) (Nilsson, 1965), one-versus-one (OneVsOne) (Hastie et al., 1998), discriminant ECOC (DECOC) (Pujol et al., 2006), ECOC optimizing node embedding (ECOCONE) (Escalera et al., 2006), dense and sparse coding (Escalera et al., 2009; Allwein et al., 2001), and so on. Among them, the OneVsAll, OneVsOne, dense and sparse coding strategies are problem-independent, whilst the DECOC and ECOCONE are problem-dependent. Generally, the length of the codeword by the OneVsAll, OneVsOne, DECOC and ECOCONE coding designs is related to the number of classes, but that by the dense and sparse coding design is relatively flexible. In this work, we design the structure of DeepECOC based on the properties of each coding strategy. The commonly used binary ECOC decoding strategies are the Hamming decoding (Nilsson, 1965) and Euclidean decoding (Hastie et al., 1998). For ternary ECOC decoding strategies, the attenuated Euclidean decoding (Pujol et al., 2008), loss-based decoding (Allwein et al., 2001), and probabilistic-based decoding (Passerini et al., 2004) are widely used. Currently, the state-of-the-art ternary ECOC decoding strategies are the discrete pessimistic beta density distribution decoding and loss-weighted decoding (Escalera et al., 2010). In this work, for the simplicity of back propagation, we directly add a Softmax layer at the top of DeepECOC for the decoding. Note that, although many sophisticated coding and decoding strategies have been proposed in recent years (Escalera et al., 2010; Zhong et al., 2012; Zhong & Cheriet, 2013), the learnability of ECOC is limited by its single-layer structure. To further exploit the advantages of ECOC, such as supervised ensemble learning and effective coding design, it’s necessary to combine its ideas with that of deep learning. ", + "bbox": [ + 174, + 604, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/86fe1fb2b334c231e08ad95e358eeac2a15ba70ca951e0e93d5b04445c7aad57.jpg", + "image_caption": [ + "Figure 1: Two coding matrices encoded with the one-versus-all (binary case) and one-versus-one (ternary case) coding strategies. " + ], + "image_footnote": [], + "bbox": [ + 240, + 101, + 730, + 191 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the literature of deep learning, there is some work that attempts to construct a deep architecture with multiple feature learning methods (Hinton & Salakhutdinov, 2006; Trigeorgis et al., 2014; Yuan et al., 2015; Zheng et al., 2015; 2014). For instance, deep autoencoder is built up by RBMs (Hinton & Salakhutdinov, 2006), and deep semi-NMF combines multiple steps of matrix factorization (Trigeorgis et al., 2014). Similarly, deep CNNs and RNNs can also be considered as deep models that learn the new representations of data layer by layer (Krizhevsky et al., 2012; Graves et al., 2009; Sak et al., 2014). The success of these existing models demonstrate that deep networks are beneficial to the representation learning tasks, especially for the large scale applications. However, as discussed in the previous section, existing deep learning models are generally initialized with unsupervised methods, such as random assignments and greedy layerwise pre-training, which result in a long training time of the deep models. In this work, we propose the DeepECOC model, which is based on the stacked ECOC modules. When pre-training DeepECOC, the ECOC modules can be learned with the available supervisory information. Intuitively, as this manner of supervised pre-training has deterministic objective, the learned value of the parameters will be very close to the best local minimum on the solution manifold. Experimental results shown in Section 4 also demonstrate this fact. ", + "bbox": [ + 174, + 243, + 825, + 465 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 DEEP ERROR-CORRECTING OUTPUT CODES (DEEPECOC) ", + "text_level": 1, + "bbox": [ + 174, + 484, + 694, + 502 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we first introduce the traditional ECOC framework, which is the important building block of DeepECOC. Then we present the learning procedures of DeepECOC in detail. ", + "bbox": [ + 173, + 516, + 823, + 545 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 THE ECOC FRAMEWORK ", + "text_level": 1, + "bbox": [ + 176, + 561, + 392, + 577 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Error correcting output codes (ECOC), which combine multiple binary classifiers to solve multiclass classification problems, are an ensemble learning framework. The ECOC methods in general consist of two steps: coding and decoding. In the coding step, the ECOC coding matrix $\\mathbf { M } \\in \\{ - 1 , 1 \\} ^ { C \\times L }$ (binary case) or $\\mathbf { M } \\in \\{ - 1 , \\mathbf { \\bar { 0 } } , 1 \\} ^ { C \\times L }$ (ternary case) is first defined or learned from the training data, where each row of $\\mathbf { M }$ is the codeword of a class, each column corresponds to a dichotomizer (binary classifier), $L$ is the length of the codewords (the number of binary classifiers), $C$ is the number of classes, symbol ‘1’ indicates positive class, $^ { \\bullet } - _ { 1 } \\bullet$ indicates negative class, and $\\cdot _ { 0 } \\cdot \\mathrm { ~ }$ indicates that a particular class is not considered by a given classifier. Then, the binary classifiers (dichotomizers) are trained according to the partition of the classes in the columns of M. Fig. 1 shows two coding matrices encoded with the one-versus-all (binary case) and one-versus-one (ternary case) coding strategies. The matrix is coded using several dichotomizers for a 4-class problem with respective codewords $\\{ y _ { 1 } , \\ldots , y _ { 4 } \\}$ . The white girds are coded by 1 (considered as positive class by the respective dichotomizer $h _ { j }$ ), the dark girds by $^ { - 1 }$ (considered as the negative class), and the gray girds by 0 (classes that are not considered by the respective dichotomizer $h _ { j }$ ). In the decoding step, the test data are predicted based on an adopted decoding strategy and the outputs of the binary classifiers. ", + "bbox": [ + 173, + 588, + 825, + 809 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In order to take the probabilistic outputs of the base classifiers as new representations of data, we adopt linear support vector machines (linear SVMs) as the binary classifiers (dichotomizers), which solve a quadratic programming problem ", + "bbox": [ + 176, + 816, + 825, + 859 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6714816b12887b68e2216505dca1b64c0b7e0e06a33c1bfbb776eff4ea362399.jpg", + "text": "$$\n\\begin{array} { c l l } { \\displaystyle \\operatorname* { m i n } _ { \\mathbf { w } , b , \\xi _ { i } } J ( \\mathbf { w } ) = \\frac { 1 } { 2 } \\| \\mathbf { w } \\| ^ { 2 } + C \\displaystyle \\sum _ { i = 1 } ^ { N } \\xi _ { i } } \\\\ { s . t . } & { \\ y _ { i } f ( \\mathbf { x } _ { i } ) \\geq 1 - \\xi _ { i } , \\xi _ { i } \\geq 0 , i = 1 , \\ldots , N } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 864, + 663, + 928 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where w and $b$ are the coefficients and bias of the binary classifier, $y _ { i } \\in \\{ + 1 , - 1 \\}$ , $\\xi _ { i }$ ’s are the slack variables, and $N$ is the number of the training data. The discriminant function can be expressed as ", + "bbox": [ + 169, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ca973cc5ef0b5540bf8cfc4cc42127753a8bc2bbc255244beafcf3eff69a3003.jpg", + "text": "$$\nf ( \\mathbf { x } ) = \\mathbf { w } ^ { T } \\mathbf { x } + b .\n$$", + "text_format": "latex", + "bbox": [ + 436, + 138, + 562, + 157 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This problem can be solved as, ", + "bbox": [ + 174, + 165, + 379, + 179 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ef02cda97eac47e4f3cba35b6b54cbc55a00dfa63fea9264644799efef4cea60.jpg", + "text": "$$\n\\mathbf { w } = \\sum _ { i = 1 } ^ { N } \\alpha _ { i } y _ { i } x _ { i } ,\n$$", + "text_format": "latex", + "bbox": [ + 441, + 176, + 555, + 220 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1b66cfa6042bfb1c55c604836a8c1f107f2b5382b407c942ff030ac77ff542cd.jpg", + "text": "$$\nb = \\frac { 1 } { N _ { S \\mathcal { V } } } \\sum _ { { \\bf x } _ { i } \\in S \\mathcal { V } , i = 1 } ^ { N _ { S \\mathcal { V } } } ( y _ { i } - { \\bf w } ^ { T } { \\bf x } _ { i } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 224, + 611, + 268 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\alpha _ { i }$ ’s are the non-negative Lagrange multipliers, $N _ { s v }$ is the number of support vectors and $\\mathit { s v }$ is the set of support vectors. The dual form of Problem (1) can be written as ", + "bbox": [ + 173, + 272, + 825, + 301 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/affa6a130ddaf9e93cdb584fde7939e6f875e663f67fb4205c233ff9ad652c33.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle \\operatorname* { m a x } _ { \\alpha } } & { \\displaystyle \\sum _ { i = 1 } ^ { N } \\alpha _ { i } - \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ { N } \\alpha _ { i } \\alpha _ { j } y _ { i } y _ { j } \\mathbf { x } _ { i } ^ { T } \\mathbf { x } _ { j } } \\\\ { = } & { \\displaystyle \\sum _ { i = 1 } ^ { N } \\alpha _ { i } - \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ { N } \\alpha _ { i } \\alpha _ { j } y _ { i } y _ { j } k ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ) } \\\\ { s . t . } & { 0 \\le \\alpha _ { i } \\le \\lambda , i = 1 , \\dots , N , } \\\\ { \\displaystyle } & { \\displaystyle \\sum _ { i = 1 } ^ { N } \\alpha _ { i } y _ { i } = 0 , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 362, + 306, + 650, + 460 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $k ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ) ~ = ~ \\mathbf { x } _ { i } ^ { T } \\mathbf { x } _ { j }$ is the linear kernel function, $\\lambda$ is a constant number, and $\\alpha =$ $\\{ \\alpha _ { 1 } , \\dotsc , \\alpha _ { N } \\}$ is the vector of Lagrange multipliers. Replacing the linear kernel function with a nonlinear kernel, such as the Gaussian kernel ", + "bbox": [ + 173, + 465, + 826, + 508 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a9f43b1e69e2d93802f029fa07d5a2d9f8605e82592b72bf02c8d4f334910d99.jpg", + "text": "$$\nk ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ) = \\exp ( - \\sigma ^ { - 1 } \\| \\mathbf { x } _ { i } - \\mathbf { x } _ { j } \\| ^ { 2 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 377, + 515, + 620, + 535 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "we can learn a nonlinear SVM, where $\\sigma$ is the parameter for the Gaussian kernel function. The discriminant function of SVMs with a nonlinear kernel can be written as ", + "bbox": [ + 171, + 540, + 825, + 569 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/23cafef4a6e1d35e5c3b659788edc242560d5a859c49e30f02152252d49d3260.jpg", + "text": "$$\nf ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { N } \\alpha _ { i } y _ { i } k ( \\mathbf { x } _ { i } , \\mathbf { x } ) + b .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 573, + 598, + 616 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Applying a decoding strategy on the outputs of the binary classifiers, the ECOC framework can be used for multi-class learning, while applying the sigmoid function on the values of the discriminant function, ECOC can be used for feature learning (Zhong & Liu, 2013). This is also the foundation of the DeepECOC model. ", + "bbox": [ + 173, + 630, + 825, + 686 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 DEEPECOC ", + "text_level": 1, + "bbox": [ + 174, + 703, + 300, + 718 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To combine the advantages of ECOC and deep learning algorithms, we build the DeepECOC architecture as follows ", + "bbox": [ + 176, + 728, + 823, + 758 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ac2f32b54e3c8bf390f1442340c7f455fa6fa98b2ff1c7f5eb6bb56cbf96b531.jpg", + "text": "$$\n\\mathbf { x } \\ { \\overset { q _ { D } } { \\longrightarrow } } \\ { \\tilde { \\mathbf { x } } } \\ { \\overset { \\mathbf { W } _ { 1 } } { \\underset { \\mathbf { b } _ { 1 } } { \\longrightarrow } } } \\ \\mathbf { h } _ { 1 } \\ { \\overset { \\mathbf { W } _ { 2 } } { \\underset { \\mathbf { b } _ { 2 } } { \\longrightarrow } } } \\ \\cdot \\cdot \\cdot \\ { \\overset { \\mathbf { W } _ { n - 1 } } { \\underset { \\mathbf { b } _ { n - 1 } } { \\longrightarrow } } } \\ \\mathbf { h } _ { n - 1 } \\ { \\overset { s o f t m a x } { \\longrightarrow } } \\ \\mathbf { y } ,\n$$", + "text_format": "latex", + "bbox": [ + 320, + 762, + 676, + 792 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the first step makes the clean input $\\mathbf { x } \\in [ 0 , 1 ] ^ { d }$ partially destroyed by means of a stochastic mapping $\\tilde { \\textbf { x } } \\sim q _ { D } ( \\tilde { \\textbf { x } } | \\textbf { x } )$ . In the corrupting process, we set a parameter called denoising rate $\\nu$ . For each input $\\mathbf { x }$ , a fixed number $\\nu d$ of components are chosen at random, and their value is forced to 0, while the others are left untouched. This operation makes the model more robust and prevent the overfitting problem in most cases (Vincent et al., 2008). Subsequently, the “corrupted” data are taken as inputs for the DeepECOC model. $\\mathbf { W } _ { 1 }$ and $\\mathbf { b } _ { 1 }$ are the weight matrix and bias vector learned from the first ECOC module. The output of the first hidden layer is denoted as ", + "bbox": [ + 173, + 800, + 825, + 900 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7536ce8b74dc3e826083ce44bf324582efe53e22aca923fa391217fb8548e474.jpg", + "text": "$$\n\\mathbf { h } _ { 1 } = s ( \\mathbf { W } _ { 1 } ^ { T } \\mathbf { x } + \\mathbf { b } _ { 1 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 426, + 906, + 570, + 925 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $s ( \\cdot )$ is the sigmoid activation function $\\begin{array} { r } { s ( x ) = \\frac { 1 } { 1 + e ^ { - x } } } \\end{array}$ . From the second layer to the $( n - 1 )$ - th layer, we use the stacked ECOC modules to learn the weight matrices and biases, which can be considered as weights between two successive layers of a deep network. Similarly, we use the output of the $\\left( k - 1 \\right)$ -th layer as the input of the $k$ -th layer, ", + "bbox": [ + 173, + 102, + 825, + 161 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6dd66ca70944530149e774873a568f1f4897bc6bc56c46f6023b79917b4c604d.jpg", + "text": "$$\n\\mathbf { h } _ { k } = s ( \\mathbf { W } _ { k } ^ { T } \\mathbf { h } _ { k - 1 } + \\mathbf { b } _ { k } ) .\n$$", + "text_format": "latex", + "bbox": [ + 411, + 167, + 584, + 186 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Here, $\\mathbf { h } _ { k }$ can be viewed as an activation output and a new representation of the input datum $\\mathbf { x }$ . ", + "bbox": [ + 173, + 193, + 790, + 208 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For example, if we adopt the OneVsAll coding strategy for one layer of the ECOC module, we first define the coding matrix $\\mathbf { M } ^ { C \\times C }$ , where $C$ is the number of classes. Then, we can train $C$ SVM classifiers to obtain the weight matrix $\\mathbf { W } = \\left\\{ \\mathbf { w } _ { 1 } , \\dots , \\mathbf { w } _ { i } , \\dots , \\mathbf { w } _ { C } \\right\\}$ and the bias $\\mathbf { b } = \\{ b _ { 1 } , \\dots , b _ { i } , \\dots , b _ { C } \\}$ . Next, we calculate the output of the first layer by using Eq. (9). Subsequently, we repeat this process layer by layer to build the DeepECOC model. It’s obvious that, if we adopt different coding strategies, we can get different kinds of DeepECOC architectures. ", + "bbox": [ + 173, + 213, + 825, + 297 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For the last layer of DeepECOC, we employ the softmax regression for the multi-class learning. Its cost function is defined as ", + "bbox": [ + 171, + 304, + 823, + 333 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/dff099fd46ad9d3a12892ac5a3a21d3a3413c02cbf2c42dd0e839841e084aa92.jpg", + "text": "$$\nJ ( \\mathbf { w } ) = - \\frac { 1 } { N } ( \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { K } \\mathbf { I } ( y _ { i } = j ) \\log \\frac { \\exp ( \\mathbf { w } _ { j } ^ { T } \\mathbf { h } _ { i } ^ { n - 1 } ) } { \\sum _ { l = 1 } ^ { K } \\exp ( \\mathbf { w } _ { l } ^ { T } \\mathbf { h } _ { i } ^ { n - 1 } ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 300, + 335, + 696, + 381 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathbf { I } ( x )$ is the indicator function, $\\mathbf { I } ( x ) = 1$ if $x$ is true, else $\\mathbf { I } ( x ) = 0$ . $y _ { i }$ is the label corresponding to $\\mathbf { x } _ { i }$ . It’s easy to compute the probability that $\\mathbf { x } _ { i }$ is classified to class $j$ , ", + "bbox": [ + 173, + 387, + 825, + 416 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/667cbea538a2c01c800af3eea5b245db892f15d4b1f191b0d5118dd71ab4f7bb.jpg", + "text": "$$\np ( y _ { i } = j | \\mathbf { x } _ { i } , \\mathbf { w } ) = \\frac { \\exp ( \\mathbf { w } _ { j } ^ { T } \\mathbf { h } _ { i } ^ { n - 1 } ) } { \\sum _ { l = 1 } ^ { K } \\exp ( \\mathbf { w } _ { l } ^ { T } \\mathbf { h } _ { i } ^ { n - 1 } ) } .\n$$", + "text_format": "latex", + "bbox": [ + 359, + 421, + 638, + 463 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Taking derivatives, one can show that the gradient of $J ( \\mathbf { w } )$ with respect to w is, ", + "bbox": [ + 174, + 468, + 697, + 483 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6e907982271e941891ccdd03e899cfa4a5deba920bfa264517873943d5beae75.jpg", + "text": "$$\n\\nabla J ( \\mathbf { w } ) = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } [ \\mathbf { x } _ { i } ( \\mathbf { I } ( y _ { i } = j ) - p ( y _ { i } = j | \\mathbf { x } _ { i } , \\mathbf { w } ) ) ] .\n$$", + "text_format": "latex", + "bbox": [ + 316, + 489, + 679, + 534 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "After the pre-training step, we use back propagation (Rumelhart et al., 1988) to fine tune the whole architecture. Moreover, we also employ a technique called “dropout” for regularization (Hinton et al., 2012). When a large feedforward neural network is trained on a small training set, dropout generally performs well on the test set. The basic idea of dropout is that each hidden node is randomly omitted from the network with a probability of $\\beta$ . In another view, dropout is a very efficient way to perform model averaging with neural networks. Through these processes, we finally obtain the DeepECOC model, which is robust and easy to be applied to multi-class classification tasks. ", + "bbox": [ + 173, + 545, + 825, + 643 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that, compared to existing deep learning algorithms, DeepECOC have some important advantages. Firstly, unlike previous deep learning algorithms, DeepECOC are built with the ECOC modules and pre-trained in a supervised learning fashion. Secondly, if we adopt ternary coding strategies, due to the natural merit of ECOC, the weights can be learned using only part of the training data. Thirdly, in contrast to the learning of the weight matrices in previous deep learning models, the binary classifiers in each ECOC module can be learned in parallel, which may greatly speed up the learning of DeepECOC. ", + "bbox": [ + 173, + 650, + 825, + 748 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 768, + 328, + 784 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To evaluate the effectiveness of the proposed method, DeepECOC, we conducted 4 parts of experiments. In the first part, we compared DeepECOC with some deep learning models and single-layer ECOC approaches on 16 data sets from the UCI machine learning repository 1. In the second part, we compared DeepECOC with traditional feature learning models, some deep learning models and single-layer ECOC approaches on the USPS handwritten digits 2, and tested DeepECOC with different number of hidden layers. In the third part, we used the MNIST handwritten digits 3 to further demonstrate the effectiveness of DeepECOC for handwritten digits recognition. Finally, the CIFAR10 data set 4 was used to demonstrate the effectiveness of DeepECOC on image classification tasks. For all the data sets, the features were normalized within $[ 0 , 1 ]$ . In the following, we report the experimental results in detail. ", + "bbox": [ + 173, + 787, + 825, + 872 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/004ca6c638bf4ca61909bc493d4630cea0cc7615fe65729f4cc8cb90c3c45669.jpg", + "table_caption": [ + "Table 1: Details of the UCI data sets (T: training samples; A: attributes; C: classes). " + ], + "table_footnote": [], + "table_body": "
Problem#ofT#ofA#ofCProblem#ofT#ofAofC
Dermatology366346Yeast1484810
Iris15043Satimage6435367
Ecoli33688Letter200001626
Wine178133Pendigits109921610
Glass21497Segmentation2310197
Thyroid21553Optdigits56206410
Vowel9901011Shuttle1450097
Balance62543Vehicle846184
", + "bbox": [ + 271, + 131, + 727, + 227 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/6677ace4e6333f58a4302ceaaaaae5f80d8d06aab3dceb92e596a37a7ba173db.jpg", + "table_caption": [ + "Table 2: Classification accuracy and standard deviation obtained by DeepECOC and the compared approaches on 16 UCI data sets. Here, DeepECOC(1) $\\sim$ DeepECOC(3) are 3 variant of DeepECOC with the ECOCONE coding design initialized by one-versus-one, one-versus-all and DECOC respectively. The best results are highlighted in boldface. " + ], + "table_footnote": [], + "table_body": "
ProblemSingleAEDAEDeepECOC (1)DeepECOC (2)DeepECOC (3）
Dermatology0.95130.9429 ±0.06710.9674 ±0.03120.9702 ±0.03540.9779 ±0.02080.9747 ±0.0318
Iris0.96000.9600 ±0.05620.9333 ±0.08890.9600 ±0.05350.9267 ±0.11090.9533 ±0.0383
Ecoli0.81470.7725 ±0.06080.8000 ±0.03620.8529 ±0.04030.8824 ±0.06260.9118 ±0.0636
Wine0.96050.9765 ±0.02640.9563 ±0.04220.9875 ±0.02640.9813 ±0.03020.9688 ±0.0329
Glass0.67620.6669 ±0.10320.6669 ±0.07150.7895 ±0.07880.7368 ±0.11400.7562 ±0.0879
Thyroid0.92100.9513 ±0.06140.9599 ±0.05670.9656 ±0.05130.9703 ±0.05400.9608 ±0.0518
Vowel0.71770.6985 ±0.07450.7101 ±0.07560.7475 ±0.09010.6010 ±0.06270.6863 ±0.0788
Balance0.82220.8036 ±0.03200.8268 ±0.05480.9137 ±0.04120.8333 ±0.03180.9167 ±0.0312
Yeast0.52170.5641 ±0.03460.5891 ±0.02720.5959 ±0.05990.5494 ±0.04340.5697 ±0.0462
Satimage0.85370.8675 ±0.0528 0.92340.8897 ±0.03040.8961 ±0.04800.8360 ±0.03900.9077 ±0.0555
Letter0.9192±0.0547 0.98310.9381 ±0.0641 0.98860.9532 ±0.03410.9247 ±0.03520.9501 ±0.0563
Pendigits0.9801±0.0123 0.9584±0.0034 0.95960.9908 ±0.0031 0.97110.9866 ±0.0107 0.95840.9899 ±0.0075 0.9711
Segmentation0.9701±0.0317 0.9785±0.0211 0.9856±0.0286 0.9867±0.0163 0.9848±0.0233 0.9911
Optdigits0.9982±0.0101 0.9953±0.0088 0.9976±0.0096 0.9988±0.0123 0.9983±0.0091 0.9993
Shuttle0.9988±0.0012 0.6987±0.0014 0.7348±0.0021 0.7561±0.0018 0.6908±0.0010 0.7195
Vehicle Mean rank0.7315 4.0938±0.0521 4.8750±0.0454 3.9375±0.0480 1.7500±0.04321 3.9375±0.0148 2.4063
", + "bbox": [ + 253, + 304, + 746, + 705 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 722, + 825, + 777 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 CLASSIFICATION ON 16 UCI MACHINE LEARNING REPOSITORY DATA SETS ", + "text_level": 1, + "bbox": [ + 176, + 785, + 738, + 801 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The detail of the UCI data sets are shown in Table 1. In these experiments, we compared DeepECOC with autoencoder (AE) (Hinton & Salakhutdinov, 2006), denoising autoencoder (DAE) (Vincent et al., 2008) and single-layer ECOC approaches (Single) (Escalera et al., 2010). We built DeepECOC with the ECOC optimizing node embedding (ECOCONE) coding method (Escalera et al., 2006). Here, since we initialized ECOCONE with 3 different coding methods, i.e. one-versus-one, oneversus-all and DECOC, DeepECOC had 3 variants. In addition, the state-of-the-art linear lossweighted (LLW) decoding strategy was used for ECOCONE. Finally, a structure with 3 hidden layers was adopted for DeepECOC, which had 0.1 denoising rate and 0.1 dropout rate: ", + "bbox": [ + 174, + 813, + 825, + 896 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 821, + 132 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/2f9608a69fc7f88180cac3b3bd23ba947988dd955afc29c4acf47acef9c3238b.jpg", + "text": "$$\n\\begin{array} { r } { \\textbf { x } \\overset { q _ { D } } { \\longrightarrow } \\tilde { \\textbf { x } } \\xrightarrow { \\textbf { W } _ { 1 } } \\textbf { h } _ { 1 } \\xrightarrow { \\textbf { W } _ { 2 } } \\textbf { h } _ { 2 } \\xrightarrow { \\textbf { W } _ { 3 } } \\textbf { h } _ { 3 } \\xrightarrow { s o f t m a x } \\textbf { y } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 336, + 137, + 658, + 164 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For the fine-tuning process, we used the stochastic gradient descent algorithm. The learning rate and epoches from different data sets are described in Table 3. The autoencoder and denoising autoencoder’s architectures are as same as DeepECOC with ECOCONE initialized by one-versus-one. For single-layer ECOC approaches, we chose the best results shown in (Escalera et al., 2010) as our compared results. For all DeepECOC models, we used support vector machines (SVMs) with RBF kernel function as base classifiers. The parameters of SVMs were set to default (Chang & Lin, 2011). ", + "bbox": [ + 173, + 167, + 825, + 265 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 2 shows the average classification accuracy and standard deviation on 16 UCI data sets. Except on the OptDigits data set, DeepECOC achieved the best results compared with autoencoder, denoising autoencoder and single-layer ECOC approaches. In fact, on the OptDigits data set, DeepECOC achieved comparative result with single-layer ECOC approaches. Among others, DeepECOC with ECOCONE (initialized by one-versus-one) coding strategy obtained the best results on 9 data sets, while DeepECOC with ECOCONE (initialized by DECOC) coding strategy obtained the best results on 5 data sets. From the mean rank values, we can see that DeepECOC with ECOCONE (initialized by one-versus-one and DECOC) strategy far surpass other compared methods. ", + "bbox": [ + 173, + 272, + 825, + 385 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/c1e198fe1eb7c53cf0311f192ffc042631023f88ac0b8b4b2c265afb0305a581.jpg", + "table_caption": [ + "Table 3: Details of the learning rate $\\eta$ and epoch on the UCI data sets. " + ], + "table_footnote": [], + "table_body": "
ProblemmEpochProblemnEpoch
Dermatology0.12000Yeast0.014000
Iris0.1400Satimage0.014000
Ecoli0.12000Letter0.018000
Wine0.12000Pendigits0.012000
Glass0.014000Segmentation0.018000
Thyroid0.1800Optdigits0.012000
Vowel0.14000Shuttle0.12000
Balance0.14000Vehicle0.14000
", + "bbox": [ + 326, + 416, + 671, + 511 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 CLASSIFICATION ON THE USPS DATA SET ", + "text_level": 1, + "bbox": [ + 174, + 536, + 504, + 550 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The USPS handwritten digits data set includes 7291 training samples and 2007 test samples from 10 classes. The size of the images is $1 6 \\times 1 6 = 2 5 6$ . Our experiments on this data set were divided into 2 parts. Firstly, we compared DeepECOC with two traditional feature learning models (principal components analysis (PCA) (Jolliffe, 2002) and marginal Fisher analysis (MFA) (Yan et al., 2007)), autoencoder (AE), denoising autoencoder (DAE), LeNet (LeCun et al., 1998), PCANet (Chan et al., 2015) and single-layer ECOC approaches. Here, PCA is an unsupervised method, MFA is a supervised method. For MFA, the number of nearest neighbors for constructing the intrinsic graph was set to 5, while that for constructing the penalty graph was set to 15. For DeepECOC, we also used 3 coding design methods in this experiment. We used batch gradient descent for the fine-tuning process, the batch size was set to 100, the learning rate was set to 1, the number of epoch was set to 40000, the denoising rate, and dropout rate were set to 0.1. We also used SVMs with RBF kernel and default parameters as base classifiers. For single-layer ECOC approaches, we adopted ECOCONE (initialized by one-versus-one) as coding design method and linear loss-weighted (LLW) decoding strategy. For the LeNet model, we used 2 convolutional layers, two pooling layers and two fully connected layers. The kernel size of the convolutional layers and pooling layers was set to $2 \\times 2$ , the stride was set to 1, the number of nodes of the first layer was set to 200, the epoch was set to 8000, the initial learning rate was set to 0.001, learning rate policy was set to “inv”, and the momentum was set to 0.9. For the PCANet model, we used two PCA-filter stages, one binary hashing stage and one blockwise histograms. The filter size, the number of filters, and the block size were set to $k _ { 1 } = k _ { 2 } = 3$ , $L _ { 1 } = L _ { 2 } = 4$ , and $7 \\times 7$ , respectively. The experimental results are shown in Fig. 2(a). ", + "bbox": [ + 173, + 560, + 825, + 840 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "From Fig. 2(a), we can see that DeepECOC with ECOCONE (initialized by one-versus-one) coding strategy achieved the best result than other methods include traditional feature learning models, existing deep learning methods and single-layer ECOC approaches. ", + "bbox": [ + 174, + 845, + 825, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In the second part, we evaluated DeepECOC with different number of hidden layers. We used 2 to 6 hidden layers in our experiments. The parameter settings were as same as the first part. Fig. 2(b) ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/69601db0f9abe1f64fba996cd93599b01ce074f8d2c7529a048c62226e3571e4.jpg", + "image_caption": [ + "Figure 2: (a) Classification accuracy obtained on the USPS data set. Here, DeepE ${ \\mathrm { C O C } } ( 1 ) \\sim$ DeepECOC(3) are 3 variant of DeepECOC with the ECOCONE coding design initialized by one-versusone, one-versus-all and DECOC respectively. (b) Classification accuracy with different numbers of hidden layers on the USPS data set. " + ], + "image_footnote": [], + "bbox": [ + 222, + 113, + 769, + 276 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/04d6c3cab8ccbbf66a6ef3819036db6e0b5b4cd06a21498a052a3052377c8fef.jpg", + "image_caption": [ + "Figure 3: Classification accuracy obtained on the MNIST data set for two architectures. " + ], + "image_footnote": [], + "bbox": [ + 194, + 375, + 790, + 537 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "shows the experimental results. We can see that DeepECOC obtained the best result when using 3 hidden layers. When the number of hidden layers is less than 3, the effectiveness of DeepECOC increases with the increasing of the number of hidden layers. Along with the number of hidden layers continues to grow, the effectiveness of DeepECOC decreases. ", + "bbox": [ + 174, + 594, + 825, + 650 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 CLASSIFICATION ON THE MNIST DATA SET ", + "text_level": 1, + "bbox": [ + 174, + 667, + 517, + 683 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "MNIST handwritten digits data set has a training set of 60,000 examples, and a test set of 10,000 examples with 784 dimensional features. We designed 2 architectures for autoencoder, denoising autoencoder and DeepECOC. The first architecture was $7 8 4 - Z _ { 1 } - Z _ { 2 } - Z _ { 3 } - 1 0$ , where $Z _ { i }$ was the number of hidden neurons designed based on some ECOC coding strategies. We designed this architecture because we wanted to make autoencoder and denoising autoencoder had the same structure with DeepECOC. The second architecture is $7 8 4 - 5 0 0 - 5 0 0 - 2 0 0 0 - 1 0$ . This architecture was used in (Hinton & Salakhutdinov, 2006). In order to make DeepECOC adapt to this structure, we used the dense and sparse coding design methods that can control the codeword length. Note that, the dense and sparse coding design methods are totally random and data-independent. The denoising rate and dropout rate were set to 0.1, the batch size was set to 100, the learning rate was set to 0.01, and the number of epoch was set to 80000. For LeNet model, we adopted the parameters as same as (LeCun et al., 1998). For PCANet model, we used two PCA-filter stages, one binary hashing stage and one blockwise histograms. In the PCANet, the filter size, the number of filters, and the block size were set to $k _ { 1 } = k _ { 2 } = 8$ , $L _ { 1 } = L _ { 2 } = 7$ , and $7 \\times 7$ , respectively. ", + "bbox": [ + 173, + 694, + 825, + 888 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Fig. 3(a) and Fig. 3(b) show the experimental results on 2 architectures. We can see that DeepECOC are comparative with existing deep learning methods on the second architecture and outperform ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/ce426d4cda7ed81d33745ba03a1463e3b4f63d2fcc9be94056189d3bc65190b4.jpg", + "table_caption": [ + "Table 4: Classification accuracy obtained on the LBP-CIFAR10 data set. The best result for each scenario is highlighted in bold face. " + ], + "table_footnote": [ + "them on the first architecture. In addition, DeepECOC with both two architectures outperform the single-layer ECOC approaches. " + ], + "table_body": "
ProblemAEDAELeNetPCANetDeepECOC(1)DeepECOC(2)DeepECOC(3)
LBP-CIFAR10 (36)0.35010.36780.32560.25690.50890.45170.4752
LBP-CIFAR10 (256)0.43520.45870.32210.25690.55880.45890.5224
", + "bbox": [ + 202, + 140, + 797, + 174 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.4 CLASSIFICATION ON THE LBP-CIFAR10 DATA SET ", + "text_level": 1, + "bbox": [ + 171, + 228, + 570, + 243 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The CIFAR-10 dataset is a relative large scale data set which consists of $6 0 0 0 0 3 2 \\times 3 2$ colour images in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images. For the purpose of reducing computational cost, we attempted to extract features of the data using an efficient local binary patterns algorithm. As a result, the representations with dimensionality 36 and 256 were adopted and the data were normalized to [0, 1] as well, called LBP-CIFAR10 (36) and LBP-CIFAR10 (256). We also used 3 hidden layers for all deep learning methods. The learning rate was set to 0.1, and the epoch was set to 4000. For the LeNet model, we used 2 convolutional layers and two fully connected layers without pooling layers. The kernel size was set to $2 \\times 2$ , the stride was set to 1, the number of node of the first fully connected layer was set to 64, the epoch was set to 4000, the initial learning rate was set to 0.01, learning rate policy was set to “inv”, and the momentum was set to 0.9. For the PCANet model, we used two PCA-filter stages, one binary hashing stage and one blockwise histograms. In the PCANet, the filter size, the number of filters, and the block size were set to $k _ { 1 } = k _ { 2 } = 3$ , $L _ { 1 } = L _ { 2 } = 4$ , and $7 \\times 7$ , respectively. The classification accuracy are reported in Table 4. ", + "bbox": [ + 174, + 256, + 825, + 449 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "From Table 4, we can easy to see that DeepECOC achieved the best results. Moreover, DeepECOC with ECOCONE (initialized by one-versus-one) coding strategy achieved the better results than autoencoder and denoising autoencoder, LeNet and PCANet. Hence, we can conclude that, DeepECOC are a general model to handle different real world applications and achieves desirable results in most cases. ", + "bbox": [ + 174, + 457, + 825, + 526 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 547, + 318, + 564 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we propose a novel deep learning model, called deep error correcting output codes (DeepECOC). DeepECOC extend traditional ECOC algorithms to a deep architecture fashion, and meanwhile, brings new elements to the deep learning area, such as supervised initialization, and automatic neglecting of part of the data during network training. Extensive experiments on 16 data sets from the UCI machine learning repository, the USPS and MNIST handwritten digits and the CIFAR-10 data set demonstrate the superiority of DeepECOC over traditional ECOC, feature learning and deep learning methods. In future work, we will further exploit the learnability of DeepECOC on large scale applications. ", + "bbox": [ + 173, + 580, + 825, + 691 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 713, + 285, + 728 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "E. L. Allwein, R. E. Schapire, and Y. Singer. Reducing multiclass to binary: A unifying approach for margin classifiers. The Journal of Machine Learning Research, 1:113–141, 2001. \nT.-H. Chan, K. Jia, S. Gao, J. Lu, Z. Zeng, and Y. Ma. PCANet: A Simple Deep Learning Baseline for Image Classification? Image Processing, IEEE Transactions on, 24(12):5017–5032, 2015. \nC.-C. Chang and C.-J. Lin. 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The work by (Zhong & Liu, 2013) shows that", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "the ECOC methods can also be used for feature learning, in either a linear or a nonlinear manner.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "However, although sophisticated coding and decoding strategies are applied (Escalera et al., 2010;", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "Zhong et al., 2012; Zhong & Cheriet, 2013), the learnability of ECOC is limited by its single-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "layer structure. 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Hence, the supervised pre-training of DeepECOC is in general very effective", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 141, + 479, + 470, + 492 + ], + "spans": [ + { + "bbox": [ + 141, + 479, + 470, + 492 + ], + "score": 1.0, + "content": "and efficient. We have conducted extensive experiments to compare DeepECOC", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 141, + 489, + 469, + 503 + ], + "spans": [ + { + "bbox": [ + 141, + 489, + 469, + 503 + ], + "score": 1.0, + "content": "with traditional ECOC, feature learning and deep learning algorithms on several", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 141, + 500, + 469, + 513 + ], + "spans": [ + { + "bbox": [ + 141, + 500, + 469, + 513 + ], + "score": 1.0, + "content": "benchmark data sets. The results demonstrate that DeepECOC perform not only", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 142, + 512, + 470, + 524 + ], + "spans": [ + { + "bbox": [ + 142, + 512, + 470, + 524 + ], + "score": 1.0, + "content": "better than traditional ECOC and feature learning algorithms, but also state-of-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 142, + 523, + 316, + 535 + ], + "spans": [ + { + "bbox": [ + 142, + 523, + 316, + 535 + ], + "score": 1.0, + "content": "the-art deep learning models in most cases.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 30.5, + "bbox_fs": [ + 141, + 337, + 470, + 535 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 557, + 206, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 556, + 208, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 208, + 573 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "Error correcting output codes (ECOC) are an ensemble learning framework to address multi-class", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "classification problems (Dietterich & Bakiri, 1995). The work by (Zhong & Liu, 2013) shows that", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "the ECOC methods can also be used for feature learning, in either a linear or a nonlinear manner.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "However, although sophisticated coding and decoding strategies are applied (Escalera et al., 2010;", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "Zhong et al., 2012; Zhong & Cheriet, 2013), the learnability of ECOC is limited by its single-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "score": 1.0, + "content": "layer structure. Therefore, to exploit the advantages of the ECOC framework, such as supervised", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "ensemble learning and effective coding design, it’s necessary to combine its ideas with that of deep", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 659, + 145, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 145, + 673 + ], + "score": 1.0, + "content": "learning.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 583, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "In recent years, many deep learning models have been proposed to handle various challenging prob-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "lems. Meantime, desirable performances in many domains have been achieved, such as image clas-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "score": 1.0, + "content": "sification and detection, document analysis and recognition, natural language processing, and video", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "analysis (Hinton & Salakhutdinov, 2006; Krizhevsky et al., 2012; Szegedy et al., 2014; Simonyan &", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "Zisserman, 2014; Zhang et al., 2015; Wang & Ji, 2015; Hong et al., 2015). Among others, (Hinton &", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 51, + "bbox_fs": [ + 105, + 677, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 83, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Salakhutdinov, 2006) presents the ground-breaking deep autoencoder that learns the weight matrices", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "by pre-training the stacked restricted Boltzmann machines (RBMs) and fine-tuning the weights using", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "gradient descent. It delivers much better representations of data than shallow feature learning algo-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "rithms, such as principal components analysis (PCA) (Jolliffe, 1986) and latent semantic analysis", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "(LSA) (Deerwester et al., 1990). In order to boost the traditional autoencoder and prevent the “over-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "fitting” problem, (Vincent et al., 2008) introduces the denosing autoencoder that corrupted the data", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "with a random noise. Recently, most of the research focuses on deep convolutional neural networks", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "(CNNs) and recurrent neural networks (RNNs), which greatly improves the state-of-the-art in the ar-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "eas of object recognition, unsegmented handwriting recognition and speech recognition (Krizhevsky", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 180, + 505, + 195 + ], + "spans": [ + { + "bbox": [ + 104, + 180, + 505, + 195 + ], + "score": 1.0, + "content": "et al., 2012; Graves et al., 2009; Sak et al., 2014). However, existing deep networks are generally ini-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "tialized with unsupervised methods, such as random assignments and greedy layerwise pre-training.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "In the case of random initialization, to obtain good results, many training data and a long training", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "time are generally used; while in the case of greedy layerwise pre-training, as the whole training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "score": 1.0, + "content": "data set needs to be used, the pre-training process is very time-consuming and difficult to find a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 235, + 169, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 169, + 248 + ], + "score": 1.0, + "content": "stable solution.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "To overcome the limitations of both traditional ECOC methods and deep learning models, and mean-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "while, take advantages of both of them, in this paper, we propose a novel deep learning model called", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "deep error-correcting output codes (DeepECOC). DeepECOC are composed of multiple stacked E-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "COC modules, each of which combines multiple binary classifiers for feature learning. Here, the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 310 + ], + "score": 1.0, + "content": "weights learned for the binary classifiers can be considered as weights between two successive lay-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "ers, while the probabilistic outputs of the combined binary classifiers as the outputs of a hidden", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 317, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 505, + 333 + ], + "score": 1.0, + "content": "layer or new representations of data. On the one hand, the ECOC modules can be learned layer by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 328, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 505, + 344 + ], + "score": 1.0, + "content": "layer using the given supervisory information, and on the other hand, based on the ternary coding", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "design, some classes of data are automatically neglected when training the binary classifiers, such", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 350, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 366 + ], + "score": 1.0, + "content": "that the weights are learned only using part of the training data. Hence, the supervised pre-training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "of DeepECOC is in general very effective and efficient. We have compared DeepECOC with tra-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "score": 1.0, + "content": "ditional ECOC, feature learning and deep learning algorithms to demonstrate the effectiveness and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 385, + 366, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 366, + 397 + ], + "score": 1.0, + "content": "superiority of DeepECOC. The results are reported in Section 4.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 108, + 401, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "The rest of this paper is organized as follows: In Section 2, we give a brief overview to related work.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "In Section 3, we present the proposed model, DeepECOC, in detail. The experimental results are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 424, + 466, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 466, + 435 + ], + "score": 1.0, + "content": "reported in Section 4, while Section 5 concludes this paper with remarks and future work.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 453, + 210, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 213, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 213, + 468 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "Traditional ECOC framework has two steps: coding and decoding. In the coding step, an E-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "COC matrix is defined or learned from data, and the binary classifiers are trained based on the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "ECOC coding; in the decoding step, the class label is given to a test sample based on a similarity", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 511, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 505, + 526 + ], + "score": 1.0, + "content": "measure between codewords and outputs of the binary classifiers. The widely used coding strate-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "gies include one-versus-all (OneVsAll) (Nilsson, 1965), one-versus-one (OneVsOne) (Hastie et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "1998), discriminant ECOC (DECOC) (Pujol et al., 2006), ECOC optimizing node embedding (E-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "score": 1.0, + "content": "COCONE) (Escalera et al., 2006), dense and sparse coding (Escalera et al., 2009; Allwein et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "2001), and so on. Among them, the OneVsAll, OneVsOne, dense and sparse coding strategies are", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "problem-independent, whilst the DECOC and ECOCONE are problem-dependent. Generally, the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 577, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 592 + ], + "score": 1.0, + "content": "length of the codeword by the OneVsAll, OneVsOne, DECOC and ECOCONE coding designs is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "related to the number of classes, but that by the dense and sparse coding design is relatively flex-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "ible. In this work, we design the structure of DeepECOC based on the properties of each coding", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "strategy. The commonly used binary ECOC decoding strategies are the Hamming decoding (Nils-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "son, 1965) and Euclidean decoding (Hastie et al., 1998). For ternary ECOC decoding strategies, the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "attenuated Euclidean decoding (Pujol et al., 2008), loss-based decoding (Allwein et al., 2001), and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "probabilistic-based decoding (Passerini et al., 2004) are widely used. Currently, the state-of-the-art", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "ternary ECOC decoding strategies are the discrete pessimistic beta density distribution decoding and", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "loss-weighted decoding (Escalera et al., 2010). In this work, for the simplicity of back propagation,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "we directly add a Softmax layer at the top of DeepECOC for the decoding. Note that, although", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "many sophisticated coding and decoding strategies have been proposed in recent years (Escalera", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "et al., 2010; Zhong et al., 2012; Zhong & Cheriet, 2013), the learnability of ECOC is limited by", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "its single-layer structure. To further exploit the advantages of ECOC, such as supervised ensemble", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 720, + 501, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 501, + 734 + ], + "score": 1.0, + "content": "learning and effective coding design, it’s necessary to combine its ideas with that of deep learning.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 43 + } + ], + "page_idx": 1, + "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 2017", + "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, + 83, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Salakhutdinov, 2006) presents the ground-breaking deep autoencoder that learns the weight matrices", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "by pre-training the stacked restricted Boltzmann machines (RBMs) and fine-tuning the weights using", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "gradient descent. It delivers much better representations of data than shallow feature learning algo-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "rithms, such as principal components analysis (PCA) (Jolliffe, 1986) and latent semantic analysis", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "(LSA) (Deerwester et al., 1990). 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However, existing deep networks are generally ini-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "tialized with unsupervised methods, such as random assignments and greedy layerwise pre-training.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "In the case of random initialization, to obtain good results, many training data and a long training", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "time are generally used; while in the case of greedy layerwise pre-training, as the whole training", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "score": 1.0, + "content": "data set needs to be used, the pre-training process is very time-consuming and difficult to find a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 235, + 169, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 169, + 248 + ], + "score": 1.0, + "content": "stable solution.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 7, + "bbox_fs": [ + 104, + 82, + 506, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 396 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 266 + ], + "score": 1.0, + "content": "To overcome the limitations of both traditional ECOC methods and deep learning models, and mean-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "while, take advantages of both of them, in this paper, we propose a novel deep learning model called", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "deep error-correcting output codes (DeepECOC). DeepECOC are composed of multiple stacked E-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "COC modules, each of which combines multiple binary classifiers for feature learning. Here, the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 296, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 505, + 310 + ], + "score": 1.0, + "content": "weights learned for the binary classifiers can be considered as weights between two successive lay-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "ers, while the probabilistic outputs of the combined binary classifiers as the outputs of a hidden", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 317, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 104, + 317, + 505, + 333 + ], + "score": 1.0, + "content": "layer or new representations of data. On the one hand, the ECOC modules can be learned layer by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 328, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 104, + 328, + 505, + 344 + ], + "score": 1.0, + "content": "layer using the given supervisory information, and on the other hand, based on the ternary coding", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "design, some classes of data are automatically neglected when training the binary classifiers, such", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 350, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 505, + 366 + ], + "score": 1.0, + "content": "that the weights are learned only using part of the training data. Hence, the supervised pre-training", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "of DeepECOC is in general very effective and efficient. We have compared DeepECOC with tra-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 387 + ], + "score": 1.0, + "content": "ditional ECOC, feature learning and deep learning algorithms to demonstrate the effectiveness and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 385, + 366, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 366, + 397 + ], + "score": 1.0, + "content": "superiority of DeepECOC. The results are reported in Section 4.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 252, + 506, + 397 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 401, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "The rest of this paper is organized as follows: In Section 2, we give a brief overview to related work.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "In Section 3, we present the proposed model, DeepECOC, in detail. The experimental results are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 424, + 466, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 466, + 435 + ], + "score": 1.0, + "content": "reported in Section 4, while Section 5 concludes this paper with remarks and future work.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 401, + 505, + 435 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 453, + 210, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 213, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 213, + 468 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 479, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 493 + ], + "score": 1.0, + "content": "Traditional ECOC framework has two steps: coding and decoding. In the coding step, an E-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "COC matrix is defined or learned from data, and the binary classifiers are trained based on the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "ECOC coding; in the decoding step, the class label is given to a test sample based on a similarity", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 511, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 505, + 526 + ], + "score": 1.0, + "content": "measure between codewords and outputs of the binary classifiers. The widely used coding strate-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "gies include one-versus-all (OneVsAll) (Nilsson, 1965), one-versus-one (OneVsOne) (Hastie et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "1998), discriminant ECOC (DECOC) (Pujol et al., 2006), ECOC optimizing node embedding (E-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 558 + ], + "score": 1.0, + "content": "COCONE) (Escalera et al., 2006), dense and sparse coding (Escalera et al., 2009; Allwein et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "2001), and so on. Among them, the OneVsAll, OneVsOne, dense and sparse coding strategies are", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 506, + 580 + ], + "score": 1.0, + "content": "problem-independent, whilst the DECOC and ECOCONE are problem-dependent. 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In this work, we design the structure of DeepECOC based on the properties of each coding", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "strategy. The commonly used binary ECOC decoding strategies are the Hamming decoding (Nils-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "son, 1965) and Euclidean decoding (Hastie et al., 1998). For ternary ECOC decoding strategies, the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "attenuated Euclidean decoding (Pujol et al., 2008), loss-based decoding (Allwein et al., 2001), and", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "probabilistic-based decoding (Passerini et al., 2004) are widely used. Currently, the state-of-the-art", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "ternary ECOC decoding strategies are the discrete pessimistic beta density distribution decoding and", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "loss-weighted decoding (Escalera et al., 2010). In this work, for the simplicity of back propagation,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "we directly add a Softmax layer at the top of DeepECOC for the decoding. 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To further exploit the advantages of ECOC, such as supervised ensemble", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 720, + 501, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 501, + 734 + ], + "score": 1.0, + "content": "learning and effective coding design, it’s necessary to combine its ideas with that of deep learning.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 479, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 147, + 80, + 447, + 152 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 147, + 80, + 447, + 152 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 80, + 447, + 152 + ], + "spans": [ + { + "bbox": [ + 147, + 80, + 447, + 152 + ], + "score": 0.935, + "type": "image", + "image_path": "86fe1fb2b334c231e08ad95e358eeac2a15ba70ca951e0e93d5b04445c7aad57.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 147, + 80, + 447, + 104.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 147, + 104.0, + 447, + 128.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 147, + 128.0, + 447, + 152.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 165, + 504, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 179 + ], + "score": 1.0, + "content": "Figure 1: Two coding matrices encoded with the one-versus-all (binary case) and one-versus-one", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 177, + 235, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 235, + 190 + ], + "score": 1.0, + "content": "(ternary case) coding strategies.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 193, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 193, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 193, + 505, + 206 + ], + "score": 1.0, + "content": "In the literature of deep learning, there is some work that attempts to construct a deep architecture", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 205, + 504, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 504, + 216 + ], + "score": 1.0, + "content": "with multiple feature learning methods (Hinton & Salakhutdinov, 2006; Trigeorgis et al., 2014; Yuan", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "et al., 2015; Zheng et al., 2015; 2014). For instance, deep autoencoder is built up by RBMs (Hinton", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "& Salakhutdinov, 2006), and deep semi-NMF combines multiple steps of matrix factorization (Tri-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "georgis et al., 2014). Similarly, deep CNNs and RNNs can also be considered as deep models that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "learn the new representations of data layer by layer (Krizhevsky et al., 2012; Graves et al., 2009; Sak", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "et al., 2014). The success of these existing models demonstrate that deep networks are beneficial to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "the representation learning tasks, especially for the large scale applications. However, as discussed", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "in the previous section, existing deep learning models are generally initialized with unsupervised", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "score": 1.0, + "content": "methods, such as random assignments and greedy layerwise pre-training, which result in a long", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "training time of the deep models. In this work, we propose the DeepECOC model, which is based", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "on the stacked ECOC modules. When pre-training DeepECOC, the ECOC modules can be learned", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "score": 1.0, + "content": "with the available supervisory information. Intuitively, as this manner of supervised pre-training", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 336, + 504, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 504, + 348 + ], + "score": 1.0, + "content": "has deterministic objective, the learned value of the parameters will be very close to the best local", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 348, + 504, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 504, + 358 + ], + "score": 1.0, + "content": "minimum on the solution manifold. Experimental results shown in Section 4 also demonstrate this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 127, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 127, + 370 + ], + "score": 1.0, + "content": "fact.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 384, + 425, + 398 + ], + "lines": [ + { + "bbox": [ + 104, + 382, + 427, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 382, + 427, + 401 + ], + "score": 1.0, + "content": "3 DEEP ERROR-CORRECTING OUTPUT CODES (DEEPECOC)", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 409, + 504, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 424 + ], + "score": 1.0, + "content": "In this section, we first introduce the traditional ECOC framework, which is the important building", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 421, + 457, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 457, + 434 + ], + "score": 1.0, + "content": "block of DeepECOC. Then we present the learning procedures of DeepECOC in detail.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 108, + 445, + 240, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 243, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 243, + 459 + ], + "score": 1.0, + "content": "3.1 THE ECOC FRAMEWORK", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "Error correcting output codes (ECOC), which combine multiple binary classifiers to solve multi-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "class classification problems, are an ensemble learning framework. The ECOC methods in gen-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "eral consist of two steps: coding and decoding. In the coding step, the ECOC coding matrix", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 496, + 507, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 183, + 511 + ], + "score": 0.93, + "content": "\\mathbf { M } \\in \\{ - 1 , 1 \\} ^ { C \\times L }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 496, + 253, + 514 + ], + "score": 1.0, + "content": "(binary case) or", + "type": "text" + }, + { + "bbox": [ + 253, + 498, + 340, + 511 + ], + "score": 0.93, + "content": "\\mathbf { M } \\in \\{ - 1 , \\mathbf { \\bar { 0 } } , 1 \\} ^ { C \\times L }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 496, + 507, + 514 + ], + "score": 1.0, + "content": "(ternary case) is first defined or learned", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 277, + 522 + ], + "score": 1.0, + "content": "from the training data, where each row of", + "type": "text" + }, + { + "bbox": [ + 277, + 511, + 290, + 520 + ], + "score": 0.29, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "is the codeword of a class, each column corresponds", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 521, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 254, + 534 + ], + "score": 1.0, + "content": "to a dichotomizer (binary classifier),", + "type": "text" + }, + { + "bbox": [ + 254, + 522, + 263, + 531 + ], + "score": 0.8, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 521, + 505, + 534 + ], + "score": 1.0, + "content": "is the length of the codewords (the number of binary classi-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 531, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 132, + 545 + ], + "score": 1.0, + "content": "fiers),", + "type": "text" + }, + { + "bbox": [ + 132, + 532, + 141, + 542 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 531, + 389, + 545 + ], + "score": 1.0, + "content": "is the number of classes, symbol ‘1’ indicates positive class,", + "type": "text" + }, + { + "bbox": [ + 389, + 532, + 405, + 542 + ], + "score": 0.37, + "content": "^ { \\bullet } - _ { 1 } \\bullet", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 531, + 505, + 545 + ], + "score": 1.0, + "content": "indicates negative class,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 104, + 542, + 125, + 556 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 126, + 543, + 137, + 553 + ], + "score": 0.67, + "content": "\\cdot _ { 0 } \\cdot \\mathrm { ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "indicates that a particular class is not considered by a given classifier. 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For instance, deep autoencoder is built up by RBMs (Hinton", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "& Salakhutdinov, 2006), and deep semi-NMF combines multiple steps of matrix factorization (Tri-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "georgis et al., 2014). Similarly, deep CNNs and RNNs can also be considered as deep models that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 260 + ], + "score": 1.0, + "content": "learn the new representations of data layer by layer (Krizhevsky et al., 2012; Graves et al., 2009; Sak", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "et al., 2014). The success of these existing models demonstrate that deep networks are beneficial to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 282 + ], + "score": 1.0, + "content": "the representation learning tasks, especially for the large scale applications. However, as discussed", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "in the previous section, existing deep learning models are generally initialized with unsupervised", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "score": 1.0, + "content": "methods, such as random assignments and greedy layerwise pre-training, which result in a long", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "training time of the deep models. In this work, we propose the DeepECOC model, which is based", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "on the stacked ECOC modules. When pre-training DeepECOC, the ECOC modules can be learned", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 339 + ], + "score": 1.0, + "content": "with the available supervisory information. Intuitively, as this manner of supervised pre-training", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 336, + 504, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 504, + 348 + ], + "score": 1.0, + "content": "has deterministic objective, the learned value of the parameters will be very close to the best local", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 348, + 504, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 504, + 358 + ], + "score": 1.0, + "content": "minimum on the solution manifold. Experimental results shown in Section 4 also demonstrate this", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 357, + 127, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 127, + 370 + ], + "score": 1.0, + "content": "fact.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 193, + 506, + 370 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 384, + 425, + 398 + ], + "lines": [ + { + "bbox": [ + 104, + 382, + 427, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 382, + 427, + 401 + ], + "score": 1.0, + "content": "3 DEEP ERROR-CORRECTING OUTPUT CODES (DEEPECOC)", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 409, + 504, + 432 + ], + "lines": [ + { + "bbox": [ + 105, + 408, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 424 + ], + "score": 1.0, + "content": "In this section, we first introduce the traditional ECOC framework, which is the important building", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 421, + 457, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 457, + 434 + ], + "score": 1.0, + "content": "block of DeepECOC. Then we present the learning procedures of DeepECOC in detail.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 408, + 506, + 434 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 445, + 240, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 243, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 243, + 459 + ], + "score": 1.0, + "content": "3.1 THE ECOC FRAMEWORK", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 505, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "Error correcting output codes (ECOC), which combine multiple binary classifiers to solve multi-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 490 + ], + "score": 1.0, + "content": "class classification problems, are an ensemble learning framework. The ECOC methods in gen-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "eral consist of two steps: coding and decoding. 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Next, we calculate the output of the first layer by using Eq. (9). Subse-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "quently, we repeat this process layer by layer to build the DeepECOC model. 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Through these processes, we finally obtain", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 499, + 491, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 491, + 511 + ], + "score": 1.0, + "content": "the DeepECOC model, which is robust and easy to be applied to multi-class classification tasks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 515, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "Note that, compared to existing deep learning algorithms, DeepECOC have some important ad-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "vantages. 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Next, we calculate the output of the first layer by using Eq. (9). Subse-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "quently, we repeat this process layer by layer to build the DeepECOC model. It’s obvious that, if we", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 463, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 463, + 238 + ], + "score": 1.0, + "content": "adopt different coding strategies, we can get different kinds of DeepECOC architectures.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 170, + 506, + 238 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 241, + 504, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "score": 1.0, + "content": "For the last layer of DeepECOC, we employ the softmax regression for the multi-class learning. 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It’s easy to compute the probability that", + "type": "text" + }, + { + "bbox": [ + 293, + 321, + 303, + 330 + ], + "score": 0.87, + "content": "\\mathbf { x } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 318, + 385, + 332 + ], + "score": 1.0, + "content": "is classified to class", + "type": "text" + }, + { + "bbox": [ + 385, + 320, + 391, + 330 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 318, + 395, + 332 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 106, + 306, + 505, + 332 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 334, + 391, + 367 + ], + "lines": [ + { + "bbox": [ + 220, + 334, + 391, + 367 + ], + "spans": [ + { + "bbox": [ + 220, + 334, + 391, + 367 + ], + "score": 0.95, + "content": "p ( y _ { i } = j | \\mathbf { x } _ { i } , \\mathbf { w } ) = \\frac { \\exp ( \\mathbf { w } _ { j } ^ { T } \\mathbf { h } _ { i } ^ { n - 1 } ) } { \\sum _ { l = 1 } ^ { K } \\exp ( \\mathbf { w } _ { l } ^ { T } \\mathbf { h } _ { i } ^ { n - 1 } ) } .", + "type": "interline_equation", + "image_path": "667cbea538a2c01c800af3eea5b245db892f15d4b1f191b0d5118dd71ab4f7bb.jpg" + } + ] + } + ], + "index": 18.5, + "virtual_lines": [ + { + "bbox": [ + 220, + 334, + 391, + 350.5 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 220, + 350.5, + 391, + 367.0 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 427, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 428, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 319, + 386 + ], + "score": 1.0, + "content": "Taking derivatives, one can show that the gradient of", + "type": "text" + }, + { + "bbox": [ + 319, + 371, + 342, + 384 + ], + "score": 0.92, + "content": "J ( \\mathbf { w } )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 369, + 428, + 386 + ], + "score": 1.0, + "content": "with respect to w is,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 369, + 428, + 386 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 388, + 416, + 423 + ], + "lines": [ + { + "bbox": [ + 194, + 388, + 416, + 423 + ], + "spans": [ + { + "bbox": [ + 194, + 388, + 416, + 423 + ], + "score": 0.93, + "content": "\\nabla J ( \\mathbf { w } ) = - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } [ \\mathbf { x } _ { i } ( \\mathbf { I } ( y _ { i } = j ) - p ( y _ { i } = j | \\mathbf { x } _ { i } , \\mathbf { w } ) ) ] .", + "type": "interline_equation", + "image_path": "6e907982271e941891ccdd03e899cfa4a5deba920bfa264517873943d5beae75.jpg" + } + ] + } + ], + "index": 21.5, + "virtual_lines": [ + { + "bbox": [ + 194, + 388, + 416, + 405.5 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 194, + 405.5, + 416, + 423.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 506, + 445 + ], + "score": 1.0, + "content": "After the pre-training step, we use back propagation (Rumelhart et al., 1988) to fine tune the whole", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "architecture. Moreover, we also employ a technique called “dropout” for regularization (Hinton", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "et al., 2012). When a large feedforward neural network is trained on a small training set, dropout", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 478 + ], + "score": 1.0, + "content": "generally performs well on the test set. The basic idea of dropout is that each hidden node is ran-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 322, + 489 + ], + "score": 1.0, + "content": "domly omitted from the network with a probability of", + "type": "text" + }, + { + "bbox": [ + 323, + 477, + 330, + 488 + ], + "score": 0.81, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 477, + 505, + 489 + ], + "score": 1.0, + "content": ". In another view, dropout is a very efficient", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "way to perform model averaging with neural networks. Through these processes, we finally obtain", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 499, + 491, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 491, + 511 + ], + "score": 1.0, + "content": "the DeepECOC model, which is robust and easy to be applied to multi-class classification tasks.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 433, + 506, + 511 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 515, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "Note that, compared to existing deep learning algorithms, DeepECOC have some important ad-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 527, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 538 + ], + "score": 1.0, + "content": "vantages. Firstly, unlike previous deep learning algorithms, DeepECOC are built with the ECOC", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 551 + ], + "score": 1.0, + "content": "modules and pre-trained in a supervised learning fashion. Secondly, if we adopt ternary coding s-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 547, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 104, + 547, + 506, + 563 + ], + "score": 1.0, + "content": "trategies, due to the natural merit of ECOC, the weights can be learned using only part of the training", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 572 + ], + "score": 1.0, + "content": "data. Thirdly, in contrast to the learning of the weight matrices in previous deep learning models,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 569, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 505, + 584 + ], + "score": 1.0, + "content": "the binary classifiers in each ECOC module can be learned in parallel, which may greatly speed up", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 581, + 220, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 220, + 594 + ], + "score": 1.0, + "content": "the learning of DeepECOC.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 515, + 506, + 594 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 609, + 201, + 621 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 201, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 201, + 623 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 624, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 625, + 504, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 504, + 637 + ], + "score": 1.0, + "content": "To evaluate the effectiveness of the proposed method, DeepECOC, we conducted 4 parts of experi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 649 + ], + "score": 1.0, + "content": "ments. In the first part, we compared DeepECOC with some deep learning models and single-layer", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 647, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 647, + 505, + 660 + ], + "score": 1.0, + "content": "ECOC approaches on 16 data sets from the UCI machine learning repository 1. In the second part,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 671 + ], + "score": 1.0, + "content": "we compared DeepECOC with traditional feature learning models, some deep learning models and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "single-layer ECOC approaches on the USPS handwritten digits 2, and tested DeepECOC with dif-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 680, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 506, + 693 + ], + "score": 1.0, + "content": "ferent number of hidden layers. In the third part, we used the MNIST handwritten digits 3 to further", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "demonstrate the effectiveness of DeepECOC for handwritten digits recognition. Finally, the CIFAR-", + "type": "text", + "cross_page": true + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "10 data set 4 was used to demonstrate the effectiveness of DeepECOC on image classification tasks.", + "type": "text", + "cross_page": true + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 346, + 607 + ], + "score": 1.0, + "content": "For all the data sets, the features were normalized within", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 347, + 594, + 367, + 606 + ], + "score": 0.46, + "content": "[ 0 , 1 ]", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 367, + 593, + 505, + 607 + ], + "score": 1.0, + "content": ". In the following, we report the", + "type": "text", + "cross_page": true + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 606, + 226, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 226, + 616 + ], + "score": 1.0, + "content": "experimental results in detail.", + "type": "text", + "cross_page": true + } + ], + "index": 33 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 625, + 506, + 693 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 166, + 104, + 445, + 180 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 137, + 89, + 471, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 88, + 473, + 102 + ], + "spans": [ + { + "bbox": [ + 137, + 88, + 473, + 102 + ], + "score": 1.0, + "content": "Table 1: Details of the UCI data sets (T: training samples; A: attributes; C: classes).", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 166, + 104, + 445, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 104, + 445, + 180 + ], + "spans": [ + { + "bbox": [ + 166, + 104, + 445, + 180 + ], + "score": 0.98, + "html": "
Problem#ofT#ofA#ofCProblem#ofT#ofAofC
Dermatology366346Yeast1484810
Iris15043Satimage6435367
Ecoli33688Letter200001626
Wine178133Pendigits109921610
Glass21497Segmentation2310197
Thyroid21553Optdigits56206410
Vowel9901011Shuttle1450097
Balance62543Vehicle846184
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ProblemSingleAEDAEDeepECOC (1)DeepECOC (2)DeepECOC (3）
Dermatology0.95130.9429 ±0.06710.9674 ±0.03120.9702 ±0.03540.9779 ±0.02080.9747 ±0.0318
Iris0.96000.9600 ±0.05620.9333 ±0.08890.9600 ±0.05350.9267 ±0.11090.9533 ±0.0383
Ecoli0.81470.7725 ±0.06080.8000 ±0.03620.8529 ±0.04030.8824 ±0.06260.9118 ±0.0636
Wine0.96050.9765 ±0.02640.9563 ±0.04220.9875 ±0.02640.9813 ±0.03020.9688 ±0.0329
Glass0.67620.6669 ±0.10320.6669 ±0.07150.7895 ±0.07880.7368 ±0.11400.7562 ±0.0879
Thyroid0.92100.9513 ±0.06140.9599 ±0.05670.9656 ±0.05130.9703 ±0.05400.9608 ±0.0518
Vowel0.71770.6985 ±0.07450.7101 ±0.07560.7475 ±0.09010.6010 ±0.06270.6863 ±0.0788
Balance0.82220.8036 ±0.03200.8268 ±0.05480.9137 ±0.04120.8333 ±0.03180.9167 ±0.0312
Yeast0.52170.5641 ±0.03460.5891 ±0.02720.5959 ±0.05990.5494 ±0.04340.5697 ±0.0462
Satimage0.85370.8675 ±0.0528 0.92340.8897 ±0.03040.8961 ±0.04800.8360 ±0.03900.9077 ±0.0555
Letter0.9192±0.0547 0.98310.9381 ±0.0641 0.98860.9532 ±0.03410.9247 ±0.03520.9501 ±0.0563
Pendigits0.9801±0.0123 0.9584±0.0034 0.95960.9908 ±0.0031 0.97110.9866 ±0.0107 0.95840.9899 ±0.0075 0.9711
Segmentation0.9701±0.0317 0.9785±0.0211 0.9856±0.0286 0.9867±0.0163 0.9848±0.0233 0.9911
Optdigits0.9982±0.0101 0.9953±0.0088 0.9976±0.0096 0.9988±0.0123 0.9983±0.0091 0.9993
Shuttle0.9988±0.0012 0.6987±0.0014 0.7348±0.0021 0.7561±0.0018 0.6908±0.0010 0.7195
Vehicle Mean rank0.7315 4.0938±0.0521 4.8750±0.0454 3.9375±0.0480 1.7500±0.04321 3.9375±0.0148 2.4063
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Finally, the CIFAR-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "10 data set 4 was used to demonstrate the effectiveness of DeepECOC on image classification tasks.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 593, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 346, + 607 + ], + "score": 1.0, + "content": "For all the data sets, the features were normalized within", + "type": "text" + }, + { + "bbox": [ + 347, + 594, + 367, + 606 + ], + "score": 0.46, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 593, + 505, + 607 + ], + "score": 1.0, + "content": ". In the following, we report the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 606, + 226, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 226, + 616 + ], + "score": 1.0, + "content": "experimental results in detail.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 622, + 452, + 635 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 454, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 454, + 635 + ], + "score": 1.0, + "content": "4.1 CLASSIFICATION ON 16 UCI MACHINE LEARNING REPOSITORY DATA SETS", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 710 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "The detail of the UCI data sets are shown in Table 1. In these experiments, we compared DeepECOC", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "with autoencoder (AE) (Hinton & Salakhutdinov, 2006), denoising autoencoder (DAE) (Vincent", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "score": 1.0, + "content": "et al., 2008) and single-layer ECOC approaches (Single) (Escalera et al., 2010). We built DeepECOC", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "with the ECOC optimizing node embedding (ECOCONE) coding method (Escalera et al., 2006).", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "Here, since we initialized ECOCONE with 3 different coding methods, i.e. one-versus-one, one-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "versus-all and DECOC, DeepECOC had 3 variants. In addition, the state-of-the-art linear loss-", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 120, + 721, + 278, + 731 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 279, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 279, + 733 + ], + "score": 1.0, + "content": "4http://www.cs.toronto.edu/∼kriz/cifar.html", + "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": "table", + "bbox": [ + 166, + 104, + 445, + 180 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 137, + 89, + 471, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 88, + 473, + 102 + ], + "spans": [ + { + "bbox": [ + 137, + 88, + 473, + 102 + ], + "score": 1.0, + "content": "Table 1: Details of the UCI data sets (T: training samples; A: attributes; C: classes).", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 166, + 104, + 445, + 180 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 104, + 445, + 180 + ], + "spans": [ + { + "bbox": [ + 166, + 104, + 445, + 180 + ], + "score": 0.98, + "html": "
Problem#ofT#ofA#ofCProblem#ofT#ofAofC
Dermatology366346Yeast1484810
Iris15043Satimage6435367
Ecoli33688Letter200001626
Wine178133Pendigits109921610
Glass21497Segmentation2310197
Thyroid21553Optdigits56206410
Vowel9901011Shuttle1450097
Balance62543Vehicle846184
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ProblemSingleAEDAEDeepECOC (1)DeepECOC (2)DeepECOC (3）
Dermatology0.95130.9429 ±0.06710.9674 ±0.03120.9702 ±0.03540.9779 ±0.02080.9747 ±0.0318
Iris0.96000.9600 ±0.05620.9333 ±0.08890.9600 ±0.05350.9267 ±0.11090.9533 ±0.0383
Ecoli0.81470.7725 ±0.06080.8000 ±0.03620.8529 ±0.04030.8824 ±0.06260.9118 ±0.0636
Wine0.96050.9765 ±0.02640.9563 ±0.04220.9875 ±0.02640.9813 ±0.03020.9688 ±0.0329
Glass0.67620.6669 ±0.10320.6669 ±0.07150.7895 ±0.07880.7368 ±0.11400.7562 ±0.0879
Thyroid0.92100.9513 ±0.06140.9599 ±0.05670.9656 ±0.05130.9703 ±0.05400.9608 ±0.0518
Vowel0.71770.6985 ±0.07450.7101 ±0.07560.7475 ±0.09010.6010 ±0.06270.6863 ±0.0788
Balance0.82220.8036 ±0.03200.8268 ±0.05480.9137 ±0.04120.8333 ±0.03180.9167 ±0.0312
Yeast0.52170.5641 ±0.03460.5891 ±0.02720.5959 ±0.05990.5494 ±0.04340.5697 ±0.0462
Satimage0.85370.8675 ±0.0528 0.92340.8897 ±0.03040.8961 ±0.04800.8360 ±0.03900.9077 ±0.0555
Letter0.9192±0.0547 0.98310.9381 ±0.0641 0.98860.9532 ±0.03410.9247 ±0.03520.9501 ±0.0563
Pendigits0.9801±0.0123 0.9584±0.0034 0.95960.9908 ±0.0031 0.97110.9866 ±0.0107 0.95840.9899 ±0.0075 0.9711
Segmentation0.9701±0.0317 0.9785±0.0211 0.9856±0.0286 0.9867±0.0163 0.9848±0.0233 0.9911
Optdigits0.9982±0.0101 0.9953±0.0088 0.9976±0.0096 0.9988±0.0123 0.9983±0.0091 0.9993
Shuttle0.9988±0.0012 0.6987±0.0014 0.7348±0.0021 0.7561±0.0018 0.6908±0.0010 0.7195
Vehicle Mean rank0.7315 4.0938±0.0521 4.8750±0.0454 3.9375±0.0480 1.7500±0.04321 3.9375±0.0148 2.4063
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In these experiments, we compared DeepECOC", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "with autoencoder (AE) (Hinton & Salakhutdinov, 2006), denoising autoencoder (DAE) (Vincent", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "score": 1.0, + "content": "et al., 2008) and single-layer ECOC approaches (Single) (Escalera et al., 2010). We built DeepECOC", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "with the ECOC optimizing node embedding (ECOCONE) coding method (Escalera et al., 2006).", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "Here, since we initialized ECOCONE with 3 different coding methods, i.e. one-versus-one, one-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "versus-all and DECOC, DeepECOC had 3 variants. In addition, the state-of-the-art linear loss-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "weighted (LLW) decoding strategy was used for ECOCONE. Finally, a structure with 3 hidden", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 454, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 454, + 107 + ], + "score": 1.0, + "content": "layers was adopted for DeepECOC, which had 0.1 denoising rate and 0.1 dropout rate:", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 644, + 506, + 711 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 503, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "weighted (LLW) decoding strategy was used for ECOCONE. Finally, a structure with 3 hidden", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 454, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 454, + 107 + ], + "score": 1.0, + "content": "layers was adopted for DeepECOC, which had 0.1 denoising rate and 0.1 dropout rate:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 109, + 403, + 130 + ], + "lines": [ + { + "bbox": [ + 206, + 109, + 403, + 130 + ], + "spans": [ + { + "bbox": [ + 206, + 109, + 403, + 130 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\textbf { x } \\overset { q _ { D } } { \\longrightarrow } \\tilde { \\textbf { x } } \\xrightarrow { \\textbf { W } _ { 1 } } \\textbf { h } _ { 1 } \\xrightarrow { \\textbf { W } _ { 2 } } \\textbf { h } _ { 2 } \\xrightarrow { \\textbf { W } _ { 3 } } \\textbf { h } _ { 3 } \\xrightarrow { s o f t m a x } \\textbf { y } . } \\end{array}", + "type": "interline_equation", + "image_path": "2f9608a69fc7f88180cac3b3bd23ba947988dd955afc29c4acf47acef9c3238b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 206, + 109, + 403, + 130 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 133, + 505, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 133, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 505, + 147 + ], + "score": 1.0, + "content": "For the fine-tuning process, we used the stochastic gradient descent algorithm. The learning rate", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 144, + 504, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 504, + 158 + ], + "score": 1.0, + "content": "and epoches from different data sets are described in Table 3. The autoencoder and denoising au-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 156, + 504, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 504, + 168 + ], + "score": 1.0, + "content": "toencoder’s architectures are as same as DeepECOC with ECOCONE initialized by one-versus-one.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 505, + 179 + ], + "score": 1.0, + "content": "For single-layer ECOC approaches, we chose the best results shown in (Escalera et al., 2010) as", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 190 + ], + "score": 1.0, + "content": "our compared results. For all DeepECOC models, we used support vector machines (SVMs) with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 188, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 505, + 201 + ], + "score": 1.0, + "content": "RBF kernel function as base classifiers. The parameters of SVMs were set to default (Chang & Lin,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 198, + 136, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 136, + 212 + ], + "score": 1.0, + "content": "2011).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 216, + 505, + 305 + ], + "lines": [ + { + "bbox": [ + 105, + 215, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 229 + ], + "score": 1.0, + "content": "Table 2 shows the average classification accuracy and standard deviation on 16 UCI data sets. Except", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "on the OptDigits data set, DeepECOC achieved the best results compared with autoencoder, denois-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 104, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "ing autoencoder and single-layer ECOC approaches. In fact, on the OptDigits data set, DeepECOC", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 104, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "achieved comparative result with single-layer ECOC approaches. Among others, DeepECOC with", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 259, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 259, + 506, + 274 + ], + "score": 1.0, + "content": "ECOCONE (initialized by one-versus-one) coding strategy obtained the best results on 9 data sets,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "while DeepECOC with ECOCONE (initialized by DECOC) coding strategy obtained the best results", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "on 5 data sets. 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The size of the images is", + "type": "text" + }, + { + "bbox": [ + 240, + 457, + 299, + 467 + ], + "score": 0.91, + "content": "1 6 \\times 1 6 = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 456, + 506, + 469 + ], + "score": 1.0, + "content": ". Our experiments on this data set were divided into", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "2 parts. 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Here, PCA is an unsupervised method, MFA is a super-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "vised method. For MFA, the number of nearest neighbors for constructing the intrinsic graph was", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "score": 1.0, + "content": "set to 5, while that for constructing the penalty graph was set to 15. For DeepECOC, we also used", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 532, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 506, + 547 + ], + "score": 1.0, + "content": "3 coding design methods in this experiment. 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The kernel size of the convolutional layers and pooling layers was set to", + "type": "text" + }, + { + "bbox": [ + 465, + 599, + 487, + 610 + ], + "score": 0.91, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 600, + 505, + 611 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "stride was set to 1, the number of nodes of the first layer was set to 200, the epoch was set to 8000,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "the initial learning rate was set to 0.001, learning rate policy was set to “inv”, and the momentum", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "was set to 0.9. 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Except", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 227, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 505, + 239 + ], + "score": 1.0, + "content": "on the OptDigits data set, DeepECOC achieved the best results compared with autoencoder, denois-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 237, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 104, + 237, + 506, + 251 + ], + "score": 1.0, + "content": "ing autoencoder and single-layer ECOC approaches. In fact, on the OptDigits data set, DeepECOC", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 104, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "achieved comparative result with single-layer ECOC approaches. Among others, DeepECOC with", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 259, + 506, + 274 + ], + "spans": [ + { + "bbox": [ + 104, + 259, + 506, + 274 + ], + "score": 1.0, + "content": "ECOCONE (initialized by one-versus-one) coding strategy obtained the best results on 9 data sets,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 284 + ], + "score": 1.0, + "content": "while DeepECOC with ECOCONE (initialized by DECOC) coding strategy obtained the best results", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "on 5 data sets. 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The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "denoising rate and dropout rate were set to 0.1, the batch size was set to 100, the learning rate was", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 104, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "set to 0.01, and the number of epoch was set to 80000. For LeNet model, we adopted the parameters", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 104, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "as same as (LeCun et al., 1998). For PCANet model, we used two PCA-filter stages, one binary", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "hashing stage and one blockwise histograms. 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We can see that DeepECOC", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "are comparative with existing deep learning methods on the second architecture and outperform", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 124, + 111, + 488, + 138 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 501, + 110 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 88, + 503, + 101 + ], + "spans": [ + { + "bbox": [ + 105, + 88, + 503, + 101 + ], + "score": 1.0, + "content": "Table 4: Classification accuracy obtained on the LBP-CIFAR10 data set. The best result for each", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 100, + 251, + 112 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 251, + 112 + ], + "score": 1.0, + "content": "scenario is highlighted in bold face.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 124, + 111, + 488, + 138 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 111, + 488, + 138 + ], + "spans": [ + { + "bbox": [ + 124, + 111, + 488, + 138 + ], + "score": 0.952, + "html": "
ProblemAEDAELeNetPCANetDeepECOC(1)DeepECOC(2)DeepECOC(3)
LBP-CIFAR10 (36)0.35010.36780.32560.25690.50890.45170.4752
LBP-CIFAR10 (256)0.43520.45870.32210.25690.55880.45890.5224
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In addition, DeepECOC with both two architectures outperform the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 234, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 234, + 168 + ], + "score": 1.0, + "content": "single-layer ECOC approaches.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 105, + 181, + 349, + 193 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 351, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 351, + 195 + ], + "score": 1.0, + "content": "4.4 CLASSIFICATION ON THE LBP-CIFAR10 DATA SET", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 389, + 216 + ], + "score": 1.0, + "content": "The CIFAR-10 dataset is a relative large scale data set which consists of", + "type": "text" + }, + { + "bbox": [ + 390, + 203, + 446, + 213 + ], + "score": 0.89, + "content": "6 0 0 0 0 3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 201, + 505, + 216 + ], + "score": 1.0, + "content": "colour images", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 212, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 227 + ], + "score": 1.0, + "content": "in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 238 + ], + "score": 1.0, + "content": "For the purpose of reducing computational cost, we attempted to extract features of the data using", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "an efficient local binary patterns algorithm. As a result, the representations with dimensionality 36", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 258 + ], + "score": 1.0, + "content": "and 256 were adopted and the data were normalized to [0, 1] as well, called LBP-CIFAR10 (36)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 256, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 271 + ], + "score": 1.0, + "content": "and LBP-CIFAR10 (256). We also used 3 hidden layers for all deep learning methods. The learning", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 268, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 280 + ], + "score": 1.0, + "content": "rate was set to 0.1, and the epoch was set to 4000. 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The kernel size was set to", + "type": "text" + }, + { + "bbox": [ + 462, + 280, + 486, + 290 + ], + "score": 0.9, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 279, + 506, + 292 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "stride was set to 1, the number of node of the first fully connected layer was set to 64, the epoch", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "was set to 4000, the initial learning rate was set to 0.01, learning rate policy was set to “inv”, and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 311, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 505, + 326 + ], + "score": 1.0, + "content": "the momentum was set to 0.9. For the PCANet model, we used two PCA-filter stages, one binary", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "score": 1.0, + "content": "hashing stage and one blockwise histograms. 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The classification", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 346, + 239, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 239, + 356 + ], + "score": 1.0, + "content": "accuracy are reported in Table 4.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "From Table 4, we can easy to see that DeepECOC achieved the best results. Moreover, DeepECOC", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "with ECOCONE (initialized by one-versus-one) coding strategy achieved the better results than", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 382, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 382, + 506, + 398 + ], + "score": 1.0, + "content": "autoencoder and denoising autoencoder, LeNet and PCANet. Hence, we can conclude that, Deep-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "ECOC are a general model to handle different real world applications and achieves desirable results", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 406, + 165, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 165, + 419 + ], + "score": 1.0, + "content": "in most cases.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 108, + 434, + 195, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 197, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 197, + 450 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 104, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "In this paper, we propose a novel deep learning model, called deep error correcting output codes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "(DeepECOC). DeepECOC extend traditional ECOC algorithms to a deep architecture fashion, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "meanwhile, brings new elements to the deep learning area, such as supervised initialization, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "automatic neglecting of part of the data during network training. Extensive experiments on 16", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "data sets from the UCI machine learning repository, the USPS and MNIST handwritten digits and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "the CIFAR-10 data set demonstrate the superiority of DeepECOC over traditional ECOC, feature", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 506, + 538 + ], + "score": 1.0, + "content": "learning and deep learning methods. 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The best result for each", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 100, + 251, + 112 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 251, + 112 + ], + "score": 1.0, + "content": "scenario is highlighted in bold face.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 124, + 111, + 488, + 138 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 111, + 488, + 138 + ], + "spans": [ + { + "bbox": [ + 124, + 111, + 488, + 138 + ], + "score": 0.952, + "html": "
ProblemAEDAELeNetPCANetDeepECOC(1)DeepECOC(2)DeepECOC(3)
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In addition, DeepECOC with both two architectures outperform the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 234, + 168 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 234, + 168 + ], + "score": 1.0, + "content": "single-layer ECOC approaches.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 105, + 181, + 349, + 193 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 351, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 351, + 195 + ], + "score": 1.0, + "content": "4.4 CLASSIFICATION ON THE LBP-CIFAR10 DATA SET", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 203, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 389, + 216 + ], + "score": 1.0, + "content": "The CIFAR-10 dataset is a relative large scale data set which consists of", + "type": "text" + }, + { + "bbox": [ + 390, + 203, + 446, + 213 + ], + "score": 0.89, + "content": "6 0 0 0 0 3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 201, + 505, + 216 + ], + "score": 1.0, + "content": "colour images", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 212, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 227 + ], + "score": 1.0, + "content": "in 10 classes, with 6000 images per class. There are 50000 training images and 10000 test images.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 223, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 506, + 238 + ], + "score": 1.0, + "content": "For the purpose of reducing computational cost, we attempted to extract features of the data using", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 248 + ], + "score": 1.0, + "content": "an efficient local binary patterns algorithm. 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For the PCANet model, we used two PCA-filter stages, one binary", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 335 + ], + "score": 1.0, + "content": "hashing stage and one blockwise histograms. 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The classification", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 346, + 239, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 239, + 356 + ], + "score": 1.0, + "content": "accuracy are reported in Table 4.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 201, + 506, + 356 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "From Table 4, we can easy to see that DeepECOC achieved the best results. Moreover, DeepECOC", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "with ECOCONE (initialized by one-versus-one) coding strategy achieved the better results than", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 382, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 382, + 506, + 398 + ], + "score": 1.0, + "content": "autoencoder and denoising autoencoder, LeNet and PCANet. Hence, we can conclude that, Deep-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "ECOC are a general model to handle different real world applications and achieves desirable results", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 406, + 165, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 165, + 419 + ], + "score": 1.0, + "content": "in most cases.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 362, + 506, + 419 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 434, + 195, + 447 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 197, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 197, + 450 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 460, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 104, + 459, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 104, + 459, + 506, + 473 + ], + "score": 1.0, + "content": "In this paper, we propose a novel deep learning model, called deep error correcting output codes", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 483 + ], + "score": 1.0, + "content": "(DeepECOC). DeepECOC extend traditional ECOC algorithms to a deep architecture fashion, and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 495 + ], + "score": 1.0, + "content": "meanwhile, brings new elements to the deep learning area, such as supervised initialization, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "automatic neglecting of part of the data during network training. Extensive experiments on 16", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 506, + 516 + ], + "score": 1.0, + "content": "data sets from the UCI machine learning repository, the USPS and MNIST handwritten digits and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "the CIFAR-10 data set demonstrate the superiority of DeepECOC over traditional ECOC, feature", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 526, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 506, + 538 + ], + "score": 1.0, + "content": "learning and deep learning methods. 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ProblemSingleAEDAEDeepECOC (1)DeepECOC (2)DeepECOC (3）
Dermatology0.95130.9429 ±0.06710.9674 ±0.03120.9702 ±0.03540.9779 ±0.02080.9747 ±0.0318
Iris0.96000.9600 ±0.05620.9333 ±0.08890.9600 ±0.05350.9267 ±0.11090.9533 ±0.0383
Ecoli0.81470.7725 ±0.06080.8000 ±0.03620.8529 ±0.04030.8824 ±0.06260.9118 ±0.0636
Wine0.96050.9765 ±0.02640.9563 ±0.04220.9875 ±0.02640.9813 ±0.03020.9688 ±0.0329
Glass0.67620.6669 ±0.10320.6669 ±0.07150.7895 ±0.07880.7368 ±0.11400.7562 ±0.0879
Thyroid0.92100.9513 ±0.06140.9599 ±0.05670.9656 ±0.05130.9703 ±0.05400.9608 ±0.0518
Vowel0.71770.6985 ±0.07450.7101 ±0.07560.7475 ±0.09010.6010 ±0.06270.6863 ±0.0788
Balance0.82220.8036 ±0.03200.8268 ±0.05480.9137 ±0.04120.8333 ±0.03180.9167 ±0.0312
Yeast0.52170.5641 ±0.03460.5891 ±0.02720.5959 ±0.05990.5494 ±0.04340.5697 ±0.0462
Satimage0.85370.8675 ±0.0528 0.92340.8897 ±0.03040.8961 ±0.04800.8360 ±0.03900.9077 ±0.0555
Letter0.9192±0.0547 0.98310.9381 ±0.0641 0.98860.9532 ±0.03410.9247 ±0.03520.9501 ±0.0563
Pendigits0.9801±0.0123 0.9584±0.0034 0.95960.9908 ±0.0031 0.97110.9866 ±0.0107 0.95840.9899 ±0.0075 0.9711
Segmentation0.9701±0.0317 0.9785±0.0211 0.9856±0.0286 0.9867±0.0163 0.9848±0.0233 0.9911
Optdigits0.9982±0.0101 0.9953±0.0088 0.9976±0.0096 0.9988±0.0123 0.9983±0.0091 0.9993
Shuttle0.9988±0.0012 0.6987±0.0014 0.7348±0.0021 0.7561±0.0018 0.6908±0.0010 0.7195
Vehicle Mean rank0.7315 4.0938±0.0521 4.8750±0.0454 3.9375±0.0480 1.7500±0.04321 3.9375±0.0148 2.4063
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Problem#ofT#ofA#ofCProblem#ofT#ofAofC
Dermatology366346Yeast1484810
Iris15043Satimage6435367
Ecoli33688Letter200001626
Wine178133Pendigits109921610
Glass21497Segmentation2310197
Thyroid21553Optdigits56206410
Vowel9901011Shuttle1450097
Balance62543Vehicle846184
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ProblemAEDAELeNetPCANetDeepECOC(1)DeepECOC(2)DeepECOC(3)
LBP-CIFAR10 (36)0.35010.36780.32560.25690.50890.45170.4752
LBP-CIFAR10 (256)0.43520.45870.32210.25690.55880.45890.5224
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a/parse/train/N2dEPuK0vom/N2dEPuK0vom.md b/parse/train/N2dEPuK0vom/N2dEPuK0vom.md new file mode 100644 index 0000000000000000000000000000000000000000..873538a4dad3acabf45db2ff8a89c120f3ac7c73 --- /dev/null +++ b/parse/train/N2dEPuK0vom/N2dEPuK0vom.md @@ -0,0 +1,510 @@ +# Counterfactual Graph Learning for Link Prediction + +Anonymous Author(s) +Affiliation +Address +email + +# Abstract + +Learning to predict missing links is important for many graph-based applications. Existing methods were designed to learn the observed association between two sets of variables: (1) the observed graph structure and (2) the existence of link between a pair of nodes. However, the causal relationship between these variables was ignored and we visit the possibility of learning it by simply asking a counterfactual question: “would the link exist or not if the observed graph structure became different?” To answer this question by causal inference, we consider the information of the node pair as context, global graph structural properties as treatment, and link existence as outcome. In this work, we propose a novel link prediction method that enhances graph learning by the counterfactual inference. It creates counterfactual links from the observed ones, and our method learns representations from both of them. Experiments on a number of benchmark datasets show that our proposed method achieves the state-of-the-art performance on link prediction. + +# 14 1 Introduction + +15 Link prediction seeks to predict the likelihood of edge existence between node pairs based on the +16 observed graph. Given the omnipresence of graph-structured data, link prediction has copious applica +17 tions such as movie recommendation (Bennett et al., 2007), chemical interaction prediction (Stanfield +18 et al., 2017), and knowledge graph completion (Kazemi and Poole, 2018). Graph machine learning +19 methods have been widely applied to solve this problem. Their standard scheme is to first learn the +20 representation vectors of nodes and then learn the association between the representations of a pair of +21 nodes and the existence of the link between them. For example, graph neural networks (GNNs) use +22 neighborhood aggregation to create the representation vectors: the representation vector of a node +23 is computed by recursively aggregating and transforming representation vectors of its neighboring +24 nodes (Kipf and Welling, 2016a; Hamilton et al., 2017; Wu et al., 2020). Then the vectors are fed +25 into a binary classification model to learn the association. GNN methods have shown predominance +26 in the task of link prediction (Kipf and Welling, 2016b; Zhang and Chen, 2018; Zhang et al., 2020a). +27 Unfortunately, the causal relationship between graph structure and link existence was largely ignored +28 in the previous work. Existing methods that learn from association only were not able to capture +29 essential factors to accurately predict missing links in the test data. Take social network as an example. +30 Suppose Alice and Adam live in the same neighborhood and they are close friends. The association +31 between neighborhood belonging and friend closeness could be too strong to discover the essential +32 factors of the friendship such as common interests or family relationship which could be the cause of +33 being living in the same neighborhood. So, our idea is to ask a counterfactual question: “would Alice +34 and Adam still be close friends if they were not living in the same neighborhood?” If a graph learning +35 model could learn the causal relationship from data by asking the counterfactual questions, it would +36 improve the performance of link prediction with the novel knowledge it captured. Generally, the +37 questions can be described as “would the link exist or not if the graph structure became different?” + +![](images/3faaa9eaa22e961dc833289ae7f8919e66d2220e8c36f98725f81166d66c30b8.jpg) + +(Alice, Adam) (Helen, Bob)Most similar with a different treatment + +Factual link: 1 Counterfactual link: 1 (a) Find counterfactual link as the most similar node pair with a different treatment. + +![](images/7c895b3ed93f45a2ff5e6a6167d1d78665e3dbc492d83aabed634a32d4e2cce4.jpg) +(b) Train a GNN-based link predictor to predict factual and counterfactual links given the corresponding treatments. + +Figure 1: The proposed CFLP learns the causal relationship between the observed graph structure (e.g., neighborhood similarity, considered as treatment variable) and link existence (considered as outcome). In this example, the link predictor would be trained to estimate the individual treatment effect (ITE) as $1 - 1 = 0$ so it looks for factors other than neighborhood to predict the factual link. + +38 As known to many, counterfactual question is a key component of causal inference and have been +39 well defined in the literature. A counterfactual question is usually framed with three factors: context +40 (as a data point), manipulation (e.g., treatment, intervention, action, strategy), and outcome (van der +41 Laan and Petersen, 2007; Johansson et al., 2016). (To simplify the language, we use “treatment” to +42 refer to the manipulation in this paper, as readers might be familiar more with the word “treatment.”) +43 Given certain data context, it asks what the outcome would have been if the treatment had not been +44 the observed value. In the scenario of link prediction, we consider the information of a pair of nodes +45 as context, graph structural properties as treatment, and link existence as outcome. Recall the social +46 network example. The context is Alice and Adam, which includes their personal attributes and +47 relationships with others on the network. The treatment is living in the same neighborhood, which can +48 be given as one attribute or identified by community detection. And the outcome is their friendship. +49 In this work, we present a counterfactual graph learning method for link prediction (CFLP) that +50 trains graph learning models to answer the counterfactual questions. Figure 1 illustrates this two-step +51 method. Suppose the treatment variable is defined as one type of global graph structure, e.g., the +52 neighborhood assignment discovered by spectral clustering or community detection algorithms. We +53 are wondering how likely the neighborhood distribution makes a difference on the link (non-)existence +54 for each pair of nodes. So, given a pair of nodes (like Alice and Adam) and the treatment value on +55 this pair (in the same neighborhood), we find a pair of nodes (like Helen and Bob) that satisfies two +56 conditions: (1) it has a different treatment (in different neighborhoods) and (2) it is the most similar +57 pair with the given pair of nodes. We call these matched pair of nodes as “counterfactual links.” Note +58 that the outcome of the counterfactual link can be either 1 or 0, depending on whether there exists an +59 edge between the matched pair of nodes. The counterfactual link provides unobserved outcome to the +60 given pair of nodes (Alice and Adam) under a counterfactual condition (in different neighborhoods). +61 After counterfactual links are created for all (positive and negative) training examples, CFLP trains +62 a link predictor (which can be GNN-based) to learn the representation vectors of nodes to predict +63 both the observed factual links and the counterfactual links. In this Alice-Adam example, the link +64 predictor is trained to estimate the individual treatment effect (ITE) of neighborhood assignment as +65 $1 - 1 = 0$ . So, the learner will try to discover the essential factors on the friendship between Alice +66 and Adam. For some other examples, if the outcome of counterfactual link is different from that of +67 the given pair of nodes, the learner will estimate the strong effect of the treatment variable. Therefore, +68 CFLP enables graph learning models to predict missing links regarding causal relationship. +69 Contributions. Our main contributions can be summarized as follows. (1) This is the first work that +70 proposes to improve link prediction by causal inference, specifically, learning to answer counterfactual +71 questions about link existence. (2) This work introduces CFLP that trains GNN-based link predictors +72 to predict both factual and counterfactual links. It learns the causal relationship between global +73 graph structure and link existence. (3) CFLP outperforms competitive baseline methods on several +74 benchmark datasets. On OGB-DDI, our CFLP achieves the state-of-the-art performance. We analyze +75 the impact of counterfactual links as well as the choice of treatment variable. This work sheds insights +76 for improving graph machine learning with causal analysis, which has not been extensively studied +77 yet, when the other direction (machine learning for causal inference) has been studied for a long time. + +# 78 2 Preliminary + +79 Notations Let $G = ( \nu , \mathcal { E } )$ be an undirected graph of $N$ nodes, where $\mathcal { V } = \{ v _ { 1 } , v _ { 2 } , \ldots , v _ { N } \}$ is +80 the set of nodes and $\mathcal { E } \subseteq \mathcal { V } \times \mathcal { V }$ is the set of observed links. We denote the adjacency matrix as +81 $\mathbf { A } \in \{ 0 , 1 \} ^ { N \times N }$ , where $A _ { i , j } = 1$ indicates nodes $v _ { i }$ and $v _ { j }$ are connected and vice versa. We denote +82 the node feature matrix as $\mathbf { \bar { X } } \in \mathbb { R } ^ { N \times F }$ , where $F$ is the number of node features and $\mathbf { x } _ { i }$ (bolded) +83 indicates the feature vector of node $v _ { i }$ (the $i$ -th row of $\mathbf { X }$ ). +84 Counterfactual Learning Let $\mathcal { X }$ be the set of contexts, $\mathcal { V }$ be the set of outcome values, and $\tau$ be +85 the set of treatments. For a context $x \in \mathcal { X }$ and a treatment $t \in \tau$ , we denote the outcome of $x$ under +86 the treatment $t$ by $Y _ { t } ( x ) \in \mathcal { V }$ . Ideally, we would need all possible outcomes of $x$ under all kinds of +87 treatments to study the causal relationships (Morgan and Winship, 2015). However, in reality, only +88 one treatment was applied and thus only one outcome was observed for a given context $x$ . When the +89 variables are specified in data, people use Neyman–Rubin casual model (BCM) to develop statistical +90 learning methods such as propensity score matching (PSM) for causal inference (Rubin, 1974, 2005). +91 In this work, we look at link prediction on graphs. Here we define the variables of counterfactual +92 learning in this scenario. Given a graph $G$ , a context is a pair of nodes $\boldsymbol { x } = ( v _ { i } , v _ { j } )$ in the graph; +93 and thus, $\chi = \mathcal { V } \times \mathcal { V }$ . The outcome variable $Y ( x )$ is naturally binary, indicating whether a link +94 exists between the node pair $x$ ; and thus, $\mathcal { V } = \{ 0 , \dot { 1 } \}$ . We study the causal effect of binary treatment +95 variable $t \in \mathcal { T } = \{ 0 , 1 \}$ , where the value of $\dot { Y } _ { 1 } ( \dot { x } ) - Y _ { 0 } ( x )$ for a particular context $x$ is of high +96 interest and known as the individualized treatment effect (ITE) (van der Laan and Petersen, 2007; +97 Weiss et al., 2015). The value of ITE indicates the causality relationship between the treatment and +98 outcome on the context. And the expected ITE given the context distribution is called averaged +99 treatment effect (ATE). i.e., $\mathrm { A T E } = \mathbb { E } _ { x \sim \mathcal { X } } \mathrm { \Pi } \mathrm { T E } ( x )$ , for a particular treatment variable. +100 However, as aforementioned, the fact that we can only observe one potential outcome under one +101 particular treatment prevents the ITE from being known (Johansson et al., 2016). In the problem +102 setting of link prediction, we refer the observed adjacency matrix as the factual outcomes A and the +103 unobserved adjacency matrix when the treatment is different as the counterfactual outcomes $\mathbf { A } ^ { C F }$ . +104 We denote $\mathbf { T } \breve { \in } \{ 0 , 1 \} ^ { \breve { N } \times N }$ as the factual treatment matrix, where $T _ { i , j }$ indicates the treatment of the +105 node pair We are int $( v _ { i } , v _ { j } )$ . We denote in (1) estima $\mathbf { T } ^ { C F }$ as the counterfactual treathe counterfactual outcomes $\mathbf { A } ^ { C F }$ matrix where via observed $T _ { i , j } ^ { C F } = 1 - T _ { i , j }$ +107 with the counterfactual adjacency matrix $\mathbf { A } ^ { C F }$ to enhance link prediction, and (3) learning the causal +108 relationship between graph structural information (treatment) and link existence (outcome). + +# 109 3 The Proposed Method + +110 In this section, we introduce CFLP, a novel counterfactual graph learning method for link prediction. +111 In Section 3.1, we define treatment variable and counterfactual outcomes/links on graph data and +112 present how to compute them (Figure 1(a)). In Section 3.2, we introduce the graph learning model +113 that learns from both the observed graph and the created counterfactual links (Figure 1(b)). + +# 114 3.1 Defining Treatment Variable and Counterfactual Links + +115 Treatment Previous work on graph machine learning (Velickovic et al., 2019; Park et al., 2020) +116 showed that the graph’s global structural information could improve the quality of representation +117 vectors of nodes learned by GNNs. This is because the message passing-based GNNs aggregate local +118 information in the algorithm of representation vector generation and the global structural information +119 is complementary with the aggregated information. Therefore, for a pair of nodes, one option of +120 defining the treatment variable is its global structural role in the graph. Without the loss of generality, +121 we use Louvain (Blondel et al., 2008), an unsupervised approach that has been widely used for +122 community detection, as an example. Louvain discovers community structure of a graph and assigns +123 each node to one community. Then we can define the binary treatment variable as whether these +124 two nodes in the pair belong to the same community. Let $c : \mathcal { V } \to \mathbb { N }$ be any graph mining/clustering +125 method that outputs the index of community/cluster/neighborhood that each node belongs to. The +126 treatment matrix $\mathbf { T }$ is defined as + +$$ +T _ { i , j } = { \left\{ \begin{array} { l l } { 1 } & { , { \mathrm { i f ~ } } c ( v _ { i } ) = c ( v _ { j } ) ; } \\ { 0 } & { , { \mathrm { o t h e r w i s e . } } } \end{array} \right. } +$$ + +127 For the choice of $c$ , we suggest methods that group nodes based on global graph structural information, +128 including but not limited to Louvain (Blondel et al., 2008), K-core (Bader and Hogue, 2003), and +129 spectral clustering ( $\mathrm { N g }$ et al., 2001). +130 Counterfactual Links As mentioned in Section 2, for each node pair (context), the observed data +131 contains only the factual treatment and outcome, meaning that the link existence for the given node +132 pair with an opposite treatment is unknown. Therefore, we use the outcome from the nearest observed +133 context as a substitute. This idea has been adopted by many methods (Johansson et al., 2016; Alaa and +134 Van Der Schaar, 2019). That is, we want to find the nearest neighbor with the opposite treatment for +135 each observed node pairs and use the nearest neighbor’s outcome as a counterfactual link. Formally, +136 $\forall ( v _ { i } , v _ { j } ) \in \mathcal { V } \times \mathcal { V }$ , we want to find its counterfactual link $( v _ { a } , v _ { b } )$ as below: + +$$ +( v _ { a } , v _ { b } ) = \underset { v _ { a } , v _ { b } \in \mathcal V } { \arg \operatorname* { m i n } } \{ d ( ( v _ { i } , v _ { j } ) , ( v _ { a } , v _ { b } ) ) \ | \ T _ { a , b } = 1 - T _ { i , j } \} , +$$ + +137 where $d ( \cdot , \cdot )$ is a metric of measuring the distance between a pair of node pairs (a pair of contexts). +138 Considering that we want to find the nearest node pair based on not only the raw node features but +139 also structural features, here we take the state-of-the-art unsupervised graph representation learning +140 method MVGRL (Hassani and Khasahmadi, 2020) to learn the node embeddings $\tilde { \mathbf { X } } \in \mathbb { R } ^ { N \times \tilde { F } }$ from +141 the observed graph. We use $\tilde { \mathbf { X } }$ to find the nearest neighbors of node pairs. Nevertheless, finding the +142 nearest neighbors by computing the distance between all pairs of node pairs is extremely inefficient, +143 which takes $O ( N ^ { 4 } )$ comparisons (as there are totally $\bar { O ( N ^ { 2 } ) }$ node pairs). Hence we approximate +144 Eq. (2) by substituting the distance between node pairs by the distance between nodes. That is, +145 $\forall ( v _ { i } , v _ { j } ) \in \mathcal { V } \times \mathcal { V }$ , we want to find its counterfactual link $( v _ { a } , v _ { b } )$ as below: + +$$ +( v _ { a } , v _ { b } ) = \underset { v _ { a } , v _ { b } \in \mathcal { V } } { \arg \operatorname* { m i n } } \{ d ( \tilde { \mathbf { x } } _ { i } , \tilde { \mathbf { x } } _ { a } ) + d ( \tilde { \mathbf { x } } _ { j } , \tilde { \mathbf { x } } _ { b } ) \ \vert \ T _ { a , b } = 1 - T _ { i , j } , d ( \tilde { \mathbf { x } } _ { i } , \tilde { \mathbf { x } } _ { a } ) + d ( \tilde { \mathbf { x } } _ { j } , \tilde { \mathbf { x } } _ { b } ) < 2 \gamma \} , +$$ + +146 where $d ( \cdot , \cdot )$ is specified as the Euclidean distance on the embedding space of $\tilde { \mathbf { X } }$ , and $\gamma$ is a hyperpa +147 rameter that defines the maximum distance that two nodes are considered as similar. Note that when +148 no node pair satisfies the above equation, we do not assign any nearest neighbor for a given node pair +149 to ensure all the neighbors are similar enough (as substitutes) in the feature space. Therefore, the +150 counterfactual treatment matrix $\mathbf { T } ^ { C F }$ and the counterfactual adjacency matrix $\mathbf { \dot { A } } ^ { C F }$ are defined as + +$$ +T _ { i , j } ^ { C F } , A _ { i , j } ^ { C F } = \left\{ \begin{array} { l l } { 1 - T _ { i , j } , A _ { a , b } } & { \mathrm { , ~ i f \exists ~ ( } v _ { a } , v _ { b } ) \in \mathcal { V } \times \mathcal { V } \mathrm { s a t i s f i e s } } \\ { T _ { i , j } , A _ { i , j } } & { \mathrm { , ~ o t h e r w i s e . } } \end{array} \right. +$$ + +It is worth noting that the node embeddings 151 $\tilde { \mathbf { X } }$ and the nearest neighbors are computed only once and do not change during the learning process. 152 $\tilde { \mathbf { X } }$ is only used for finding the nearest neighbors. + +153 Learning from Counterfactual Distributions Let $P ^ { F }$ be the factual distribution of the observed +154 contexts and treatments, and $P ^ { C F }$ be the counterfactual distribution that is composed of the observed +155 156 $\{ ( v _ { i } , v _ { j } , T _ { i , j } ^ { F } ) \} _ { i , j = 1 } ^ { N }$ ite treatments. We define the empirical factual distrib, and define the empirical counterfactual distribution $\hat { P } ^ { C F } \sim P ^ { C F }$ $\hat { P } ^ { F } \sim P ^ { F }$ as $\hat { P } ^ { C F } =$ ${ \hat { P } } ^ { F } = { }$ +157 $\{ ( v _ { i } , v _ { j } , T _ { i , j _ { . } } ^ { C F } ) \} _ { i , j = 1 } ^ { N }$ . Unlike traditional link prediction methods that take only ${ \hat { P } } ^ { F }$ as input and use +158 the observed outcomes $\mathbf { A }$ as the training target, the idea of counterfactual graph learning is to take +159 advantage of the counterfactual distribution by having $\hat { P } ^ { C F }$ as a complementary input and use the +160 counterfactual outcomes $\mathbf { A } ^ { C F }$ as the training target for the counterfactual data samples. + +# 61 3.2 The Counterfactual Graph Learning Model + +162 In this subsection, we present the design of our model as well as the training method. The input of +163 the model in CFLP includes (1) the observed graph data $\mathbf { A }$ and raw feature matrix $\mathbf { X }$ , (2) the factual +164 treatments $\mathbf { T } ^ { F }$ and counterfactual treatments $\bar { \mathbf { T } } ^ { C F }$ , and (3) the counterfactual graph data $\mathbf { A } ^ { C F }$ . The +165 output contains link prediction logits in $\widehat { \mathbf A }$ and $\widehat { \mathbf { A } } ^ { C F }$ for the factual and counterfactual adjacency +166 matrices $\mathbf { A }$ and $\mathbf { A } ^ { C F }$ , respectively. +167 Graph Learning Model The model consist of two trainable components: a graph encoder $f$ and a +168 link decoder $g$ . The graph encoder generates representation vectors of nodes from graph data $G$ . And +169 the link decoder projects the representation vectors of node pairs into the link prediction logits. The +170 choice of the graph encoder $f$ can be any end-to-end GNN model. Without the loss of generality, here +171 we use the commonly used graph convolutional network (GCN) (Kipf and Welling, 2016a). Each +172 layer of GCN is defined as + +$$ +\mathbf { H } ^ { ( l ) } = f ^ { ( l ) } ( \mathbf { A } , \mathbf { H } ^ { ( l - 1 ) } ; \mathbf { W } ^ { ( l ) } ) = \sigma ( \tilde { \mathbf { D } } ^ { - \frac { 1 } { 2 } } \tilde { \mathbf { A } } \tilde { \mathbf { D } } ^ { - \frac { 1 } { 2 } } \mathbf { H } ^ { ( l - 1 ) } \mathbf { W } ^ { ( l ) } ) , +$$ + +173 where $l$ is the layer index, $\tilde { \mathbf { A } } = \mathbf { A } + \mathbf { I }$ is the adjacency matrix with added self-loops, $\tilde { \bf D }$ is the +174 diagonal degree matrix $\begin{array} { r } { \tilde { D } _ { i i } = \sum _ { j } \tilde { A } _ { i j } } \end{array}$ , $\mathbf { H } ^ { ( 0 ) } = \mathbf { X }$ , $\mathbf { W } ^ { ( l ) }$ is the learnable weight matrix at the $l$ -th +175 layer, and $\sigma ( \cdot )$ denotes a nonlinear activation such as ReLU. We denote $\mathbf { Z } = f ( \mathbf { A } , \mathbf { X } ) \in \mathbb { R } ^ { N \times H }$ +176 as the output from the encoder’s last layer, i.e., the $H$ -dimensional representation vectors of nodes. +177 Following previous work (Zhang et al., 2020a), we compute the representation of a node pair as +178 the Hadamard product of the vectors of the two nodes. That is, the representation for the node pair +179 $( v _ { i } , v _ { j } )$ is $\mathbf { z } _ { i } \odot \mathbf { \bar { z } } _ { j } \in \mathbb { R } ^ { H }$ , where $\odot$ stands for the Hadamard product. +180 For the link decoder that predicts whether a link exists between a pair of nodes, we opt for simplicity +181 and adopt a simple decoder based on multi-layer perceptron (MLP), given the representations of node +182 pairs and their treatments. That is, the decoder $g$ is defined as + +$$ +\begin{array} { r l } & { \widehat { \mathbf { A } } = g ( \mathbf { Z } , \mathbf { T } ) , \mathrm { ~ w h e r e ~ } \widehat { A } _ { i , j } = \mathbf { M } \mathrm { L P } \big ( [ \mathbf { z } _ { i } \odot \mathbf { z } _ { j } , T _ { i , j } ] \big ) , } \\ & { \widehat { \mathbf { A } } ^ { C F } = g ( \mathbf { Z } , \mathbf { T } ^ { C F } ) , \mathrm { ~ w h e r e ~ } \widehat { A } _ { i , j } ^ { C F } = \mathbf { M } \mathrm { L P } \big ( [ \mathbf { z } _ { i } \odot \mathbf { z } _ { j } , T _ { i , j } ^ { C F } ] \big ) , } \end{array} +$$ + +183 where $[ \cdot , \cdot ]$ stands for the concatenation of vectors. + +184 During the training process, data samples from the empirical factual distribution ${ \hat { P } } ^ { F }$ and the em +185 pirical counterfactual distribution $\hat { P } ^ { C F }$ are fed into decoder $g$ and optimized towards $\mathbf { A }$ and $\mathbf { A } ^ { C F }$ , +186 respectively. That is, for the two distributions, the loss functions are as follows: + +$$ +\begin{array} { l } { { \mathcal { L } _ { F } = \displaystyle \frac { 1 } { N ^ { 2 } } \sum _ { i = 1 } ^ { N } \sum _ { j = 1 } ^ { N } A _ { i , j } \cdot \log \widehat { A } _ { i , j } + ( 1 - A _ { i , j } ) \cdot \log ( 1 - \widehat { A } _ { i , j } ) , } } \\ { { \mathcal { L } _ { C F } = \displaystyle \frac { 1 } { N ^ { 2 } } \sum _ { i = 1 } ^ { N } \sum _ { j = 1 } ^ { N } A _ { i , j } ^ { C F } \cdot \log \widehat { A } _ { i , j } ^ { C F } + ( 1 - A _ { i , j } ^ { C F } ) \cdot \log ( 1 - \widehat { A } _ { i , j } ^ { C F } ) . } } \end{array} +$$ + +187 Balancing Counterfactual Learning In the training process, the above loss minimizations train the +188 model on both the empirical factual distribution $\hat { P } ^ { F } \sim P ^ { F }$ and empirical counterfactual distribution +189 $\hat { P } ^ { C F } \sim P ^ { C F }$ that are not necessarily equal – the training examples (node pairs) do not have to be +190 aligned. However, at the stage of inference, the test data contains only observed (factual) samples. +191 Such a gap between the training and test data distributions exposes the model in the risk of covariant +192 shift, which is a common issue in counterfactual learning (Johansson et al., 2016; Assaad et al., 2021). +193 To force the distributions of representations of factual distributions and counterfactual distributions to +194 be similar, we use the discrepancy distance (Mansour et al., 2009; Johansson et al., 2016) as another +195 objective to regularize the representation learning. That is, we use the following loss term to minimize +196 the distance between the learned representations from ${ \hat { P } } ^ { F }$ and $\hat { P } ^ { C F }$ : + +$$ +\mathcal { L } _ { d i s c } = \mathrm { d i s c } ( \hat { P } _ { f } ^ { F } , \hat { P } _ { f } ^ { C F } ) , \mathrm { w h e r e \ d i s c } ( P , Q ) = | | P - Q | | _ { F } , +$$ + +where 97 $| | \cdot | | _ { F }$ denotes the Frobenius Norm, and $\hat { P } _ { f } ^ { F }$ and $\hat { P } _ { f } ^ { C F }$ denote the node pair representations 98 learned by graph encoder $f$ from factual distribution and counterfactual distribution, respectively. + +199 Training During the training of CFLP, we want the model to be optimized towards three targets: +200 (1) accurate link prediction on the observed outcomes (Eq. (8)), (2) accurate estimation on the +201 counterfactual outcomes (Eq. (9)), and (3) regularization on the representation spaces learned from +202 ${ \hat { P } } ^ { F }$ and $\hat { P } ^ { C F }$ (Eq. (10)). Therefore, the overall training loss of our proposed CFLP is + +$$ +\mathcal { L } = \mathcal { L } _ { F } + \alpha \cdot \mathcal { L } _ { C F } + \beta \cdot \mathcal { L } _ { d i s c } , +$$ + +203 where $\alpha$ and $\beta$ are hyperparameters to control the weights of counterfactual link prediction (outcome +204 estimation) loss and discrepancy loss. +205 Summary Algorithm 1 summarizes +206 the whole process of CFLP. The first +207 step is to compute the factual and coun +208 terfactual treatments T, $\mathbf { T } ^ { C F }$ as well +209 as the counterfactual outcomes $\mathbf { A } ^ { C F }$ +210 Then, the second step trains the graph +211 learning model on both the observed +212 factual data and created counterfactual +213 data with the integrated loss function +214 (Eq. (11)). Note that the discrepancy +215 loss (Eq. (10)) is computed on the rep +216 resentations of node pairs learned by +217 the graph encoder $f$ , so the decoder +218 $g$ is trained with data from both ${ \hat { P } } ^ { F }$ +219 and $\hat { P } ^ { C F }$ without balancing the con +220 straints. Therefore, after the model is +221 sufficiently trained, we freeze the graph +222 encoder $f$ and fine-tune $g$ with only the +223 factual data. Finally, after the decoder +224 is sufficiently fine-tuned, we output the +225 link prediction logits for both the fac +226 tual and counterfactual adjacency ma +227 trices. + +Table 1: Statistics of datasets used in the experiments. +Algorithm 1: CFLP: Counterfactual graph learning for link prediction + +
DatasetCORACITESEERPUBMEDFACEBOOKOGB-DDI
# nodes2,7083,32719,7174,0394,267
# links5,2784,55244,32488,2341,334,889
# validation node pairs1,0549108,86417,646235,371
# test node pairs2,1101,82017,72835,292229,088
+ +Input : $f$ , g, A, X, n_epochs, n_epoch_f t +1 Compute T by Eq. (1) ; +2 Compute $\mathbf { T } ^ { C F }$ , $\mathbf { A } ^ { C F }$ by Eqs. (3) and (4) ; $^ { \prime * }$ model training \*/ +3 Initialize $\Theta _ { f }$ in $f$ and $\Theta _ { g }$ in $g$ ; +4 for epoch in range(n_epochs) do +5 $\mathbf { Z } = f ( \mathbf { A } , \mathbf { X } )$ ; +6 Get $\widehat { \mathbf A }$ and $\widehat { \mathbf { A } } ^ { C F }$ via $g$ with Eqs. (6) and (7) ; +7 Update $\Theta _ { f }$ and $\Theta _ { g }$ with $\mathcal { L }$ ; // (11) +8 end $^ { \prime * }$ decoder fine-tuning \*/ +9 Freeze $\Theta _ { f }$ and re-initialize $\Theta _ { g }$ ; +10 $\mathbf { Z } = f ( \mathbf { A } , \mathbf { X } )$ ; +11 for epoch in range(n_epochs_f t) do +12 Get $\widehat { \mathbf A }$ via $g$ with Eq. (6) ; +13 Update $\Theta _ { g }$ with $\mathcal { L } _ { F }$ ; // Eq. (8) +14 end $^ { \prime * }$ model inferencing \*/ +15 $\mathbf { Z } = f ( \mathbf { A } , \mathbf { X } )$ ; +16 Get $\widehat { \mathbf A }$ and $\widehat { \mathbf { A } } ^ { C F }$ via $g$ with Eqs. (6) and (7) ; Output : $\widehat { \mathbf A }$ for link prediction, $\widehat { \mathbf { A } } ^ { C F }$ + +28 Complexity The complexity of the + +first step (finding counterfactual links with nearest neighbors) is proportional to the number of node pairs. When $\gamma$ is set as a small value to obtain indeed similar node pairs, this step (Eq. (3)) uses constant time. Moreover, the computation in Eq. (3) can be parallelized. Therefore, the time complexity is $O ( N ^ { 2 } / C )$ where $C$ is the number of processes. For the complexity of the second step (training counterfactual learning model), the GNN encoder has time complexity of $O ( L H ^ { 2 } N + L H | \mathcal { E } | )$ (Wu et al., 2020), where $L$ is the number of GNN layers and $H$ is the size of node representations. Given that we sample the same number of non-existing links as that of observed links during training, the complexity of a three-layer MLP decoder is $O ( ( ( \bar { H } + 1 ) \cdot d _ { h } + d _ { h } \cdot 1 ) | \mathcal { E } | ) =$ $O ( d _ { h } ( H + 2 ) | \mathcal { E } | )$ , where $d _ { h }$ is the number of neurons in the hidden layer. Therefore, the second step has linear time complexity w.r.t. the sum of node and edge counts. + +239 Limitations First, as mentioned above, the computation of finding counterfactual links has a worst +240 case complexity of $O ( N ^ { 2 } )$ . Second, CFLP performs counterfactual prediction with only a single +241 treatment; however, there are quite a few kinds of graph structural information that can be considered +242 as treatments. Future work can leverage the rich structural information by bundled treatments (Zou +243 et al., 2020) in counterfactual graph learning. + +# 4 Experiments + +# 4.1 Experimental Setup + +246 We conduct experiments on five benchmark datasets including citation networks (CORA, CITESEER, +247 PUBMED (Yang et al., 2016)), social network (FACEBOOK (McAuley and Leskovec, 2012)), and +248 drug-drug interaction network (OGB-DDI (Wishart et al., 2018)) from the Open Graph Benchmark +249 (OGB) (Hu et al., 2020). For the first four datasets, we randomly select $10 \% / 2 0 \%$ of the links and +250 the same numbers of disconnected node pairs as validation/test samples. The links in the validation +251 and test sets are masked off from the training graph. For OGB-DDI, we used the OGB official +252 train/validation/test splits. Statistics for the datasets are given in Table 1 and details are in Appendix. +253 We use K-core (Bader and Hogue, 2003) clusters as the default treatment variable. We evaluate CFLP +254 on three commonly used GNN encoders: GCN (Kipf and Welling, 2016a), GSAGE (Hamilton et al., +255 2017), and JKNet (Xu et al., 2018). We compare the link prediction performance of CFLP against +256 Node2Vec (Grover and Leskovec, 2016), MVGRL (Hassani and Khasahmadi, 2020), VGAE (Kipf +257 and Welling, 2016b), SEAL (Zhang and Chen, 2018), LGLP (Cai et al., 2021), and GNNs with MLP +258 decoder. We report averaged test performance and their standard deviation over 20 runs with different +259 random parameter initializations. Other than the most commonly used of Area Under ROC Curve +260 (AUC), we report Hits $@ 2 0$ (one of the primary metrics on OGB leaderboard) as a more challenging +261 metric, as it expects models to rank positive edges higher than nearly all negative edges. +262 Besides performance comparison on link prediction, we will answer two questions to suggest a way +263 of choosing a treatment variable for creating counterfactual links: (Q1) Does CFLP sufficiently learn +264 the observed averaged treatment effect (ATE) derived from the counterfactual links? (Q2) What is the +265 relationship between the estimated ATE learned in the method and the prediction performance? If the +266 answer to Q1 is yes, then the answer to Q2 will indicate how to choose treatment based on observed +267 ATE. To answer the Q1, we calculate the observed ATE $( \widehat { \mathrm { A I E } } _ { o b s } )$ by comparing the observed links +268 in A and created counterfactual links $\mathbf { A } ^ { C F }$ dthat have opposite treatments. And we calculate the +269 estimated ATE $\widehat { ( \mathrm { A I E } _ { e s t } ) }$ by comparing the predicted links in $\widehat { \mathbf A }$ and predicted counterfactual links +270 $\widehat { \mathbf { A } } ^ { C F }$ . Formally, $\widehat { \mathrm { A I E } } _ { o b s }$ and $\widehat { \mathrm { A T E } } _ { e s t }$ are defined as + +Table 2: Link prediction performances measured by Hits $@ 2 0$ . Best performance and best baseline performance are marked with bold and underline, respectively. + +
CORACITESEERPUBMEDFACEBOOKOGB-DDI
Node2Vec49.96±2.5147.78±1.7239.19±1.0224.24±3.0223.26±2.09
MVGRL19.53±2.6414.07±0.7914.19±0.8514.43±0.3310.02±1.01
VGAE45.91±3.3844.04±4.8623.73±1.6137.01±0.6311.71±1.96
SEAL51.35±2.2640.90±3.6828.45±3.8140.89±5.7030.56±3.86
LGLP62.98±0.5657.43±3.71137.86±2.13
GCN49.06±1.7255.56±1.3221.84±3.8753.89±2.1437.07±5.07
GSAGE53.54±2.9653.67±2.9439.13±4.4145.51±3.2253.90±4.74
JKNet48.21±3.8655.60±2.1725.64±4.1152.25±1.4860.56±8.69
Our proposed CFLP with different graph encoders
CFLP w/GCN60.34±2.3359.45±2.3034.12±2.7253.95±2.2952.51±1.09
CFLP w/GSAGE57.33±1.7353.05±2.0743.07±2.3647.28±3.0075.49±4.33
CFLP w/JKNet65.57±1.0568.09±1.4944.90±2.0055.22±1.2986.08±1.98
+ +$$ +\begin{array} { r l } & { \widehat { \mathrm { A T E } } _ { o b s } = \displaystyle \frac { 1 } { N ^ { 2 } } \sum _ { i = 1 } ^ { N } \sum _ { j = 1 } ^ { N } \left\{ \mathbf { T } \odot \left( \mathbf { A } - \mathbf { A } ^ { C F } \right) + \left( \mathbf { 1 } _ { N \times N } - \mathbf { T } \right) \odot \left( \mathbf { A } ^ { C F } - \mathbf { A } \right) \right\} _ { i , j } . } \\ & { \widehat { \mathrm { A T E } } _ { e s t } = \displaystyle \frac { 1 } { N ^ { 2 } } \sum _ { i = 1 } ^ { N } \sum _ { j = 1 } ^ { N } \left\{ \mathbf { T } \odot \left( \widehat { \mathbf { A } } - \widehat { \mathbf { A } } ^ { C F } \right) + \left( \mathbf { 1 } _ { N \times N } - \mathbf { T } \right) \odot \left( \widehat { \mathbf { A } } ^ { C F } - \widehat { \mathbf { A } } \right) \right\} _ { i , j } . } \end{array} +$$ + +271 The treatment variables we will investigate are usually graph clustering or community detection meth +272 ods, such as $\mathbf { K }$ -core (Bader and Hogue, 2003), stochastic block model (SBM) (Karrer and Newman, +273 2011), spectral clustering (SpecC) $\mathrm { N g }$ et al., 2001), propagation clustering (PropC) (Raghavan et al., +274 2007), Louvain (Blondel et al., 2008), common neighbors (CommN), Katz index, and hierarchical +275 clustering (Ward) (Ward Jr, 1963). We use JKNet $\mathrm { { X u } }$ et al., 2018) as the default graph encoder. + +Implementation details and supplementary experimental results (e.g., sensitivity on $\gamma$ , ablation study on $\mathcal { L } _ { C F }$ and $\mathcal { L } _ { d i s c , }$ ) can be found in Appendix. Source code is available in supplementary material. + +# 4.2 Experimental Results + +279 Link Prediction Tables 2 and 3 show the link prediction performance of Hits $\textcircled{ a} 2 0$ and AUC by +280 all methods. LGLP on PUBMED and OGB-DDI are missing due to the out of memory error when +281 running the code package from the authors. We observe that our CFLP on different graph encoders +282 achieve similar or better performances compared with baselines. The only exception is the AUC on +283 FACEBOOK where most methods have close-to-perfect AUC. As AUC is a relatively easier metric +284 comparing with Hits $\textcircled{ a} 2 0$ , most methods achieved good performance on AUC. We observe that CFLP +285 with JKNet almost consistently achieves the best performance and outperforms baselines significantly +286 on Hits $@ 2 0$ . Specifically, compared with the best baseline, CFLP improves relatively by $1 6 . 4 \%$ and +287 $0 . 8 \%$ on Hits $\textcircled{ a} 2 0$ and AUC, respectively. It is worth noting that CFLP with JKNet achieves the +288 state-of-the-art performance on the official leaderboard1 of OGB-DDI. +289 Figure 2 shows the AUC performance of CFLP on +290 CORA with different combinations of $\alpha$ and $\beta$ . We +291 observe that the performance is the poorest when +292 $\alpha = \beta = 0$ and gradually improves and gets stable +293 as $\alpha$ and $\beta$ increase, showing that CFLP is robust +294 to the hyperparameters $\alpha$ and $\beta$ . + +Table 3: Link prediction performances measured by AUC. Best performance and best baseline performance are marked with bold and underline, respectively. + +
CORACITESEERPUBMEDFACEBOOKOGB-DDI
Node2Vec84.49±0.4980.00±0.6880.32±0.2986.49±4.3290.83±0.02
MVGRL75.07±3.6361.20±0.5580.78±1.2879.83±0.3081.45±0.99
VGAE88.68±0.4085.35±0.6095.80±0.1398.66±0.0493.08±0.15
SEAL92.55±0.5085.82±0.4496.36±0.2899.60±0.0297.85±0.17
LGLP91.30±0.0589.41±0.1398.51±0.01
GCN90.25±0.5371.47±1.4096.33±0.8099.43±0.0299.82±0.05
GSAGE90.24±0.3487.38±1.3996.78±0.1199.29±0.0499.93±0.02
JKNet89.05±0.6788.58±1.7896.58±0.2399.43±0.0299.94±0.01
Our proposed CFLP with different graph encoders
CFLP w/GCN92.55±0.5089.65±0.2096.99±0.0899.38±0.0199.44±0.05
CFLPw/GSAGE92.61±0.5291.84±0.2097.01±0.0199.34±0.1099.83±0.05
CFLP w/JKNet93.05±0.2492.12±0.4797.53±0.1799.31±0.0499.94±0.01
+ +ATE with Different Treatments Tables 4 and 5 show the link prediction performance, $\widehat { \mathrm { A T E } } _ { o b s }$ , and $\widehat { \mathrm { A I E } } _ { e s t }$ of CFLP (with JKNet) when using different treatments. The treatments in Tables 4 and 5 are sorted by the Hits $\textcircled{ a} 2 0$ performance. Bigger ATE indicates stronger causal relationship between the treatment and outcome, and vice versa. We observe: (1) $\widehat { \mathrm { A I E } } _ { e s t }$ values are generally close to $\widehat { \mathrm { A I E } } _ { o b s }$ showing that CFLP was sufficiently trained to learn the causal relationship between graph structure information and link existence; (2) $\widehat { \mathrm { A I E } } _ { o b s }$ and $\widehat { \mathrm { A I E } } _ { e s t }$ are both negatively correlated with the link prediction performance, showing that we can pick a proper treatment prior to training a model with CFLP. Using the treatment that has the weakest causal relationship with link existence is likely to train the model to capture more essential factors on the outcome, in a way similar to denoising the unrelated information from the representations. + +![](images/47e57d6501e7ce1ae9dd51fb0b8af84b88b44a8f67aca2501688ea437b8662eb.jpg) +Figure 2: AUC performance of CFLP on CORA w.r.t different combanitions of $\alpha$ and $\beta$ . + +# 5 Related Work + +Link Prediction With its wide applications, link prediction has draw attention from many research communities including statistical machine learning and data mining. Stochastic generative methods based on stochastic block models (SBM) are developed to generate links (Mehta et al., 2019). In data mining, matrix factorization (Menon and Elkan, 2011), heuristic methods (Philip et al., 2010; Martínez et al., 2016), and graph embedding methods (Cui et al., 2018) have been applied to predict links in the graph. Heuristic methods compute the similarity score of nodes based on their neighborhoods. These + +Table 4: Results of CFLP with different treatments on CORA. (sorted by Hits $@ 2 0$ ) + +
Hits@20ATEobsATEest
K-core65.6±1.10.0020.013±0.003
SBM64.2±1.10.0060.023±0.015
CommN62.3±1.60.0070.053±0.021
PropC61.7±1.40.0370.059±0.065
Ward61.2±2.30.0010.033±0.012
SpecC59.3±2.80.0020.033±0.011
Louvain57.6±1.80.0250.138±0.091
Katz56.6±3.40.7400.802±0.041
+ +Table 5: Results of CFLP with different treatments on CITESEER. (sorted by Hits $@ 2 0$ ) + +
Hits@20ATEobsATEest
SBM71.6 ±1.90.0040.005 ±0.001
K-core68.1±1.50.0020.010±0.002
Ward67.0±1.70.0030.037±0.009
PropC64.6±3.60.1410.232±0.113
Louvain63.3±2.50.1260.151±0.078
SpecC59.9±1.30.0090.166±0.034
Katz57.3±0.50.2450.224±0.037
CommN56.8±4.90.6780.195±0.034
+ +316 methods can be generally categorized into first-order, second-order, and high-order heuristics based +317 on the maximum distance of the neighbors. Graph embedding methods learn latent node features via +318 embedding lookup and use them for link prediction (Perozzi et al., 2014; Tang et al., 2015; Grover +319 and Leskovec, 2016; Wang et al., 2016). + +In the past few years, GNNs have showed promising results on various graph-based tasks with their ability of learning from features and custom aggregations on structures, (Kipf and Welling, 2016a; Hamilton et al., 2017; Xu et al., 2018; Wu et al., 2020). With node pair representations and an attached MLP or inner-product decoder, GNNs can be used for link prediction (Zhang et al., 2020a). For example, VGAE used GCN to learn node representations and reconstruct the graph structure (Kipf and Welling, 2016b). SEAL extracted a local subgraph around each target node pair and then learned graph representation from local subgraph for link prediction (Zhang and Chen, 2018). Following the scheme of SEAL, Cai and Ji (2020) proposed to improve local subgraph representation learning by multi-scale graph representation learning. And LGLP inverted the local subgraphs to line graphs before learning representations (Cai et al., 2021). However, very limited work has studied to use causal inference for improving link prediction. + +Counterfactual Prediction As a mean of learning the causality between treatment and outcome, counterfactual prediction has been used for a variety of applicaitons such as recommender systems (Wang et al., 2020; Xu et al., 2020), health care (Alaa and van der Schaar, 2017), vision-language tasks (Zhang et al., 2020b; Parvaneh et al., 2020), and decision making (Coston et al., 2020; Pitis et al., 2020; Kusner et al., 2017). To infer the causal relationships, previous work usually estimated the ITE via function fitting models (Gelman and Hill, 2006; Chipman et al., 2010; Wager and Athey, 2018; Assaad et al., 2021) which estimated the transductive ITE. Peysakhovich et al. (2019) and Zou et al. (2020) studied counterfactual prediction with multiple agents and bundled treatments, respectively. Pawlowski et al. (2020) proposed a deep structural causal model for tractable counterfactual inference. + +Causal Inference Causal inference methods usually re-weighted samples based on propensity score (Rosenbaum and Rubin, 1983; Austin, 2011; Kuang et al., 2017a,b) to remove confounding bias from binary treatments. Recently, several works studied about learning treatment invariant representation to predict the counterfactual outcomes (Hassanpour and Greiner, 2019b,a; Shalit et al., 2017; Yao et al., 2018; Bica et al., 2020; Hassanpour and Greiner, 2019a; Li and Fu, 2017). When part of unobserved outcomes may mislead the counterfactual prediction, Louizos et al. (2017) attempted to infer the outcomes from proxies, and Hartford et al. (2017) introduced instrumental variable. SITE preserved local similarity to balance the distributions of control and treated groups (Yao et al., 2018). Yoon et al. (2018) estimated ITE with generative adversarial networks (GANs). Assaad et al. (2021) discussed the trade-off between achieving balance and predictive power. + +# 6 Conclusion + +In this work, we presented a counterfactual graph learning method for link prediction (CFLP). We introduced the idea of counterfactual prediction to improve link prediction on graphs. CFLP accurately predicted the missing links by exploring the causal relationship between global graph structure and link existence. Extensive experiments demonstrated that CFLP achieved the state-ofthe-art performance on benchmark datasets. + +References +57 Ahmed Alaa and Mihaela Van Der Schaar. 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If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] + +3. If you ran experiments... + +(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] + +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] in Section 4.1 and Appendix +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] in Appendix + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] +(b) Did you mention the license of the assets? [N/A] +(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/train/N2dEPuK0vom/N2dEPuK0vom_content_list.json b/parse/train/N2dEPuK0vom/N2dEPuK0vom_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..0482c27c98799d3116bf06b18f35eaf78cd2e187 --- /dev/null +++ b/parse/train/N2dEPuK0vom/N2dEPuK0vom_content_list.json @@ -0,0 +1,1193 @@ +[ + { + "type": "text", + "text": "Counterfactual Graph Learning for Link Prediction ", + "text_level": 1, + "bbox": [ + 183, + 122, + 815, + 148 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 200, + 580, + 256 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 292, + 535, + 309 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Learning to predict missing links is important for many graph-based applications. Existing methods were designed to learn the observed association between two sets of variables: (1) the observed graph structure and (2) the existence of link between a pair of nodes. However, the causal relationship between these variables was ignored and we visit the possibility of learning it by simply asking a counterfactual question: “would the link exist or not if the observed graph structure became different?” To answer this question by causal inference, we consider the information of the node pair as context, global graph structural properties as treatment, and link existence as outcome. In this work, we propose a novel link prediction method that enhances graph learning by the counterfactual inference. It creates counterfactual links from the observed ones, and our method learns representations from both of them. Experiments on a number of benchmark datasets show that our proposed method achieves the state-of-the-art performance on link prediction. ", + "bbox": [ + 148, + 327, + 766, + 506 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "14 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 539, + 312, + 556 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "15 Link prediction seeks to predict the likelihood of edge existence between node pairs based on the \n16 observed graph. Given the omnipresence of graph-structured data, link prediction has copious applica \n17 tions such as movie recommendation (Bennett et al., 2007), chemical interaction prediction (Stanfield \n18 et al., 2017), and knowledge graph completion (Kazemi and Poole, 2018). Graph machine learning \n19 methods have been widely applied to solve this problem. Their standard scheme is to first learn the \n20 representation vectors of nodes and then learn the association between the representations of a pair of \n21 nodes and the existence of the link between them. For example, graph neural networks (GNNs) use \n22 neighborhood aggregation to create the representation vectors: the representation vector of a node \n23 is computed by recursively aggregating and transforming representation vectors of its neighboring \n24 nodes (Kipf and Welling, 2016a; Hamilton et al., 2017; Wu et al., 2020). Then the vectors are fed \n25 into a binary classification model to learn the association. GNN methods have shown predominance \n26 in the task of link prediction (Kipf and Welling, 2016b; Zhang and Chen, 2018; Zhang et al., 2020a). \n27 Unfortunately, the causal relationship between graph structure and link existence was largely ignored \n28 in the previous work. Existing methods that learn from association only were not able to capture \n29 essential factors to accurately predict missing links in the test data. Take social network as an example. \n30 Suppose Alice and Adam live in the same neighborhood and they are close friends. The association \n31 between neighborhood belonging and friend closeness could be too strong to discover the essential \n32 factors of the friendship such as common interests or family relationship which could be the cause of \n33 being living in the same neighborhood. So, our idea is to ask a counterfactual question: “would Alice \n34 and Adam still be close friends if they were not living in the same neighborhood?” If a graph learning \n35 model could learn the causal relationship from data by asking the counterfactual questions, it would \n36 improve the performance of link prediction with the novel knowledge it captured. Generally, the \n37 questions can be described as “would the link exist or not if the graph structure became different?” ", + "bbox": [ + 147, + 573, + 825, + 738 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 744, + 825, + 897 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/3faaa9eaa22e961dc833289ae7f8919e66d2220e8c36f98725f81166d66c30b8.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 235, + 101, + 348, + 164 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(Alice, Adam) (Helen, Bob)Most similar with a different treatment ", + "bbox": [ + 183, + 171, + 418, + 194 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Factual link: 1 Counterfactual link: 1 (a) Find counterfactual link as the most similar node pair with a different treatment. ", + "bbox": [ + 179, + 198, + 423, + 208 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/7c895b3ed93f45a2ff5e6a6167d1d78665e3dbc492d83aabed634a32d4e2cce4.jpg", + "image_caption": [ + "(b) Train a GNN-based link predictor to predict factual and counterfactual links given the corresponding treatments. " + ], + "image_footnote": [], + "bbox": [ + 450, + 98, + 812, + 205 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 217, + 434, + 243 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Figure 1: The proposed CFLP learns the causal relationship between the observed graph structure (e.g., neighborhood similarity, considered as treatment variable) and link existence (considered as outcome). In this example, the link predictor would be trained to estimate the individual treatment effect (ITE) as $1 - 1 = 0$ so it looks for factors other than neighborhood to predict the factual link. ", + "bbox": [ + 173, + 256, + 825, + 311 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "38 As known to many, counterfactual question is a key component of causal inference and have been \n39 well defined in the literature. A counterfactual question is usually framed with three factors: context \n40 (as a data point), manipulation (e.g., treatment, intervention, action, strategy), and outcome (van der \n41 Laan and Petersen, 2007; Johansson et al., 2016). (To simplify the language, we use “treatment” to \n42 refer to the manipulation in this paper, as readers might be familiar more with the word “treatment.”) \n43 Given certain data context, it asks what the outcome would have been if the treatment had not been \n44 the observed value. In the scenario of link prediction, we consider the information of a pair of nodes \n45 as context, graph structural properties as treatment, and link existence as outcome. Recall the social \n46 network example. The context is Alice and Adam, which includes their personal attributes and \n47 relationships with others on the network. The treatment is living in the same neighborhood, which can \n48 be given as one attribute or identified by community detection. And the outcome is their friendship. \n49 In this work, we present a counterfactual graph learning method for link prediction (CFLP) that \n50 trains graph learning models to answer the counterfactual questions. Figure 1 illustrates this two-step \n51 method. Suppose the treatment variable is defined as one type of global graph structure, e.g., the \n52 neighborhood assignment discovered by spectral clustering or community detection algorithms. We \n53 are wondering how likely the neighborhood distribution makes a difference on the link (non-)existence \n54 for each pair of nodes. So, given a pair of nodes (like Alice and Adam) and the treatment value on \n55 this pair (in the same neighborhood), we find a pair of nodes (like Helen and Bob) that satisfies two \n56 conditions: (1) it has a different treatment (in different neighborhoods) and (2) it is the most similar \n57 pair with the given pair of nodes. We call these matched pair of nodes as “counterfactual links.” Note \n58 that the outcome of the counterfactual link can be either 1 or 0, depending on whether there exists an \n59 edge between the matched pair of nodes. The counterfactual link provides unobserved outcome to the \n60 given pair of nodes (Alice and Adam) under a counterfactual condition (in different neighborhoods). \n61 After counterfactual links are created for all (positive and negative) training examples, CFLP trains \n62 a link predictor (which can be GNN-based) to learn the representation vectors of nodes to predict \n63 both the observed factual links and the counterfactual links. In this Alice-Adam example, the link \n64 predictor is trained to estimate the individual treatment effect (ITE) of neighborhood assignment as \n65 $1 - 1 = 0$ . So, the learner will try to discover the essential factors on the friendship between Alice \n66 and Adam. For some other examples, if the outcome of counterfactual link is different from that of \n67 the given pair of nodes, the learner will estimate the strong effect of the treatment variable. Therefore, \n68 CFLP enables graph learning models to predict missing links regarding causal relationship. \n69 Contributions. Our main contributions can be summarized as follows. (1) This is the first work that \n70 proposes to improve link prediction by causal inference, specifically, learning to answer counterfactual \n71 questions about link existence. (2) This work introduces CFLP that trains GNN-based link predictors \n72 to predict both factual and counterfactual links. It learns the causal relationship between global \n73 graph structure and link existence. (3) CFLP outperforms competitive baseline methods on several \n74 benchmark datasets. On OGB-DDI, our CFLP achieves the state-of-the-art performance. We analyze \n75 the impact of counterfactual links as well as the choice of treatment variable. This work sheds insights \n76 for improving graph machine learning with causal analysis, which has not been extensively studied \n77 yet, when the other direction (machine learning for causal inference) has been studied for a long time. ", + "bbox": [ + 145, + 345, + 825, + 498 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 503, + 826, + 780 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 786, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "78 2 Preliminary ", + "text_level": 1, + "bbox": [ + 147, + 89, + 305, + 107 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "79 Notations Let $G = ( \\nu , \\mathcal { E } )$ be an undirected graph of $N$ nodes, where $\\mathcal { V } = \\{ v _ { 1 } , v _ { 2 } , \\ldots , v _ { N } \\}$ is \n80 the set of nodes and $\\mathcal { E } \\subseteq \\mathcal { V } \\times \\mathcal { V }$ is the set of observed links. We denote the adjacency matrix as \n81 $\\mathbf { A } \\in \\{ 0 , 1 \\} ^ { N \\times N }$ , where $A _ { i , j } = 1$ indicates nodes $v _ { i }$ and $v _ { j }$ are connected and vice versa. We denote \n82 the node feature matrix as $\\mathbf { \\bar { X } } \\in \\mathbb { R } ^ { N \\times F }$ , where $F$ is the number of node features and $\\mathbf { x } _ { i }$ (bolded) \n83 indicates the feature vector of node $v _ { i }$ (the $i$ -th row of $\\mathbf { X }$ ). \n84 Counterfactual Learning Let $\\mathcal { X }$ be the set of contexts, $\\mathcal { V }$ be the set of outcome values, and $\\tau$ be \n85 the set of treatments. For a context $x \\in \\mathcal { X }$ and a treatment $t \\in \\tau$ , we denote the outcome of $x$ under \n86 the treatment $t$ by $Y _ { t } ( x ) \\in \\mathcal { V }$ . Ideally, we would need all possible outcomes of $x$ under all kinds of \n87 treatments to study the causal relationships (Morgan and Winship, 2015). However, in reality, only \n88 one treatment was applied and thus only one outcome was observed for a given context $x$ . When the \n89 variables are specified in data, people use Neyman–Rubin casual model (BCM) to develop statistical \n90 learning methods such as propensity score matching (PSM) for causal inference (Rubin, 1974, 2005). \n91 In this work, we look at link prediction on graphs. Here we define the variables of counterfactual \n92 learning in this scenario. Given a graph $G$ , a context is a pair of nodes $\\boldsymbol { x } = ( v _ { i } , v _ { j } )$ in the graph; \n93 and thus, $\\chi = \\mathcal { V } \\times \\mathcal { V }$ . The outcome variable $Y ( x )$ is naturally binary, indicating whether a link \n94 exists between the node pair $x$ ; and thus, $\\mathcal { V } = \\{ 0 , \\dot { 1 } \\}$ . We study the causal effect of binary treatment \n95 variable $t \\in \\mathcal { T } = \\{ 0 , 1 \\}$ , where the value of $\\dot { Y } _ { 1 } ( \\dot { x } ) - Y _ { 0 } ( x )$ for a particular context $x$ is of high \n96 interest and known as the individualized treatment effect (ITE) (van der Laan and Petersen, 2007; \n97 Weiss et al., 2015). The value of ITE indicates the causality relationship between the treatment and \n98 outcome on the context. And the expected ITE given the context distribution is called averaged \n99 treatment effect (ATE). i.e., $\\mathrm { A T E } = \\mathbb { E } _ { x \\sim \\mathcal { X } } \\mathrm { \\Pi } \\mathrm { T E } ( x )$ , for a particular treatment variable. \n100 However, as aforementioned, the fact that we can only observe one potential outcome under one \n101 particular treatment prevents the ITE from being known (Johansson et al., 2016). In the problem \n102 setting of link prediction, we refer the observed adjacency matrix as the factual outcomes A and the \n103 unobserved adjacency matrix when the treatment is different as the counterfactual outcomes $\\mathbf { A } ^ { C F }$ . \n104 We denote $\\mathbf { T } \\breve { \\in } \\{ 0 , 1 \\} ^ { \\breve { N } \\times N }$ as the factual treatment matrix, where $T _ { i , j }$ indicates the treatment of the \n105 node pair We are int $( v _ { i } , v _ { j } )$ . We denote in (1) estima $\\mathbf { T } ^ { C F }$ as the counterfactual treathe counterfactual outcomes $\\mathbf { A } ^ { C F }$ matrix where via observed $T _ { i , j } ^ { C F } = 1 - T _ { i , j }$ \n107 with the counterfactual adjacency matrix $\\mathbf { A } ^ { C F }$ to enhance link prediction, and (3) learning the causal \n108 relationship between graph structural information (treatment) and link existence (outcome). ", + "bbox": [ + 147, + 119, + 825, + 190 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 204, + 825, + 303 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 308, + 825, + 433 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 439, + 826, + 569 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "109 3 The Proposed Method ", + "text_level": 1, + "bbox": [ + 150, + 587, + 390, + 603 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "110 In this section, we introduce CFLP, a novel counterfactual graph learning method for link prediction. \n111 In Section 3.1, we define treatment variable and counterfactual outcomes/links on graph data and \n112 present how to compute them (Figure 1(a)). In Section 3.2, we introduce the graph learning model \n113 that learns from both the observed graph and the created counterfactual links (Figure 1(b)). ", + "bbox": [ + 142, + 617, + 825, + 672 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "114 3.1 Defining Treatment Variable and Counterfactual Links ", + "text_level": 1, + "bbox": [ + 147, + 688, + 596, + 703 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "115 Treatment Previous work on graph machine learning (Velickovic et al., 2019; Park et al., 2020) \n116 showed that the graph’s global structural information could improve the quality of representation \n117 vectors of nodes learned by GNNs. This is because the message passing-based GNNs aggregate local \n118 information in the algorithm of representation vector generation and the global structural information \n119 is complementary with the aggregated information. Therefore, for a pair of nodes, one option of \n120 defining the treatment variable is its global structural role in the graph. Without the loss of generality, \n121 we use Louvain (Blondel et al., 2008), an unsupervised approach that has been widely used for \n122 community detection, as an example. Louvain discovers community structure of a graph and assigns \n123 each node to one community. Then we can define the binary treatment variable as whether these \n124 two nodes in the pair belong to the same community. Let $c : \\mathcal { V } \\to \\mathbb { N }$ be any graph mining/clustering \n125 method that outputs the index of community/cluster/neighborhood that each node belongs to. The \n126 treatment matrix $\\mathbf { T }$ is defined as ", + "bbox": [ + 140, + 713, + 825, + 878 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/dc3c79b2d80107e67277a07a7a76492a11d01edf7725985274d8948658d86a87.jpg", + "text": "$$\nT _ { i , j } = { \\left\\{ \\begin{array} { l l } { 1 } & { , { \\mathrm { i f ~ } } c ( v _ { i } ) = c ( v _ { j } ) ; } \\\\ { 0 } & { , { \\mathrm { o t h e r w i s e . } } } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 397, + 880, + 599, + 915 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "127 For the choice of $c$ , we suggest methods that group nodes based on global graph structural information, \n128 including but not limited to Louvain (Blondel et al., 2008), K-core (Bader and Hogue, 2003), and \n129 spectral clustering ( $\\mathrm { N g }$ et al., 2001). \n130 Counterfactual Links As mentioned in Section 2, for each node pair (context), the observed data \n131 contains only the factual treatment and outcome, meaning that the link existence for the given node \n132 pair with an opposite treatment is unknown. Therefore, we use the outcome from the nearest observed \n133 context as a substitute. This idea has been adopted by many methods (Johansson et al., 2016; Alaa and \n134 Van Der Schaar, 2019). That is, we want to find the nearest neighbor with the opposite treatment for \n135 each observed node pairs and use the nearest neighbor’s outcome as a counterfactual link. Formally, \n136 $\\forall ( v _ { i } , v _ { j } ) \\in \\mathcal { V } \\times \\mathcal { V }$ , we want to find its counterfactual link $( v _ { a } , v _ { b } )$ as below: ", + "bbox": [ + 140, + 92, + 826, + 133 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 147, + 825, + 246 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c86c7ef6f36ed73dd303b2e20ff00e2cd6cd5a3e8b0a380427b9177074902f80.jpg", + "text": "$$\n( v _ { a } , v _ { b } ) = \\underset { v _ { a } , v _ { b } \\in \\mathcal V } { \\arg \\operatorname* { m i n } } \\{ d ( ( v _ { i } , v _ { j } ) , ( v _ { a } , v _ { b } ) ) \\ | \\ T _ { a , b } = 1 - T _ { i , j } \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 251, + 694, + 277 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "137 where $d ( \\cdot , \\cdot )$ is a metric of measuring the distance between a pair of node pairs (a pair of contexts). \n138 Considering that we want to find the nearest node pair based on not only the raw node features but \n139 also structural features, here we take the state-of-the-art unsupervised graph representation learning \n140 method MVGRL (Hassani and Khasahmadi, 2020) to learn the node embeddings $\\tilde { \\mathbf { X } } \\in \\mathbb { R } ^ { N \\times \\tilde { F } }$ from \n141 the observed graph. We use $\\tilde { \\mathbf { X } }$ to find the nearest neighbors of node pairs. Nevertheless, finding the \n142 nearest neighbors by computing the distance between all pairs of node pairs is extremely inefficient, \n143 which takes $O ( N ^ { 4 } )$ comparisons (as there are totally $\\bar { O ( N ^ { 2 } ) }$ node pairs). Hence we approximate \n144 Eq. (2) by substituting the distance between node pairs by the distance between nodes. That is, \n145 $\\forall ( v _ { i } , v _ { j } ) \\in \\mathcal { V } \\times \\mathcal { V }$ , we want to find its counterfactual link $( v _ { a } , v _ { b } )$ as below: ", + "bbox": [ + 140, + 284, + 825, + 414 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c7535c67a2682290c3018dd79ba783a829748a1ee1c3fa999b701f4d138c3cc9.jpg", + "text": "$$\n( v _ { a } , v _ { b } ) = \\underset { v _ { a } , v _ { b } \\in \\mathcal { V } } { \\arg \\operatorname* { m i n } } \\{ d ( \\tilde { \\mathbf { x } } _ { i } , \\tilde { \\mathbf { x } } _ { a } ) + d ( \\tilde { \\mathbf { x } } _ { j } , \\tilde { \\mathbf { x } } _ { b } ) \\ \\vert \\ T _ { a , b } = 1 - T _ { i , j } , d ( \\tilde { \\mathbf { x } } _ { i } , \\tilde { \\mathbf { x } } _ { a } ) + d ( \\tilde { \\mathbf { x } } _ { j } , \\tilde { \\mathbf { x } } _ { b } ) < 2 \\gamma \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 191, + 419, + 787, + 445 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "146 where $d ( \\cdot , \\cdot )$ is specified as the Euclidean distance on the embedding space of $\\tilde { \\mathbf { X } }$ , and $\\gamma$ is a hyperpa \n147 rameter that defines the maximum distance that two nodes are considered as similar. Note that when \n148 no node pair satisfies the above equation, we do not assign any nearest neighbor for a given node pair \n149 to ensure all the neighbors are similar enough (as substitutes) in the feature space. Therefore, the \n150 counterfactual treatment matrix $\\mathbf { T } ^ { C F }$ and the counterfactual adjacency matrix $\\mathbf { \\dot { A } } ^ { C F }$ are defined as ", + "bbox": [ + 140, + 454, + 825, + 525 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/cc4a01daac557afd01de2aa7f50822881d26d4fc9efa2734d6947cf854d99dec.jpg", + "text": "$$\nT _ { i , j } ^ { C F } , A _ { i , j } ^ { C F } = \\left\\{ \\begin{array} { l l } { 1 - T _ { i , j } , A _ { a , b } } & { \\mathrm { , ~ i f \\exists ~ ( } v _ { a } , v _ { b } ) \\in \\mathcal { V } \\times \\mathcal { V } \\mathrm { s a t i s f i e s } } \\\\ { T _ { i , j } , A _ { i , j } } & { \\mathrm { , ~ o t h e r w i s e . } } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 259, + 530, + 679, + 566 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "It is worth noting that the node embeddings 151 $\\tilde { \\mathbf { X } }$ and the nearest neighbors are computed only once and do not change during the learning process. 152 $\\tilde { \\mathbf { X } }$ is only used for finding the nearest neighbors. ", + "bbox": [ + 142, + 574, + 830, + 606 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "153 Learning from Counterfactual Distributions Let $P ^ { F }$ be the factual distribution of the observed \n154 contexts and treatments, and $P ^ { C F }$ be the counterfactual distribution that is composed of the observed \n155 156 $\\{ ( v _ { i } , v _ { j } , T _ { i , j } ^ { F } ) \\} _ { i , j = 1 } ^ { N }$ ite treatments. We define the empirical factual distrib, and define the empirical counterfactual distribution $\\hat { P } ^ { C F } \\sim P ^ { C F }$ $\\hat { P } ^ { F } \\sim P ^ { F }$ as $\\hat { P } ^ { C F } =$ ${ \\hat { P } } ^ { F } = { }$ \n157 $\\{ ( v _ { i } , v _ { j } , T _ { i , j _ { . } } ^ { C F } ) \\} _ { i , j = 1 } ^ { N }$ . Unlike traditional link prediction methods that take only ${ \\hat { P } } ^ { F }$ as input and use \n158 the observed outcomes $\\mathbf { A }$ as the training target, the idea of counterfactual graph learning is to take \n159 advantage of the counterfactual distribution by having $\\hat { P } ^ { C F }$ as a complementary input and use the \n160 counterfactual outcomes $\\mathbf { A } ^ { C F }$ as the training target for the counterfactual data samples. ", + "bbox": [ + 143, + 618, + 826, + 742 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "61 3.2 The Counterfactual Graph Learning Model ", + "text_level": 1, + "bbox": [ + 155, + 757, + 517, + 772 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "162 In this subsection, we present the design of our model as well as the training method. The input of \n163 the model in CFLP includes (1) the observed graph data $\\mathbf { A }$ and raw feature matrix $\\mathbf { X }$ , (2) the factual \n164 treatments $\\mathbf { T } ^ { F }$ and counterfactual treatments $\\bar { \\mathbf { T } } ^ { C F }$ , and (3) the counterfactual graph data $\\mathbf { A } ^ { C F }$ . The \n165 output contains link prediction logits in $\\widehat { \\mathbf A }$ and $\\widehat { \\mathbf { A } } ^ { C F }$ for the factual and counterfactual adjacency \n166 matrices $\\mathbf { A }$ and $\\mathbf { A } ^ { C F }$ , respectively. \n167 Graph Learning Model The model consist of two trainable components: a graph encoder $f$ and a \n168 link decoder $g$ . The graph encoder generates representation vectors of nodes from graph data $G$ . And \n169 the link decoder projects the representation vectors of node pairs into the link prediction logits. The \n170 choice of the graph encoder $f$ can be any end-to-end GNN model. Without the loss of generality, here \n171 we use the commonly used graph convolutional network (GCN) (Kipf and Welling, 2016a). Each \n172 layer of GCN is defined as ", + "bbox": [ + 142, + 782, + 825, + 856 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 868, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 90, + 828, + 133 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c15d56749f60bbfee272f89e4f0513cc0bd697603430dc562ca515ac5df7ea3e.jpg", + "text": "$$\n\\mathbf { H } ^ { ( l ) } = f ^ { ( l ) } ( \\mathbf { A } , \\mathbf { H } ^ { ( l - 1 ) } ; \\mathbf { W } ^ { ( l ) } ) = \\sigma ( \\tilde { \\mathbf { D } } ^ { - \\frac { 1 } { 2 } } \\tilde { \\mathbf { A } } \\tilde { \\mathbf { D } } ^ { - \\frac { 1 } { 2 } } \\mathbf { H } ^ { ( l - 1 ) } \\mathbf { W } ^ { ( l ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 289, + 138, + 705, + 160 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "173 where $l$ is the layer index, $\\tilde { \\mathbf { A } } = \\mathbf { A } + \\mathbf { I }$ is the adjacency matrix with added self-loops, $\\tilde { \\bf D }$ is the \n174 diagonal degree matrix $\\begin{array} { r } { \\tilde { D } _ { i i } = \\sum _ { j } \\tilde { A } _ { i j } } \\end{array}$ , $\\mathbf { H } ^ { ( 0 ) } = \\mathbf { X }$ , $\\mathbf { W } ^ { ( l ) }$ is the learnable weight matrix at the $l$ -th \n175 layer, and $\\sigma ( \\cdot )$ denotes a nonlinear activation such as ReLU. We denote $\\mathbf { Z } = f ( \\mathbf { A } , \\mathbf { X } ) \\in \\mathbb { R } ^ { N \\times H }$ \n176 as the output from the encoder’s last layer, i.e., the $H$ -dimensional representation vectors of nodes. \n177 Following previous work (Zhang et al., 2020a), we compute the representation of a node pair as \n178 the Hadamard product of the vectors of the two nodes. That is, the representation for the node pair \n179 $( v _ { i } , v _ { j } )$ is $\\mathbf { z } _ { i } \\odot \\mathbf { \\bar { z } } _ { j } \\in \\mathbb { R } ^ { H }$ , where $\\odot$ stands for the Hadamard product. \n180 For the link decoder that predicts whether a link exists between a pair of nodes, we opt for simplicity \n181 and adopt a simple decoder based on multi-layer perceptron (MLP), given the representations of node \n182 pairs and their treatments. That is, the decoder $g$ is defined as ", + "bbox": [ + 140, + 166, + 825, + 272 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 276, + 825, + 319 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/76e4d2ea2c85138cdff83101185eb4b984ee48118e83761e4e513ba2d3a43cc3.jpg", + "text": "$$\n\\begin{array} { r l } & { \\widehat { \\mathbf { A } } = g ( \\mathbf { Z } , \\mathbf { T } ) , \\mathrm { ~ w h e r e ~ } \\widehat { A } _ { i , j } = \\mathbf { M } \\mathrm { L P } \\big ( [ \\mathbf { z } _ { i } \\odot \\mathbf { z } _ { j } , T _ { i , j } ] \\big ) , } \\\\ & { \\widehat { \\mathbf { A } } ^ { C F } = g ( \\mathbf { Z } , \\mathbf { T } ^ { C F } ) , \\mathrm { ~ w h e r e ~ } \\widehat { A } _ { i , j } ^ { C F } = \\mathbf { M } \\mathrm { L P } \\big ( [ \\mathbf { z } _ { i } \\odot \\mathbf { z } _ { j } , T _ { i , j } ^ { C F } ] \\big ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 300, + 324, + 696, + 369 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "183 where $[ \\cdot , \\cdot ]$ stands for the concatenation of vectors. ", + "bbox": [ + 142, + 375, + 503, + 390 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "184 During the training process, data samples from the empirical factual distribution ${ \\hat { P } } ^ { F }$ and the em \n185 pirical counterfactual distribution $\\hat { P } ^ { C F }$ are fed into decoder $g$ and optimized towards $\\mathbf { A }$ and $\\mathbf { A } ^ { C F }$ , \n186 respectively. That is, for the two distributions, the loss functions are as follows: ", + "bbox": [ + 142, + 397, + 826, + 443 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/64cc1514d642d976ad0bef9981a4003dc68208c107c8153e8c30462343e4c05e.jpg", + "text": "$$\n\\begin{array} { l } { { \\mathcal { L } _ { F } = \\displaystyle \\frac { 1 } { N ^ { 2 } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } A _ { i , j } \\cdot \\log \\widehat { A } _ { i , j } + ( 1 - A _ { i , j } ) \\cdot \\log ( 1 - \\widehat { A } _ { i , j } ) , } } \\\\ { { \\mathcal { L } _ { C F } = \\displaystyle \\frac { 1 } { N ^ { 2 } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } A _ { i , j } ^ { C F } \\cdot \\log \\widehat { A } _ { i , j } ^ { C F } + ( 1 - A _ { i , j } ^ { C F } ) \\cdot \\log ( 1 - \\widehat { A } _ { i , j } ^ { C F } ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 276, + 448, + 720, + 541 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "187 Balancing Counterfactual Learning In the training process, the above loss minimizations train the \n188 model on both the empirical factual distribution $\\hat { P } ^ { F } \\sim P ^ { F }$ and empirical counterfactual distribution \n189 $\\hat { P } ^ { C F } \\sim P ^ { C F }$ that are not necessarily equal – the training examples (node pairs) do not have to be \n190 aligned. However, at the stage of inference, the test data contains only observed (factual) samples. \n191 Such a gap between the training and test data distributions exposes the model in the risk of covariant \n192 shift, which is a common issue in counterfactual learning (Johansson et al., 2016; Assaad et al., 2021). \n193 To force the distributions of representations of factual distributions and counterfactual distributions to \n194 be similar, we use the discrepancy distance (Mansour et al., 2009; Johansson et al., 2016) as another \n195 objective to regularize the representation learning. That is, we use the following loss term to minimize \n196 the distance between the learned representations from ${ \\hat { P } } ^ { F }$ and $\\hat { P } ^ { C F }$ : ", + "bbox": [ + 140, + 551, + 825, + 640 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 645, + 825, + 704 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/94888a9eb7e87db31fb986221ebf747a63088d7e625d53a57540f4eef79846f9.jpg", + "text": "$$\n\\mathcal { L } _ { d i s c } = \\mathrm { d i s c } ( \\hat { P } _ { f } ^ { F } , \\hat { P } _ { f } ^ { C F } ) , \\mathrm { w h e r e \\ d i s c } ( P , Q ) = | | P - Q | | _ { F } ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 709, + 694, + 731 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where 97 $| | \\cdot | | _ { F }$ denotes the Frobenius Norm, and $\\hat { P } _ { f } ^ { F }$ and $\\hat { P } _ { f } ^ { C F }$ denote the node pair representations 98 learned by graph encoder $f$ from factual distribution and counterfactual distribution, respectively. ", + "bbox": [ + 155, + 737, + 825, + 770 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "199 Training During the training of CFLP, we want the model to be optimized towards three targets: \n200 (1) accurate link prediction on the observed outcomes (Eq. (8)), (2) accurate estimation on the \n201 counterfactual outcomes (Eq. (9)), and (3) regularization on the representation spaces learned from \n202 ${ \\hat { P } } ^ { F }$ and $\\hat { P } ^ { C F }$ (Eq. (10)). Therefore, the overall training loss of our proposed CFLP is ", + "bbox": [ + 140, + 782, + 825, + 843 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/078fdbb54d2c71e7df43b931ea21d479681fe4690318cc993778fbc478e43833.jpg", + "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { F } + \\alpha \\cdot \\mathcal { L } _ { C F } + \\beta \\cdot \\mathcal { L } _ { d i s c } ,\n$$", + "text_format": "latex", + "bbox": [ + 392, + 849, + 606, + 864 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "203 where $\\alpha$ and $\\beta$ are hyperparameters to control the weights of counterfactual link prediction (outcome \n204 estimation) loss and discrepancy loss. \n205 Summary Algorithm 1 summarizes \n206 the whole process of CFLP. The first \n207 step is to compute the factual and coun \n208 terfactual treatments T, $\\mathbf { T } ^ { C F }$ as well \n209 as the counterfactual outcomes $\\mathbf { A } ^ { C F }$ \n210 Then, the second step trains the graph \n211 learning model on both the observed \n212 factual data and created counterfactual \n213 data with the integrated loss function \n214 (Eq. (11)). Note that the discrepancy \n215 loss (Eq. (10)) is computed on the rep \n216 resentations of node pairs learned by \n217 the graph encoder $f$ , so the decoder \n218 $g$ is trained with data from both ${ \\hat { P } } ^ { F }$ \n219 and $\\hat { P } ^ { C F }$ without balancing the con \n220 straints. Therefore, after the model is \n221 sufficiently trained, we freeze the graph \n222 encoder $f$ and fine-tune $g$ with only the \n223 factual data. Finally, after the decoder \n224 is sufficiently fine-tuned, we output the \n225 link prediction logits for both the fac \n226 tual and counterfactual adjacency ma \n227 trices. ", + "bbox": [ + 140, + 871, + 831, + 901 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/357a0a855aee81c267638bc8da32ce884531a28e82f7eb8f101740cd9e34f27b.jpg", + "table_caption": [ + "Table 1: Statistics of datasets used in the experiments. ", + "Algorithm 1: CFLP: Counterfactual graph learning for link prediction " + ], + "table_footnote": [], + "table_body": "
DatasetCORACITESEERPUBMEDFACEBOOKOGB-DDI
# nodes2,7083,32719,7174,0394,267
# links5,2784,55244,32488,2341,334,889
# validation node pairs1,0549108,86417,646235,371
# test node pairs2,1101,82017,72835,292229,088
", + "bbox": [ + 227, + 109, + 772, + 189 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 217, + 426, + 536 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Input : $f$ , g, A, X, n_epochs, n_epoch_f t \n1 Compute T by Eq. (1) ; \n2 Compute $\\mathbf { T } ^ { C F }$ , $\\mathbf { A } ^ { C F }$ by Eqs. (3) and (4) ; $^ { \\prime * }$ model training \\*/ \n3 Initialize $\\Theta _ { f }$ in $f$ and $\\Theta _ { g }$ in $g$ ; \n4 for epoch in range(n_epochs) do \n5 $\\mathbf { Z } = f ( \\mathbf { A } , \\mathbf { X } )$ ; \n6 Get $\\widehat { \\mathbf A }$ and $\\widehat { \\mathbf { A } } ^ { C F }$ via $g$ with Eqs. (6) and (7) ; \n7 Update $\\Theta _ { f }$ and $\\Theta _ { g }$ with $\\mathcal { L }$ ; // (11) \n8 end $^ { \\prime * }$ decoder fine-tuning \\*/ \n9 Freeze $\\Theta _ { f }$ and re-initialize $\\Theta _ { g }$ ; \n10 $\\mathbf { Z } = f ( \\mathbf { A } , \\mathbf { X } )$ ; \n11 for epoch in range(n_epochs_f t) do \n12 Get $\\widehat { \\mathbf A }$ via $g$ with Eq. (6) ; \n13 Update $\\Theta _ { g }$ with $\\mathcal { L } _ { F }$ ; // Eq. (8) \n14 end $^ { \\prime * }$ model inferencing \\*/ \n15 $\\mathbf { Z } = f ( \\mathbf { A } , \\mathbf { X } )$ ; \n16 Get $\\widehat { \\mathbf A }$ and $\\widehat { \\mathbf { A } } ^ { C F }$ via $g$ with Eqs. (6) and (7) ; Output : $\\widehat { \\mathbf A }$ for link prediction, $\\widehat { \\mathbf { A } } ^ { C F }$ ", + "bbox": [ + 441, + 242, + 808, + 555 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "28 Complexity The complexity of the ", + "bbox": [ + 156, + 554, + 426, + 568 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "first step (finding counterfactual links with nearest neighbors) is proportional to the number of node pairs. When $\\gamma$ is set as a small value to obtain indeed similar node pairs, this step (Eq. (3)) uses constant time. Moreover, the computation in Eq. (3) can be parallelized. Therefore, the time complexity is $O ( N ^ { 2 } / C )$ where $C$ is the number of processes. For the complexity of the second step (training counterfactual learning model), the GNN encoder has time complexity of $O ( L H ^ { 2 } N + L H | \\mathcal { E } | )$ (Wu et al., 2020), where $L$ is the number of GNN layers and $H$ is the size of node representations. Given that we sample the same number of non-existing links as that of observed links during training, the complexity of a three-layer MLP decoder is $O ( ( ( \\bar { H } + 1 ) \\cdot d _ { h } + d _ { h } \\cdot 1 ) | \\mathcal { E } | ) =$ $O ( d _ { h } ( H + 2 ) | \\mathcal { E } | )$ , where $d _ { h }$ is the number of neurons in the hidden layer. Therefore, the second step has linear time complexity w.r.t. the sum of node and edge counts. ", + "bbox": [ + 158, + 566, + 825, + 705 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "239 Limitations First, as mentioned above, the computation of finding counterfactual links has a worst \n240 case complexity of $O ( N ^ { 2 } )$ . Second, CFLP performs counterfactual prediction with only a single \n241 treatment; however, there are quite a few kinds of graph structural information that can be considered \n242 as treatments. Future work can leverage the rich structural information by bundled treatments (Zou \n243 et al., 2020) in counterfactual graph learning. ", + "bbox": [ + 147, + 722, + 825, + 791 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 811, + 312, + 829 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Experimental Setup ", + "text_level": 1, + "bbox": [ + 173, + 843, + 351, + 858 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "246 We conduct experiments on five benchmark datasets including citation networks (CORA, CITESEER, \n247 PUBMED (Yang et al., 2016)), social network (FACEBOOK (McAuley and Leskovec, 2012)), and \n248 drug-drug interaction network (OGB-DDI (Wishart et al., 2018)) from the Open Graph Benchmark \n249 (OGB) (Hu et al., 2020). For the first four datasets, we randomly select $10 \\% / 2 0 \\%$ of the links and \n250 the same numbers of disconnected node pairs as validation/test samples. The links in the validation \n251 and test sets are masked off from the training graph. For OGB-DDI, we used the OGB official \n252 train/validation/test splits. Statistics for the datasets are given in Table 1 and details are in Appendix. \n253 We use K-core (Bader and Hogue, 2003) clusters as the default treatment variable. We evaluate CFLP \n254 on three commonly used GNN encoders: GCN (Kipf and Welling, 2016a), GSAGE (Hamilton et al., \n255 2017), and JKNet (Xu et al., 2018). We compare the link prediction performance of CFLP against \n256 Node2Vec (Grover and Leskovec, 2016), MVGRL (Hassani and Khasahmadi, 2020), VGAE (Kipf \n257 and Welling, 2016b), SEAL (Zhang and Chen, 2018), LGLP (Cai et al., 2021), and GNNs with MLP \n258 decoder. We report averaged test performance and their standard deviation over 20 runs with different \n259 random parameter initializations. Other than the most commonly used of Area Under ROC Curve \n260 (AUC), we report Hits $@ 2 0$ (one of the primary metrics on OGB leaderboard) as a more challenging \n261 metric, as it expects models to rank positive edges higher than nearly all negative edges. \n262 Besides performance comparison on link prediction, we will answer two questions to suggest a way \n263 of choosing a treatment variable for creating counterfactual links: (Q1) Does CFLP sufficiently learn \n264 the observed averaged treatment effect (ATE) derived from the counterfactual links? (Q2) What is the \n265 relationship between the estimated ATE learned in the method and the prediction performance? If the \n266 answer to Q1 is yes, then the answer to Q2 will indicate how to choose treatment based on observed \n267 ATE. To answer the Q1, we calculate the observed ATE $( \\widehat { \\mathrm { A I E } } _ { o b s } )$ by comparing the observed links \n268 in A and created counterfactual links $\\mathbf { A } ^ { C F }$ dthat have opposite treatments. And we calculate the \n269 estimated ATE $\\widehat { ( \\mathrm { A I E } _ { e s t } ) }$ by comparing the predicted links in $\\widehat { \\mathbf A }$ and predicted counterfactual links \n270 $\\widehat { \\mathbf { A } } ^ { C F }$ . Formally, $\\widehat { \\mathrm { A I E } } _ { o b s }$ and $\\widehat { \\mathrm { A T E } } _ { e s t }$ are defined as ", + "bbox": [ + 143, + 869, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/4cf158ba2bdb7c16c18a4b287e77dbaf09961d918248ae0eda94e04002d1e261.jpg", + "table_caption": [ + "Table 2: Link prediction performances measured by Hits $@ 2 0$ . Best performance and best baseline performance are marked with bold and underline, respectively. " + ], + "table_footnote": [], + "table_body": "
CORACITESEERPUBMEDFACEBOOKOGB-DDI
Node2Vec49.96±2.5147.78±1.7239.19±1.0224.24±3.0223.26±2.09
MVGRL19.53±2.6414.07±0.7914.19±0.8514.43±0.3310.02±1.01
VGAE45.91±3.3844.04±4.8623.73±1.6137.01±0.6311.71±1.96
SEAL51.35±2.2640.90±3.6828.45±3.8140.89±5.7030.56±3.86
LGLP62.98±0.5657.43±3.71137.86±2.13
GCN49.06±1.7255.56±1.3221.84±3.8753.89±2.1437.07±5.07
GSAGE53.54±2.9653.67±2.9439.13±4.4145.51±3.2253.90±4.74
JKNet48.21±3.8655.60±2.1725.64±4.1152.25±1.4860.56±8.69
Our proposed CFLP with different graph encoders
CFLP w/GCN60.34±2.3359.45±2.3034.12±2.7253.95±2.2952.51±1.09
CFLP w/GSAGE57.33±1.7353.05±2.0743.07±2.3647.28±3.0075.49±4.33
CFLP w/JKNet65.57±1.0568.09±1.4944.90±2.0055.22±1.2986.08±1.98
", + "bbox": [ + 205, + 123, + 789, + 310 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 323, + 826, + 503 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 508, + 825, + 643 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/e8ca76c9f3ab30ef5e54bc10251ffc0f8d8f372522fa3a4a6991a9571004056d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\widehat { \\mathrm { A T E } } _ { o b s } = \\displaystyle \\frac { 1 } { N ^ { 2 } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } \\left\\{ \\mathbf { T } \\odot \\left( \\mathbf { A } - \\mathbf { A } ^ { C F } \\right) + \\left( \\mathbf { 1 } _ { N \\times N } - \\mathbf { T } \\right) \\odot \\left( \\mathbf { A } ^ { C F } - \\mathbf { A } \\right) \\right\\} _ { i , j } . } \\\\ & { \\widehat { \\mathrm { A T E } } _ { e s t } = \\displaystyle \\frac { 1 } { N ^ { 2 } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } \\left\\{ \\mathbf { T } \\odot \\left( \\widehat { \\mathbf { A } } - \\widehat { \\mathbf { A } } ^ { C F } \\right) + \\left( \\mathbf { 1 } _ { N \\times N } - \\mathbf { T } \\right) \\odot \\left( \\widehat { \\mathbf { A } } ^ { C F } - \\widehat { \\mathbf { A } } \\right) \\right\\} _ { i , j } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 645, + 759, + 737 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "271 The treatment variables we will investigate are usually graph clustering or community detection meth \n272 ods, such as $\\mathbf { K }$ -core (Bader and Hogue, 2003), stochastic block model (SBM) (Karrer and Newman, \n273 2011), spectral clustering (SpecC) $\\mathrm { N g }$ et al., 2001), propagation clustering (PropC) (Raghavan et al., \n274 2007), Louvain (Blondel et al., 2008), common neighbors (CommN), Katz index, and hierarchical \n275 clustering (Ward) (Ward Jr, 1963). We use JKNet $\\mathrm { { X u } }$ et al., 2018) as the default graph encoder. ", + "bbox": [ + 140, + 738, + 825, + 808 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Implementation details and supplementary experimental results (e.g., sensitivity on $\\gamma$ , ablation study on $\\mathcal { L } _ { C F }$ and $\\mathcal { L } _ { d i s c , }$ ) can be found in Appendix. Source code is available in supplementary material. ", + "bbox": [ + 163, + 814, + 823, + 843 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 Experimental Results ", + "text_level": 1, + "bbox": [ + 174, + 857, + 362, + 872 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "279 Link Prediction Tables 2 and 3 show the link prediction performance of Hits $\\textcircled{ a} 2 0$ and AUC by \n280 all methods. LGLP on PUBMED and OGB-DDI are missing due to the out of memory error when \n281 running the code package from the authors. We observe that our CFLP on different graph encoders \n282 achieve similar or better performances compared with baselines. The only exception is the AUC on \n283 FACEBOOK where most methods have close-to-perfect AUC. As AUC is a relatively easier metric \n284 comparing with Hits $\\textcircled{ a} 2 0$ , most methods achieved good performance on AUC. We observe that CFLP \n285 with JKNet almost consistently achieves the best performance and outperforms baselines significantly \n286 on Hits $@ 2 0$ . Specifically, compared with the best baseline, CFLP improves relatively by $1 6 . 4 \\%$ and \n287 $0 . 8 \\%$ on Hits $\\textcircled{ a} 2 0$ and AUC, respectively. It is worth noting that CFLP with JKNet achieves the \n288 state-of-the-art performance on the official leaderboard1 of OGB-DDI. \n289 Figure 2 shows the AUC performance of CFLP on \n290 CORA with different combinations of $\\alpha$ and $\\beta$ . We \n291 observe that the performance is the poorest when \n292 $\\alpha = \\beta = 0$ and gradually improves and gets stable \n293 as $\\alpha$ and $\\beta$ increase, showing that CFLP is robust \n294 to the hyperparameters $\\alpha$ and $\\beta$ . ", + "bbox": [ + 148, + 882, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/c3140843cf53cdb1f8d4e88df6a694b80ec7dc0faf7d33b1e29a99fe2ffa8231.jpg", + "table_caption": [ + "Table 3: Link prediction performances measured by AUC. Best performance and best baseline performance are marked with bold and underline, respectively. " + ], + "table_footnote": [], + "table_body": "
CORACITESEERPUBMEDFACEBOOKOGB-DDI
Node2Vec84.49±0.4980.00±0.6880.32±0.2986.49±4.3290.83±0.02
MVGRL75.07±3.6361.20±0.5580.78±1.2879.83±0.3081.45±0.99
VGAE88.68±0.4085.35±0.6095.80±0.1398.66±0.0493.08±0.15
SEAL92.55±0.5085.82±0.4496.36±0.2899.60±0.0297.85±0.17
LGLP91.30±0.0589.41±0.1398.51±0.01
GCN90.25±0.5371.47±1.4096.33±0.8099.43±0.0299.82±0.05
GSAGE90.24±0.3487.38±1.3996.78±0.1199.29±0.0499.93±0.02
JKNet89.05±0.6788.58±1.7896.58±0.2399.43±0.0299.94±0.01
Our proposed CFLP with different graph encoders
CFLP w/GCN92.55±0.5089.65±0.2096.99±0.0899.38±0.0199.44±0.05
CFLPw/GSAGE92.61±0.5291.84±0.2097.01±0.0199.34±0.1099.83±0.05
CFLP w/JKNet93.05±0.2492.12±0.4797.53±0.1799.31±0.0499.94±0.01
", + "bbox": [ + 205, + 123, + 789, + 310 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 328, + 825, + 440 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 445, + 504, + 529 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "ATE with Different Treatments Tables 4 and 5 show the link prediction performance, $\\widehat { \\mathrm { A T E } } _ { o b s }$ , and $\\widehat { \\mathrm { A I E } } _ { e s t }$ of CFLP (with JKNet) when using different treatments. The treatments in Tables 4 and 5 are sorted by the Hits $\\textcircled{ a} 2 0$ performance. Bigger ATE indicates stronger causal relationship between the treatment and outcome, and vice versa. We observe: (1) $\\widehat { \\mathrm { A I E } } _ { e s t }$ values are generally close to $\\widehat { \\mathrm { A I E } } _ { o b s }$ showing that CFLP was sufficiently trained to learn the causal relationship between graph structure information and link existence; (2) $\\widehat { \\mathrm { A I E } } _ { o b s }$ and $\\widehat { \\mathrm { A I E } } _ { e s t }$ are both negatively correlated with the link prediction performance, showing that we can pick a proper treatment prior to training a model with CFLP. Using the treatment that has the weakest causal relationship with link existence is likely to train the model to capture more essential factors on the outcome, in a way similar to denoising the unrelated information from the representations. ", + "bbox": [ + 166, + 546, + 504, + 675 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/47e57d6501e7ce1ae9dd51fb0b8af84b88b44a8f67aca2501688ea437b8662eb.jpg", + "image_caption": [ + "Figure 2: AUC performance of CFLP on CORA w.r.t different combanitions of $\\alpha$ and $\\beta$ . " + ], + "image_footnote": [], + "bbox": [ + 532, + 465, + 823, + 623 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 161, + 678, + 825, + 747 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 Related Work ", + "text_level": 1, + "bbox": [ + 163, + 767, + 320, + 785 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Link Prediction With its wide applications, link prediction has draw attention from many research communities including statistical machine learning and data mining. Stochastic generative methods based on stochastic block models (SBM) are developed to generate links (Mehta et al., 2019). In data mining, matrix factorization (Menon and Elkan, 2011), heuristic methods (Philip et al., 2010; Martínez et al., 2016), and graph embedding methods (Cui et al., 2018) have been applied to predict links in the graph. Heuristic methods compute the similarity score of nodes based on their neighborhoods. These ", + "bbox": [ + 173, + 800, + 825, + 885 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/38282d4d5d2044496487e0616c791cc5c0f38e03be6ad67dfe6799333acc720f.jpg", + "table_caption": [ + "Table 4: Results of CFLP with different treatments on CORA. (sorted by Hits $@ 2 0$ ) " + ], + "table_footnote": [], + "table_body": "
Hits@20ATEobsATEest
K-core65.6±1.10.0020.013±0.003
SBM64.2±1.10.0060.023±0.015
CommN62.3±1.60.0070.053±0.021
PropC61.7±1.40.0370.059±0.065
Ward61.2±2.30.0010.033±0.012
SpecC59.3±2.80.0020.033±0.011
Louvain57.6±1.80.0250.138±0.091
Katz56.6±3.40.7400.802±0.041
", + "bbox": [ + 174, + 123, + 486, + 256 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/7a4142aaec59ac598bafa7b5a812120de440dfe0d9c662e25ce88666da424c98.jpg", + "table_caption": [ + "Table 5: Results of CFLP with different treatments on CITESEER. (sorted by Hits $@ 2 0$ ) " + ], + "table_footnote": [], + "table_body": "
Hits@20ATEobsATEest
SBM71.6 ±1.90.0040.005 ±0.001
K-core68.1±1.50.0020.010±0.002
Ward67.0±1.70.0030.037±0.009
PropC64.6±3.60.1410.232±0.113
Louvain63.3±2.50.1260.151±0.078
SpecC59.9±1.30.0090.166±0.034
Katz57.3±0.50.2450.224±0.037
CommN56.8±4.90.6780.195±0.034
", + "bbox": [ + 508, + 123, + 826, + 256 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "316 methods can be generally categorized into first-order, second-order, and high-order heuristics based \n317 on the maximum distance of the neighbors. Graph embedding methods learn latent node features via \n318 embedding lookup and use them for link prediction (Perozzi et al., 2014; Tang et al., 2015; Grover \n319 and Leskovec, 2016; Wang et al., 2016). ", + "bbox": [ + 142, + 282, + 825, + 338 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In the past few years, GNNs have showed promising results on various graph-based tasks with their ability of learning from features and custom aggregations on structures, (Kipf and Welling, 2016a; Hamilton et al., 2017; Xu et al., 2018; Wu et al., 2020). With node pair representations and an attached MLP or inner-product decoder, GNNs can be used for link prediction (Zhang et al., 2020a). For example, VGAE used GCN to learn node representations and reconstruct the graph structure (Kipf and Welling, 2016b). SEAL extracted a local subgraph around each target node pair and then learned graph representation from local subgraph for link prediction (Zhang and Chen, 2018). Following the scheme of SEAL, Cai and Ji (2020) proposed to improve local subgraph representation learning by multi-scale graph representation learning. And LGLP inverted the local subgraphs to line graphs before learning representations (Cai et al., 2021). However, very limited work has studied to use causal inference for improving link prediction. ", + "bbox": [ + 173, + 344, + 825, + 496 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Counterfactual Prediction As a mean of learning the causality between treatment and outcome, counterfactual prediction has been used for a variety of applicaitons such as recommender systems (Wang et al., 2020; Xu et al., 2020), health care (Alaa and van der Schaar, 2017), vision-language tasks (Zhang et al., 2020b; Parvaneh et al., 2020), and decision making (Coston et al., 2020; Pitis et al., 2020; Kusner et al., 2017). To infer the causal relationships, previous work usually estimated the ITE via function fitting models (Gelman and Hill, 2006; Chipman et al., 2010; Wager and Athey, 2018; Assaad et al., 2021) which estimated the transductive ITE. Peysakhovich et al. (2019) and Zou et al. (2020) studied counterfactual prediction with multiple agents and bundled treatments, respectively. Pawlowski et al. (2020) proposed a deep structural causal model for tractable counterfactual inference. ", + "bbox": [ + 171, + 512, + 826, + 636 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Causal Inference Causal inference methods usually re-weighted samples based on propensity score (Rosenbaum and Rubin, 1983; Austin, 2011; Kuang et al., 2017a,b) to remove confounding bias from binary treatments. Recently, several works studied about learning treatment invariant representation to predict the counterfactual outcomes (Hassanpour and Greiner, 2019b,a; Shalit et al., 2017; Yao et al., 2018; Bica et al., 2020; Hassanpour and Greiner, 2019a; Li and Fu, 2017). When part of unobserved outcomes may mislead the counterfactual prediction, Louizos et al. (2017) attempted to infer the outcomes from proxies, and Hartford et al. (2017) introduced instrumental variable. SITE preserved local similarity to balance the distributions of control and treated groups (Yao et al., 2018). Yoon et al. (2018) estimated ITE with generative adversarial networks (GANs). Assaad et al. (2021) discussed the trade-off between achieving balance and predictive power. ", + "bbox": [ + 173, + 652, + 826, + 791 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 Conclusion ", + "text_level": 1, + "bbox": [ + 166, + 810, + 297, + 827 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we presented a counterfactual graph learning method for link prediction (CFLP). We introduced the idea of counterfactual prediction to improve link prediction on graphs. CFLP accurately predicted the missing links by exploring the causal relationship between global graph structure and link existence. Extensive experiments demonstrated that CFLP achieved the state-ofthe-art performance on benchmark datasets. ", + "bbox": [ + 169, + 842, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "References \n57 Ahmed Alaa and Mihaela Van Der Schaar. 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", + "bbox": [ + 155, + 69, + 828, + 922 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 69, + 828, + 917 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 47, + 828, + 917 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 90, + 826, + 654 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Checklist ", + "text_level": 1, + "bbox": [ + 171, + 670, + 254, + 685 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 696, + 339, + 710 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] at the end of Section 3.2 \n(c) Did you discuss any potential negative societal impacts of your work? [No] \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 238, + 715, + 825, + 806 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 214, + 810, + 493, + 825 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ", + "bbox": [ + 236, + 829, + 738, + 861 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 214, + 864, + 393, + 878 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] ", + "bbox": [ + 236, + 883, + 823, + 911 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] in Section 4.1 and Appendix \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] in Appendix ", + "bbox": [ + 236, + 92, + 825, + 180 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 210, + 184, + 823, + 199 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] \n(b) Did you mention the license of the assets? [N/A] \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 238, + 203, + 823, + 310 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 214, + 314, + 705, + 329 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ", + "bbox": [ + 238, + 333, + 825, + 422 + ], + "page_idx": 13 + } +] \ No newline at end of file diff --git a/parse/train/N2dEPuK0vom/N2dEPuK0vom_middle.json b/parse/train/N2dEPuK0vom/N2dEPuK0vom_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..d4b0221c18c0f4afe8fd781b199f16e2d6e57b7c --- /dev/null +++ b/parse/train/N2dEPuK0vom/N2dEPuK0vom_middle.json @@ -0,0 +1,52295 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 112, + 97, + 499, + 118 + ], + "lines": [ + { + "bbox": [ + 112, + 97, + 500, + 119 + ], + "spans": [ + { + "bbox": [ + 112, + 97, + 500, + 119 + ], + "score": 1.0, + "content": "Counterfactual Graph Learning for Link Prediction", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 259, + 159, + 355, + 203 + ], + "lines": [ + { + "bbox": [ + 258, + 159, + 356, + 171 + ], + "spans": [ + { + "bbox": [ + 258, + 159, + 356, + 171 + ], + "score": 1.0, + "content": "Anonymous Author(s)", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 283, + 169, + 328, + 182 + ], + "spans": [ + { + "bbox": [ + 283, + 169, + 328, + 182 + ], + "score": 1.0, + "content": "Affiliation", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 287, + 180, + 324, + 193 + ], + "spans": [ + { + "bbox": [ + 287, + 180, + 324, + 193 + ], + "score": 1.0, + "content": "Address", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 290, + 193, + 320, + 203 + ], + "spans": [ + { + "bbox": [ + 290, + 193, + 320, + 203 + ], + "score": 1.0, + "content": "email", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 283, + 232, + 328, + 245 + ], + "lines": [ + { + "bbox": [ + 281, + 231, + 331, + 246 + ], + "spans": [ + { + "bbox": [ + 281, + 231, + 331, + 246 + ], + "score": 1.0, + "content": "Abstract", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 91, + 259, + 469, + 401 + ], + "lines": [ + { + "bbox": [ + 141, + 259, + 470, + 272 + ], + "spans": [ + { + "bbox": [ + 141, + 259, + 470, + 272 + ], + "score": 1.0, + "content": "Learning to predict missing links is important for many graph-based applications.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 271, + 469, + 281 + ], + "spans": [ + { + "bbox": [ + 142, + 271, + 469, + 281 + ], + "score": 1.0, + "content": "Existing methods were designed to learn the observed association between two sets", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 142, + 281, + 470, + 293 + ], + "spans": [ + { + "bbox": [ + 142, + 281, + 470, + 293 + ], + "score": 1.0, + "content": "of variables: (1) the observed graph structure and (2) the existence of link between a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 292, + 470, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 292, + 470, + 304 + ], + "score": 1.0, + "content": "pair of nodes. 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In this work, we propose a novel link prediction method that enhances", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 358, + 470, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 358, + 470, + 370 + ], + "score": 1.0, + "content": "graph learning by the counterfactual inference. It creates counterfactual links", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 368, + 470, + 381 + ], + "spans": [ + { + "bbox": [ + 141, + 368, + 470, + 381 + ], + "score": 1.0, + "content": "from the observed ones, and our method learns representations from both of them.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 379, + 470, + 391 + ], + "spans": [ + { + "bbox": [ + 141, + 379, + 470, + 391 + ], + "score": 1.0, + "content": "Experiments on a number of benchmark datasets show that our proposed method", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 390, + 383, + 402 + ], + "spans": [ + { + "bbox": [ + 141, + 390, + 383, + 402 + ], + "score": 1.0, + "content": "achieves the state-of-the-art performance on link prediction.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12, + "bbox_fs": [ + 141, + 259, + 470, + 402 + ] + }, + { + "type": "title", + "bbox": [ + 91, + 427, + 191, + 441 + ], + "lines": [ + { + "bbox": [ + 87, + 426, + 192, + 443 + ], + "spans": [ + { + "bbox": [ + 87, + 426, + 192, + 443 + ], + "score": 1.0, + "content": "14 1 Introduction", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "index", + "bbox": [ + 90, + 454, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 90, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 90, + 456, + 100, + 465 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 105, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "Link prediction seeks to predict the likelihood of edge existence between node pairs based on the", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 90, + 467, + 100, + 476 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "score": 1.0, + "content": "observed graph. Given the omnipresence of graph-structured data, link prediction has copious applica-", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 89, + 478, + 100, + 487 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 105, + 476, + 505, + 488 + ], + "score": 1.0, + "content": "tions such as movie recommendation (Bennett et al., 2007), chemical interaction prediction (Stanfield", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 485, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 89, + 489, + 100, + 498 + ], + "score": 1.0, + "content": "18", + "type": "text" + }, + { + "bbox": [ + 104, + 485, + 506, + 501 + ], + "score": 1.0, + "content": "et al., 2017), and knowledge graph completion (Kazemi and Poole, 2018). Graph machine learning", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 89, + 500, + 100, + 510 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 105, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "methods have been widely applied to solve this problem. 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Existing methods that learn from association only were not able to capture", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 88, + 614, + 99, + 624 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 105, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "essential factors to accurately predict missing links in the test data. Take social network as an example.", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 88, + 624, + 100, + 636 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 106, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "Suppose Alice and Adam live in the same neighborhood and they are close friends. 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In this example, the link predictor would be trained to estimate the individual treatment", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 236, + 502, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 167, + 248 + ], + "score": 1.0, + "content": "effect (ITE) as", + "type": "text" + }, + { + "bbox": [ + 167, + 236, + 209, + 246 + ], + "score": 0.9, + "content": "1 - 1 = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 236, + 502, + 248 + ], + "score": 1.0, + "content": "so it looks for factors other than neighborhood to predict the factual link.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 89, + 274, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 89, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 89, + 276, + 99, + 286 + ], + "score": 1.0, + "content": "38", + "type": "text" + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "As known to many, counterfactual question is a key component of causal inference and have been", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 88, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 88, + 286, + 100, + 298 + ], + "score": 1.0, + "content": "39", + "type": "text" + }, + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "well defined in the literature. A counterfactual question is usually framed with three factors: context", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 88, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 88, + 298, + 99, + 308 + ], + "score": 1.0, + "content": "40", + "type": "text" + }, + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "(as a data point), manipulation (e.g., treatment, intervention, action, strategy), and outcome (van der", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 88, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 88, + 309, + 99, + 319 + ], + "score": 1.0, + "content": "41", + "type": "text" + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "Laan and Petersen, 2007; Johansson et al., 2016). (To simplify the language, we use “treatment” to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 88, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 88, + 320, + 100, + 330 + ], + "score": 1.0, + "content": "42", + "type": "text" + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "refer to the manipulation in this paper, as readers might be familiar more with the word “treatment.”)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 88, + 328, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 88, + 331, + 99, + 340 + ], + "score": 1.0, + "content": "43", + "type": "text" + }, + { + "bbox": [ + 106, + 328, + 505, + 341 + ], + "score": 1.0, + "content": "Given certain data context, it asks what the outcome would have been if the treatment had not been", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 88, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 88, + 342, + 100, + 352 + ], + "score": 1.0, + "content": "44", + "type": "text" + }, + { + "bbox": [ + 106, + 340, + 505, + 351 + ], + "score": 1.0, + "content": "the observed value. In the scenario of link prediction, we consider the information of a pair of nodes", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 88, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 88, + 353, + 100, + 362 + ], + "score": 1.0, + "content": "45", + "type": "text" + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "as context, graph structural properties as treatment, and link existence as outcome. Recall the social", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 88, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 88, + 363, + 100, + 373 + ], + "score": 1.0, + "content": "46", + "type": "text" + }, + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "network example. The context is Alice and Adam, which includes their personal attributes and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 88, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 88, + 374, + 100, + 384 + ], + "score": 1.0, + "content": "47", + "type": "text" + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "relationships with others on the network. The treatment is living in the same neighborhood, which can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 88, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 88, + 386, + 99, + 395 + ], + "score": 1.0, + "content": "48", + "type": "text" + }, + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "be given as one attribute or identified by community detection. And the outcome is their friendship.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 89, + 399, + 506, + 618 + ], + "lines": [ + { + "bbox": [ + 89, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 89, + 402, + 100, + 411 + ], + "score": 1.0, + "content": "49", + "type": "text" + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "In this work, we present a counterfactual graph learning method for link prediction (CFLP) that", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 89, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 89, + 413, + 100, + 423 + ], + "score": 1.0, + "content": "50", + "type": "text" + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "trains graph learning models to answer the counterfactual questions. Figure 1 illustrates this two-step", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 89, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 89, + 424, + 99, + 433 + ], + "score": 1.0, + "content": "51", + "type": "text" + }, + { + "bbox": [ + 104, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "method. Suppose the treatment variable is defined as one type of global graph structure, e.g., the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 89, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 89, + 434, + 100, + 444 + ], + "score": 1.0, + "content": "52", + "type": "text" + }, + { + "bbox": [ + 104, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "neighborhood assignment discovered by spectral clustering or community detection algorithms. We", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 89, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 89, + 446, + 100, + 455 + ], + "score": 1.0, + "content": "53", + "type": "text" + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "are wondering how likely the neighborhood distribution makes a difference on the link (non-)existence", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 89, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 89, + 456, + 100, + 466 + ], + "score": 1.0, + "content": "54", + "type": "text" + }, + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "for each pair of nodes. So, given a pair of nodes (like Alice and Adam) and the treatment value on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 89, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 89, + 467, + 100, + 477 + ], + "score": 1.0, + "content": "55", + "type": "text" + }, + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "this pair (in the same neighborhood), we find a pair of nodes (like Helen and Bob) that satisfies two", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 89, + 477, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 89, + 478, + 100, + 488 + ], + "score": 1.0, + "content": "56", + "type": "text" + }, + { + "bbox": [ + 106, + 477, + 506, + 488 + ], + "score": 1.0, + "content": "conditions: (1) it has a different treatment (in different neighborhoods) and (2) it is the most similar", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 89, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 89, + 489, + 100, + 499 + ], + "score": 1.0, + "content": "57", + "type": "text" + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "pair with the given pair of nodes. We call these matched pair of nodes as “counterfactual links.” Note", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 89, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 89, + 500, + 100, + 509 + ], + "score": 1.0, + "content": "58", + "type": "text" + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "that the outcome of the counterfactual link can be either 1 or 0, depending on whether there exists an", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 89, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 89, + 511, + 100, + 521 + ], + "score": 1.0, + "content": "59", + "type": "text" + }, + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "edge between the matched pair of nodes. The counterfactual link provides unobserved outcome to the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 89, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 89, + 522, + 100, + 531 + ], + "score": 1.0, + "content": "60", + "type": "text" + }, + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "given pair of nodes (Alice and Adam) under a counterfactual condition (in different neighborhoods).", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 89, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 89, + 533, + 99, + 542 + ], + "score": 1.0, + "content": "61", + "type": "text" + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "After counterfactual links are created for all (positive and negative) training examples, CFLP trains", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 89, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 89, + 543, + 100, + 553 + ], + "score": 1.0, + "content": "62", + "type": "text" + }, + { + "bbox": [ + 104, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "a link predictor (which can be GNN-based) to learn the representation vectors of nodes to predict", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 89, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 89, + 555, + 100, + 564 + ], + "score": 1.0, + "content": "63", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "both the observed factual links and the counterfactual links. In this Alice-Adam example, the link", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 89, + 562, + 507, + 577 + ], + "spans": [ + { + "bbox": [ + 89, + 565, + 100, + 575 + ], + "score": 1.0, + "content": "64", + "type": "text" + }, + { + "bbox": [ + 104, + 562, + 507, + 577 + ], + "score": 1.0, + "content": "predictor is trained to estimate the individual treatment effect (ITE) of neighborhood assignment as", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 88, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 88, + 576, + 101, + 586 + ], + "score": 1.0, + "content": "65", + "type": "text" + }, + { + "bbox": [ + 106, + 574, + 147, + 585 + ], + "score": 0.88, + "content": "1 - 1 = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 574, + 505, + 586 + ], + "score": 1.0, + "content": ". So, the learner will try to discover the essential factors on the friendship between Alice", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 89, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 89, + 588, + 100, + 596 + ], + "score": 1.0, + "content": "66", + "type": "text" + }, + { + "bbox": [ + 105, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "and Adam. For some other examples, if the outcome of counterfactual link is different from that of", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 89, + 596, + 507, + 609 + ], + "spans": [ + { + "bbox": [ + 89, + 599, + 100, + 608 + ], + "score": 1.0, + "content": "67", + "type": "text" + }, + { + "bbox": [ + 105, + 596, + 507, + 609 + ], + "score": 1.0, + "content": "the given pair of nodes, the learner will estimate the strong effect of the treatment variable. Therefore,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 89, + 606, + 473, + 621 + ], + "spans": [ + { + "bbox": [ + 89, + 609, + 100, + 619 + ], + "score": 1.0, + "content": "68", + "type": "text" + }, + { + "bbox": [ + 105, + 606, + 473, + 621 + ], + "score": 1.0, + "content": "CFLP enables graph learning models to predict missing links regarding causal relationship.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 90, + 623, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 89, + 623, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 89, + 626, + 100, + 635 + ], + "score": 1.0, + "content": "69", + "type": "text" + }, + { + "bbox": [ + 106, + 623, + 506, + 635 + ], + "score": 1.0, + "content": "Contributions. Our main contributions can be summarized as follows. (1) This is the first work that", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 89, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 89, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "70", + "type": "text" + }, + { + "bbox": [ + 105, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "proposes to improve link prediction by causal inference, specifically, learning to answer counterfactual", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 89, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 89, + 647, + 99, + 657 + ], + "score": 1.0, + "content": "71", + "type": "text" + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "questions about link existence. (2) This work introduces CFLP that trains GNN-based link predictors", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 89, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 89, + 658, + 100, + 668 + ], + "score": 1.0, + "content": "72", + "type": "text" + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "to predict both factual and counterfactual links. It learns the causal relationship between global", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 89, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 89, + 669, + 100, + 679 + ], + "score": 1.0, + "content": "73", + "type": "text" + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "graph structure and link existence. (3) CFLP outperforms competitive baseline methods on several", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 88, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 88, + 680, + 101, + 690 + ], + "score": 1.0, + "content": "74", + "type": "text" + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "benchmark datasets. On OGB-DDI, our CFLP achieves the state-of-the-art performance. We analyze", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 89, + 689, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 89, + 691, + 100, + 700 + ], + "score": 1.0, + "content": "75", + "type": "text" + }, + { + "bbox": [ + 105, + 689, + 505, + 700 + ], + "score": 1.0, + "content": "the impact of counterfactual links as well as the choice of treatment variable. This work sheds insights", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 89, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 89, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "76", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "for improving graph machine learning with causal analysis, which has not been extensively studied", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 89, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 89, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "77", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "yet, when the other direction (machine learning for causal inference) has been studied for a long time.", + "type": "text" + } + ], + "index": 59 + } + ], + "index": 55 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, 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"content": "(b) Train a GNN-based link predictor to predict factual and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 273, + 182, + 478, + 194 + ], + "spans": [ + { + "bbox": [ + 273, + 182, + 478, + 194 + ], + "score": 1.0, + "content": "counterfactual links given the corresponding treatments.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + } + ], + "index": 11.25 + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 266, + 193 + ], + "lines": [], + "index": 12.5, + "bbox_fs": [ + 105, + 172, + 268, + 193 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 203, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 505, + 215 + ], + "score": 1.0, + "content": "Figure 1: The proposed CFLP learns the causal relationship between the observed graph structure", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 104, + 214, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 506, + 226 + ], + "score": 1.0, + "content": "(e.g., neighborhood similarity, considered as treatment variable) and link existence (considered as", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 226, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 505, + 237 + ], + "score": 1.0, + "content": "outcome). In this example, the link predictor would be trained to estimate the individual treatment", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 236, + 502, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 167, + 248 + ], + "score": 1.0, + "content": "effect (ITE) as", + "type": "text" + }, + { + "bbox": [ + 167, + 236, + 209, + 246 + ], + "score": 0.9, + "content": "1 - 1 = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 236, + 502, + 248 + ], + "score": 1.0, + "content": "so it looks for factors other than neighborhood to predict the factual link.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 104, + 203, + 506, + 248 + ] + }, + { + "type": "index", + "bbox": [ + 89, + 274, + 505, + 395 + ], + "lines": [ + { + "bbox": [ + 89, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 89, + 276, + 99, + 286 + ], + "score": 1.0, + "content": "38", + "type": "text" + }, + { + "bbox": [ + 105, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "As known to many, counterfactual question is a key component of causal inference and have been", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 88, + 286, + 100, + 298 + ], + "score": 1.0, + "content": "39", + "type": "text" + }, + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "well defined in the literature. A counterfactual question is usually framed with three factors: context", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 88, + 298, + 99, + 308 + ], + "score": 1.0, + "content": "40", + "type": "text" + }, + { + "bbox": [ + 105, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "(as a data point), manipulation (e.g., treatment, intervention, action, strategy), and outcome (van der", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 88, + 309, + 99, + 319 + ], + "score": 1.0, + "content": "41", + "type": "text" + }, + { + "bbox": [ + 105, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "Laan and Petersen, 2007; Johansson et al., 2016). (To simplify the language, we use “treatment” to", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 318, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 88, + 320, + 100, + 330 + ], + "score": 1.0, + "content": "42", + "type": "text" + }, + { + "bbox": [ + 106, + 318, + 505, + 330 + ], + "score": 1.0, + "content": "refer to the manipulation in this paper, as readers might be familiar more with the word “treatment.”)", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 328, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 88, + 331, + 99, + 340 + ], + "score": 1.0, + "content": "43", + "type": "text" + }, + { + "bbox": [ + 106, + 328, + 505, + 341 + ], + "score": 1.0, + "content": "Given certain data context, it asks what the outcome would have been if the treatment had not been", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 88, + 342, + 100, + 352 + ], + "score": 1.0, + "content": "44", + "type": "text" + }, + { + "bbox": [ + 106, + 340, + 505, + 351 + ], + "score": 1.0, + "content": "the observed value. In the scenario of link prediction, we consider the information of a pair of nodes", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 351, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 88, + 353, + 100, + 362 + ], + "score": 1.0, + "content": "45", + "type": "text" + }, + { + "bbox": [ + 105, + 351, + 505, + 363 + ], + "score": 1.0, + "content": "as context, graph structural properties as treatment, and link existence as outcome. Recall the social", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 362, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 88, + 363, + 100, + 373 + ], + "score": 1.0, + "content": "46", + "type": "text" + }, + { + "bbox": [ + 105, + 362, + 505, + 374 + ], + "score": 1.0, + "content": "network example. The context is Alice and Adam, which includes their personal attributes and", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 372, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 88, + 374, + 100, + 384 + ], + "score": 1.0, + "content": "47", + "type": "text" + }, + { + "bbox": [ + 105, + 372, + 505, + 385 + ], + "score": 1.0, + "content": "relationships with others on the network. The treatment is living in the same neighborhood, which can", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 88, + 386, + 99, + 395 + ], + "score": 1.0, + "content": "48", + "type": "text" + }, + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "be given as one attribute or identified by community detection. And the outcome is their friendship.", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 400, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 89, + 402, + 100, + 411 + ], + "score": 1.0, + "content": "49", + "type": "text" + }, + { + "bbox": [ + 105, + 400, + 506, + 412 + ], + "score": 1.0, + "content": "In this work, we present a counterfactual graph learning method for link prediction (CFLP) that", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 410, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 89, + 413, + 100, + 423 + ], + "score": 1.0, + "content": "50", + "type": "text" + }, + { + "bbox": [ + 105, + 410, + 506, + 424 + ], + "score": 1.0, + "content": "trains graph learning models to answer the counterfactual questions. Figure 1 illustrates this two-step", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 89, + 424, + 99, + 433 + ], + "score": 1.0, + "content": "51", + "type": "text" + }, + { + "bbox": [ + 104, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "method. Suppose the treatment variable is defined as one type of global graph structure, e.g., the", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 89, + 434, + 100, + 444 + ], + "score": 1.0, + "content": "52", + "type": "text" + }, + { + "bbox": [ + 104, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "neighborhood assignment discovered by spectral clustering or community detection algorithms. We", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 89, + 446, + 100, + 455 + ], + "score": 1.0, + "content": "53", + "type": "text" + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "are wondering how likely the neighborhood distribution makes a difference on the link (non-)existence", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 89, + 456, + 100, + 466 + ], + "score": 1.0, + "content": "54", + "type": "text" + }, + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "for each pair of nodes. So, given a pair of nodes (like Alice and Adam) and the treatment value on", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 465, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 89, + 467, + 100, + 477 + ], + "score": 1.0, + "content": "55", + "type": "text" + }, + { + "bbox": [ + 105, + 465, + 506, + 477 + ], + "score": 1.0, + "content": "this pair (in the same neighborhood), we find a pair of nodes (like Helen and Bob) that satisfies two", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 477, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 89, + 478, + 100, + 488 + ], + "score": 1.0, + "content": "56", + "type": "text" + }, + { + "bbox": [ + 106, + 477, + 506, + 488 + ], + "score": 1.0, + "content": "conditions: (1) it has a different treatment (in different neighborhoods) and (2) it is the most similar", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 89, + 489, + 100, + 499 + ], + "score": 1.0, + "content": "57", + "type": "text" + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "pair with the given pair of nodes. We call these matched pair of nodes as “counterfactual links.” Note", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 497, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 89, + 500, + 100, + 509 + ], + "score": 1.0, + "content": "58", + "type": "text" + }, + { + "bbox": [ + 105, + 497, + 506, + 511 + ], + "score": 1.0, + "content": "that the outcome of the counterfactual link can be either 1 or 0, depending on whether there exists an", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 89, + 511, + 100, + 521 + ], + "score": 1.0, + "content": "59", + "type": "text" + }, + { + "bbox": [ + 105, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "edge between the matched pair of nodes. The counterfactual link provides unobserved outcome to the", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 520, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 89, + 522, + 100, + 531 + ], + "score": 1.0, + "content": "60", + "type": "text" + }, + { + "bbox": [ + 105, + 520, + 506, + 532 + ], + "score": 1.0, + "content": "given pair of nodes (Alice and Adam) under a counterfactual condition (in different neighborhoods).", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 531, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 89, + 533, + 99, + 542 + ], + "score": 1.0, + "content": "61", + "type": "text" + }, + { + "bbox": [ + 105, + 531, + 506, + 543 + ], + "score": 1.0, + "content": "After counterfactual links are created for all (positive and negative) training examples, CFLP trains", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 541, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 89, + 543, + 100, + 553 + ], + "score": 1.0, + "content": "62", + "type": "text" + }, + { + "bbox": [ + 104, + 541, + 506, + 554 + ], + "score": 1.0, + "content": "a link predictor (which can be GNN-based) to learn the representation vectors of nodes to predict", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 89, + 555, + 100, + 564 + ], + "score": 1.0, + "content": "63", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "both the observed factual links and the counterfactual links. In this Alice-Adam example, the link", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 562, + 507, + 577 + ], + "spans": [ + { + "bbox": [ + 89, + 565, + 100, + 575 + ], + "score": 1.0, + "content": "64", + "type": "text" + }, + { + "bbox": [ + 104, + 562, + 507, + 577 + ], + "score": 1.0, + "content": "predictor is trained to estimate the individual treatment effect (ITE) of neighborhood assignment as", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 574, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 88, + 576, + 101, + 586 + ], + "score": 1.0, + "content": "65", + "type": "text" + }, + { + "bbox": [ + 106, + 574, + 147, + 585 + ], + "score": 0.88, + "content": "1 - 1 = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 574, + 505, + 586 + ], + "score": 1.0, + "content": ". So, the learner will try to discover the essential factors on the friendship between Alice", + "type": "text" + } + ], + "index": 47, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 89, + 588, + 100, + 596 + ], + "score": 1.0, + "content": "66", + "type": "text" + }, + { + "bbox": [ + 105, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "and Adam. For some other examples, if the outcome of counterfactual link is different from that of", + "type": "text" + } + ], + "index": 48, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 596, + 507, + 609 + ], + "spans": [ + { + "bbox": [ + 89, + 599, + 100, + 608 + ], + "score": 1.0, + "content": "67", + "type": "text" + }, + { + "bbox": [ + 105, + 596, + 507, + 609 + ], + "score": 1.0, + "content": "the given pair of nodes, the learner will estimate the strong effect of the treatment variable. Therefore,", + "type": "text" + } + ], + "index": 49, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 606, + 473, + 621 + ], + "spans": [ + { + "bbox": [ + 89, + 609, + 100, + 619 + ], + "score": 1.0, + "content": "68", + "type": "text" + }, + { + "bbox": [ + 105, + 606, + 473, + 621 + ], + "score": 1.0, + "content": "CFLP enables graph learning models to predict missing links regarding causal relationship.", + "type": "text" + } + ], + "index": 50, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 623, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 89, + 626, + 100, + 635 + ], + "score": 1.0, + "content": "69", + "type": "text" + }, + { + "bbox": [ + 106, + 623, + 506, + 635 + ], + "score": 1.0, + "content": "Contributions. Our main contributions can be summarized as follows. (1) This is the first work that", + "type": "text" + } + ], + "index": 51, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 635, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 89, + 636, + 100, + 646 + ], + "score": 1.0, + "content": "70", + "type": "text" + }, + { + "bbox": [ + 105, + 635, + 506, + 647 + ], + "score": 1.0, + "content": "proposes to improve link prediction by causal inference, specifically, learning to answer counterfactual", + "type": "text" + } + ], + "index": 52, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 89, + 647, + 99, + 657 + ], + "score": 1.0, + "content": "71", + "type": "text" + }, + { + "bbox": [ + 105, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "questions about link existence. 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This work sheds insights", + "type": "text" + } + ], + "index": 57, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 89, + 702, + 100, + 712 + ], + "score": 1.0, + "content": "76", + "type": "text" + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "for improving graph machine learning with causal analysis, which has not been extensively studied", + "type": "text" + } + ], + "index": 58, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 89, + 713, + 100, + 722 + ], + "score": 1.0, + "content": "77", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "yet, when the other direction (machine learning for causal inference) has been studied for a long time.", + "type": "text" + } + ], + "index": 59, + "is_list_start_line": true + } + ], + "index": 25, + "bbox_fs": [ + 88, + 274, + 506, + 396 + ] + }, + { + "type": "index", + "bbox": [ + 89, + 399, + 506, + 618 + ], + "lines": [], + "index": 40.5, + "bbox_fs": [ + 88, + 400, + 507, + 621 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 90, + 623, + 505, + 722 + ], + "lines": [], + "index": 55, + "bbox_fs": [ + 88, + 623, + 506, + 723 + ], + "lines_deleted": true + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 90, + 71, + 187, + 85 + ], + "lines": [ + { + "bbox": [ + 86, + 68, + 189, + 89 + ], + "spans": [ + { + "bbox": [ + 86, + 68, + 189, + 89 + ], + "score": 1.0, + "content": "78 2 Preliminary", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 90, + 95, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 89, + 95, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 89, + 97, + 100, + 106 + ], + "score": 1.0, + "content": "79", + "type": "text" + }, + { + "bbox": [ + 105, + 95, + 174, + 108 + ], + "score": 1.0, + "content": "Notations Let", + "type": "text" + }, + { + "bbox": [ + 174, + 95, + 223, + 106 + ], + "score": 0.92, + "content": "G = ( \\nu , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 95, + 334, + 108 + ], + "score": 1.0, + "content": "be an undirected graph of", + "type": "text" + }, + { + "bbox": [ + 334, + 96, + 344, + 105 + ], + "score": 0.84, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 95, + 403, + 108 + ], + "score": 1.0, + "content": "nodes, where", + "type": "text" + }, + { + "bbox": [ + 403, + 95, + 494, + 107 + ], + "score": 0.93, + "content": "\\mathcal { V } = \\{ v _ { 1 } , v _ { 2 } , \\ldots , v _ { N } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 95, + 506, + 108 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 89, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 89, + 108, + 100, + 117 + ], + "score": 1.0, + "content": "80", + "type": "text" + }, + { + "bbox": [ + 105, + 105, + 192, + 119 + ], + "score": 1.0, + "content": "the set of nodes and", + "type": "text" + }, + { + "bbox": [ + 192, + 106, + 241, + 117 + ], + "score": 0.9, + "content": "\\mathcal { E } \\subseteq \\mathcal { V } \\times \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "is the set of observed links. 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The outcome variable", + "type": "text" + }, + { + "bbox": [ + 296, + 266, + 319, + 277 + ], + "score": 0.9, + "content": "Y ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 266, + 506, + 278 + ], + "score": 1.0, + "content": "is naturally binary, indicating whether a link", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 89, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 89, + 279, + 100, + 289 + ], + "score": 1.0, + "content": "94", + "type": "text" + }, + { + "bbox": [ + 105, + 276, + 221, + 289 + ], + "score": 1.0, + "content": "exists between the node pair", + "type": "text" + }, + { + "bbox": [ + 221, + 279, + 228, + 287 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 276, + 270, + 289 + ], + "score": 1.0, + "content": "; and thus,", + "type": "text" + }, + { + "bbox": [ + 270, + 277, + 316, + 289 + ], + "score": 0.92, + "content": "\\mathcal { V } = \\{ 0 , \\dot { 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 276, + 506, + 289 + ], + "score": 1.0, + "content": ". We study the causal effect of binary treatment", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 89, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 89, + 290, + 100, + 299 + ], + "score": 1.0, + "content": "95", + "type": "text" + }, + { + "bbox": [ + 106, + 288, + 142, + 300 + ], + "score": 1.0, + "content": "variable", + "type": "text" + }, + { + "bbox": [ + 142, + 288, + 209, + 300 + ], + "score": 0.93, + "content": "t \\in \\mathcal { T } = \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 288, + 293, + 300 + ], + "score": 1.0, + "content": ", where the value of", + "type": "text" + }, + { + "bbox": [ + 294, + 288, + 355, + 300 + ], + "score": 0.92, + "content": "\\dot { Y } _ { 1 } ( \\dot { x } ) - Y _ { 0 } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 288, + 454, + 300 + ], + "score": 1.0, + "content": "for a particular context", + "type": "text" + }, + { + "bbox": [ + 454, + 290, + 461, + 298 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "is of high", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 88, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 88, + 300, + 100, + 311 + ], + "score": 1.0, + "content": "96", + "type": "text" + }, + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "interest and known as the individualized treatment effect (ITE) (van der Laan and Petersen, 2007;", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 89, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 89, + 312, + 99, + 321 + ], + "score": 1.0, + "content": "97", + "type": "text" + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "Weiss et al., 2015). The value of ITE indicates the causality relationship between the treatment and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 89, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 89, + 322, + 100, + 332 + ], + "score": 1.0, + "content": "98", + "type": "text" + }, + { + "bbox": [ + 104, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "outcome on the context. And the expected ITE given the context distribution is called averaged", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 89, + 331, + 449, + 344 + ], + "spans": [ + { + "bbox": [ + 89, + 334, + 100, + 343 + ], + "score": 1.0, + "content": "99", + "type": "text" + }, + { + "bbox": [ + 105, + 332, + 218, + 343 + ], + "score": 1.0, + "content": "treatment effect (ATE). i.e.,", + "type": "text" + }, + { + "bbox": [ + 218, + 331, + 307, + 344 + ], + "score": 0.91, + "content": "\\mathrm { A T E } = \\mathbb { E } _ { x \\sim \\mathcal { X } } \\mathrm { \\Pi } \\mathrm { T E } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 332, + 449, + 343 + ], + "score": 1.0, + "content": ", for a particular treatment variable.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 86, + 348, + 506, + 451 + ], + "lines": [ + { + "bbox": [ + 86, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 86, + 350, + 99, + 359 + ], + "score": 1.0, + "content": "100", + "type": "text" + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "However, as aforementioned, the fact that we can only observe one potential outcome under one", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 85, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 85, + 361, + 99, + 371 + ], + "score": 1.0, + "content": "101", + "type": "text" + }, + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "score": 1.0, + "content": "particular treatment prevents the ITE from being known (Johansson et al., 2016). In the problem", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 85, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 85, + 371, + 100, + 382 + ], + "score": 1.0, + "content": "102", + "type": "text" + }, + { + "bbox": [ + 105, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "setting of link prediction, we refer the observed adjacency matrix as the factual outcomes A and the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 85, + 378, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 85, + 382, + 100, + 393 + ], + "score": 1.0, + "content": "103", + "type": "text" + }, + { + "bbox": [ + 103, + 378, + 480, + 394 + ], + "score": 1.0, + "content": "unobserved adjacency matrix when the treatment is different as the counterfactual outcomes", + "type": "text" + }, + { + "bbox": [ + 480, + 380, + 502, + 391 + ], + "score": 0.89, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 378, + 507, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 85, + 389, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 85, + 393, + 100, + 403 + ], + "score": 1.0, + "content": "104", + "type": "text" + }, + { + "bbox": [ + 104, + 389, + 150, + 405 + ], + "score": 1.0, + "content": "We denote", + "type": "text" + }, + { + "bbox": [ + 151, + 390, + 216, + 403 + ], + "score": 0.94, + "content": "\\mathbf { T } \\breve { \\in } \\{ 0 , 1 \\} ^ { \\breve { N } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 389, + 370, + 405 + ], + "score": 1.0, + "content": "as the factual treatment matrix, where", + "type": "text" + }, + { + "bbox": [ + 370, + 392, + 386, + 404 + ], + "score": 0.91, + "content": "T _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 389, + 507, + 405 + ], + "score": 1.0, + "content": "indicates the treatment of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 85, + 401, + 504, + 432 + ], + "spans": [ + { + "bbox": [ + 85, + 405, + 100, + 428 + ], + "score": 1.0, + "content": "105", + "type": "text" + }, + { + "bbox": [ + 103, + 401, + 147, + 432 + ], + "score": 1.0, + "content": "node pair We are int", + "type": "text" + }, + { + "bbox": [ + 148, + 404, + 177, + 416 + ], + "score": 0.92, + "content": "( v _ { i } , v _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 401, + 228, + 432 + ], + "score": 1.0, + "content": ". We denote in (1) estima", + "type": "text" + }, + { + "bbox": [ + 228, + 402, + 250, + 414 + ], + "score": 0.88, + "content": "\\mathbf { T } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 401, + 358, + 432 + ], + "score": 1.0, + "content": "as the counterfactual treathe counterfactual outcomes", + "type": "text" + }, + { + "bbox": [ + 358, + 416, + 381, + 427 + ], + "score": 0.87, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 401, + 435, + 432 + ], + "score": 1.0, + "content": "matrix where via observed", + "type": "text" + }, + { + "bbox": [ + 436, + 402, + 504, + 417 + ], + "score": 0.92, + "content": "T _ { i , j } ^ { C F } = 1 - T _ { i , j }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 426, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 85, + 429, + 100, + 440 + ], + "score": 1.0, + "content": "107", + "type": "text" + }, + { + "bbox": [ + 104, + 426, + 268, + 442 + ], + "score": 1.0, + "content": "with the counterfactual adjacency matrix", + "type": "text" + }, + { + "bbox": [ + 268, + 427, + 290, + 438 + ], + "score": 0.89, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 426, + 506, + 442 + ], + "score": 1.0, + "content": "to enhance link prediction, and (3) learning the causal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 439, + 473, + 451 + ], + "spans": [ + { + "bbox": [ + 86, + 441, + 100, + 450 + ], + "score": 1.0, + "content": "108", + "type": "text" + }, + { + "bbox": [ + 105, + 439, + 473, + 451 + ], + "score": 1.0, + "content": "relationship between graph structural information (treatment) and link existence (outcome).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 92, + 465, + 239, + 478 + ], + "lines": [ + { + "bbox": [ + 87, + 463, + 241, + 480 + ], + "spans": [ + { + "bbox": [ + 87, + 463, + 241, + 480 + ], + "score": 1.0, + "content": "109 3 The Proposed Method", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 87, + 489, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 86, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 86, + 490, + 100, + 501 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "In this section, we introduce CFLP, a novel counterfactual graph learning method for link prediction.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 86, + 501, + 99, + 511 + ], + "score": 1.0, + "content": "111", + "type": "text" + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "In Section 3.1, we define treatment variable and counterfactual outcomes/links on graph data and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 85, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 85, + 513, + 100, + 523 + ], + "score": 1.0, + "content": "112", + "type": "text" + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "present how to compute them (Figure 1(a)). In Section 3.2, we introduce the graph learning model", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 85, + 520, + 472, + 535 + ], + "spans": [ + { + "bbox": [ + 85, + 523, + 100, + 533 + ], + "score": 1.0, + "content": "113", + "type": "text" + }, + { + "bbox": [ + 105, + 520, + 472, + 535 + ], + "score": 1.0, + "content": "that learns from both the observed graph and the created counterfactual links (Figure 1(b)).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 90, + 545, + 365, + 557 + ], + "lines": [ + { + "bbox": [ + 88, + 544, + 366, + 559 + ], + "spans": [ + { + "bbox": [ + 88, + 544, + 366, + 559 + ], + "score": 1.0, + "content": "114 3.1 Defining Treatment Variable and Counterfactual Links", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 86, + 565, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 86, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 86, + 567, + 100, + 576 + ], + "score": 1.0, + "content": "115", + "type": "text" + }, + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "Treatment Previous work on graph machine learning (Velickovic et al., 2019; Park et al., 2020)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 577, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 86, + 579, + 100, + 588 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 106, + 577, + 505, + 588 + ], + "score": 1.0, + "content": "showed that the graph’s global structural information could improve the quality of representation", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 86, + 590, + 100, + 599 + ], + "score": 1.0, + "content": "117", + "type": "text" + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "vectors of nodes learned by GNNs. 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Therefore, for a pair of nodes, one option of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 86, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 86, + 621, + 100, + 632 + ], + "score": 1.0, + "content": "120", + "type": "text" + }, + { + "bbox": [ + 104, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "defining the treatment variable is its global structural role in the graph. 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Louvain discovers community structure of a graph and assigns", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 86, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 86, + 655, + 100, + 664 + ], + "score": 1.0, + "content": "123", + "type": "text" + }, + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "each node to one community. Then we can define the binary treatment variable as whether these", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 86, + 661, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 86, + 666, + 100, + 675 + ], + "score": 1.0, + "content": "124", + "type": "text" + }, + { + "bbox": [ + 104, + 661, + 334, + 678 + ], + "score": 1.0, + "content": "two nodes in the pair belong to the same community. Let", + "type": "text" + }, + { + "bbox": [ + 335, + 664, + 379, + 674 + ], + "score": 0.91, + "content": "c : \\mathcal { V } \\to \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 661, + 506, + 678 + ], + "score": 1.0, + "content": "be any graph mining/clustering", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 86, + 675, + 505, + 686 + ], + "spans": [ + { + "bbox": [ + 86, + 677, + 100, + 686 + ], + "score": 1.0, + "content": "125", + "type": "text" + }, + { + "bbox": [ + 106, + 675, + 505, + 686 + ], + "score": 1.0, + "content": "method that outputs the index of community/cluster/neighborhood that each node belongs to. 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Ideally, we would need all possible outcomes of", + "type": "text" + }, + { + "bbox": [ + 424, + 186, + 431, + 194 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 184, + 506, + 196 + ], + "score": 1.0, + "content": "under all kinds of", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 195, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 89, + 198, + 99, + 207 + ], + "score": 1.0, + "content": "87", + "type": "text" + }, + { + "bbox": [ + 105, + 195, + 506, + 208 + ], + "score": 1.0, + "content": "treatments to study the causal relationships (Morgan and Winship, 2015). However, in reality, only", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 207, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 89, + 208, + 100, + 218 + ], + "score": 1.0, + "content": "88", + "type": "text" + }, + { + "bbox": [ + 106, + 207, + 453, + 218 + ], + "score": 1.0, + "content": "one treatment was applied and thus only one outcome was observed for a given context", + "type": "text" + }, + { + "bbox": [ + 453, + 208, + 460, + 216 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 207, + 505, + 218 + ], + "score": 1.0, + "content": ". When the", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 217, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 89, + 219, + 100, + 228 + ], + "score": 1.0, + "content": "89", + "type": "text" + }, + { + "bbox": [ + 105, + 217, + 506, + 229 + ], + "score": 1.0, + "content": "variables are specified in data, people use Neyman–Rubin casual model (BCM) to develop statistical", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 227, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 89, + 230, + 100, + 239 + ], + "score": 1.0, + "content": "90", + "type": "text" + }, + { + "bbox": [ + 104, + 227, + 506, + 240 + ], + "score": 1.0, + "content": "learning methods such as propensity score matching (PSM) for causal inference (Rubin, 1974, 2005).", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 89, + 246, + 99, + 255 + ], + "score": 1.0, + "content": "91", + "type": "text" + }, + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "In this work, we look at link prediction on graphs. 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Given a graph", + "type": "text" + }, + { + "bbox": [ + 272, + 256, + 281, + 266 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 255, + 399, + 267 + ], + "score": 1.0, + "content": ", a context is a pair of nodes", + "type": "text" + }, + { + "bbox": [ + 399, + 255, + 450, + 267 + ], + "score": 0.93, + "content": "\\boldsymbol { x } = ( v _ { i } , v _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 255, + 505, + 267 + ], + "score": 1.0, + "content": "in the graph;", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 266, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 89, + 268, + 100, + 277 + ], + "score": 1.0, + "content": "93", + "type": "text" + }, + { + "bbox": [ + 104, + 266, + 146, + 278 + ], + "score": 1.0, + "content": "and thus,", + "type": "text" + }, + { + "bbox": [ + 147, + 266, + 198, + 277 + ], + "score": 0.92, + "content": "\\chi = \\mathcal { V } \\times \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 266, + 296, + 278 + ], + "score": 1.0, + "content": ". The outcome variable", + "type": "text" + }, + { + "bbox": [ + 296, + 266, + 319, + 277 + ], + "score": 0.9, + "content": "Y ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 266, + 506, + 278 + ], + "score": 1.0, + "content": "is naturally binary, indicating whether a link", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 276, + 506, + 289 + ], + "spans": [ + { + "bbox": [ + 89, + 279, + 100, + 289 + ], + "score": 1.0, + "content": "94", + "type": "text" + }, + { + "bbox": [ + 105, + 276, + 221, + 289 + ], + "score": 1.0, + "content": "exists between the node pair", + "type": "text" + }, + { + "bbox": [ + 221, + 279, + 228, + 287 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 276, + 270, + 289 + ], + "score": 1.0, + "content": "; and thus,", + "type": "text" + }, + { + "bbox": [ + 270, + 277, + 316, + 289 + ], + "score": 0.92, + "content": "\\mathcal { V } = \\{ 0 , \\dot { 1 } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 276, + 506, + 289 + ], + "score": 1.0, + "content": ". We study the causal effect of binary treatment", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 89, + 290, + 100, + 299 + ], + "score": 1.0, + "content": "95", + "type": "text" + }, + { + "bbox": [ + 106, + 288, + 142, + 300 + ], + "score": 1.0, + "content": "variable", + "type": "text" + }, + { + "bbox": [ + 142, + 288, + 209, + 300 + ], + "score": 0.93, + "content": "t \\in \\mathcal { T } = \\{ 0 , 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 288, + 293, + 300 + ], + "score": 1.0, + "content": ", where the value of", + "type": "text" + }, + { + "bbox": [ + 294, + 288, + 355, + 300 + ], + "score": 0.92, + "content": "\\dot { Y } _ { 1 } ( \\dot { x } ) - Y _ { 0 } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 288, + 454, + 300 + ], + "score": 1.0, + "content": "for a particular context", + "type": "text" + }, + { + "bbox": [ + 454, + 290, + 461, + 298 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "is of high", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 299, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 88, + 300, + 100, + 311 + ], + "score": 1.0, + "content": "96", + "type": "text" + }, + { + "bbox": [ + 106, + 299, + 505, + 311 + ], + "score": 1.0, + "content": "interest and known as the individualized treatment effect (ITE) (van der Laan and Petersen, 2007;", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 309, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 89, + 312, + 99, + 321 + ], + "score": 1.0, + "content": "97", + "type": "text" + }, + { + "bbox": [ + 106, + 309, + 505, + 321 + ], + "score": 1.0, + "content": "Weiss et al., 2015). The value of ITE indicates the causality relationship between the treatment and", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 89, + 322, + 100, + 332 + ], + "score": 1.0, + "content": "98", + "type": "text" + }, + { + "bbox": [ + 104, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "outcome on the context. And the expected ITE given the context distribution is called averaged", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 331, + 449, + 344 + ], + "spans": [ + { + "bbox": [ + 89, + 334, + 100, + 343 + ], + "score": 1.0, + "content": "99", + "type": "text" + }, + { + "bbox": [ + 105, + 332, + 218, + 343 + ], + "score": 1.0, + "content": "treatment effect (ATE). i.e.,", + "type": "text" + }, + { + "bbox": [ + 218, + 331, + 307, + 344 + ], + "score": 0.91, + "content": "\\mathrm { A T E } = \\mathbb { E } _ { x \\sim \\mathcal { X } } \\mathrm { \\Pi } \\mathrm { T E } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 332, + 449, + 343 + ], + "score": 1.0, + "content": ", for a particular treatment variable.", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 86, + 350, + 99, + 359 + ], + "score": 1.0, + "content": "100", + "type": "text" + }, + { + "bbox": [ + 105, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "However, as aforementioned, the fact that we can only observe one potential outcome under one", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 85, + 361, + 99, + 371 + ], + "score": 1.0, + "content": "101", + "type": "text" + }, + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "score": 1.0, + "content": "particular treatment prevents the ITE from being known (Johansson et al., 2016). In the problem", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 370, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 85, + 371, + 100, + 382 + ], + "score": 1.0, + "content": "102", + "type": "text" + }, + { + "bbox": [ + 105, + 370, + 505, + 382 + ], + "score": 1.0, + "content": "setting of link prediction, we refer the observed adjacency matrix as the factual outcomes A and the", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 378, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 85, + 382, + 100, + 393 + ], + "score": 1.0, + "content": "103", + "type": "text" + }, + { + "bbox": [ + 103, + 378, + 480, + 394 + ], + "score": 1.0, + "content": "unobserved adjacency matrix when the treatment is different as the counterfactual outcomes", + "type": "text" + }, + { + "bbox": [ + 480, + 380, + 502, + 391 + ], + "score": 0.89, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 378, + 507, + 394 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 389, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 85, + 393, + 100, + 403 + ], + "score": 1.0, + "content": "104", + "type": "text" + }, + { + "bbox": [ + 104, + 389, + 150, + 405 + ], + "score": 1.0, + "content": "We denote", + "type": "text" + }, + { + "bbox": [ + 151, + 390, + 216, + 403 + ], + "score": 0.94, + "content": "\\mathbf { T } \\breve { \\in } \\{ 0 , 1 \\} ^ { \\breve { N } \\times N }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 389, + 370, + 405 + ], + "score": 1.0, + "content": "as the factual treatment matrix, where", + "type": "text" + }, + { + "bbox": [ + 370, + 392, + 386, + 404 + ], + "score": 0.91, + "content": "T _ { i , j }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 389, + 507, + 405 + ], + "score": 1.0, + "content": "indicates the treatment of the", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 401, + 504, + 432 + ], + "spans": [ + { + "bbox": [ + 85, + 405, + 100, + 428 + ], + "score": 1.0, + "content": "105", + "type": "text" + }, + { + "bbox": [ + 103, + 401, + 147, + 432 + ], + "score": 1.0, + "content": "node pair We are int", + "type": "text" + }, + { + "bbox": [ + 148, + 404, + 177, + 416 + ], + "score": 0.92, + "content": "( v _ { i } , v _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 401, + 228, + 432 + ], + "score": 1.0, + "content": ". We denote in (1) estima", + "type": "text" + }, + { + "bbox": [ + 228, + 402, + 250, + 414 + ], + "score": 0.88, + "content": "\\mathbf { T } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 401, + 358, + 432 + ], + "score": 1.0, + "content": "as the counterfactual treathe counterfactual outcomes", + "type": "text" + }, + { + "bbox": [ + 358, + 416, + 381, + 427 + ], + "score": 0.87, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 401, + 435, + 432 + ], + "score": 1.0, + "content": "matrix where via observed", + "type": "text" + }, + { + "bbox": [ + 436, + 402, + 504, + 417 + ], + "score": 0.92, + "content": "T _ { i , j } ^ { C F } = 1 - T _ { i , j }", + "type": "inline_equation" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 426, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 85, + 429, + 100, + 440 + ], + "score": 1.0, + "content": "107", + "type": "text" + }, + { + "bbox": [ + 104, + 426, + 268, + 442 + ], + "score": 1.0, + "content": "with the counterfactual adjacency matrix", + "type": "text" + }, + { + "bbox": [ + 268, + 427, + 290, + 438 + ], + "score": 0.89, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 426, + 506, + 442 + ], + "score": 1.0, + "content": "to enhance link prediction, and (3) learning the causal", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 439, + 473, + 451 + ], + "spans": [ + { + "bbox": [ + 86, + 441, + 100, + 450 + ], + "score": 1.0, + "content": "108", + "type": "text" + }, + { + "bbox": [ + 105, + 439, + 473, + 451 + ], + "score": 1.0, + "content": "relationship between graph structural information (treatment) and link existence (outcome).", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + } + ], + "index": 3, + "bbox_fs": [ + 89, + 95, + 507, + 152 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 162, + 505, + 240 + ], + "lines": [], + "index": 9, + "bbox_fs": [ + 89, + 163, + 506, + 240 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 88, + 244, + 505, + 343 + ], + "lines": [], + "index": 17, + "bbox_fs": [ + 88, + 243, + 506, + 344 + ], + "lines_deleted": true + }, + { + "type": "index", + "bbox": [ + 86, + 348, + 506, + 451 + ], + "lines": [], + "index": 25.5, + "bbox_fs": [ + 85, + 348, + 507, + 451 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 92, + 465, + 239, + 478 + ], + "lines": [ + { + "bbox": [ + 87, + 463, + 241, + 480 + ], + "spans": [ + { + "bbox": [ + 87, + 463, + 241, + 480 + ], + "score": 1.0, + "content": "109 3 The Proposed Method", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "index", + "bbox": [ + 87, + 489, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 86, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 86, + 490, + 100, + 501 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "In this section, we introduce CFLP, a novel counterfactual graph learning method for link prediction.", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 86, + 501, + 99, + 511 + ], + "score": 1.0, + "content": "111", + "type": "text" + }, + { + "bbox": [ + 105, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "In Section 3.1, we define treatment variable and counterfactual outcomes/links on graph data and", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 85, + 513, + 100, + 523 + ], + "score": 1.0, + "content": "112", + "type": "text" + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "present how to compute them (Figure 1(a)). In Section 3.2, we introduce the graph learning model", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 520, + 472, + 535 + ], + "spans": [ + { + "bbox": [ + 85, + 523, + 100, + 533 + ], + "score": 1.0, + "content": "113", + "type": "text" + }, + { + "bbox": [ + 105, + 520, + 472, + 535 + ], + "score": 1.0, + "content": "that learns from both the observed graph and the created counterfactual links (Figure 1(b)).", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + } + ], + "index": 32.5, + "bbox_fs": [ + 85, + 489, + 506, + 535 + ] + }, + { + "type": "title", + "bbox": [ + 90, + 545, + 365, + 557 + ], + "lines": [ + { + "bbox": [ + 88, + 544, + 366, + 559 + ], + "spans": [ + { + "bbox": [ + 88, + 544, + 366, + 559 + ], + "score": 1.0, + "content": "114 3.1 Defining Treatment Variable and Counterfactual Links", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "index", + "bbox": [ + 86, + 565, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 86, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 86, + 567, + 100, + 576 + ], + "score": 1.0, + "content": "115", + "type": "text" + }, + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "Treatment Previous work on graph machine learning (Velickovic et al., 2019; Park et al., 2020)", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 577, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 86, + 579, + 100, + 588 + ], + "score": 1.0, + "content": "116", + "type": "text" + }, + { + "bbox": [ + 106, + 577, + 505, + 588 + ], + "score": 1.0, + "content": "showed that the graph’s global structural information could improve the quality of representation", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 86, + 590, + 100, + 599 + ], + "score": 1.0, + "content": "117", + "type": "text" + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "vectors of nodes learned by GNNs. 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Therefore, for a pair of nodes, one option of", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 86, + 621, + 100, + 632 + ], + "score": 1.0, + "content": "120", + "type": "text" + }, + { + "bbox": [ + 104, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "defining the treatment variable is its global structural role in the graph. 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Therefore, we use the outcome from the nearest observed", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 86, + 150, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 86, + 153, + 100, + 161 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 105, + 150, + 505, + 162 + ], + "score": 1.0, + "content": "context as a substitute. This idea has been adopted by many methods (Johansson et al., 2016; Alaa and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 86, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 86, + 163, + 99, + 172 + ], + "score": 1.0, + "content": "134", + "type": "text" + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "Van Der Schaar, 2019). That is, we want to find the nearest neighbor with the opposite treatment for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 86, + 171, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 86, + 174, + 100, + 183 + ], + "score": 1.0, + "content": "135", + "type": "text" + }, + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "score": 1.0, + "content": "each observed node pairs and use the nearest neighbor’s outcome as a counterfactual link. Formally,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 87, + 183, + 411, + 195 + ], + "spans": [ + { + "bbox": [ + 87, + 185, + 99, + 194 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 107, + 183, + 181, + 195 + ], + "score": 0.95, + "content": "\\forall ( v _ { i } , v _ { j } ) \\in \\mathcal { V } \\times \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 183, + 338, + 195 + ], + "score": 1.0, + "content": ", we want to find its counterfactual link", + "type": "text" + }, + { + "bbox": [ + 339, + 183, + 370, + 195 + ], + "score": 0.92, + "content": "( v _ { a } , v _ { b } )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 183, + 411, + 195 + ], + "score": 1.0, + "content": "as below:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 199, + 425, + 220 + ], + "lines": [ + { + "bbox": [ + 186, + 199, + 425, + 220 + ], + "spans": [ + { + "bbox": [ + 186, + 199, + 425, + 220 + ], + "score": 0.92, + "content": "( v _ { a } , v _ { b } ) = \\underset { v _ { a } , v _ { b } \\in \\mathcal V } { \\arg \\operatorname* { m i n } } \\{ d ( ( v _ { i } , v _ { j } ) , ( v _ { a } , v _ { b } ) ) \\ | \\ T _ { a , b } = 1 - T _ { i , j } \\} ,", + "type": "interline_equation", + "image_path": "c86c7ef6f36ed73dd303b2e20ff00e2cd6cd5a3e8b0a380427b9177074902f80.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 186, + 199, + 425, + 220 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 225, + 505, + 328 + ], + "lines": [ + { + "bbox": [ + 86, + 225, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 86, + 227, + 99, + 236 + ], + "score": 1.0, + "content": "137", + "type": "text" + }, + { + "bbox": [ + 105, + 225, + 133, + 238 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 225, + 157, + 237 + ], + "score": 0.92, + "content": "d ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 225, + 506, + 238 + ], + "score": 1.0, + "content": "is a metric of measuring the distance between a pair of node pairs (a pair of contexts).", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 235, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 86, + 238, + 100, + 248 + ], + "score": 1.0, + "content": "138", + "type": "text" + }, + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "score": 1.0, + "content": "Considering that we want to find the nearest node pair based on not only the raw node features but", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 245, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 86, + 249, + 100, + 258 + ], + "score": 1.0, + "content": "139", + "type": "text" + }, + { + "bbox": [ + 105, + 245, + 506, + 261 + ], + "score": 1.0, + "content": "also structural features, here we take the state-of-the-art unsupervised graph representation learning", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 86, + 257, + 507, + 273 + ], + "spans": [ + { + "bbox": [ + 86, + 263, + 99, + 272 + ], + "score": 1.0, + "content": "140", + "type": "text" + }, + { + "bbox": [ + 105, + 257, + 432, + 273 + ], + "score": 1.0, + "content": "method MVGRL (Hassani and Khasahmadi, 2020) to learn the node embeddings", + "type": "text" + }, + { + "bbox": [ + 433, + 258, + 482, + 271 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { X } } \\in \\mathbb { R } ^ { N \\times \\tilde { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 257, + 507, + 273 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 86, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 86, + 275, + 99, + 284 + ], + "score": 1.0, + "content": "141", + "type": "text" + }, + { + "bbox": [ + 106, + 273, + 219, + 285 + ], + "score": 1.0, + "content": "the observed graph. We use", + "type": "text" + }, + { + "bbox": [ + 219, + 271, + 229, + 282 + ], + "score": 0.82, + "content": "\\tilde { \\mathbf { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "to find the nearest neighbors of node pairs. Nevertheless, finding the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 86, + 284, + 506, + 296 + ], + "spans": [ + { + "bbox": [ + 86, + 285, + 100, + 296 + ], + "score": 1.0, + "content": "142", + "type": "text" + }, + { + "bbox": [ + 105, + 284, + 506, + 296 + ], + "score": 1.0, + "content": "nearest neighbors by computing the distance between all pairs of node pairs is extremely inefficient,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 86, + 294, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 86, + 297, + 100, + 306 + ], + "score": 1.0, + "content": "143", + "type": "text" + }, + { + "bbox": [ + 105, + 294, + 157, + 307 + ], + "score": 1.0, + "content": "which takes", + "type": "text" + }, + { + "bbox": [ + 157, + 294, + 187, + 307 + ], + "score": 0.92, + "content": "O ( N ^ { 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 294, + 325, + 307 + ], + "score": 1.0, + "content": "comparisons (as there are totally", + "type": "text" + }, + { + "bbox": [ + 325, + 294, + 355, + 307 + ], + "score": 0.93, + "content": "\\bar { O ( N ^ { 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 294, + 506, + 307 + ], + "score": 1.0, + "content": "node pairs). Hence we approximate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 86, + 304, + 507, + 318 + ], + "spans": [ + { + "bbox": [ + 86, + 307, + 100, + 317 + ], + "score": 1.0, + "content": "144", + "type": "text" + }, + { + "bbox": [ + 105, + 304, + 507, + 318 + ], + "score": 1.0, + "content": "Eq. (2) by substituting the distance between node pairs by the distance between nodes. That is,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 316, + 411, + 329 + ], + "spans": [ + { + "bbox": [ + 86, + 318, + 100, + 328 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 106, + 316, + 181, + 329 + ], + "score": 0.91, + "content": "\\forall ( v _ { i } , v _ { j } ) \\in \\mathcal { V } \\times \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 316, + 338, + 329 + ], + "score": 1.0, + "content": ", we want to find its counterfactual link", + "type": "text" + }, + { + "bbox": [ + 339, + 316, + 370, + 328 + ], + "score": 0.92, + "content": "( v _ { a } , v _ { b } )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 316, + 411, + 329 + ], + "score": 1.0, + "content": "as below:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 332, + 482, + 353 + ], + "lines": [ + { + "bbox": [ + 117, + 332, + 482, + 353 + ], + "spans": [ + { + "bbox": [ + 117, + 332, + 482, + 353 + ], + "score": 0.92, + "content": "( v _ { a } , v _ { b } ) = \\underset { v _ { a } , v _ { b } \\in \\mathcal { V } } { \\arg \\operatorname* { m i n } } \\{ d ( \\tilde { \\mathbf { x } } _ { i } , \\tilde { \\mathbf { x } } _ { a } ) + d ( \\tilde { \\mathbf { x } } _ { j } , \\tilde { \\mathbf { x } } _ { b } ) \\ \\vert \\ T _ { a , b } = 1 - T _ { i , j } , d ( \\tilde { \\mathbf { x } } _ { i } , \\tilde { \\mathbf { x } } _ { a } ) + d ( \\tilde { \\mathbf { x } } _ { j } , \\tilde { \\mathbf { x } } _ { b } ) < 2 \\gamma \\} ,", + "type": "interline_equation", + "image_path": "c7535c67a2682290c3018dd79ba783a829748a1ee1c3fa999b701f4d138c3cc9.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 117, + 332, + 482, + 353 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 360, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 86, + 359, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 86, + 362, + 100, + 371 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 104, + 359, + 132, + 373 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 360, + 156, + 372 + ], + "score": 0.93, + "content": "d ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 359, + 415, + 373 + ], + "score": 1.0, + "content": "is specified as the Euclidean distance on the embedding space of", + "type": "text" + }, + { + "bbox": [ + 415, + 359, + 425, + 371 + ], + "score": 0.85, + "content": "\\tilde { \\mathbf { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 359, + 444, + 373 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 445, + 362, + 452, + 372 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 359, + 506, + 373 + ], + "score": 1.0, + "content": "is a hyperpa-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 86, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 86, + 373, + 99, + 382 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "rameter that defines the maximum distance that two nodes are considered as similar. Note that when", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 86, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 86, + 385, + 100, + 393 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "no node pair satisfies the above equation, we do not assign any nearest neighbor for a given node pair", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 86, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 86, + 395, + 100, + 405 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "to ensure all the neighbors are similar enough (as substitutes) in the feature space. Therefore, the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 86, + 401, + 503, + 417 + ], + "spans": [ + { + "bbox": [ + 86, + 406, + 100, + 415 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 104, + 401, + 234, + 417 + ], + "score": 1.0, + "content": "counterfactual treatment matrix", + "type": "text" + }, + { + "bbox": [ + 235, + 403, + 256, + 414 + ], + "score": 0.9, + "content": "\\mathbf { T } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 401, + 419, + 417 + ], + "score": 1.0, + "content": "and the counterfactual adjacency matrix", + "type": "text" + }, + { + "bbox": [ + 420, + 403, + 442, + 414 + ], + "score": 0.9, + "content": "\\mathbf { \\dot { A } } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 401, + 503, + 417 + ], + "score": 1.0, + "content": "are defined as", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 420, + 416, + 449 + ], + "lines": [ + { + "bbox": [ + 159, + 420, + 416, + 449 + ], + "spans": [ + { + "bbox": [ + 159, + 420, + 416, + 449 + ], + "score": 0.9, + "content": "T _ { i , j } ^ { C F } , A _ { i , j } ^ { C F } = \\left\\{ \\begin{array} { l l } { 1 - T _ { i , j } , A _ { a , b } } & { \\mathrm { , ~ i f \\exists ~ ( } v _ { a } , v _ { b } ) \\in \\mathcal { V } \\times \\mathcal { V } \\mathrm { s a t i s f i e s } } \\\\ { T _ { i , j } , A _ { i , j } } & { \\mathrm { , ~ o t h e r w i s e . } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "cc4a01daac557afd01de2aa7f50822881d26d4fc9efa2734d6947cf854d99dec.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 159, + 420, + 416, + 429.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 159, + 429.6666666666667, + 416, + 439.33333333333337 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 159, + 439.33333333333337, + 416, + 449.00000000000006 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 455, + 508, + 480 + ], + "lines": [ + { + "bbox": [ + 84, + 454, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 84, + 454, + 279, + 469 + ], + "score": 1.0, + "content": "It is worth noting that the node embeddings 151", + "type": "text" + }, + { + "bbox": [ + 279, + 454, + 289, + 466 + ], + "score": 0.84, + "content": "\\tilde { \\mathbf { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 454, + 506, + 469 + ], + "score": 1.0, + "content": "and the nearest neighbors are computed only once and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 84, + 467, + 474, + 481 + ], + "spans": [ + { + "bbox": [ + 84, + 468, + 278, + 481 + ], + "score": 1.0, + "content": "do not change during the learning process. 152", + "type": "text" + }, + { + "bbox": [ + 279, + 467, + 289, + 478 + ], + "score": 0.81, + "content": "\\tilde { \\mathbf { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 468, + 474, + 481 + ], + "score": 1.0, + "content": "is only used for finding the nearest neighbors.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 88, + 490, + 506, + 588 + ], + "lines": [ + { + "bbox": [ + 87, + 490, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 87, + 493, + 99, + 502 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 104, + 490, + 320, + 503 + ], + "score": 1.0, + "content": "Learning from Counterfactual Distributions Let", + "type": "text" + }, + { + "bbox": [ + 320, + 490, + 335, + 501 + ], + "score": 0.88, + "content": "P ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 490, + 505, + 503 + ], + "score": 1.0, + "content": "be the factual distribution of the observed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 499, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 86, + 504, + 100, + 514 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 104, + 499, + 220, + 515 + ], + "score": 1.0, + "content": "contexts and treatments, and", + "type": "text" + }, + { + "bbox": [ + 220, + 501, + 241, + 512 + ], + "score": 0.92, + "content": "P ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 499, + 506, + 515 + ], + "score": 1.0, + "content": "be the counterfactual distribution that is composed of the observed", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 513, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 86, + 514, + 107, + 549 + ], + "score": 1.0, + "content": "155 156", + "type": "text" + }, + { + "bbox": [ + 107, + 526, + 186, + 541 + ], + "score": 0.92, + "content": "\\{ ( v _ { i } , v _ { j } , T _ { i , j } ^ { F } ) \\} _ { i , j = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 514, + 402, + 549 + ], + "score": 1.0, + "content": "ite treatments. 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Unlike traditional link prediction methods that take only", + "type": "text" + }, + { + "bbox": [ + 422, + 540, + 437, + 551 + ], + "score": 0.88, + "content": "{ \\hat { P } } ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 538, + 507, + 559 + ], + "score": 1.0, + "content": "as input and use", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 86, + 554, + 100, + 564 + ], + "score": 1.0, + "content": "158", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 201, + 565 + ], + "score": 1.0, + "content": "the observed outcomes", + "type": "text" + }, + { + "bbox": [ + 201, + 553, + 212, + 563 + ], + "score": 0.37, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "as the training target, the idea of counterfactual graph learning is to take", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 562, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 86, + 567, + 100, + 577 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 104, + 562, + 327, + 579 + ], + "score": 1.0, + "content": "advantage of the counterfactual distribution by having", + "type": "text" + }, + { + "bbox": [ + 327, + 564, + 349, + 575 + ], + "score": 0.89, + "content": "\\hat { P } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 562, + 506, + 579 + ], + "score": 1.0, + "content": "as a complementary input and use the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 86, + 574, + 459, + 590 + ], + "spans": [ + { + "bbox": [ + 86, + 578, + 100, + 588 + ], + "score": 1.0, + "content": "160", + "type": "text" + }, + { + "bbox": [ + 104, + 574, + 207, + 590 + ], + "score": 1.0, + "content": "counterfactual outcomes", + "type": "text" + }, + { + "bbox": [ + 207, + 575, + 229, + 586 + ], + "score": 0.88, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 574, + 459, + 590 + ], + "score": 1.0, + "content": "as the training target for the counterfactual data samples.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 95, + 600, + 317, + 612 + ], + "lines": [ + { + "bbox": [ + 92, + 599, + 317, + 614 + ], + "spans": [ + { + "bbox": [ + 92, + 599, + 317, + 614 + ], + "score": 1.0, + "content": "61 3.2 The Counterfactual Graph Learning Model", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 87, + 620, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 86, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 86, + 622, + 99, + 632 + ], + "score": 1.0, + "content": "162", + "type": "text" + }, + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "In this subsection, we present the design of our model as well as the training method. 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We use", + "type": "text" + }, + { + "bbox": [ + 219, + 271, + 229, + 282 + ], + "score": 0.82, + "content": "\\tilde { \\mathbf { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "to find the nearest neighbors of node pairs. 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Hence we approximate", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 304, + 507, + 318 + ], + "spans": [ + { + "bbox": [ + 86, + 307, + 100, + 317 + ], + "score": 1.0, + "content": "144", + "type": "text" + }, + { + "bbox": [ + 105, + 304, + 507, + 318 + ], + "score": 1.0, + "content": "Eq. (2) by substituting the distance between node pairs by the distance between nodes. That is,", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 316, + 411, + 329 + ], + "spans": [ + { + "bbox": [ + 86, + 318, + 100, + 328 + ], + "score": 1.0, + "content": "145", + "type": "text" + }, + { + "bbox": [ + 106, + 316, + 181, + 329 + ], + "score": 0.91, + "content": "\\forall ( v _ { i } , v _ { j } ) \\in \\mathcal { V } \\times \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 316, + 338, + 329 + ], + "score": 1.0, + "content": ", we want to find its counterfactual link", + "type": "text" + }, + { + "bbox": [ + 339, + 316, + 370, + 328 + ], + "score": 0.92, + "content": "( v _ { a } , v _ { b } )", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 316, + 411, + 329 + ], + "score": 1.0, + "content": "as below:", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + } + ], + "index": 15, + "bbox_fs": [ + 86, + 225, + 507, + 329 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 332, + 482, + 353 + ], + "lines": [ + { + "bbox": [ + 117, + 332, + 482, + 353 + ], + "spans": [ + { + "bbox": [ + 117, + 332, + 482, + 353 + ], + "score": 0.92, + "content": "( v _ { a } , v _ { b } ) = \\underset { v _ { a } , v _ { b } \\in \\mathcal { V } } { \\arg \\operatorname* { m i n } } \\{ d ( \\tilde { \\mathbf { x } } _ { i } , \\tilde { \\mathbf { x } } _ { a } ) + d ( \\tilde { \\mathbf { x } } _ { j } , \\tilde { \\mathbf { x } } _ { b } ) \\ \\vert \\ T _ { a , b } = 1 - T _ { i , j } , d ( \\tilde { \\mathbf { x } } _ { i } , \\tilde { \\mathbf { x } } _ { a } ) + d ( \\tilde { \\mathbf { x } } _ { j } , \\tilde { \\mathbf { x } } _ { b } ) < 2 \\gamma \\} ,", + "type": "interline_equation", + "image_path": "c7535c67a2682290c3018dd79ba783a829748a1ee1c3fa999b701f4d138c3cc9.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 117, + 332, + 482, + 353 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "index", + "bbox": [ + 86, + 360, + 505, + 416 + ], + "lines": [ + { + "bbox": [ + 86, + 359, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 86, + 362, + 100, + 371 + ], + "score": 1.0, + "content": "146", + "type": "text" + }, + { + "bbox": [ + 104, + 359, + 132, + 373 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 360, + 156, + 372 + ], + "score": 0.93, + "content": "d ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 359, + 415, + 373 + ], + "score": 1.0, + "content": "is specified as the Euclidean distance on the embedding space of", + "type": "text" + }, + { + "bbox": [ + 415, + 359, + 425, + 371 + ], + "score": 0.85, + "content": "\\tilde { \\mathbf { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 359, + 444, + 373 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 445, + 362, + 452, + 372 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 359, + 506, + 373 + ], + "score": 1.0, + "content": "is a hyperpa-", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 86, + 373, + 99, + 382 + ], + "score": 1.0, + "content": "147", + "type": "text" + }, + { + "bbox": [ + 105, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "rameter that defines the maximum distance that two nodes are considered as similar. Note that when", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 86, + 385, + 100, + 393 + ], + "score": 1.0, + "content": "148", + "type": "text" + }, + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "no node pair satisfies the above equation, we do not assign any nearest neighbor for a given node pair", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 86, + 395, + 100, + 405 + ], + "score": 1.0, + "content": "149", + "type": "text" + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "to ensure all the neighbors are similar enough (as substitutes) in the feature space. Therefore, the", + "type": "text" + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 401, + 503, + 417 + ], + "spans": [ + { + "bbox": [ + 86, + 406, + 100, + 415 + ], + "score": 1.0, + "content": "150", + "type": "text" + }, + { + "bbox": [ + 104, + 401, + 234, + 417 + ], + "score": 1.0, + "content": "counterfactual treatment matrix", + "type": "text" + }, + { + "bbox": [ + 235, + 403, + 256, + 414 + ], + "score": 0.9, + "content": "\\mathbf { T } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 401, + 419, + 417 + ], + "score": 1.0, + "content": "and the counterfactual adjacency matrix", + "type": "text" + }, + { + "bbox": [ + 420, + 403, + 442, + 414 + ], + "score": 0.9, + "content": "\\mathbf { \\dot { A } } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 401, + 503, + 417 + ], + "score": 1.0, + "content": "are defined as", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + } + ], + "index": 23, + "bbox_fs": [ + 86, + 359, + 506, + 417 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 420, + 416, + 449 + ], + "lines": [ + { + "bbox": [ + 159, + 420, + 416, + 449 + ], + "spans": [ + { + "bbox": [ + 159, + 420, + 416, + 449 + ], + "score": 0.9, + "content": "T _ { i , j } ^ { C F } , A _ { i , j } ^ { C F } = \\left\\{ \\begin{array} { l l } { 1 - T _ { i , j } , A _ { a , b } } & { \\mathrm { , ~ i f \\exists ~ ( } v _ { a } , v _ { b } ) \\in \\mathcal { V } \\times \\mathcal { V } \\mathrm { s a t i s f i e s } } \\\\ { T _ { i , j } , A _ { i , j } } & { \\mathrm { , ~ o t h e r w i s e . } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "cc4a01daac557afd01de2aa7f50822881d26d4fc9efa2734d6947cf854d99dec.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 159, + 420, + 416, + 429.6666666666667 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 159, + 429.6666666666667, + 416, + 439.33333333333337 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 159, + 439.33333333333337, + 416, + 449.00000000000006 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 455, + 508, + 480 + ], + "lines": [ + { + "bbox": [ + 84, + 454, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 84, + 454, + 279, + 469 + ], + "score": 1.0, + "content": "It is worth noting that the node embeddings 151", + "type": "text" + }, + { + "bbox": [ + 279, + 454, + 289, + 466 + ], + "score": 0.84, + "content": "\\tilde { \\mathbf { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 454, + 506, + 469 + ], + "score": 1.0, + "content": "and the nearest neighbors are computed only once and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 84, + 467, + 474, + 481 + ], + "spans": [ + { + "bbox": [ + 84, + 468, + 278, + 481 + ], + "score": 1.0, + "content": "do not change during the learning process. 152", + "type": "text" + }, + { + "bbox": [ + 279, + 467, + 289, + 478 + ], + "score": 0.81, + "content": "\\tilde { \\mathbf { X } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 468, + 474, + 481 + ], + "score": 1.0, + "content": "is only used for finding the nearest neighbors.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 84, + 454, + 506, + 481 + ] + }, + { + "type": "index", + "bbox": [ + 88, + 490, + 506, + 588 + ], + "lines": [ + { + "bbox": [ + 87, + 490, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 87, + 493, + 99, + 502 + ], + "score": 1.0, + "content": "153", + "type": "text" + }, + { + "bbox": [ + 104, + 490, + 320, + 503 + ], + "score": 1.0, + "content": "Learning from Counterfactual Distributions Let", + "type": "text" + }, + { + "bbox": [ + 320, + 490, + 335, + 501 + ], + "score": 0.88, + "content": "P ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 490, + 505, + 503 + ], + "score": 1.0, + "content": "be the factual distribution of the observed", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 499, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 86, + 504, + 100, + 514 + ], + "score": 1.0, + "content": "154", + "type": "text" + }, + { + "bbox": [ + 104, + 499, + 220, + 515 + ], + "score": 1.0, + "content": "contexts and treatments, and", + "type": "text" + }, + { + "bbox": [ + 220, + 501, + 241, + 512 + ], + "score": 0.92, + "content": "P ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 499, + 506, + 515 + ], + "score": 1.0, + "content": "be the counterfactual distribution that is composed of the observed", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 513, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 86, + 514, + 107, + 549 + ], + "score": 1.0, + "content": "155 156", + "type": "text" + }, + { + "bbox": [ + 107, + 526, + 186, + 541 + ], + "score": 0.92, + "content": "\\{ ( v _ { i } , v _ { j } , T _ { i , j } ^ { F } ) \\} _ { i , j = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 514, + 402, + 549 + ], + "score": 1.0, + "content": "ite treatments. We define the empirical factual distrib, and define the empirical counterfactual distribution", + "type": "text" + }, + { + "bbox": [ + 402, + 525, + 458, + 537 + ], + "score": 0.91, + "content": "\\hat { P } ^ { C F } \\sim P ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 513, + 465, + 524 + ], + "score": 0.91, + "content": "\\hat { P } ^ { F } \\sim P ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 514, + 471, + 549 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 471, + 525, + 505, + 538 + ], + "score": 0.87, + "content": "\\hat { P } ^ { C F } =", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 513, + 505, + 525 + ], + "score": 0.88, + "content": "{ \\hat { P } } ^ { F } = { }", + "type": "inline_equation" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 538, + 507, + 559 + ], + "spans": [ + { + "bbox": [ + 86, + 542, + 100, + 553 + ], + "score": 1.0, + "content": "157", + "type": "text" + }, + { + "bbox": [ + 107, + 541, + 191, + 555 + ], + "score": 0.91, + "content": "\\{ ( v _ { i } , v _ { j } , T _ { i , j _ { . } } ^ { C F } ) \\} _ { i , j = 1 } ^ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 538, + 422, + 559 + ], + "score": 1.0, + "content": ". Unlike traditional link prediction methods that take only", + "type": "text" + }, + { + "bbox": [ + 422, + 540, + 437, + 551 + ], + "score": 0.88, + "content": "{ \\hat { P } } ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 538, + 507, + 559 + ], + "score": 1.0, + "content": "as input and use", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 86, + 554, + 100, + 564 + ], + "score": 1.0, + "content": "158", + "type": "text" + }, + { + "bbox": [ + 105, + 552, + 201, + 565 + ], + "score": 1.0, + "content": "the observed outcomes", + "type": "text" + }, + { + "bbox": [ + 201, + 553, + 212, + 563 + ], + "score": 0.37, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "as the training target, the idea of counterfactual graph learning is to take", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 562, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 86, + 567, + 100, + 577 + ], + "score": 1.0, + "content": "159", + "type": "text" + }, + { + "bbox": [ + 104, + 562, + 327, + 579 + ], + "score": 1.0, + "content": "advantage of the counterfactual distribution by having", + "type": "text" + }, + { + "bbox": [ + 327, + 564, + 349, + 575 + ], + "score": 0.89, + "content": "\\hat { P } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 562, + 506, + 579 + ], + "score": 1.0, + "content": "as a complementary input and use the", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 574, + 459, + 590 + ], + "spans": [ + { + "bbox": [ + 86, + 578, + 100, + 588 + ], + "score": 1.0, + "content": "160", + "type": "text" + }, + { + "bbox": [ + 104, + 574, + 207, + 590 + ], + "score": 1.0, + "content": "counterfactual outcomes", + "type": "text" + }, + { + "bbox": [ + 207, + 575, + 229, + 586 + ], + "score": 0.88, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 574, + 459, + 590 + ], + "score": 1.0, + "content": "as the training target for the counterfactual data samples.", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + } + ], + "index": 34, + "bbox_fs": [ + 86, + 490, + 507, + 590 + ] + }, + { + "type": "title", + "bbox": [ + 95, + 600, + 317, + 612 + ], + "lines": [ + { + "bbox": [ + 92, + 599, + 317, + 614 + ], + "spans": [ + { + "bbox": [ + 92, + 599, + 317, + 614 + ], + "score": 1.0, + "content": "61 3.2 The Counterfactual Graph Learning Model", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "index", + "bbox": [ + 87, + 620, + 505, + 678 + ], + "lines": [ + { + "bbox": [ + 86, + 621, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 86, + 622, + 99, + 632 + ], + "score": 1.0, + "content": "162", + "type": "text" + }, + { + "bbox": [ + 106, + 621, + 505, + 632 + ], + "score": 1.0, + "content": "In this subsection, we present the design of our model as well as the training method. The input of", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 86, + 633, + 100, + 642 + ], + "score": 1.0, + "content": "163", + "type": "text" + }, + { + "bbox": [ + 106, + 631, + 331, + 643 + ], + "score": 1.0, + "content": "the model in CFLP includes (1) the observed graph data", + "type": "text" + }, + { + "bbox": [ + 331, + 632, + 341, + 641 + ], + "score": 0.26, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 631, + 434, + 643 + ], + "score": 1.0, + "content": "and raw feature matrix", + "type": "text" + }, + { + "bbox": [ + 434, + 631, + 444, + 641 + ], + "score": 0.38, + "content": "\\mathbf { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 631, + 505, + 643 + ], + "score": 1.0, + "content": ", (2) the factual", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 640, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 86, + 644, + 99, + 653 + ], + "score": 1.0, + "content": "164", + "type": "text" + }, + { + "bbox": [ + 105, + 640, + 150, + 655 + ], + "score": 1.0, + "content": "treatments", + "type": "text" + }, + { + "bbox": [ + 150, + 641, + 165, + 652 + ], + "score": 0.88, + "content": "\\mathbf { T } ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 640, + 286, + 655 + ], + "score": 1.0, + "content": "and counterfactual treatments", + "type": "text" + }, + { + "bbox": [ + 286, + 641, + 308, + 652 + ], + "score": 0.9, + "content": "\\bar { \\mathbf { T } } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 640, + 460, + 655 + ], + "score": 1.0, + "content": ", and (3) the counterfactual graph data", + "type": "text" + }, + { + "bbox": [ + 460, + 641, + 483, + 653 + ], + "score": 0.89, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 640, + 506, + 655 + ], + "score": 1.0, + "content": ". 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We denote", + "type": "text" + }, + { + "bbox": [ + 404, + 158, + 504, + 172 + ], + "score": 0.92, + "content": "\\mathbf { Z } = f ( \\mathbf { A } , \\mathbf { X } ) \\in \\mathbb { R } ^ { N \\times H }", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 85, + 170, + 507, + 183 + ], + "spans": [ + { + "bbox": [ + 85, + 172, + 100, + 183 + ], + "score": 1.0, + "content": "176", + "type": "text" + }, + { + "bbox": [ + 105, + 170, + 312, + 183 + ], + "score": 1.0, + "content": "as the output from the encoder’s last layer, i.e., the", + "type": "text" + }, + { + "bbox": [ + 312, + 171, + 322, + 181 + ], + "score": 0.82, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 170, + 507, + 183 + ], + "score": 1.0, + "content": "-dimensional representation vectors of nodes.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 85, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 85, + 183, + 100, + 193 + ], + "score": 1.0, + "content": "177", + "type": "text" + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "Following previous work (Zhang et al., 2020a), we compute the representation of a node pair as", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 85, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 85, + 194, + 100, + 204 + ], + "score": 1.0, + "content": "178", + "type": "text" + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "score": 1.0, + "content": "the Hadamard product of the vectors of the two nodes. That is, the representation for the node pair", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 201, + 382, + 217 + ], + "spans": [ + { + "bbox": [ + 86, + 205, + 100, + 216 + ], + "score": 1.0, + "content": "179", + "type": "text" + }, + { + "bbox": [ + 106, + 203, + 136, + 216 + ], + "score": 0.92, + "content": "( v _ { i } , v _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 201, + 146, + 217 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 146, + 203, + 204, + 216 + ], + "score": 0.93, + "content": "\\mathbf { z } _ { i } \\odot \\mathbf { \\bar { z } } _ { j } \\in \\mathbb { R } ^ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 201, + 235, + 217 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 235, + 204, + 244, + 214 + ], + "score": 0.82, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 201, + 382, + 217 + ], + "score": 1.0, + "content": "stands for the Hadamard product.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 86, + 219, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 86, + 218, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 86, + 221, + 100, + 231 + ], + "score": 1.0, + "content": "180", + "type": "text" + }, + { + "bbox": [ + 103, + 218, + 505, + 233 + ], + "score": 1.0, + "content": "For the link decoder that predicts whether a link exists between a pair of nodes, we opt for simplicity", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 86, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 86, + 233, + 99, + 241 + ], + "score": 1.0, + "content": "181", + "type": "text" + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "and adopt a simple decoder based on multi-layer perceptron (MLP), given the representations of node", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 86, + 241, + 354, + 254 + ], + "spans": [ + { + "bbox": [ + 86, + 243, + 100, + 253 + ], + "score": 1.0, + "content": "182", + "type": "text" + }, + { + "bbox": [ + 105, + 241, + 294, + 254 + ], + "score": 1.0, + "content": "pairs and their treatments. That is, the decoder", + "type": "text" + }, + { + "bbox": [ + 294, + 244, + 300, + 253 + ], + "score": 0.82, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 241, + 354, + 254 + ], + "score": 1.0, + "content": "is defined as", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 257, + 426, + 293 + ], + "lines": [ + { + "bbox": [ + 184, + 257, + 426, + 293 + ], + "spans": [ + { + "bbox": [ + 184, + 257, + 426, + 293 + ], + "score": 0.73, + "content": "\\begin{array} { r l } & { \\widehat { \\mathbf { A } } = g ( \\mathbf { Z } , \\mathbf { T } ) , \\mathrm { ~ w h e r e ~ } \\widehat { A } _ { i , j } = \\mathbf { M } \\mathrm { L P } \\big ( [ \\mathbf { z } _ { i } \\odot \\mathbf { z } _ { j } , T _ { i , j } ] \\big ) , } \\\\ & { \\widehat { \\mathbf { A } } ^ { C F } = g ( \\mathbf { Z } , \\mathbf { T } ^ { C F } ) , \\mathrm { ~ w h e r e ~ } \\widehat { A } _ { i , j } ^ { C F } = \\mathbf { M } \\mathrm { L P } \\big ( [ \\mathbf { z } _ { i } \\odot \\mathbf { z } _ { j } , T _ { i , j } ^ { C F } ] \\big ) , } \\end{array}", + "type": "interline_equation", + "image_path": "76e4d2ea2c85138cdff83101185eb4b984ee48118e83761e4e513ba2d3a43cc3.jpg" + } + ] + } + ], + "index": 14.5, + "virtual_lines": [ + { + "bbox": [ + 184, + 257, + 426, + 275.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 184, + 275.0, + 426, + 293.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 297, + 308, + 309 + ], + "lines": [ + { + "bbox": [ + 85, + 295, + 310, + 310 + ], + "spans": [ + { + "bbox": [ + 85, + 295, + 133, + 310 + ], + "score": 1.0, + "content": "183 where", + "type": "text" + }, + { + "bbox": [ + 133, + 297, + 150, + 309 + ], + "score": 0.89, + "content": "[ \\cdot , \\cdot ]", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 295, + 310, + 310 + ], + "score": 1.0, + "content": "stands for the concatenation of vectors.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 87, + 315, + 506, + 351 + ], + "lines": [ + { + "bbox": [ + 87, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 87, + 318, + 100, + 326 + ], + "score": 1.0, + "content": "184", + "type": "text" + }, + { + "bbox": [ + 103, + 315, + 438, + 327 + ], + "score": 1.0, + "content": "During the training process, data samples from the empirical factual distribution", + "type": "text" + }, + { + "bbox": [ + 439, + 314, + 453, + 325 + ], + "score": 0.89, + "content": "{ \\hat { P } } ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 315, + 506, + 327 + ], + "score": 1.0, + "content": "and the em-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 87, + 326, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 87, + 330, + 100, + 339 + ], + "score": 1.0, + "content": "185", + "type": "text" + }, + { + "bbox": [ + 103, + 326, + 244, + 341 + ], + "score": 1.0, + "content": "pirical counterfactual distribution", + "type": "text" + }, + { + "bbox": [ + 244, + 326, + 266, + 338 + ], + "score": 0.9, + "content": "\\hat { P } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 326, + 349, + 341 + ], + "score": 1.0, + "content": "are fed into decoder", + "type": "text" + }, + { + "bbox": [ + 350, + 330, + 357, + 340 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 326, + 452, + 341 + ], + "score": 1.0, + "content": "and optimized towards", + "type": "text" + }, + { + "bbox": [ + 452, + 328, + 461, + 338 + ], + "score": 0.41, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 326, + 480, + 341 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 480, + 327, + 502, + 339 + ], + "score": 0.87, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 326, + 506, + 341 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 86, + 339, + 425, + 352 + ], + "spans": [ + { + "bbox": [ + 86, + 341, + 100, + 351 + ], + "score": 1.0, + "content": "186", + "type": "text" + }, + { + "bbox": [ + 105, + 339, + 425, + 352 + ], + "score": 1.0, + "content": "respectively. That is, for the two distributions, the loss functions are as follows:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 169, + 355, + 441, + 429 + ], + "lines": [ + { + "bbox": [ + 169, + 355, + 441, + 429 + ], + "spans": [ + { + "bbox": [ + 169, + 355, + 441, + 429 + ], + "score": 0.96, + "content": "\\begin{array} { l } { { \\mathcal { L } _ { F } = \\displaystyle \\frac { 1 } { N ^ { 2 } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } A _ { i , j } \\cdot \\log \\widehat { A } _ { i , j } + ( 1 - A _ { i , j } ) \\cdot \\log ( 1 - \\widehat { A } _ { i , j } ) , } } \\\\ { { \\mathcal { L } _ { C F } = \\displaystyle \\frac { 1 } { N ^ { 2 } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } A _ { i , j } ^ { C F } \\cdot \\log \\widehat { A } _ { i , j } ^ { C F } + ( 1 - A _ { i , j } ^ { C F } ) \\cdot \\log ( 1 - \\widehat { A } _ { i , j } ^ { C F } ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "64cc1514d642d976ad0bef9981a4003dc68208c107c8153e8c30462343e4c05e.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 169, + 355, + 441, + 379.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 169, + 379.6666666666667, + 441, + 404.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 169, + 404.33333333333337, + 441, + 429.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 86, + 437, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 86, + 436, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 86, + 439, + 100, + 449 + ], + "score": 1.0, + "content": "187", + "type": "text" + }, + { + "bbox": [ + 104, + 436, + 506, + 450 + ], + "score": 1.0, + "content": "Balancing Counterfactual Learning In the training process, the above loss minimizations train the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 85, + 448, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 85, + 451, + 100, + 462 + ], + "score": 1.0, + "content": "188", + "type": "text" + }, + { + "bbox": [ + 105, + 448, + 298, + 463 + ], + "score": 1.0, + "content": "model on both the empirical factual distribution", + "type": "text" + }, + { + "bbox": [ + 298, + 448, + 340, + 460 + ], + "score": 0.92, + "content": "\\hat { P } ^ { F } \\sim P ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 448, + 506, + 463 + ], + "score": 1.0, + "content": "and empirical counterfactual distribution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 85, + 459, + 507, + 478 + ], + "spans": [ + { + "bbox": [ + 85, + 464, + 100, + 474 + ], + "score": 1.0, + "content": "189", + "type": "text" + }, + { + "bbox": [ + 107, + 461, + 162, + 473 + ], + "score": 0.9, + "content": "\\hat { P } ^ { C F } \\sim P ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 459, + 507, + 478 + ], + "score": 1.0, + "content": "that are not necessarily equal – the training examples (node pairs) do not have to be", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 86, + 472, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 86, + 476, + 100, + 485 + ], + "score": 1.0, + "content": "190", + "type": "text" + }, + { + "bbox": [ + 105, + 472, + 506, + 486 + ], + "score": 1.0, + "content": "aligned. However, at the stage of inference, the test data contains only observed (factual) samples.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 85, + 484, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 85, + 486, + 100, + 496 + ], + "score": 1.0, + "content": "191", + "type": "text" + }, + { + "bbox": [ + 106, + 484, + 506, + 496 + ], + "score": 1.0, + "content": "Such a gap between the training and test data distributions exposes the model in the risk of covariant", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 86, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 86, + 497, + 100, + 507 + ], + "score": 1.0, + "content": "192", + "type": "text" + }, + { + "bbox": [ + 105, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "shift, which is a common issue in counterfactual learning (Johansson et al., 2016; Assaad et al., 2021).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 86, + 511, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 86, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 86, + 513, + 100, + 523 + ], + "score": 1.0, + "content": "193", + "type": "text" + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "To force the distributions of representations of factual distributions and counterfactual distributions to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 86, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 86, + 524, + 100, + 534 + ], + "score": 1.0, + "content": "194", + "type": "text" + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "be similar, we use the discrepancy distance (Mansour et al., 2009; Johansson et al., 2016) as another", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 86, + 536, + 100, + 545 + ], + "score": 1.0, + "content": "195", + "type": "text" + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "objective to regularize the representation learning. That is, we use the following loss term to minimize", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 86, + 544, + 383, + 558 + ], + "spans": [ + { + "bbox": [ + 86, + 548, + 100, + 557 + ], + "score": 1.0, + "content": "196", + "type": "text" + }, + { + "bbox": [ + 104, + 544, + 323, + 558 + ], + "score": 1.0, + "content": "the distance between the learned representations from", + "type": "text" + }, + { + "bbox": [ + 324, + 545, + 338, + 556 + ], + "score": 0.9, + "content": "{ \\hat { P } } ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 544, + 357, + 558 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 544, + 378, + 556 + ], + "score": 0.93, + "content": "\\hat { P } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 544, + 383, + 558 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 562, + 425, + 579 + ], + "lines": [ + { + "bbox": [ + 186, + 562, + 425, + 579 + ], + "spans": [ + { + "bbox": [ + 186, + 562, + 425, + 579 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { d i s c } = \\mathrm { d i s c } ( \\hat { P } _ { f } ^ { F } , \\hat { P } _ { f } ^ { C F } ) , \\mathrm { w h e r e \\ d i s c } ( P , Q ) = | | P - Q | | _ { F } ,", + "type": "interline_equation", + "image_path": "94888a9eb7e87db31fb986221ebf747a63088d7e625d53a57540f4eef79846f9.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 186, + 562, + 425, + 579 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 95, + 584, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 92, + 583, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 92, + 583, + 133, + 600 + ], + "score": 1.0, + "content": "where 97", + "type": "text" + }, + { + "bbox": [ + 134, + 586, + 159, + 598 + ], + "score": 0.92, + "content": "| | \\cdot | | _ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 583, + 298, + 600 + ], + "score": 1.0, + "content": "denotes the Frobenius Norm, and", + "type": "text" + }, + { + "bbox": [ + 299, + 584, + 314, + 600 + ], + "score": 0.92, + "content": "\\hat { P } _ { f } ^ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 583, + 333, + 600 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 333, + 584, + 354, + 600 + ], + "score": 0.93, + "content": "\\hat { P } _ { f } ^ { C F }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 583, + 506, + 600 + ], + "score": 1.0, + "content": "denote the node pair representations", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 91, + 597, + 496, + 611 + ], + "spans": [ + { + "bbox": [ + 91, + 597, + 210, + 611 + ], + "score": 1.0, + "content": "98 learned by graph encoder", + "type": "text" + }, + { + "bbox": [ + 210, + 599, + 217, + 610 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 597, + 496, + 611 + ], + "score": 1.0, + "content": "from factual distribution and counterfactual distribution, respectively.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 86, + 620, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 86, + 621, + 507, + 634 + ], + "spans": [ + { + "bbox": [ + 86, + 623, + 100, + 633 + ], + "score": 1.0, + "content": "199", + "type": "text" + }, + { + "bbox": [ + 104, + 621, + 507, + 634 + ], + "score": 1.0, + "content": "Training During the training of CFLP, we want the model to be optimized towards three targets:", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 85, + 632, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 85, + 634, + 100, + 644 + ], + "score": 1.0, + "content": "200", + "type": "text" + }, + { + "bbox": [ + 105, + 632, + 506, + 644 + ], + "score": 1.0, + "content": "(1) accurate link prediction on the observed outcomes (Eq. 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DatasetCORACITESEERPUBMEDFACEBOOKOGB-DDI
# nodes2,7083,32719,7174,0394,267
# links5,2784,55244,32488,2341,334,889
# validation node pairs1,0549108,86417,646235,371
# test node pairs2,1101,82017,72835,292229,088
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DatasetCORACITESEERPUBMEDFACEBOOKOGB-DDI
# nodes2,7083,32719,7174,0394,267
# links5,2784,55244,32488,2341,334,889
# validation node pairs1,0549108,86417,646235,371
# test node pairs2,1101,82017,72835,292229,088
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Therefore, the second step", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 547, + 373, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 373, + 560 + ], + "score": 1.0, + "content": "has linear time complexity w.r.t. the sum of node and edge counts.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 55.5, + "bbox_fs": [ + 105, + 449, + 506, + 560 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 572, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 89, + 571, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 89, + 574, + 100, + 583 + ], + "score": 1.0, + "content": "239", + "type": "text" + }, + { + "bbox": [ + 105, + 571, + 506, + 583 + ], + "score": 1.0, + "content": "Limitations First, as mentioned above, the computation of finding counterfactual links has a worst-", + "type": "text" + } + ], + "index": 61, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 88, + 585, + 100, + 594 + ], + "score": 1.0, + "content": "240", + "type": "text" + }, + { + "bbox": [ + 105, + 582, + 186, + 595 + ], + "score": 1.0, + "content": "case complexity of", + "type": "text" + }, + { + "bbox": [ + 186, + 582, + 216, + 595 + ], + "score": 0.93, + "content": "O ( N ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 582, + 506, + 595 + ], + "score": 1.0, + "content": ". Second, CFLP performs counterfactual prediction with only a single", + "type": "text" + } + ], + "index": 62, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 89, + 597, + 99, + 604 + ], + "score": 1.0, + "content": "241", + "type": "text" + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "treatment; however, there are quite a few kinds of graph structural information that can be considered", + "type": "text" + } + ], + "index": 63, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 88, + 607, + 100, + 616 + ], + "score": 1.0, + "content": "242", + "type": "text" + }, + { + "bbox": [ + 106, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "as treatments. Future work can leverage the rich structural information by bundled treatments (Zou", + "type": "text" + } + ], + "index": 64, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 614, + 288, + 630 + ], + "spans": [ + { + "bbox": [ + 88, + 618, + 100, + 627 + ], + "score": 1.0, + "content": "243", + "type": "text" + }, + { + "bbox": [ + 105, + 614, + 288, + 630 + ], + "score": 1.0, + "content": "et al., 2020) in counterfactual graph learning.", + "type": "text" + } + ], + "index": 65, + "is_list_start_line": true + } + ], + "index": 63, + "bbox_fs": [ + 88, + 571, + 506, + 630 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 643, + 191, + 657 + ], + "lines": [ + { + "bbox": [ + 103, + 641, + 193, + 660 + ], + "spans": [ + { + "bbox": [ + 103, + 641, + 193, + 660 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 66 + } + ], + "index": 66 + }, + { + "type": "title", + "bbox": [ + 106, + 668, + 215, + 680 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 217, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 217, + 684 + ], + "score": 1.0, + "content": "4.1 Experimental Setup", + "type": "text" + } + ], + "index": 67 + } + ], + "index": 67 + }, + { + "type": "index", + "bbox": [ + 88, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 86, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 86, + 690, + 100, + 700 + ], + "score": 1.0, + "content": "246", + "type": "text" + }, + { + "bbox": [ + 104, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "We conduct experiments on five benchmark datasets including citation networks (CORA, CITESEER,", + "type": "text" + } + ], + "index": 68, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 86, + 702, + 100, + 710 + ], + "score": 1.0, + "content": "247", + "type": "text" + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "PUBMED (Yang et al., 2016)), social network (FACEBOOK (McAuley and Leskovec, 2012)), and", + "type": "text" + } + ], + "index": 69, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 87, + 713, + 100, + 721 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 105, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "drug-drug interaction network (OGB-DDI (Wishart et al., 2018)) from the Open Graph Benchmark", + "type": "text" + } + ], + "index": 70, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 257, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 86, + 259, + 100, + 268 + ], + "score": 1.0, + "content": "249", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 257, + 397, + 268 + ], + "score": 1.0, + "content": "(OGB) (Hu et al., 2020). For the first four datasets, we randomly select", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 397, + 257, + 439, + 267 + ], + "score": 0.91, + "content": "10 \\% / 2 0 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 439, + 257, + 505, + 268 + ], + "score": 1.0, + "content": "of the links and", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 268, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 85, + 269, + 100, + 279 + ], + "score": 1.0, + "content": "250", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 268, + 505, + 279 + ], + "score": 1.0, + "content": "the same numbers of disconnected node pairs as validation/test samples. The links in the validation", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 279, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 85, + 280, + 100, + 290 + ], + "score": 1.0, + "content": "251", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 279, + 505, + 290 + ], + "score": 1.0, + "content": "and test sets are masked off from the training graph. For OGB-DDI, we used the OGB official", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 289, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 86, + 291, + 99, + 300 + ], + "score": 1.0, + "content": "252", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 289, + 506, + 302 + ], + "score": 1.0, + "content": "train/validation/test splits. Statistics for the datasets are given in Table 1 and details are in Appendix.", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 85, + 302, + 100, + 312 + ], + "score": 1.0, + "content": "253", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "We use K-core (Bader and Hogue, 2003) clusters as the default treatment variable. We evaluate CFLP", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 311, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 85, + 313, + 100, + 323 + ], + "score": 1.0, + "content": "254", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 311, + 506, + 324 + ], + "score": 1.0, + "content": "on three commonly used GNN encoders: GCN (Kipf and Welling, 2016a), GSAGE (Hamilton et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 85, + 324, + 100, + 334 + ], + "score": 1.0, + "content": "255", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "2017), and JKNet (Xu et al., 2018). We compare the link prediction performance of CFLP against", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 331, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 85, + 335, + 100, + 345 + ], + "score": 1.0, + "content": "256", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 331, + 506, + 345 + ], + "score": 1.0, + "content": "Node2Vec (Grover and Leskovec, 2016), MVGRL (Hassani and Khasahmadi, 2020), VGAE (Kipf", + "type": "text", + "cross_page": true + } + ], + "index": 12, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 85, + 345, + 100, + 356 + ], + "score": 1.0, + "content": "257", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "score": 1.0, + "content": "and Welling, 2016b), SEAL (Zhang and Chen, 2018), LGLP (Cai et al., 2021), and GNNs with MLP", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 86, + 357, + 100, + 366 + ], + "score": 1.0, + "content": "258", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "decoder. We report averaged test performance and their standard deviation over 20 runs with different", + "type": "text", + "cross_page": true + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 366, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 86, + 368, + 99, + 377 + ], + "score": 1.0, + "content": "259", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "score": 1.0, + "content": "random parameter initializations. Other than the most commonly used of Area Under ROC Curve", + "type": "text", + "cross_page": true + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 375, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 86, + 379, + 100, + 388 + ], + "score": 1.0, + "content": "260", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 375, + 196, + 391 + ], + "score": 1.0, + "content": "(AUC), we report Hits", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 196, + 377, + 216, + 388 + ], + "score": 0.62, + "content": "@ 2 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 216, + 375, + 506, + 391 + ], + "score": 1.0, + "content": "(one of the primary metrics on OGB leaderboard) as a more challenging", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 388, + 459, + 400 + ], + "spans": [ + { + "bbox": [ + 85, + 390, + 99, + 399 + ], + "score": 1.0, + "content": "261", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 388, + 459, + 400 + ], + "score": 1.0, + "content": "metric, as it expects models to rank positive edges higher than nearly all negative edges.", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 402, + 505, + 418 + ], + "spans": [ + { + "bbox": [ + 86, + 406, + 99, + 416 + ], + "score": 1.0, + "content": "262", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 402, + 505, + 418 + ], + "score": 1.0, + "content": "Besides performance comparison on link prediction, we will answer two questions to suggest a way", + "type": "text", + "cross_page": true + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 86, + 417, + 99, + 426 + ], + "score": 1.0, + "content": "263", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "of choosing a treatment variable for creating counterfactual links: (Q1) Does CFLP sufficiently learn", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 426, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 86, + 428, + 99, + 437 + ], + "score": 1.0, + "content": "264", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 426, + 506, + 438 + ], + "score": 1.0, + "content": "the observed averaged treatment effect (ATE) derived from the counterfactual links? (Q2) What is the", + "type": "text", + "cross_page": true + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 86, + 439, + 99, + 448 + ], + "score": 1.0, + "content": "265", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "relationship between the estimated ATE learned in the method and the prediction performance? If the", + "type": "text", + "cross_page": true + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 86, + 450, + 99, + 460 + ], + "score": 1.0, + "content": "266", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "answer to Q1 is yes, then the answer to Q2 will indicate how to choose treatment based on observed", + "type": "text", + "cross_page": true + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 459, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 86, + 463, + 99, + 472 + ], + "score": 1.0, + "content": "267", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 461, + 333, + 473 + ], + "score": 1.0, + "content": "ATE. To answer the Q1, we calculate the observed ATE", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 333, + 459, + 369, + 472 + ], + "score": 0.88, + "content": "( \\widehat { \\mathrm { A I E } } _ { o b s } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 369, + 461, + 505, + 473 + ], + "score": 1.0, + "content": "by comparing the observed links", + "type": "text", + "cross_page": true + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 469, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 86, + 474, + 99, + 483 + ], + "score": 1.0, + "content": "268", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 469, + 263, + 485 + ], + "score": 1.0, + "content": "in A and created counterfactual links", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 263, + 471, + 286, + 482 + ], + "score": 0.89, + "content": "\\mathbf { A } ^ { C F }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 286, + 469, + 506, + 485 + ], + "score": 1.0, + "content": "dthat have opposite treatments. And we calculate the", + "type": "text", + "cross_page": true + } + ], + "index": 24, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 483, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 86, + 487, + 99, + 497 + ], + "score": 1.0, + "content": "269", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 484, + 169, + 498 + ], + "score": 1.0, + "content": "estimated ATE", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 170, + 483, + 206, + 496 + ], + "score": 0.88, + "content": "\\widehat { ( \\mathrm { A I E } _ { e s t } ) }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 206, + 484, + 353, + 498 + ], + "score": 1.0, + "content": "by comparing the predicted links in", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 354, + 483, + 364, + 495 + ], + "score": 0.81, + "content": "\\widehat { \\mathbf A }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 364, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "and predicted counterfactual links", + "type": "text", + "cross_page": true + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 495, + 314, + 512 + ], + "spans": [ + { + "bbox": [ + 86, + 500, + 99, + 509 + ], + "score": 1.0, + "content": "270", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 496, + 129, + 508 + ], + "score": 0.87, + "content": "\\widehat { \\mathbf { A } } ^ { C F }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 129, + 495, + 174, + 512 + ], + "score": 1.0, + "content": ". 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CORACITESEERPUBMEDFACEBOOKOGB-DDI
Node2Vec49.96±2.5147.78±1.7239.19±1.0224.24±3.0223.26±2.09
MVGRL19.53±2.6414.07±0.7914.19±0.8514.43±0.3310.02±1.01
VGAE45.91±3.3844.04±4.8623.73±1.6137.01±0.6311.71±1.96
SEAL51.35±2.2640.90±3.6828.45±3.8140.89±5.7030.56±3.86
LGLP62.98±0.5657.43±3.71137.86±2.13
GCN49.06±1.7255.56±1.3221.84±3.8753.89±2.1437.07±5.07
GSAGE53.54±2.9653.67±2.9439.13±4.4145.51±3.2253.90±4.74
JKNet48.21±3.8655.60±2.1725.64±4.1152.25±1.4860.56±8.69
Our proposed CFLP with different graph encoders
CFLP w/GCN60.34±2.3359.45±2.3034.12±2.7253.95±2.2952.51±1.09
CFLP w/GSAGE57.33±1.7353.05±2.0743.07±2.3647.28±3.0075.49±4.33
CFLP w/JKNet65.57±1.0568.09±1.4944.90±2.0055.22±1.2986.08±1.98
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Statistics for the datasets are given in Table 1 and details are in Appendix.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 85, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 85, + 302, + 100, + 312 + ], + "score": 1.0, + "content": "253", + "type": "text" + }, + { + "bbox": [ + 106, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "We use K-core (Bader and Hogue, 2003) clusters as the default treatment variable. 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CORACITESEERPUBMEDFACEBOOKOGB-DDI
Node2Vec49.96±2.5147.78±1.7239.19±1.0224.24±3.0223.26±2.09
MVGRL19.53±2.6414.07±0.7914.19±0.8514.43±0.3310.02±1.01
VGAE45.91±3.3844.04±4.8623.73±1.6137.01±0.6311.71±1.96
SEAL51.35±2.2640.90±3.6828.45±3.8140.89±5.7030.56±3.86
LGLP62.98±0.5657.43±3.71137.86±2.13
GCN49.06±1.7255.56±1.3221.84±3.8753.89±2.1437.07±5.07
GSAGE53.54±2.9653.67±2.9439.13±4.4145.51±3.2253.90±4.74
JKNet48.21±3.8655.60±2.1725.64±4.1152.25±1.4860.56±8.69
Our proposed CFLP with different graph encoders
CFLP w/GCN60.34±2.3359.45±2.3034.12±2.7253.95±2.2952.51±1.09
CFLP w/GSAGE57.33±1.7353.05±2.0743.07±2.3647.28±3.0075.49±4.33
CFLP w/JKNet65.57±1.0568.09±1.4944.90±2.0055.22±1.2986.08±1.98
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CORACITESEERPUBMEDFACEBOOKOGB-DDI
Node2Vec84.49±0.4980.00±0.6880.32±0.2986.49±4.3290.83±0.02
MVGRL75.07±3.6361.20±0.5580.78±1.2879.83±0.3081.45±0.99
VGAE88.68±0.4085.35±0.6095.80±0.1398.66±0.0493.08±0.15
SEAL92.55±0.5085.82±0.4496.36±0.2899.60±0.0297.85±0.17
LGLP91.30±0.0589.41±0.1398.51±0.01
GCN90.25±0.5371.47±1.4096.33±0.8099.43±0.0299.82±0.05
GSAGE90.24±0.3487.38±1.3996.78±0.1199.29±0.0499.93±0.02
JKNet89.05±0.6788.58±1.7896.58±0.2399.43±0.0299.94±0.01
Our proposed CFLP with different graph encoders
CFLP w/GCN92.55±0.5089.65±0.2096.99±0.0899.38±0.0199.44±0.05
CFLPw/GSAGE92.61±0.5291.84±0.2097.01±0.0199.34±0.1099.83±0.05
CFLP w/JKNet93.05±0.2492.12±0.4797.53±0.1799.31±0.0499.94±0.01
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CORACITESEERPUBMEDFACEBOOKOGB-DDI
Node2Vec84.49±0.4980.00±0.6880.32±0.2986.49±4.3290.83±0.02
MVGRL75.07±3.6361.20±0.5580.78±1.2879.83±0.3081.45±0.99
VGAE88.68±0.4085.35±0.6095.80±0.1398.66±0.0493.08±0.15
SEAL92.55±0.5085.82±0.4496.36±0.2899.60±0.0297.85±0.17
LGLP91.30±0.0589.41±0.1398.51±0.01
GCN90.25±0.5371.47±1.4096.33±0.8099.43±0.0299.82±0.05
GSAGE90.24±0.3487.38±1.3996.78±0.1199.29±0.0499.93±0.02
JKNet89.05±0.6788.58±1.7896.58±0.2399.43±0.0299.94±0.01
Our proposed CFLP with different graph encoders
CFLP w/GCN92.55±0.5089.65±0.2096.99±0.0899.38±0.0199.44±0.05
CFLPw/GSAGE92.61±0.5291.84±0.2097.01±0.0199.34±0.1099.83±0.05
CFLP w/JKNet93.05±0.2492.12±0.4797.53±0.1799.31±0.0499.94±0.01
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However, very limited work has studied to use", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 382, + 294, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 294, + 393 + ], + "score": 1.0, + "content": "causal inference for improving link prediction.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 105, + 406, + 506, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 506, + 419 + ], + "score": 1.0, + "content": "Counterfactual Prediction As a mean of learning the causality between treatment and outcome,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "counterfactual prediction has been used for a variety of applicaitons such as recommender sys-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "tems (Wang et al., 2020; Xu et al., 2020), health care (Alaa and van der Schaar, 2017), vision-language", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "tasks (Zhang et al., 2020b; Parvaneh et al., 2020), and decision making (Coston et al., 2020; Pitis et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "2020; Kusner et al., 2017). To infer the causal relationships, previous work usually estimated the ITE", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "via function fitting models (Gelman and Hill, 2006; Chipman et al., 2010; Wager and Athey, 2018;", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "Assaad et al., 2021) which estimated the transductive ITE. Peysakhovich et al. (2019) and Zou et al.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "(2020) studied counterfactual prediction with multiple agents and bundled treatments, respectively.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 491, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 507, + 505 + ], + "score": 1.0, + "content": "Pawlowski et al. (2020) proposed a deep structural causal model for tractable counterfactual inference.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 506, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "Causal Inference Causal inference methods usually re-weighted samples based on propensity", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 526, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 506, + 541 + ], + "score": 1.0, + "content": "score (Rosenbaum and Rubin, 1983; Austin, 2011; Kuang et al., 2017a,b) to remove confounding", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "bias from binary treatments. 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(2017) attempted", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 582, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 593 + ], + "score": 1.0, + "content": "to infer the outcomes from proxies, and Hartford et al. (2017) introduced instrumental variable. SITE", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "preserved local similarity to balance the distributions of control and treated groups (Yao et al., 2018).", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 107, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "Yoon et al. (2018) estimated ITE with generative adversarial networks (GANs). Assaad et al. (2021)", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 614, + 397, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 397, + 628 + ], + "score": 1.0, + "content": "discussed the trade-off between achieving balance and predictive power.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 48.5 + }, + { + "type": "title", + "bbox": [ + 102, + 642, + 182, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 185, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 185, + 658 + ], + "score": 1.0, + "content": "6 Conclusion", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 54 + }, + { + "type": "text", + "bbox": [ + 104, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 507, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 507, + 680 + ], + "score": 1.0, + "content": "In this work, we presented a counterfactual graph learning method for link prediction (CFLP).", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "We introduced the idea of counterfactual prediction to improve link prediction on graphs. CFLP", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "accurately predicted the missing links by exploring the causal relationship between global graph", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "structure and link existence. 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However, very limited work has studied to use", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 382, + 294, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 294, + 393 + ], + "score": 1.0, + "content": "causal inference for improving link prediction.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 272, + 506, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 406, + 506, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 506, + 419 + ], + "score": 1.0, + "content": "Counterfactual Prediction As a mean of learning the causality between treatment and outcome,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "counterfactual prediction has been used for a variety of applicaitons such as recommender sys-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 441 + ], + "score": 1.0, + "content": "tems (Wang et al., 2020; Xu et al., 2020), health care (Alaa and van der Schaar, 2017), vision-language", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 505, + 451 + ], + "score": 1.0, + "content": "tasks (Zhang et al., 2020b; Parvaneh et al., 2020), and decision making (Coston et al., 2020; Pitis et al.,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "2020; Kusner et al., 2017). To infer the causal relationships, previous work usually estimated the ITE", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 472 + ], + "score": 1.0, + "content": "via function fitting models (Gelman and Hill, 2006; Chipman et al., 2010; Wager and Athey, 2018;", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 506, + 483 + ], + "score": 1.0, + "content": "Assaad et al., 2021) which estimated the transductive ITE. Peysakhovich et al. (2019) and Zou et al.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "(2020) studied counterfactual prediction with multiple agents and bundled treatments, respectively.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 491, + 507, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 507, + 505 + ], + "score": 1.0, + "content": "Pawlowski et al. (2020) proposed a deep structural causal model for tractable counterfactual inference.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 405, + 507, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 517, + 506, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "Causal Inference Causal inference methods usually re-weighted samples based on propensity", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 526, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 506, + 541 + ], + "score": 1.0, + "content": "score (Rosenbaum and Rubin, 1983; Austin, 2011; Kuang et al., 2017a,b) to remove confounding", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "bias from binary treatments. Recently, several works studied about learning treatment invariant", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "representation to predict the counterfactual outcomes (Hassanpour and Greiner, 2019b,a; Shalit et al.,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "2017; Yao et al., 2018; Bica et al., 2020; Hassanpour and Greiner, 2019a; Li and Fu, 2017). When part", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 583 + ], + "score": 1.0, + "content": "of unobserved outcomes may mislead the counterfactual prediction, Louizos et al. (2017) attempted", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 582, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 593 + ], + "score": 1.0, + "content": "to infer the outcomes from proxies, and Hartford et al. (2017) introduced instrumental variable. SITE", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "preserved local similarity to balance the distributions of control and treated groups (Yao et al., 2018).", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 107, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 107, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "Yoon et al. (2018) estimated ITE with generative adversarial networks (GANs). Assaad et al. (2021)", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 614, + 397, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 397, + 628 + ], + "score": 1.0, + "content": "discussed the trade-off between achieving balance and predictive power.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 48.5, + "bbox_fs": [ + 104, + 515, + 506, + 628 + ] + }, + { + "type": "title", + "bbox": [ + 102, + 642, + 182, + 655 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 185, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 185, + 658 + ], + "score": 1.0, + "content": "6 Conclusion", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 54 + }, + { + "type": "text", + "bbox": [ + 104, + 667, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 507, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 507, + 680 + ], + "score": 1.0, + "content": "In this work, we presented a counterfactual graph learning method for link prediction (CFLP).", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "We introduced the idea of counterfactual prediction to improve link prediction on graphs. CFLP", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 701 + ], + "score": 1.0, + "content": "accurately predicted the missing links by exploring the causal relationship between global graph", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "structure and link existence. 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DatasetCORACITESEERPUBMEDFACEBOOKOGB-DDI
# nodes2,7083,32719,7174,0394,267
# links5,2784,55244,32488,2341,334,889
# validation node pairs1,0549108,86417,646235,371
# test node pairs2,1101,82017,72835,292229,088
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CORACITESEERPUBMEDFACEBOOKOGB-DDI
Node2Vec49.96±2.5147.78±1.7239.19±1.0224.24±3.0223.26±2.09
MVGRL19.53±2.6414.07±0.7914.19±0.8514.43±0.3310.02±1.01
VGAE45.91±3.3844.04±4.8623.73±1.6137.01±0.6311.71±1.96
SEAL51.35±2.2640.90±3.6828.45±3.8140.89±5.7030.56±3.86
LGLP62.98±0.5657.43±3.71137.86±2.13
GCN49.06±1.7255.56±1.3221.84±3.8753.89±2.1437.07±5.07
GSAGE53.54±2.9653.67±2.9439.13±4.4145.51±3.2253.90±4.74
JKNet48.21±3.8655.60±2.1725.64±4.1152.25±1.4860.56±8.69
Our proposed CFLP with different graph encoders
CFLP w/GCN60.34±2.3359.45±2.3034.12±2.7253.95±2.2952.51±1.09
CFLP w/GSAGE57.33±1.7353.05±2.0743.07±2.3647.28±3.0075.49±4.33
CFLP w/JKNet65.57±1.0568.09±1.4944.90±2.0055.22±1.2986.08±1.98
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CORACITESEERPUBMEDFACEBOOKOGB-DDI
Node2Vec84.49±0.4980.00±0.6880.32±0.2986.49±4.3290.83±0.02
MVGRL75.07±3.6361.20±0.5580.78±1.2879.83±0.3081.45±0.99
VGAE88.68±0.4085.35±0.6095.80±0.1398.66±0.0493.08±0.15
SEAL92.55±0.5085.82±0.4496.36±0.2899.60±0.0297.85±0.17
LGLP91.30±0.0589.41±0.1398.51±0.01
GCN90.25±0.5371.47±1.4096.33±0.8099.43±0.0299.82±0.05
GSAGE90.24±0.3487.38±1.3996.78±0.1199.29±0.0499.93±0.02
JKNet89.05±0.6788.58±1.7896.58±0.2399.43±0.0299.94±0.01
Our proposed CFLP with different graph encoders
CFLP w/GCN92.55±0.5089.65±0.2096.99±0.0899.38±0.0199.44±0.05
CFLPw/GSAGE92.61±0.5291.84±0.2097.01±0.0199.34±0.1099.83±0.05
CFLP w/JKNet93.05±0.2492.12±0.4797.53±0.1799.31±0.0499.94±0.01
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Hits@20ATEobsATEest
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SBM64.2±1.10.0060.023±0.015
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Ward61.2±2.30.0010.033±0.012
SpecC59.3±2.80.0020.033±0.011
Louvain57.6±1.80.0250.138±0.091
Katz56.6±3.40.7400.802±0.041
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Hits@20ATEobsATEest
SBM71.6 ±1.90.0040.005 ±0.001
K-core68.1±1.50.0020.010±0.002
Ward67.0±1.70.0030.037±0.009
PropC64.6±3.60.1410.232±0.113
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SpecC59.9±1.30.0090.166±0.034
Katz57.3±0.50.2450.224±0.037
CommN56.8±4.90.6780.195±0.034
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topology of a point cloud whilst having strong theoretical guarantees. Most current approaches to integrating topological information into machine learning implicitly map persistence diagrams to a Hilbert space, resulting in deformation of the underlying metric structure whilst also generally requiring prior knowledge about the true topology of the space. In this paper we give an algorithm for Fuzzy c-Means (FCM) clustering directly on the space of persistence diagrams,22 enabling unsupervised learning that automatically captures the topological structure of data, with no prior knowledge or additional processing of persistence diagrams. We prove the same convergence guarantees as traditional FCM clustering: every convergent subsequence of iterates tends to a local minimum or saddle point. We end by presenting experiments where our fuzzy topological clustering algorithm allows for unsupervised top- $k$ candidate selection in settings where (i) the properties of persistence diagrams make them the natural choice over geometric equivalents, and (ii) the probabilistic membership values let us rank candidates in settings where verifying candidate suitability is expensive: lattice structure classification in materials science and pre-trained model selection in machine learning. + +# 1 INTRODUCTION + +Persistence diagrams, a concise representation of the topology of a point cloud with strong theoretical guarantees, have emerged as a new tool in the field of data analysis (Edelsbrunner & Harer, 2010). Persistence diagrams have been successfully used to analyse problems ranging from financial crashes $\mathrm { \textregistered { 1 } } \mathrm { \overline { { G i d e a \ \& \ \mathrm { K a t z } } } \mathrm { \tilde { \textmu } \mathrm { \overline { { 2 0 1 8 } } \mathrm { \AA } } } }$ to protein binding (Kovacev-Nikolic et al., 2014), but the non-Hilbertian nature of the space of persistence diagrams means it is difficult to directly use persistence diagrams for machine learning. In order to better integrate diagrams into machine learning workflows, efforts have been made to map them into a more manageable form; primarily through embeddings into finite feature vectors, functional summaries, or by defining a positive-definite kernel on diagram space. In all cases, this explicitly or implicitly embeds diagrams into a Hilbert space which deforms the metric structure, potentially losing important information. With the exception of Topological Autoencoders, techniques to integrate these persistence-based summaries as topological regularisers and loss functions currently require prior knowledge about the correct topology of the dataset, which is clearly not feasible in most scenarios. + +Against this background, we give an algorithm to perform Fuzzy c-Means (FCM) clustering (Bezdek, 1980) directly on collections of persistence diagrams, giving an important unsupervised learning algorithm and enabling learning from persistence diagrams without deforming the metric structure. We perform the convergence analysis for our algorithm, giving the same guarantees as traditional FCM clustering: that every convergent subsequence of iterates tends to a local minimum or saddle point. We demonstrate the value of our fuzzy clustering algorithm by using it to cluster datasets that benefit from both the topological and fuzzy nature of our algorithm. We apply our technique in two settings: lattice structures in materials science and the decision boundaries of CNNs. A key property for machine learning in materials science has been identified as “invariance to the basis symmetries of physics [...] rotation, reflection, translation” (Schmidt et al., 2019). Geometric clustering algorithms do not have this invariance, but persistence diagrams do, making them ideally suited for this application; we can cluster transformed lattice structure datasets where geometric equivalents fail. In addition to this, our probabilistic membership values allow us to rank the top- $k$ most likely lattices assigned to a cluster. This is particularly important in materials science, as further investigation requires expensive laboratory time and expertise. Our second application is inspired by Ramamurthy et al. (2019), who show that models perform better on tasks if they have topologically similar decision boundaries. We use our algorithm to cluster models and tasks by the persistence diagrams of their decision boundaries. Not only is our algorithm able to successfully cluster models to the correct task, based just on the topology of its decision boundary, but we show that higher membership values imply better performance on unseen tasks. + +# 1.1 RELATED WORK + +Means of persistence diagrams. Our work relies on the existence of statistics in the space of persistence diagrams. Mileyko et al. $\mathbb { Q } 0 1 1 )$ first showed that means and expectations are well-defined in the space of persistence diagrams. Specifically, they showed that the Fréchet mean, an extension of means onto metric spaces, is well-defined under weak assumptions on the space of persistence diagrams. Turner et al. (2012) then developed an algorithm to compute the Fréchet mean. We adapt the algorithm by Turner et al. to the weighted case, but the combinatoric nature of their algorithm makes it computationally intense. Lacombe et al. $\textcircled { 2 0 1 8 }$ framed the computation of means and barycentres in the space of persistence diagram as an optimal transport problem, allowing them to use the Sinkhorn algorithm $( \overline { { \mathbb { C } \mathrm { u t u r i } \ \& \ \mathrm { D o u c e t } } } , \overline { { 2 0 1 4 } } )$ for fast computation of approximate solutions. The vectorisation of the diagram required by the algorithm by Lacombe et al. makes it unsuitable for integration into our work, as we remain in the space of persistence diagrams. Techniques to speed up the matching problem fundamental to our computation have also been proposed by Vidal et al. $\dot { \overline { { ( 2 0 2 0 ) } } }$ and Kerber et al. (2017). + +Learning with persistence-based summaries. Integrating diagrams into machine learning workflows remained challenging even with well-defined means, as the space is non-Hilbertian (Turner & Spreemann, 2019). As such, efforts have been made to map diagrams into a Hilbert space; primarily either by embedding into finite feature vectors (Kališnik, 2018; Fabio & Ferri, 2015; Chepushtanova et al., 2015) or functional summaries (Bubenik, 2015; Rieck et al., 2019), or by defining a positivedefinite kernel on diagram space (Reininghaus et al., 2015; Carrière et al., 2017; Le & Yamada, 2018). These vectorisations have been integrated into deep learning either by learning parameters for the embedding (Hofer et al., 2017; Carrière et al., 2020; Kim et al., 2020; Zhao & Wang, 2019; Zielinski ´ et al., 2019), or as part of a topological loss or regulariser (Chen et al., 2018; Gabrielsson et al., 2020; Clough et al., 2020; Moor et al., 2019). However, the embeddings used in these techniques deform the metric structure of persistence diagram space (Bubenik & Wagner, 2019; Wagner, 2019; Carrière & Bauer, 2019), potentially leading to the loss of important information. Furthermore, these techniques generally require prior knowledge of a ‘correct’ target topology which cannot plausibly be known in most scenarios. In comparison, our algorithm acts in the space of persistence diagrams so it does not deform the structure of diagram space via embeddings, and is entirely unsupervised, requiring no prior knowledge about the topology. + +Hard clustering. Maroulas et al. (2017) gave an algorithm for hard clustering persistence diagrams based on the algorithm by Turner et al. Lacombe et al. $\textcircled { 2 0 1 8 }$ gave an alternate implementation of hard clustering based on their algorithm for barycentre computation, providing a computational speed-up over previous the work by Maroulas et al. The primary advantages of our work over previous work on hard clustering are as follows. + +(i) The probabilistic membership values allow us to rank datasets in the cluster, enabling top- $k$ candidate selection in settings where verifying correctness is expensive. The value provided by this fuzzy information is demonstrated in the experiments. +(ii) The fuzzy membership values provide information about proximity to all clusters, whereas hard labelling loses most of that information. In our experiments we demonstrate that this additional information can be utilised in practice. +(iii) The weighted cost function makes the convergence analysis (which we provide) entirely nontrivial in comparison to the non-fuzzy case. We consider this convergence analysis a primary contribution of our paper. +(iv) Fuzzy membership values have been shown to be more robust to noise than discrete labels (Klawonn, 2004). + +(v) Unlike hard clustering, fuzzy clustering is analytically differentiable, allowing integration of the fuzzy clustering step into deep learning methods (Wilder et al., 2019). + +Geometric equivalents. The most similar unsupervised learning technique to our algorithm is Wasserstein Barycentre Clustering (WBC). It clusters datasets of point clouds by the Wasserstein distance between the point clouds, rather than the Wasserstein distance between their persistence diagrams. We compare our algorithm experimentally to WBC using ADMM $\pmb { \mathbb { T e } } \ \& \ \bot \pmb { \mathbb { I } } \ \boxed { 2 0 1 4 } )$ , Bregman ADMM (Ye et al., 2017), Subgradient Descent (Cuturi & Doucet, 2014), Iterative Bregman Projection (Benamou et al., 2015), and full linear programming $\mathbb { \overline { { \mathbb { L } i \ \& \ \operatorname { W a n g } } } } , \mathbb { \underline { { 2 0 0 8 } } } )$ . Each of these algorithms computes or approximates the Wasserstein barycentre in different ways. Theoretically, fuzzy discrete distribution clustering (d. A. T. de Carvalho et al., $\textcircled { 2 0 1 5 } )$ is similar to our algorithm, but the addition of the diagonal in the persistence diagram makes our work distinct. + +# 1.2 OUR CONTRIBUTIONS + +1. Our main contribution is an algorithm for Fuzzy c-Means clustering of persistence diagrams, along with the convergence analysis. Given a collection of persistence diagrams $\mathbb { D } _ { 1 } , \ldots , \mathbb { D } _ { n }$ , we alternatively calculate cluster centres $\mathbb { M } _ { 1 } , \ldots , \mathbb { M } _ { c }$ and membership values $r _ { j k } \in [ 0 , 1 ]$ which denote the degree to which diagram $\mathbb { D } _ { j }$ is associated with cluster $\mathbb { M } _ { k }$ . We prove Theorem $^ { 1 , }$ showing that every convergent subsequence of these alternative update steps tends to a local minimum or saddle point of the cost function. This is the same convergence guarantee provided by traditional FCM clustering $( \mathbf { \mathbb { B } e z d e k \ e t \ a l . } ] \ [ \mathbf { \mathbb { 9 } 8 7 } )$ , but requires additional work as the space of persistence diagrams with the Wasserstein distance has far weaker theoretical properties than Euclidean space. + +2. Updating the cluster centres requires computing the weighted Fréchet mean. We extend the algorithm given by $\boxed { \mathrm { T u r n e r ~ e t ~ a l . } } \textcircled { 2 0 1 2 }$ to the weighted case, justifying our addition of weights by extending their proof to show that the updated algorithm converges. + +3. We implement our algorithm in Python, available in the supplementary materials. It works with persistence diagrams from commonly used open-source libraries for Topological Data Analysis $\mathbf { \Gamma } _ { \mathrm { ( T D A ) , } } ^ { \mathrm { ~ \tiny ~ 1 ~ } } \mathfrak { s }$ o is available for easy integration into current workflows, offering a powerful unsupervised learning algorithm to data science practitioners using TDA. + +4. We demonstrate the application of our algorithm to settings where (i) the properties of persistence diagrams makes clustering them the natural choice over geometric equivalents and (ii) the probabilistic membership values can be used to rank candidates for top- $k$ selection. Our algorithm classifies transformed lattice structures from materials science where geometric equivalents fail, whilst giving probabilistic rankings to help prioritise expensive further investigation. We also cluster the persistence diagrams of decision boundaries and labelled datasets, showing that our fuzzy clustering captures information about model performance on unseen tasks. + +# 2 TOPOLOGICAL PRELIMINARIES + +Topological Data Analysis emerged from the study of algebraic topology, providing a toolkit to fully describe the topology of a dataset. We offer a quick summary below; for more comprehensive details see Edelsbrunner & Harer $\bigoplus \iiiint \iiiint \lVert 0 1 0 \rVert$ . A set of points in $\mathbb { R } ^ { d }$ are indicative of the shape of the distribution they are sampled from. By connecting points that are pairwise within $\epsilon > 0$ distance of each other, we can create an approximation of the distribution called the Vietoris-Rips complex $\mathrm { ( V i e t o r i s , \textmu 9 2 7 ) }$ . Specifically, we add the convex hull of any collection of points that are pairwise at most $\epsilon$ apart to the $\epsilon$ -Vietoris-Rips complex. However, choosing an $\epsilon$ remains problematic; too low a value and key points can remain disconnected, too high a value and the points become fully connected. To overcome this we use persistence: we consider the approximation over all values of $\epsilon$ simultaneously, and study how the topology of that approximation evolves as $\epsilon$ grows large. We call the collection of complexes for all $\epsilon$ a filtration. + +For each $\epsilon$ , we compute the $p$ -homology group. This tells us the topology of the $\epsilon$ -Vietoris-Rips complex: the 0-homology counts the number of connected components, the 1-homology counts the number of holes, the 2-homology counts the number of voids, etc. (Edelsbrunner et al., 2000) + +![](images/64ff89fa891ae43bb690e18da28297235b2205ad0db31d4eaf8f06107e1c4f55.jpg) +Figure 1: An example Vietoris-Rips filtration with its persistence diagram. We only add convex hulls when points are pairwise $\epsilon$ apart, so there is a hole at $\epsilon = 1$ whilst the diagonals of the square are not close enough to fill the hole in. This hole can be seen as a point at (1, 1.42) in the 1-PH persistence diagram. + +The $p$ -persistent homology ( $\dot { p }$ -PH) group is created by summing the $p$ -homology groups over all ✏. This results in a $p$ -PH group that summarises information about the topology of the dataset at all granularities. If a topological feature, such as a connected component or hole, persists throughout a large range of granularities, then it’s more likely to be a feature of the distribution. If it only persists for a short amount of time, then it’s more likely to be noise (Cohen-Steiner et al., 2007). We can stably map a $p$ -PH group into a multiset in the extended plane called a persistence diagram (Chazal et al., $\bar { 2 0 1 2 } )$ . Each topological feature has a birth and death: a feature is born when it enters the complex, and dies when the complex grows enough to destroy it. For example, in Figure 1(a), a feature is born at $\epsilon = 1$ when four lines form a hole. This feature dies at $\epsilon = 1 . 4 2$ when the hole is filled in. This is shown in the persistence diagram in Figure 1(b) as a point at $( 1 , 1 . 4 2 )$ in 1-PH. By computing the birth/death points for each topological feature in the filtration, we get a complete picture of the topology of the point cloud at all granularities (Zomorodian & Carlsson, $\overline { { 2 0 0 5 } } )$ . The persistence diagram is the collection of birth/death points, along with the diagonal $\overline { { \Delta } } \equiv \{ ( a , a ) : a \in \mathbb { R } \}$ with infinite multiplicity, added in order to make the space of persistence diagrams complete (Mileyko et al., 2011). + +# 3 ALGORITHMIC DESIGN + +# 3.1 CLUSTERING PERSISTENCE DIAGRAMS + +In order to cluster we need a distance on the space of persistence diagrams. We use the 2-Wasserstein $L _ { 2 }$ metric as it is stable on persistence diagrams of finite point clouds (Chazal et al., 2012). The Wasserstein distance is an optimal transport metric that has found applications across machine learning. In the Euclidean case, it quantifies the smallest distance between optimally matched points. Given diagrams $\mathbb { D } _ { 1 } , \mathbb { D } _ { 2 }$ , the distance is + +$$ +W _ { 2 } ( \mathbb { D } _ { 1 } , \mathbb { D } _ { 2 } ) = \left( \operatorname* { i n f } _ { \gamma : \mathbb { D } _ { 1 } \to \mathbb { D } _ { 2 } } \sum _ { x \in \mathbb { D } _ { 1 } } | | x - \gamma ( x ) | | _ { 2 } ^ { 2 } \right) ^ { 1 / 2 } , +$$ + +where the infimum is taken over all bijections $\gamma : { \mathbb { D } } _ { 1 } \to { \mathbb { D } } _ { 2 }$ . Note that as we added the diagonal with infinite multiplicity to each diagram, these bijections exist. If an off-diagonal point is matched to the diagonal the transportation cost is simply the shortest distance to the diagonal. In fact, the closer a point is to the diagonal, the more likely it is to be noise (Cohen-Steiner et al., $\boxed { 2 0 0 7 }$ , so this ensures our distance is not overly affected by noise. + +We work in the space $\mathfrak { D } _ { \mathrm { L } ^ { 2 } } = \{ \mathbb { D } : W _ { 2 } ( \mathbb { D } , \Delta ) < \infty \} , ^ { 2 }$ as this leads to a geodesic space with known structure $\left\{ { \mathrm { I u r n e r ~ e t ~ a l . } } \right\} \left[ { \mathrm { 2 0 1 2 } } \right)$ . Given a collection of persistence diagrams $\{ \mathbb { D } _ { j } \} _ { j = 1 } ^ { n } \subset \mathcal { D } _ { \mathrm { L } ^ { 2 } }$ and a fixed number of clusters $c$ , we wish to find cluster centres $\{ \mathbb { M } _ { k } \} _ { k = 1 } ^ { c } \subset \mathcal { D } _ { \mathrm { L } ^ { 2 } }$ , along with membership values $r _ { j k } \in [ 0 , 1 ]$ that denote the extent to which $\mathbb { D } _ { j }$ is associated with cluster $\mathbb { M } _ { k }$ . We follow probabilistic fuzzy clustering, so that $\textstyle \sum _ { k } r _ { j k } = 1$ for each $j$ . + +We extend the FCM algorithm originally proposed by $\mathbf { B e z d e k } | \left( \mathbf { \overline { { 1 9 8 0 } } } \right) $ . Our $r _ { j k }$ is the same as traditional FCM clustering, adapted with the Wasserstein distance. That is, + +$$ +r _ { j k } = \left( \sum _ { l = 1 } ^ { c } \frac { W _ { 2 } ( \mathbb { M } _ { k } , \mathbb { D } _ { j } ) } { W _ { 2 } ( \mathbb { M } _ { l } , \mathbb { D } _ { j } ) } \right) ^ { - 1 } . +$$ + +To update $\mathbb { M } _ { k }$ , we compute the weighted Fréchet mean $\hat { \mathbb { D } }$ of the persistence diagrams $\{ \mathbb { D } _ { j } \} _ { j = 1 } ^ { n }$ with the weights $\{ r _ { j k } ^ { 2 } \} _ { j = 1 } ^ { n }$ . Specifically, + +$$ +\mathbb { M } _ { k } \longleftarrow \operatorname { a r g m i n } _ { \hat { \mathbb { D } } } \sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( \hat { \mathbb { D } } , \mathbb { D } _ { j } ) ^ { 2 } , \operatorname { f o r } k = 1 , \dots , c . +$$ + +As the weighted Fréchet mean extends weighted centroids to general metric spaces, this gives our fuzzy cluster centres. The computation of the weighted Fréchet mean is covered in Section 3.2. By alternatively updating (1) and (2) we get a sequence of iterates. Theorem 1, proven in Appendix A, provides the same convergence guarantees as traditional FCM clustering. + +Theorem 1. Every convergent subsequence of the sequence of iterates obtained by alternatively updating membership values and cluster centres with $( l )$ and (2) tends to a local minimum or saddle point of the cost function $\begin{array} { r } { J ( R , { \mathbb M } ) = \sum _ { j = 1 } ^ { n } \sum _ { k = 1 } ^ { c } r _ { j k } ^ { 2 } W _ { 2 } ( { \mathbb M } _ { k } , { \mathbb D } _ { j } ) ^ { 2 } } \end{array}$ . + +Observe that we only guarantee the convergence of subsequences of iterates. This is the same as traditional FCM clustering, so we follow the same approach to a stopping condition and run our algorithm for a fixed number of iterations. The entire algorithm is displayed in Algorithm 1. + +# Algorithm 1 FPDCluster + +
Input Diagrams D = {Dj}𝑗=1, number of clusters c, maximum iterations MAXITER Output Cluster centres M = {Mk}ε=1, membership values R = {rjk}
1:D = ADDDIAGONALS(D)7: end for
2:M=INITCENTRES(D)8: end for
3: for countin1..MAXITER do9: for k in1..c do
4: for j in1..n do10: M ← WFRECHETMEAN(D,Rk)
5: for k in1..c do11: end for
W2(Mk,D)-1 12: end for
(∑i-1 W) rjk← 6:13: return M, R
+ +# 3.2 COMPUTING THE WEIGHTED FRÉCHET MEAN + +Turner et al. $\textcircled { | 2 0 1 2 } )$ give an algorithm for the computation of Fréchet means. In this section we extend their algorithm and proof of convergence to the weighted case. In Algorithm 1 we add copies of the diagonal to ensure that each diagram has the same cardinality; denote this cardinality as $m$ . To compute the weighted Fréchet mean, we need to find $\mathbb { M } _ { k } = \bar { \{ \boldsymbol { y } ^ { ( i ) } \} } _ { i = 1 } ^ { m }$ that minimises the Fréchet function in (2). Implicit to the Wasserstein distance is a bijection $\gamma _ { j } : y ^ { ( i ) } \mapsto x _ { j } ^ { ( i ) }$ for each $j$ . Supposing we know these bijections, we can rearrange the Fréchet function into the form $\begin{array} { r } { F ( \mathbb { M } _ { k } ) = \sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( \mathbb { M } _ { k } , \mathbb { D } _ { j } ) ^ { 2 } = \sum _ { i = 1 } ^ { m } \sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } | | y ^ { ( i ) } - x _ { j } ^ { ( i ) } | | ^ { 2 } . } \end{array}$ + +![](images/0fb7e8cdd0bd53b2279667088b74f62883e5fb4f3b13a84dfffccc1f17940832.jpg) +Figure 2: Our algorithm successfully clustered the persistence diagrams, finding cluster centres that represent the topology of the datasets. The cluster centres have zero, one, or two significant off-diagonal points, representing zero, one, or two holes in the datasets. + +In this form, the summand is minimised for $\boldsymbol y ^ { ( i ) }$ by the weighted Euclidean centroid of the points $\{ x _ { j } ^ { ( i ) } \} _ { j = 1 } ^ { n }$ . Therefore to compute the weighted Fréchet mean, we need to find the correct bijections. We start by using the Hungarian algorithm to find an optimal matching between $\mathbb { M } _ { k }$ and each $\mathbb { D } _ { j }$ . Given a $\mathbb { D } _ { j }$ , for each point $\boldsymbol { y } ^ { ( i ) } \in \mathbb { M } _ { k }$ , the Hungarian algorithm will assign an optimally matched point $x _ { j } ^ { ( i ) } \in \mathbb { D } _ { j }$ . Specifically, we find matched points + +$$ +\left[ x _ { j } ^ { ( i ) } \right] _ { i = 1 } ^ { m } \longleftarrow \mathrm { H u n g a r i a n } ( \mathbb { M } _ { k } , \mathbb { D } _ { j } ) , \mathrm { ~ f o r ~ e a c h ~ } j = 1 , \ldots , n . +$$ + +Now, for each $\boldsymbol { y } ^ { ( i ) } \in \mathbb { M } _ { k }$ we need to find the weighted average of the matched poin ts h x(i) in . However, some of these points could be copies of the diagonal, so we need to consider three distinct cases: that each matched point is off-diagonal, that each one is a copy of the diagonal, or that the points are a mixture of both. We start by partitioning $1 , \ldots , n$ into the indices of the offdiagonal points $\mathcal { J } _ { \mathrm { O D } } ^ { ( i ) } = \left\{ j : x _ { j } ^ { ( i ) } \neq \Delta \right\}$ and the indices of the diagonal points $\mathcal { J } _ { \mathrm { D } } ^ { ( i ) } = \left\{ j : x _ { j } ^ { ( i ) } = \Delta \right\}$ for each $i = 1 , \ldots , m$ . Now, if ${ \mathfrak { I } } _ { \mathrm { O D } } = \ \varnothing$ then $\boldsymbol y ^ { ( i ) }$ is a copy of the diagonal. If not, let $w =$ $\begin{array} { r } { \left( \sum _ { j \in \mathcal { J } _ { \mathrm { o D } } ^ { ( i ) } } r _ { j k } ^ { 2 } \right) ^ { - 1 } \sum _ { j \in \mathcal { J } _ { \mathrm { o D } } ^ { ( i ) } } r _ { j k } ^ { 2 } x _ { j } ^ { ( i ) } } \end{array}$ 1 P j2J(i)OD r2jkx(i)j be the weighted mean of the off-diagonal points. If $\mathcal { J } _ { \mathrm { D } } ^ { ( i ) } = \emptyset$ , then $y ^ { ( i ) } = w$ . Otherwise, let $w _ { \Delta }$ be the point on the diagonal closest to $w$ . Then our update is + +$$ +y ^ { ( i ) } \longleftarrow \frac { \sum _ { j \in \mathcal { J } _ { \mathrm { 0 D } } ^ { ( i ) } } r _ { j k } ^ { 2 } x _ { j } ^ { ( i ) } + \sum _ { j \in \mathcal { J } _ { \mathrm { D } } ^ { ( i ) } } r _ { j k } ^ { 2 } w _ { \Delta } } { \sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } } , \mathrm { ~ f o r ~ } i = 1 , \dots , m . +$$ + +We alternate between (3) and (4) until the matching remains the same. Theorem 2, proving that this algorithm converges to a local minimum of the Fréchet function, is proven in Appendix B. Also in Appendix $\mathbf { B }$ is Algorithm 2, giving an overview of the entire computation. + +Theorem 2. Given diagrams $\mathbb { D } _ { j }$ , membership values , and the Fréchet function $F ( { \hat { \mathbb { D } } } ) \mathbf { \Sigma } =$ $\textstyle \sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( { \hat { \mathbb { D } } } , \mathbb { D } _ { j } ) ^ { 2 }$ , then $\mathbb { M } _ { k } \ = \ \{ y ^ { ( i ) } \} _ { i = 1 } ^ { m }$ is a local minimum of $F$ if and only if there is $a$ unique optimal pairing from $\mathbb { M } _ { k }$ to each of the $\mathbb { D } _ { j }$ and each $\boldsymbol y ^ { ( i ) }$ is updated via (4). + +![](images/3445a13ce1545f247cd45442b94aace5714885dd37d873b1a7c6954ed5ad08ac.jpg) +Figure 3: Heatmaps showing average number of iterations for fuzzy clustering of persistence diagrams (left) and the weighted Fréchet mean computation (right) to converge. Convergence of the FCM algorithm is determined when the cost function is stable to within $\pm 0 . 5 \%$ . Convergence experiments were carried out on randomly generated persistence diagrams with three repeats. + +# 4 EXPERIMENTS + +# 4.1 SYNTHETIC DATA + +Exemplar clustering. We start by demonstrating our algorithm on a simple synthetic dataset designed to highlight its ability to cluster based on the topology of the underlying datasets. We produce three datasets of noise, three datasets of a ring, and three datasets of figure-of-eights, all shown in Figure 2(a). In Figure 2(b) we show the corresponding 1-PH persistence diagrams. Note that the persistence diagrams have either zero, one, or two significant off-diagonal points, corresponding to zero, one, or two holes in the datasets. We then use our algorithm to cluster the nine persistence diagrams into three clusters. Having only been given the list of diagrams, the number of clusters, and the maximum number of iterations, our algorithm successfully clusters the diagrams based on their topology. Figure 2(c) shows that the cluster centres have zero, one, or two off-diagonal points: our algorithm has found cluster centres that reflect the topological features of the datasets. Because we are reducing the cardinality and dimensionality of datasets by mapping into persistence diagrams, we also demonstrate a speed-up of at least an order of magnitude over Wasserstein barycentre clustering methods. Details of these timing experiments are in Appendix C.1.3 + +Empirical behaviour. Figure 4 shows the results of experiments run to determine the empirical performance of our algorithm. We give theoretical guarantees that every convergent subsequence will tend to a local minimum, but in practice it remains important that our algorithm will converge, and within a reasonable timeframe. To this end we ran experiments on a total of 825 randomly generated persistence diagrams, recording the number of iterations and cost functions for both the FCM clustering and the weighted Fréchet mean (WFM) computation. We considered the FCM to have converged when the cost function remained within $\pm 0 . 5 \%$ . As explained in Section 3.2, the WFM converges when the matching stays the same. Our experiments showed that the FCM clustering consistently converges within ${ < } 5$ iterations, regardless of the number of diagrams and points per diagram (note that the time per iteration increases as the number of points/diagrams increases, even if the number of iterations remains stable). We had no experiments in which the algorithm did not converge. The WFM computation requires more iterations as both number of diagrams and number of points per diagram increases, but we once again experienced no failures to converge in each of our experiments. In general, running the algorithm offered no difficulties on a laptop, and we believe the algorithm is ready for use by practitioners in the TDA community. + +![](images/1af76cbcab2ee1ea6421350136a2706e44ee85ab360b8d60afaf3f80eda85385.jpg) +Figure 4: Cubic structures (top) and carbon allotropes (bottom). Our algorithm can cluster transformed lattice structures where comparable geometric algorithms fail. + +# 4.2 LATTICE STRUCTURES + +A key property for machine learning in materials science has been identified as “invariance to the basis symmetries of physics [...] rotation, reflection, translation” (Schmidt et al., 2019). Removing the need for a standardised coordinate system allows machine learning methods to be applied to a broader range of existing coordinate datasets generated by experimental methods (e.g., x-ray diffraction and scanning electron microscope imaging) and computational methods (e.g., density functional theory). Persistence diagrams, which capture affine transformation-invariant properties of datasets, are ideally suited for application in this domain. Our algorithm enables successful unsupervised learning on these datasets for the first time. Additionally, the fuzzy membership values allow top- $k$ ranking of candidates suggested by our algorithm. This is particularly important in materials science, where further investigation of materials can be extremely costly. + +The large majority of solids are comprised of lattices: regularly repeating unit cells of atoms. This lattice structure directly determines the properties of a material $\dot { ( \mathbb { H } \mathrm { u l l ~ } \& \mathrm { ~ B a c o n } ) } \dot { \mathbb { Z } } ^ { 0 1 1 } \mathrm { 1 } \mathrm { 1 } \mathrm { p }$ and it has been predicted that machine learning will reveal presently unknown links between structure and property by identifying new trends across materials $( \dot { \overline { { { \mathrm { M e r e d i g } } } } } , \dot { \underline { { 2 0 1 9 } } } , \dot { \overline { { { \mathrm { W e i ~ e t ~ a l . } } } } } , \dot { \underline { { 2 0 1 9 } } } )$ . We apply our algorithm to two examples of lattice structures from materials science: cubic structures and carbon allotropes. Cubic structures are important because they are ubiquitous. The most common lattice structures, particularly amongst pure metals, are face-centred cubic (FCC) structures and body-centred cubic (BCC) structures $\bar { ( \mathrm { P u t n i s } , \mathrm { } \textmu { 9 9 2 } ) }$ , shown in Figures 4(a) and 4(b). Carbon allotropes, such as graphene and diamond, are widely anticipated to revolutionise electronics and optoelectronics (Wang et al., $\boxed { 2 0 1 6 }$ . We focus on the carbon allotropes diamond and cis-hinged polydiacetylene, shown in Figures 4(c) and 4(d). + +We use atomic positions for the unit-cells of iron mp-150 and iron mp-13 from the Materials Project $\underline { { \mathrm { ( J a i n ~ e t ~ a l . } } } , \overline { { 2 0 1 3 } } )$ , representing BCC and FCC structures respectively, for our first experiment. For our second experiment we use diamond and cis-hinged polydiacetylene unit-cell atomic positions from the Samara Carbon Allotrope Database $( \overline { { { \mathrm { H o f f m a n n ~ e t ~ a l . } } } } ) , \overline { { { 2 0 1 6 } } } )$ . We simulate distinct collections of lattices by transforming the atomic coordinates, with no information about bonds given to the algorithms. The properties of persistence diagrams mean that we can successfully cluster the atomic coordinates derived from the same base unit-cell regardless of the transformations applied to the coordinate system, fulfilling the key property identified above (we consider a clustering successful when all datasets from the same lattice structure have their highest membership values for the same cluster). In comparison, we run Wasserstein barycentre clustering on the same datasets using several state-of-the-art algorithms for barycentre computation and approximation. Each can only successfully cluster the cubic structures after reflection, and none of them successfully cluster the carbon allotropes after any transformations. We give specific values for these results in Appendix C.2. + +# 4.3 DECISION BOUNDARIES + +Learnt models have been shown to perform better on datasets which have a similar persistence diagram to the model’s decision boundary $( \mathrm { \mathbb { R a m a m u r t h y ~ e t ~ a l . } } , \mathrm { \mathbb { 2 0 1 9 } } )$ . In fact, topological complexity has been shown to correlate with generalisation ability (Guss & Salakhutdinov, 2018; Gabrielsson & Carlsson, 2019; Rieck et al., 2018). We utilise our algorithm to cluster the topology of models and tasks, showing that high membership values imply better performance on tasks. Specifically, given a dataset with $n$ classes, we fix one class to define $n - 1$ tasks: binary classification of the fixed class vs each of the remaining classes. On each of these tasks, we train a model. We compute the decision boundary of the model $f$ , defined as $( x _ { 1 } , \ldots , x _ { m } , f ( x ) )$ where $f ( x )$ is the model’s prediction for $x = ( x _ { i } ) _ { i }$ , and the decision boundary of the tasks, defined via the labelled dataset as $( x _ { 1 } , \ldots , x _ { m } , y )$ where $y$ is the true label. We compute the 1-persistence diagrams of the tasks’ and models’ decision boundaries and cluster them to obtain membership values and cluster centres. To view task and model proximity through our clustering, we find the cluster centre with the highest membership value for each task, and consider the models closest to that cluster centre. Note that in general you cannot do this with hard clustering: most of the time a path will not exist from task to cluster centre to model, because each task/model is only associated with a single cluster. This contrasts with fuzzy clustering, where you have information about how close each model/task is to each cluster centre. We further discuss why this does not work for hard clustering in Appendix C.3. + +To assess the ability of our model/task clustering, we performed the above experiment on three different datasets: MNIST (LeCun et al., $\textcircled { 2 0 1 0 }$ , FashionMNIST $( \overline { { \mathrm { B i a o ~ e t ~ a l . } } } , \overline { { 2 0 1 7 } } )$ , and KuzushijiMNIST (Clanuwat et al., $\boxed { 2 0 1 8 }$ . We repeat each experiment three times. Our goal is to evaluate whether or not the clustering is capturing information about model performance on tasks, so as a baseline we use the average performance of all models on a fixed task, averaged over all tasks. Recall that we are just using the topology of the decision boundary: no information about model performance on the task was used until we assessed the ability of the clustering to capture information about the tasks. We start by verifying what happens if we use the model closest to the cluster centre associated with the task (i.e., top-1). We see a significant increase in performance, indicating that the topological fuzzy clustering has selected the model trained on the task: this means the clustering is working. We also compute the top-3 and top-2 performance change over average. We still see a statistically significant increase in performance over average performance, indicating that our membership values are capturing information about model performance on unseen tasks. These results are shown in Table 1. By successfully matching models to tasks using fuzzy clustering, we offer a new technique for future model marketplaces, demonstrating the value of topological clustering when analysing decision boundaries. + +Table 1: Performance increase/decrease over average task performance when using learnt fuzzy membership values for model selection. The increase in performance demonstrates that our fuzzy clustering automatically clusters models near tasks they perform well on. + +
Performance change vs random model selection (%)
Top-3Top-2Top-1
MNIST+6.17±2.18+10.81±1.88+20.88±4.08
FashionMNIST+16.46±4.00+21.94±4.73+23.30±8.72
Kuzushiji+6.61±1.78+11.18±2.45+21.89±5.54
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URL https://doi.org/10.1007/s00454-004-1146-y. \ No newline at end of file diff --git a/parse/train/Q1aiM7sCi1/Q1aiM7sCi1_content_list.json b/parse/train/Q1aiM7sCi1/Q1aiM7sCi1_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..8893f687ccd37313705945588fdaa0a02f80a821 --- /dev/null +++ b/parse/train/Q1aiM7sCi1/Q1aiM7sCi1_content_list.json @@ -0,0 +1,1493 @@ +[ + { + "type": "text", + "text": "FUZZY C-MEANS CLUSTERING FOR PERSISTENCE DIAGRAMS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 776, + 145 + ], + "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, + 250 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Persistence diagrams concisely represent the topology of a point cloud whilst having strong theoretical guarantees. Most current approaches to integrating topological information into machine learning implicitly map persistence diagrams to a Hilbert space, resulting in deformation of the underlying metric structure whilst also generally requiring prior knowledge about the true topology of the space. In this paper we give an algorithm for Fuzzy c-Means (FCM) clustering directly on the space of persistence diagrams,22 enabling unsupervised learning that automatically captures the topological structure of data, with no prior knowledge or additional processing of persistence diagrams. We prove the same convergence guarantees as traditional FCM clustering: every convergent subsequence of iterates tends to a local minimum or saddle point. We end by presenting experiments where our fuzzy topological clustering algorithm allows for unsupervised top- $k$ candidate selection in settings where (i) the properties of persistence diagrams make them the natural choice over geometric equivalents, and (ii) the probabilistic membership values let us rank candidates in settings where verifying candidate suitability is expensive: lattice structure classification in materials science and pre-trained model selection in machine learning. ", + "bbox": [ + 233, + 265, + 766, + 501 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 525, + 336, + 540 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Persistence diagrams, a concise representation of the topology of a point cloud with strong theoretical guarantees, have emerged as a new tool in the field of data analysis (Edelsbrunner & Harer, 2010). Persistence diagrams have been successfully used to analyse problems ranging from financial crashes $\\mathrm { \\textregistered { 1 } } \\mathrm { \\overline { { G i d e a \\ \\& \\ \\mathrm { K a t z } } } \\mathrm { \\tilde { \\textmu } \\mathrm { \\overline { { 2 0 1 8 } } \\mathrm { \\AA } } } }$ to protein binding (Kovacev-Nikolic et al., 2014), but the non-Hilbertian nature of the space of persistence diagrams means it is difficult to directly use persistence diagrams for machine learning. In order to better integrate diagrams into machine learning workflows, efforts have been made to map them into a more manageable form; primarily through embeddings into finite feature vectors, functional summaries, or by defining a positive-definite kernel on diagram space. In all cases, this explicitly or implicitly embeds diagrams into a Hilbert space which deforms the metric structure, potentially losing important information. With the exception of Topological Autoencoders, techniques to integrate these persistence-based summaries as topological regularisers and loss functions currently require prior knowledge about the correct topology of the dataset, which is clearly not feasible in most scenarios. ", + "bbox": [ + 174, + 556, + 825, + 736 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Against this background, we give an algorithm to perform Fuzzy c-Means (FCM) clustering (Bezdek, 1980) directly on collections of persistence diagrams, giving an important unsupervised learning algorithm and enabling learning from persistence diagrams without deforming the metric structure. We perform the convergence analysis for our algorithm, giving the same guarantees as traditional FCM clustering: that every convergent subsequence of iterates tends to a local minimum or saddle point. We demonstrate the value of our fuzzy clustering algorithm by using it to cluster datasets that benefit from both the topological and fuzzy nature of our algorithm. We apply our technique in two settings: lattice structures in materials science and the decision boundaries of CNNs. A key property for machine learning in materials science has been identified as “invariance to the basis symmetries of physics [...] rotation, reflection, translation” (Schmidt et al., 2019). Geometric clustering algorithms do not have this invariance, but persistence diagrams do, making them ideally suited for this application; we can cluster transformed lattice structure datasets where geometric equivalents fail. In addition to this, our probabilistic membership values allow us to rank the top- $k$ most likely lattices assigned to a cluster. This is particularly important in materials science, as further investigation requires expensive laboratory time and expertise. Our second application is inspired by Ramamurthy et al. (2019), who show that models perform better on tasks if they have topologically similar decision boundaries. We use our algorithm to cluster models and tasks by the persistence diagrams of their decision boundaries. Not only is our algorithm able to successfully cluster models to the correct task, based just on the topology of its decision boundary, but we show that higher membership values imply better performance on unseen tasks. ", + "bbox": [ + 173, + 743, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 202 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1.1 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 220, + 326, + 234 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Means of persistence diagrams. Our work relies on the existence of statistics in the space of persistence diagrams. Mileyko et al. $\\mathbb { Q } 0 1 1 )$ first showed that means and expectations are well-defined in the space of persistence diagrams. Specifically, they showed that the Fréchet mean, an extension of means onto metric spaces, is well-defined under weak assumptions on the space of persistence diagrams. Turner et al. (2012) then developed an algorithm to compute the Fréchet mean. We adapt the algorithm by Turner et al. to the weighted case, but the combinatoric nature of their algorithm makes it computationally intense. Lacombe et al. $\\textcircled { 2 0 1 8 }$ framed the computation of means and barycentres in the space of persistence diagram as an optimal transport problem, allowing them to use the Sinkhorn algorithm $( \\overline { { \\mathbb { C } \\mathrm { u t u r i } \\ \\& \\ \\mathrm { D o u c e t } } } , \\overline { { 2 0 1 4 } } )$ for fast computation of approximate solutions. The vectorisation of the diagram required by the algorithm by Lacombe et al. makes it unsuitable for integration into our work, as we remain in the space of persistence diagrams. Techniques to speed up the matching problem fundamental to our computation have also been proposed by Vidal et al. $\\dot { \\overline { { ( 2 0 2 0 ) } } }$ and Kerber et al. (2017). ", + "bbox": [ + 173, + 247, + 825, + 429 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Learning with persistence-based summaries. Integrating diagrams into machine learning workflows remained challenging even with well-defined means, as the space is non-Hilbertian (Turner & Spreemann, 2019). As such, efforts have been made to map diagrams into a Hilbert space; primarily either by embedding into finite feature vectors (Kališnik, 2018; Fabio & Ferri, 2015; Chepushtanova et al., 2015) or functional summaries (Bubenik, 2015; Rieck et al., 2019), or by defining a positivedefinite kernel on diagram space (Reininghaus et al., 2015; Carrière et al., 2017; Le & Yamada, 2018). These vectorisations have been integrated into deep learning either by learning parameters for the embedding (Hofer et al., 2017; Carrière et al., 2020; Kim et al., 2020; Zhao & Wang, 2019; Zielinski ´ et al., 2019), or as part of a topological loss or regulariser (Chen et al., 2018; Gabrielsson et al., 2020; Clough et al., 2020; Moor et al., 2019). However, the embeddings used in these techniques deform the metric structure of persistence diagram space (Bubenik & Wagner, 2019; Wagner, 2019; Carrière & Bauer, 2019), potentially leading to the loss of important information. Furthermore, these techniques generally require prior knowledge of a ‘correct’ target topology which cannot plausibly be known in most scenarios. In comparison, our algorithm acts in the space of persistence diagrams so it does not deform the structure of diagram space via embeddings, and is entirely unsupervised, requiring no prior knowledge about the topology. ", + "bbox": [ + 173, + 435, + 826, + 657 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Hard clustering. Maroulas et al. (2017) gave an algorithm for hard clustering persistence diagrams based on the algorithm by Turner et al. Lacombe et al. $\\textcircled { 2 0 1 8 }$ gave an alternate implementation of hard clustering based on their algorithm for barycentre computation, providing a computational speed-up over previous the work by Maroulas et al. The primary advantages of our work over previous work on hard clustering are as follows. ", + "bbox": [ + 174, + 664, + 825, + 734 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(i) The probabilistic membership values allow us to rank datasets in the cluster, enabling top- $k$ candidate selection in settings where verifying correctness is expensive. The value provided by this fuzzy information is demonstrated in the experiments. \n(ii) The fuzzy membership values provide information about proximity to all clusters, whereas hard labelling loses most of that information. In our experiments we demonstrate that this additional information can be utilised in practice. \n(iii) The weighted cost function makes the convergence analysis (which we provide) entirely nontrivial in comparison to the non-fuzzy case. We consider this convergence analysis a primary contribution of our paper. \n(iv) Fuzzy membership values have been shown to be more robust to noise than discrete labels (Klawonn, 2004). ", + "bbox": [ + 179, + 748, + 826, + 925 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "(v) Unlike hard clustering, fuzzy clustering is analytically differentiable, allowing integration of the fuzzy clustering step into deep learning methods (Wilder et al., 2019). ", + "bbox": [ + 178, + 103, + 823, + 133 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Geometric equivalents. The most similar unsupervised learning technique to our algorithm is Wasserstein Barycentre Clustering (WBC). It clusters datasets of point clouds by the Wasserstein distance between the point clouds, rather than the Wasserstein distance between their persistence diagrams. We compare our algorithm experimentally to WBC using ADMM $\\pmb { \\mathbb { T e } } \\ \\& \\ \\bot \\pmb { \\mathbb { I } } \\ \\boxed { 2 0 1 4 } )$ , Bregman ADMM (Ye et al., 2017), Subgradient Descent (Cuturi & Doucet, 2014), Iterative Bregman Projection (Benamou et al., 2015), and full linear programming $\\mathbb { \\overline { { \\mathbb { L } i \\ \\& \\ \\operatorname { W a n g } } } } , \\mathbb { \\underline { { 2 0 0 8 } } } )$ . Each of these algorithms computes or approximates the Wasserstein barycentre in different ways. Theoretically, fuzzy discrete distribution clustering (d. A. T. de Carvalho et al., $\\textcircled { 2 0 1 5 } )$ is similar to our algorithm, but the addition of the diagonal in the persistence diagram makes our work distinct. ", + "bbox": [ + 173, + 145, + 825, + 271 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1.2 OUR CONTRIBUTIONS ", + "text_level": 1, + "bbox": [ + 176, + 289, + 366, + 303 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "1. Our main contribution is an algorithm for Fuzzy c-Means clustering of persistence diagrams, along with the convergence analysis. Given a collection of persistence diagrams $\\mathbb { D } _ { 1 } , \\ldots , \\mathbb { D } _ { n }$ , we alternatively calculate cluster centres $\\mathbb { M } _ { 1 } , \\ldots , \\mathbb { M } _ { c }$ and membership values $r _ { j k } \\in [ 0 , 1 ]$ which denote the degree to which diagram $\\mathbb { D } _ { j }$ is associated with cluster $\\mathbb { M } _ { k }$ . We prove Theorem $^ { 1 , }$ showing that every convergent subsequence of these alternative update steps tends to a local minimum or saddle point of the cost function. This is the same convergence guarantee provided by traditional FCM clustering $( \\mathbf { \\mathbb { B } e z d e k \\ e t \\ a l . } ] \\ [ \\mathbf { \\mathbb { 9 } 8 7 } )$ , but requires additional work as the space of persistence diagrams with the Wasserstein distance has far weaker theoretical properties than Euclidean space. ", + "bbox": [ + 173, + 315, + 825, + 426 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2. Updating the cluster centres requires computing the weighted Fréchet mean. We extend the algorithm given by $\\boxed { \\mathrm { T u r n e r ~ e t ~ a l . } } \\textcircled { 2 0 1 2 }$ to the weighted case, justifying our addition of weights by extending their proof to show that the updated algorithm converges. ", + "bbox": [ + 174, + 434, + 825, + 476 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3. We implement our algorithm in Python, available in the supplementary materials. It works with persistence diagrams from commonly used open-source libraries for Topological Data Analysis $\\mathbf { \\Gamma } _ { \\mathrm { ( T D A ) , } } ^ { \\mathrm { ~ \\tiny ~ 1 ~ } } \\mathfrak { s }$ o is available for easy integration into current workflows, offering a powerful unsupervised learning algorithm to data science practitioners using TDA. ", + "bbox": [ + 174, + 482, + 825, + 537 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4. We demonstrate the application of our algorithm to settings where (i) the properties of persistence diagrams makes clustering them the natural choice over geometric equivalents and (ii) the probabilistic membership values can be used to rank candidates for top- $k$ selection. Our algorithm classifies transformed lattice structures from materials science where geometric equivalents fail, whilst giving probabilistic rankings to help prioritise expensive further investigation. We also cluster the persistence diagrams of decision boundaries and labelled datasets, showing that our fuzzy clustering captures information about model performance on unseen tasks. ", + "bbox": [ + 173, + 545, + 825, + 643 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 TOPOLOGICAL PRELIMINARIES ", + "text_level": 1, + "bbox": [ + 176, + 665, + 464, + 680 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Topological Data Analysis emerged from the study of algebraic topology, providing a toolkit to fully describe the topology of a dataset. We offer a quick summary below; for more comprehensive details see Edelsbrunner & Harer $\\bigoplus \\iiiint \\iiiint \\lVert 0 1 0 \\rVert$ . A set of points in $\\mathbb { R } ^ { d }$ are indicative of the shape of the distribution they are sampled from. By connecting points that are pairwise within $\\epsilon > 0$ distance of each other, we can create an approximation of the distribution called the Vietoris-Rips complex $\\mathrm { ( V i e t o r i s , \\textmu 9 2 7 ) }$ . Specifically, we add the convex hull of any collection of points that are pairwise at most $\\epsilon$ apart to the $\\epsilon$ -Vietoris-Rips complex. However, choosing an $\\epsilon$ remains problematic; too low a value and key points can remain disconnected, too high a value and the points become fully connected. To overcome this we use persistence: we consider the approximation over all values of $\\epsilon$ simultaneously, and study how the topology of that approximation evolves as $\\epsilon$ grows large. We call the collection of complexes for all $\\epsilon$ a filtration. ", + "bbox": [ + 173, + 695, + 826, + 849 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For each $\\epsilon$ , we compute the $p$ -homology group. This tells us the topology of the $\\epsilon$ -Vietoris-Rips complex: the 0-homology counts the number of connected components, the 1-homology counts the number of holes, the 2-homology counts the number of voids, etc. (Edelsbrunner et al., 2000) ", + "bbox": [ + 176, + 856, + 825, + 898 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/64ff89fa891ae43bb690e18da28297235b2205ad0db31d4eaf8f06107e1c4f55.jpg", + "image_caption": [ + "Figure 1: An example Vietoris-Rips filtration with its persistence diagram. We only add convex hulls when points are pairwise $\\epsilon$ apart, so there is a hole at $\\epsilon = 1$ whilst the diagonals of the square are not close enough to fill the hole in. This hole can be seen as a point at (1, 1.42) in the 1-PH persistence diagram. " + ], + "image_footnote": [], + "bbox": [ + 240, + 101, + 815, + 294 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The $p$ -persistent homology ( $\\dot { p }$ -PH) group is created by summing the $p$ -homology groups over all ✏. This results in a $p$ -PH group that summarises information about the topology of the dataset at all granularities. If a topological feature, such as a connected component or hole, persists throughout a large range of granularities, then it’s more likely to be a feature of the distribution. If it only persists for a short amount of time, then it’s more likely to be noise (Cohen-Steiner et al., 2007). We can stably map a $p$ -PH group into a multiset in the extended plane called a persistence diagram (Chazal et al., $\\bar { 2 0 1 2 } )$ . Each topological feature has a birth and death: a feature is born when it enters the complex, and dies when the complex grows enough to destroy it. For example, in Figure 1(a), a feature is born at $\\epsilon = 1$ when four lines form a hole. This feature dies at $\\epsilon = 1 . 4 2$ when the hole is filled in. This is shown in the persistence diagram in Figure 1(b) as a point at $( 1 , 1 . 4 2 )$ in 1-PH. By computing the birth/death points for each topological feature in the filtration, we get a complete picture of the topology of the point cloud at all granularities (Zomorodian & Carlsson, $\\overline { { 2 0 0 5 } } )$ . The persistence diagram is the collection of birth/death points, along with the diagonal $\\overline { { \\Delta } } \\equiv \\{ ( a , a ) : a \\in \\mathbb { R } \\}$ with infinite multiplicity, added in order to make the space of persistence diagrams complete (Mileyko et al., 2011). ", + "bbox": [ + 173, + 398, + 826, + 609 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 ALGORITHMIC DESIGN ", + "text_level": 1, + "bbox": [ + 176, + 638, + 397, + 655 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 CLUSTERING PERSISTENCE DIAGRAMS ", + "text_level": 1, + "bbox": [ + 176, + 676, + 482, + 690 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In order to cluster we need a distance on the space of persistence diagrams. We use the 2-Wasserstein $L _ { 2 }$ metric as it is stable on persistence diagrams of finite point clouds (Chazal et al., 2012). The Wasserstein distance is an optimal transport metric that has found applications across machine learning. In the Euclidean case, it quantifies the smallest distance between optimally matched points. Given diagrams $\\mathbb { D } _ { 1 } , \\mathbb { D } _ { 2 }$ , the distance is ", + "bbox": [ + 174, + 705, + 826, + 776 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/044279cb12c1257d323f24efd8909559ccd2be79bc799137572edd50a868f363.jpg", + "text": "$$\nW _ { 2 } ( \\mathbb { D } _ { 1 } , \\mathbb { D } _ { 2 } ) = \\left( \\operatorname* { i n f } _ { \\gamma : \\mathbb { D } _ { 1 } \\to \\mathbb { D } _ { 2 } } \\sum _ { x \\in \\mathbb { D } _ { 1 } } | | x - \\gamma ( x ) | | _ { 2 } ^ { 2 } \\right) ^ { 1 / 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 331, + 791, + 665, + 838 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the infimum is taken over all bijections $\\gamma : { \\mathbb { D } } _ { 1 } \\to { \\mathbb { D } } _ { 2 }$ . Note that as we added the diagonal with infinite multiplicity to each diagram, these bijections exist. If an off-diagonal point is matched to the diagonal the transportation cost is simply the shortest distance to the diagonal. In fact, the closer a point is to the diagonal, the more likely it is to be noise (Cohen-Steiner et al., $\\boxed { 2 0 0 7 }$ , so this ensures our distance is not overly affected by noise. ", + "bbox": [ + 173, + 853, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We work in the space $\\mathfrak { D } _ { \\mathrm { L } ^ { 2 } } = \\{ \\mathbb { D } : W _ { 2 } ( \\mathbb { D } , \\Delta ) < \\infty \\} , ^ { 2 }$ as this leads to a geodesic space with known structure $\\left\\{ { \\mathrm { I u r n e r ~ e t ~ a l . } } \\right\\} \\left[ { \\mathrm { 2 0 1 2 } } \\right)$ . Given a collection of persistence diagrams $\\{ \\mathbb { D } _ { j } \\} _ { j = 1 } ^ { n } \\subset \\mathcal { D } _ { \\mathrm { L } ^ { 2 } }$ and a fixed number of clusters $c$ , we wish to find cluster centres $\\{ \\mathbb { M } _ { k } \\} _ { k = 1 } ^ { c } \\subset \\mathcal { D } _ { \\mathrm { L } ^ { 2 } }$ , along with membership values $r _ { j k } \\in [ 0 , 1 ]$ that denote the extent to which $\\mathbb { D } _ { j }$ is associated with cluster $\\mathbb { M } _ { k }$ . We follow probabilistic fuzzy clustering, so that $\\textstyle \\sum _ { k } r _ { j k } = 1$ for each $j$ . ", + "bbox": [ + 173, + 101, + 825, + 176 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We extend the FCM algorithm originally proposed by $\\mathbf { B e z d e k } | \\left( \\mathbf { \\overline { { 1 9 8 0 } } } \\right) $ . Our $r _ { j k }$ is the same as traditional FCM clustering, adapted with the Wasserstein distance. That is, ", + "bbox": [ + 174, + 181, + 823, + 210 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/2d89a6075efb8ca2a17e9a94e5ce2dc0a5a92f6c81cce12dbd24305474747ea0.jpg", + "text": "$$\nr _ { j k } = \\left( \\sum _ { l = 1 } ^ { c } \\frac { W _ { 2 } ( \\mathbb { M } _ { k } , \\mathbb { D } _ { j } ) } { W _ { 2 } ( \\mathbb { M } _ { l } , \\mathbb { D } _ { j } ) } \\right) ^ { - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 217, + 606, + 262 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To update $\\mathbb { M } _ { k }$ , we compute the weighted Fréchet mean $\\hat { \\mathbb { D } }$ of the persistence diagrams $\\{ \\mathbb { D } _ { j } \\} _ { j = 1 } ^ { n }$ with the weights $\\{ r _ { j k } ^ { 2 } \\} _ { j = 1 } ^ { n }$ . Specifically, ", + "bbox": [ + 169, + 276, + 825, + 311 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e971481cdc3ef3c9785b66b8e3ba62bb6fb0a41d9f54d8692810ebbf43b1f35c.jpg", + "text": "$$\n\\mathbb { M } _ { k } \\longleftarrow \\operatorname { a r g m i n } _ { \\hat { \\mathbb { D } } } \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( \\hat { \\mathbb { D } } , \\mathbb { D } _ { j } ) ^ { 2 } , \\operatorname { f o r } k = 1 , \\dots , c .\n$$", + "text_format": "latex", + "bbox": [ + 313, + 318, + 681, + 361 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As the weighted Fréchet mean extends weighted centroids to general metric spaces, this gives our fuzzy cluster centres. The computation of the weighted Fréchet mean is covered in Section 3.2. By alternatively updating (1) and (2) we get a sequence of iterates. Theorem 1, proven in Appendix A, provides the same convergence guarantees as traditional FCM clustering. ", + "bbox": [ + 173, + 373, + 826, + 430 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 1. Every convergent subsequence of the sequence of iterates obtained by alternatively updating membership values and cluster centres with $( l )$ and (2) tends to a local minimum or saddle point of the cost function $\\begin{array} { r } { J ( R , { \\mathbb M } ) = \\sum _ { j = 1 } ^ { n } \\sum _ { k = 1 } ^ { c } r _ { j k } ^ { 2 } W _ { 2 } ( { \\mathbb M } _ { k } , { \\mathbb D } _ { j } ) ^ { 2 } } \\end{array}$ . ", + "bbox": [ + 173, + 434, + 825, + 479 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Observe that we only guarantee the convergence of subsequences of iterates. This is the same as traditional FCM clustering, so we follow the same approach to a stopping condition and run our algorithm for a fixed number of iterations. The entire algorithm is displayed in Algorithm 1. ", + "bbox": [ + 176, + 489, + 823, + 531 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1 FPDCluster ", + "text_level": 1, + "bbox": [ + 174, + 547, + 341, + 560 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/727766a169bea9ca3741654e9e120f89d4e770783a4cc91d2f9b6f9acb44b181.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Input Diagrams D = {Dj}𝑗=1, number of clusters c, maximum iterations MAXITER Output Cluster centres M = {Mk}ε=1, membership values R = {rjk}
1:D = ADDDIAGONALS(D)7: end for
2:M=INITCENTRES(D)8: end for
3: for countin1..MAXITER do9: for k in1..c do
4: for j in1..n do10: M ← WFRECHETMEAN(D,Rk)
5: for k in1..c do11: end for
W2(Mk,D)-1 12: end for
(∑i-1 W) rjk← 6:13: return M, R
", + "bbox": [ + 178, + 563, + 820, + 722 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 COMPUTING THE WEIGHTED FRÉCHET MEAN", + "text_level": 1, + "bbox": [ + 174, + 746, + 529, + 761 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Turner et al. $\\textcircled { | 2 0 1 2 } )$ give an algorithm for the computation of Fréchet means. In this section we extend their algorithm and proof of convergence to the weighted case. In Algorithm 1 we add copies of the diagonal to ensure that each diagram has the same cardinality; denote this cardinality as $m$ . To compute the weighted Fréchet mean, we need to find $\\mathbb { M } _ { k } = \\bar { \\{ \\boldsymbol { y } ^ { ( i ) } \\} } _ { i = 1 } ^ { m }$ that minimises the Fréchet function in (2). Implicit to the Wasserstein distance is a bijection $\\gamma _ { j } : y ^ { ( i ) } \\mapsto x _ { j } ^ { ( i ) }$ for each $j$ . Supposing we know these bijections, we can rearrange the Fréchet function into the form $\\begin{array} { r } { F ( \\mathbb { M } _ { k } ) = \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( \\mathbb { M } _ { k } , \\mathbb { D } _ { j } ) ^ { 2 } = \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } | | y ^ { ( i ) } - x _ { j } ^ { ( i ) } | | ^ { 2 } . } \\end{array}$ ", + "bbox": [ + 173, + 771, + 825, + 848 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/0fb7e8cdd0bd53b2279667088b74f62883e5fb4f3b13a84dfffccc1f17940832.jpg", + "image_caption": [ + "Figure 2: Our algorithm successfully clustered the persistence diagrams, finding cluster centres that represent the topology of the datasets. The cluster centres have zero, one, or two significant off-diagonal points, representing zero, one, or two holes in the datasets. " + ], + "image_footnote": [], + "bbox": [ + 199, + 117, + 753, + 287 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 378, + 823, + 415 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this form, the summand is minimised for $\\boldsymbol y ^ { ( i ) }$ by the weighted Euclidean centroid of the points $\\{ x _ { j } ^ { ( i ) } \\} _ { j = 1 } ^ { n }$ . Therefore to compute the weighted Fréchet mean, we need to find the correct bijections. We start by using the Hungarian algorithm to find an optimal matching between $\\mathbb { M } _ { k }$ and each $\\mathbb { D } _ { j }$ . Given a $\\mathbb { D } _ { j }$ , for each point $\\boldsymbol { y } ^ { ( i ) } \\in \\mathbb { M } _ { k }$ , the Hungarian algorithm will assign an optimally matched point $x _ { j } ^ { ( i ) } \\in \\mathbb { D } _ { j }$ . Specifically, we find matched points ", + "bbox": [ + 173, + 420, + 826, + 505 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/813445b60502ff54fa3b6a1d8cbe68cbd361766f4b3ffbff4acd6ad2442e5edc.jpg", + "text": "$$\n\\left[ x _ { j } ^ { ( i ) } \\right] _ { i = 1 } ^ { m } \\longleftarrow \\mathrm { H u n g a r i a n } ( \\mathbb { M } _ { k } , \\mathbb { D } _ { j } ) , \\mathrm { ~ f o r ~ e a c h ~ } j = 1 , \\ldots , n .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 522, + 696, + 553 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Now, for each $\\boldsymbol { y } ^ { ( i ) } \\in \\mathbb { M } _ { k }$ we need to find the weighted average of the matched poin ts h x(i) in . However, some of these points could be copies of the diagonal, so we need to consider three distinct cases: that each matched point is off-diagonal, that each one is a copy of the diagonal, or that the points are a mixture of both. We start by partitioning $1 , \\ldots , n$ into the indices of the offdiagonal points $\\mathcal { J } _ { \\mathrm { O D } } ^ { ( i ) } = \\left\\{ j : x _ { j } ^ { ( i ) } \\neq \\Delta \\right\\}$ and the indices of the diagonal points $\\mathcal { J } _ { \\mathrm { D } } ^ { ( i ) } = \\left\\{ j : x _ { j } ^ { ( i ) } = \\Delta \\right\\}$ for each $i = 1 , \\ldots , m$ . Now, if ${ \\mathfrak { I } } _ { \\mathrm { O D } } = \\ \\varnothing$ then $\\boldsymbol y ^ { ( i ) }$ is a copy of the diagonal. If not, let $w =$ $\\begin{array} { r } { \\left( \\sum _ { j \\in \\mathcal { J } _ { \\mathrm { o D } } ^ { ( i ) } } r _ { j k } ^ { 2 } \\right) ^ { - 1 } \\sum _ { j \\in \\mathcal { J } _ { \\mathrm { o D } } ^ { ( i ) } } r _ { j k } ^ { 2 } x _ { j } ^ { ( i ) } } \\end{array}$ 1 P j2J(i)OD r2jkx(i)j be the weighted mean of the off-diagonal points. If $\\mathcal { J } _ { \\mathrm { D } } ^ { ( i ) } = \\emptyset$ , then $y ^ { ( i ) } = w$ . Otherwise, let $w _ { \\Delta }$ be the point on the diagonal closest to $w$ . Then our update is ", + "bbox": [ + 173, + 570, + 826, + 720 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c0292e45f5f942bb00f61a98644e345db81e60b2b62814cc75c0e94929e654d0.jpg", + "text": "$$\ny ^ { ( i ) } \\longleftarrow \\frac { \\sum _ { j \\in \\mathcal { J } _ { \\mathrm { 0 D } } ^ { ( i ) } } r _ { j k } ^ { 2 } x _ { j } ^ { ( i ) } + \\sum _ { j \\in \\mathcal { J } _ { \\mathrm { D } } ^ { ( i ) } } r _ { j k } ^ { 2 } w _ { \\Delta } } { \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } } , \\mathrm { ~ f o r ~ } i = 1 , \\dots , m .\n$$", + "text_format": "latex", + "bbox": [ + 289, + 736, + 707, + 781 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We alternate between (3) and (4) until the matching remains the same. Theorem 2, proving that this algorithm converges to a local minimum of the Fréchet function, is proven in Appendix B. Also in Appendix $\\mathbf { B }$ is Algorithm 2, giving an overview of the entire computation. ", + "bbox": [ + 173, + 805, + 825, + 849 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 2. Given diagrams $\\mathbb { D } _ { j }$ , membership values , and the Fréchet function $F ( { \\hat { \\mathbb { D } } } ) \\mathbf { \\Sigma } =$ $\\textstyle \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( { \\hat { \\mathbb { D } } } , \\mathbb { D } _ { j } ) ^ { 2 }$ , then $\\mathbb { M } _ { k } \\ = \\ \\{ y ^ { ( i ) } \\} _ { i = 1 } ^ { m }$ is a local minimum of $F$ if and only if there is $a$ unique optimal pairing from $\\mathbb { M } _ { k }$ to each of the $\\mathbb { D } _ { j }$ and each $\\boldsymbol y ^ { ( i ) }$ is updated via (4). ", + "bbox": [ + 173, + 859, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/3445a13ce1545f247cd45442b94aace5714885dd37d873b1a7c6954ed5ad08ac.jpg", + "image_caption": [ + "Figure 3: Heatmaps showing average number of iterations for fuzzy clustering of persistence diagrams (left) and the weighted Fréchet mean computation (right) to converge. Convergence of the FCM algorithm is determined when the cost function is stable to within $\\pm 0 . 5 \\%$ . Convergence experiments were carried out on randomly generated persistence diagrams with three repeats. " + ], + "image_footnote": [], + "bbox": [ + 199, + 103, + 797, + 287 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 410, + 326, + 425 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 SYNTHETIC DATA ", + "text_level": 1, + "bbox": [ + 176, + 449, + 334, + 463 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Exemplar clustering. We start by demonstrating our algorithm on a simple synthetic dataset designed to highlight its ability to cluster based on the topology of the underlying datasets. We produce three datasets of noise, three datasets of a ring, and three datasets of figure-of-eights, all shown in Figure 2(a). In Figure 2(b) we show the corresponding 1-PH persistence diagrams. Note that the persistence diagrams have either zero, one, or two significant off-diagonal points, corresponding to zero, one, or two holes in the datasets. We then use our algorithm to cluster the nine persistence diagrams into three clusters. Having only been given the list of diagrams, the number of clusters, and the maximum number of iterations, our algorithm successfully clusters the diagrams based on their topology. Figure 2(c) shows that the cluster centres have zero, one, or two off-diagonal points: our algorithm has found cluster centres that reflect the topological features of the datasets. Because we are reducing the cardinality and dimensionality of datasets by mapping into persistence diagrams, we also demonstrate a speed-up of at least an order of magnitude over Wasserstein barycentre clustering methods. Details of these timing experiments are in Appendix C.1.3 ", + "bbox": [ + 173, + 481, + 825, + 662 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Empirical behaviour. Figure 4 shows the results of experiments run to determine the empirical performance of our algorithm. We give theoretical guarantees that every convergent subsequence will tend to a local minimum, but in practice it remains important that our algorithm will converge, and within a reasonable timeframe. To this end we ran experiments on a total of 825 randomly generated persistence diagrams, recording the number of iterations and cost functions for both the FCM clustering and the weighted Fréchet mean (WFM) computation. We considered the FCM to have converged when the cost function remained within $\\pm 0 . 5 \\%$ . As explained in Section 3.2, the WFM converges when the matching stays the same. Our experiments showed that the FCM clustering consistently converges within ${ < } 5$ iterations, regardless of the number of diagrams and points per diagram (note that the time per iteration increases as the number of points/diagrams increases, even if the number of iterations remains stable). We had no experiments in which the algorithm did not converge. The WFM computation requires more iterations as both number of diagrams and number of points per diagram increases, but we once again experienced no failures to converge in each of our experiments. In general, running the algorithm offered no difficulties on a laptop, and we believe the algorithm is ready for use by practitioners in the TDA community. ", + "bbox": [ + 173, + 667, + 826, + 876 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/1af76cbcab2ee1ea6421350136a2706e44ee85ab360b8d60afaf3f80eda85385.jpg", + "image_caption": [ + "Figure 4: Cubic structures (top) and carbon allotropes (bottom). Our algorithm can cluster transformed lattice structures where comparable geometric algorithms fail. " + ], + "image_footnote": [], + "bbox": [ + 232, + 93, + 769, + 313 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 LATTICE STRUCTURES ", + "text_level": 1, + "bbox": [ + 176, + 402, + 370, + 415 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "A key property for machine learning in materials science has been identified as “invariance to the basis symmetries of physics [...] rotation, reflection, translation” (Schmidt et al., 2019). Removing the need for a standardised coordinate system allows machine learning methods to be applied to a broader range of existing coordinate datasets generated by experimental methods (e.g., x-ray diffraction and scanning electron microscope imaging) and computational methods (e.g., density functional theory). Persistence diagrams, which capture affine transformation-invariant properties of datasets, are ideally suited for application in this domain. Our algorithm enables successful unsupervised learning on these datasets for the first time. Additionally, the fuzzy membership values allow top- $k$ ranking of candidates suggested by our algorithm. This is particularly important in materials science, where further investigation of materials can be extremely costly. ", + "bbox": [ + 173, + 438, + 825, + 577 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The large majority of solids are comprised of lattices: regularly repeating unit cells of atoms. This lattice structure directly determines the properties of a material $\\dot { ( \\mathbb { H } \\mathrm { u l l ~ } \\& \\mathrm { ~ B a c o n } ) } \\dot { \\mathbb { Z } } ^ { 0 1 1 } \\mathrm { 1 } \\mathrm { 1 } \\mathrm { p }$ and it has been predicted that machine learning will reveal presently unknown links between structure and property by identifying new trends across materials $( \\dot { \\overline { { { \\mathrm { M e r e d i g } } } } } , \\dot { \\underline { { 2 0 1 9 } } } , \\dot { \\overline { { { \\mathrm { W e i ~ e t ~ a l . } } } } } , \\dot { \\underline { { 2 0 1 9 } } } )$ . We apply our algorithm to two examples of lattice structures from materials science: cubic structures and carbon allotropes. Cubic structures are important because they are ubiquitous. The most common lattice structures, particularly amongst pure metals, are face-centred cubic (FCC) structures and body-centred cubic (BCC) structures $\\bar { ( \\mathrm { P u t n i s } , \\mathrm { } \\textmu { 9 9 2 } ) }$ , shown in Figures 4(a) and 4(b). Carbon allotropes, such as graphene and diamond, are widely anticipated to revolutionise electronics and optoelectronics (Wang et al., $\\boxed { 2 0 1 6 }$ . We focus on the carbon allotropes diamond and cis-hinged polydiacetylene, shown in Figures 4(c) and 4(d). ", + "bbox": [ + 173, + 583, + 825, + 736 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We use atomic positions for the unit-cells of iron mp-150 and iron mp-13 from the Materials Project $\\underline { { \\mathrm { ( J a i n ~ e t ~ a l . } } } , \\overline { { 2 0 1 3 } } )$ , representing BCC and FCC structures respectively, for our first experiment. For our second experiment we use diamond and cis-hinged polydiacetylene unit-cell atomic positions from the Samara Carbon Allotrope Database $( \\overline { { { \\mathrm { H o f f m a n n ~ e t ~ a l . } } } } ) , \\overline { { { 2 0 1 6 } } } )$ . We simulate distinct collections of lattices by transforming the atomic coordinates, with no information about bonds given to the algorithms. The properties of persistence diagrams mean that we can successfully cluster the atomic coordinates derived from the same base unit-cell regardless of the transformations applied to the coordinate system, fulfilling the key property identified above (we consider a clustering successful when all datasets from the same lattice structure have their highest membership values for the same cluster). In comparison, we run Wasserstein barycentre clustering on the same datasets using several state-of-the-art algorithms for barycentre computation and approximation. Each can only successfully cluster the cubic structures after reflection, and none of them successfully cluster the carbon allotropes after any transformations. We give specific values for these results in Appendix C.2. ", + "bbox": [ + 173, + 743, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 DECISION BOUNDARIES ", + "text_level": 1, + "bbox": [ + 174, + 103, + 380, + 117 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Learnt models have been shown to perform better on datasets which have a similar persistence diagram to the model’s decision boundary $( \\mathrm { \\mathbb { R a m a m u r t h y ~ e t ~ a l . } } , \\mathrm { \\mathbb { 2 0 1 9 } } )$ . In fact, topological complexity has been shown to correlate with generalisation ability (Guss & Salakhutdinov, 2018; Gabrielsson & Carlsson, 2019; Rieck et al., 2018). We utilise our algorithm to cluster the topology of models and tasks, showing that high membership values imply better performance on tasks. Specifically, given a dataset with $n$ classes, we fix one class to define $n - 1$ tasks: binary classification of the fixed class vs each of the remaining classes. On each of these tasks, we train a model. We compute the decision boundary of the model $f$ , defined as $( x _ { 1 } , \\ldots , x _ { m } , f ( x ) )$ where $f ( x )$ is the model’s prediction for $x = ( x _ { i } ) _ { i }$ , and the decision boundary of the tasks, defined via the labelled dataset as $( x _ { 1 } , \\ldots , x _ { m } , y )$ where $y$ is the true label. We compute the 1-persistence diagrams of the tasks’ and models’ decision boundaries and cluster them to obtain membership values and cluster centres. To view task and model proximity through our clustering, we find the cluster centre with the highest membership value for each task, and consider the models closest to that cluster centre. Note that in general you cannot do this with hard clustering: most of the time a path will not exist from task to cluster centre to model, because each task/model is only associated with a single cluster. This contrasts with fuzzy clustering, where you have information about how close each model/task is to each cluster centre. We further discuss why this does not work for hard clustering in Appendix C.3. ", + "bbox": [ + 173, + 135, + 825, + 372 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To assess the ability of our model/task clustering, we performed the above experiment on three different datasets: MNIST (LeCun et al., $\\textcircled { 2 0 1 0 }$ , FashionMNIST $( \\overline { { \\mathrm { B i a o ~ e t ~ a l . } } } , \\overline { { 2 0 1 7 } } )$ , and KuzushijiMNIST (Clanuwat et al., $\\boxed { 2 0 1 8 }$ . We repeat each experiment three times. Our goal is to evaluate whether or not the clustering is capturing information about model performance on tasks, so as a baseline we use the average performance of all models on a fixed task, averaged over all tasks. Recall that we are just using the topology of the decision boundary: no information about model performance on the task was used until we assessed the ability of the clustering to capture information about the tasks. We start by verifying what happens if we use the model closest to the cluster centre associated with the task (i.e., top-1). We see a significant increase in performance, indicating that the topological fuzzy clustering has selected the model trained on the task: this means the clustering is working. We also compute the top-3 and top-2 performance change over average. We still see a statistically significant increase in performance over average performance, indicating that our membership values are capturing information about model performance on unseen tasks. These results are shown in Table 1. By successfully matching models to tasks using fuzzy clustering, we offer a new technique for future model marketplaces, demonstrating the value of topological clustering when analysing decision boundaries. ", + "bbox": [ + 173, + 378, + 825, + 599 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/1c73e0ad76b9b042b9e50105fe260ad58ad4947360dc3352f925bfa95680484c.jpg", + "table_caption": [ + "Table 1: Performance increase/decrease over average task performance when using learnt fuzzy membership values for model selection. The increase in performance demonstrates that our fuzzy clustering automatically clusters models near tasks they perform well on. " + ], + "table_footnote": [], + "table_body": "
Performance change vs random model selection (%)
Top-3Top-2Top-1
MNIST+6.17±2.18+10.81±1.88+20.88±4.08
FashionMNIST+16.46±4.00+21.94±4.73+23.30±8.72
Kuzushiji+6.61±1.78+11.18±2.45+21.89±5.54
", + "bbox": [ + 245, + 678, + 751, + 770 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 827, + 328, + 843 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have developed FCM clustering on the space of persistence diagrams, adding an important class of unsupervised learning to Topological Data Analysis’ toolkit. Our topological fuzzy clustering paves the way for applications in materials science and pre-trained model marketplaces, where we envisage our algorithm being used as part of automated materials discovery and model selection. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 287, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Ulrich Bauer. Ripser: efficient computation of vietoris-rips persistence barcodes, August 2019. Preprint. 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In order to better integrate diagrams into machine learning workflows, efforts", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "have been made to map them into a more manageable form; primarily through embeddings into", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 531 + ], + "score": 1.0, + "content": "finite feature vectors, functional summaries, or by defining a positive-definite kernel on diagram", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "space. In all cases, this explicitly or implicitly embeds diagrams into a Hilbert space which deforms", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 552 + ], + "score": 1.0, + "content": "the metric structure, potentially losing important information. 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We demonstrate the value of our fuzzy clustering algorithm by using it to cluster datasets", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "that benefit from both the topological and fuzzy nature of our algorithm. We apply our technique", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "in two settings: lattice structures in materials science and the decision boundaries of CNNs. 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Most current approaches to integrating", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 232, + 469, + 245 + ], + "spans": [ + { + "bbox": [ + 141, + 232, + 469, + 245 + ], + "score": 1.0, + "content": "topological information into machine learning implicitly map persistence diagrams", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 243, + 470, + 256 + ], + "spans": [ + { + "bbox": [ + 141, + 243, + 470, + 256 + ], + "score": 1.0, + "content": "to a Hilbert space, resulting in deformation of the underlying metric structure whilst", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 254, + 469, + 266 + ], + "spans": [ + { + "bbox": [ + 141, + 254, + 469, + 266 + ], + "score": 1.0, + "content": "also generally requiring prior knowledge about the true topology of the space. In", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 265, + 469, + 278 + ], + "spans": [ + { + "bbox": [ + 141, + 265, + 469, + 278 + ], + "score": 1.0, + "content": "this paper we give an algorithm for Fuzzy c-Means (FCM) clustering directly on the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 276, + 469, + 289 + ], + "spans": [ + { + "bbox": [ + 141, + 276, + 469, + 289 + ], + "score": 1.0, + "content": "space of persistence diagrams,22 enabling unsupervised learning that automatically", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 288, + 469, + 299 + ], + "spans": [ + { + "bbox": [ + 141, + 288, + 469, + 299 + ], + "score": 1.0, + "content": "captures the topological structure of data, with no prior knowledge or additional", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 311 + ], + "score": 1.0, + "content": "processing of persistence diagrams. We prove the same convergence guarantees", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 308, + 470, + 322 + ], + "spans": [ + { + "bbox": [ + 141, + 308, + 470, + 322 + ], + "score": 1.0, + "content": "as traditional FCM clustering: every convergent subsequence of iterates tends to a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 319, + 469, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 319, + 469, + 333 + ], + "score": 1.0, + "content": "local minimum or saddle point. We end by presenting experiments where our fuzzy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 330, + 470, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 330, + 383, + 344 + ], + "score": 1.0, + "content": "topological clustering algorithm allows for unsupervised top-", + "type": "text" + }, + { + "bbox": [ + 384, + 331, + 390, + 340 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 330, + 470, + 344 + ], + "score": 1.0, + "content": "candidate selection", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 342, + 469, + 354 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 469, + 354 + ], + "score": 1.0, + "content": "in settings where (i) the properties of persistence diagrams make them the natural", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "spans": [ + { + "bbox": [ + 141, + 353, + 470, + 365 + ], + "score": 1.0, + "content": "choice over geometric equivalents, and (ii) the probabilistic membership values let", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 364, + 470, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 364, + 470, + 376 + ], + "score": 1.0, + "content": "us rank candidates in settings where verifying candidate suitability is expensive:", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 375, + 469, + 386 + ], + "spans": [ + { + "bbox": [ + 141, + 375, + 469, + 386 + ], + "score": 1.0, + "content": "lattice structure classification in materials science and pre-trained model selection", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 385, + 226, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 385, + 226, + 399 + ], + "score": 1.0, + "content": "in machine learning.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 13, + "bbox_fs": [ + 141, + 210, + 470, + 399 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 416, + 206, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 208, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 208, + 432 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "Persistence diagrams, a concise representation of the topology of a point cloud with strong theoretical", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 451, + 504, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 504, + 464 + ], + "score": 1.0, + "content": "guarantees, have emerged as a new tool in the field of data analysis (Edelsbrunner & Harer, 2010).", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "Persistence diagrams have been successfully used to analyse problems ranging from financial crashes", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 192, + 486 + ], + "score": 0.65, + "content": "\\mathrm { \\textregistered { 1 } } \\mathrm { \\overline { { G i d e a \\ \\& \\ \\mathrm { K a t z } } } \\mathrm { \\tilde { \\textmu } \\mathrm { \\overline { { 2 0 1 8 } } \\mathrm { \\AA } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "to protein binding (Kovacev-Nikolic et al., 2014), but the non-Hilbertian nature", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 485, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 496 + ], + "score": 1.0, + "content": "of the space of persistence diagrams means it is difficult to directly use persistence diagrams for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 504, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 504, + 507 + ], + "score": 1.0, + "content": "machine learning. 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Our second application is inspired by", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "Ramamurthy et al. (2019), who show that models perform better on tasks if they have topologically", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "similar decision boundaries. We use our algorithm to cluster models and tasks by the persistence", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "diagrams of their decision boundaries. Not only is our algorithm able to successfully cluster models", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "to the correct task, based just on the topology of its decision boundary, but we show that higher", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 357, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 357, + 160 + ], + "score": 1.0, + "content": "membership values imply better performance on unseen tasks.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 588, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "most likely lattices assigned to a cluster. This is particularly important in materials science, as further", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "investigation requires expensive laboratory time and expertise. Our second application is inspired by", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "Ramamurthy et al. (2019), who show that models perform better on tasks if they have topologically", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "similar decision boundaries. 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Our work relies on the existence of statistics in the space of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 250, + 219 + ], + "score": 1.0, + "content": "persistence diagrams. Mileyko et al.", + "type": "text" + }, + { + "bbox": [ + 250, + 207, + 277, + 220 + ], + "score": 0.26, + "content": "\\mathbb { Q } 0 1 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "first showed that means and expectations are well-defined", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "in the space of persistence diagrams. Specifically, they showed that the Fréchet mean, an extension", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "score": 1.0, + "content": "of means onto metric spaces, is well-defined under weak assumptions on the space of persistence", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "diagrams. Turner et al. (2012) then developed an algorithm to compute the Fréchet mean. We adapt", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "the algorithm by Turner et al. to the weighted case, but the combinatoric nature of their algorithm", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 315, + 275 + ], + "score": 1.0, + "content": "makes it computationally intense. Lacombe et al.", + "type": "text" + }, + { + "bbox": [ + 315, + 262, + 343, + 275 + ], + "score": 0.48, + "content": "\\textcircled { 2 0 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "framed the computation of means and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "barycentres in the space of persistence diagram as an optimal transport problem, allowing them to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 284, + 507, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 218, + 297 + ], + "score": 1.0, + "content": "use the Sinkhorn algorithm", + "type": "text" + }, + { + "bbox": [ + 218, + 284, + 318, + 297 + ], + "score": 0.26, + "content": "( \\overline { { \\mathbb { C } \\mathrm { u t u r i } \\ \\& \\ \\mathrm { D o u c e t } } } , \\overline { { 2 0 1 4 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 284, + 507, + 297 + ], + "score": 1.0, + "content": "for fast computation of approximate solutions.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "The vectorisation of the diagram required by the algorithm by Lacombe et al. makes it unsuitable for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "integration into our work, as we remain in the space of persistence diagrams. Techniques to speed up", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 477, + 330 + ], + "score": 1.0, + "content": "the matching problem fundamental to our computation have also been proposed by Vidal et al.", + "type": "text" + }, + { + "bbox": [ + 478, + 317, + 505, + 330 + ], + "score": 0.25, + "content": "\\dot { \\overline { { ( 2 0 2 0 ) } } }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 327, + 204, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 204, + 341 + ], + "score": 1.0, + "content": "and Kerber et al. (2017).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 506, + 521 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "Learning with persistence-based summaries. Integrating diagrams into machine learning work-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "flows remained challenging even with well-defined means, as the space is non-Hilbertian (Turner &", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 107, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "Spreemann, 2019). As such, efforts have been made to map diagrams into a Hilbert space; primarily", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "either by embedding into finite feature vectors (Kališnik, 2018; Fabio & Ferri, 2015; Chepushtanova", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "et al., 2015) or functional summaries (Bubenik, 2015; Rieck et al., 2019), or by defining a positive-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 399, + 504, + 412 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 504, + 412 + ], + "score": 1.0, + "content": "definite kernel on diagram space (Reininghaus et al., 2015; Carrière et al., 2017; Le & Yamada, 2018).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "These vectorisations have been integrated into deep learning either by learning parameters for the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "embedding (Hofer et al., 2017; Carrière et al., 2020; Kim et al., 2020; Zhao & Wang, 2019; Zielinski ´", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 432, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 504, + 446 + ], + "score": 1.0, + "content": "et al., 2019), or as part of a topological loss or regulariser (Chen et al., 2018; Gabrielsson et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "2020; Clough et al., 2020; Moor et al., 2019). However, the embeddings used in these techniques", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "deform the metric structure of persistence diagram space (Bubenik & Wagner, 2019; Wagner, 2019;", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "score": 1.0, + "content": "Carrière & Bauer, 2019), potentially leading to the loss of important information. Furthermore, these", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "techniques generally require prior knowledge of a ‘correct’ target topology which cannot plausibly", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "be known in most scenarios. 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The primary advantages of our work over previous", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 569, + 263, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 263, + 582 + ], + "score": 1.0, + "content": "work on hard clustering are as follows.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 110, + 593, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 113, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 113, + 594, + 497, + 606 + ], + "score": 1.0, + "content": "(i) The probabilistic membership values allow us to rank datasets in the cluster, enabling top-", + "type": "text" + }, + { + "bbox": [ + 497, + 594, + 504, + 604 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 127, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 127, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "candidate selection in settings where verifying correctness is expensive. The value provided by", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 127, + 614, + 363, + 628 + ], + "spans": [ + { + "bbox": [ + 127, + 614, + 363, + 628 + ], + "score": 1.0, + "content": "this fuzzy information is demonstrated in the experiments.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 110, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 110, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "(ii) The fuzzy membership values provide information about proximity to all clusters, whereas hard", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 128, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 128, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "labelling loses most of that information. In our experiments we demonstrate that this additional", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 128, + 654, + 285, + 667 + ], + "spans": [ + { + "bbox": [ + 128, + 654, + 285, + 667 + ], + "score": 1.0, + "content": "information can be utilised in practice.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 108, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 108, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "(iii) The weighted cost function makes the convergence analysis (which we provide) entirely non-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 128, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 128, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "trivial in comparison to the non-fuzzy case. We consider this convergence analysis a primary", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 128, + 693, + 232, + 705 + ], + "spans": [ + { + "bbox": [ + 128, + 693, + 232, + 705 + ], + "score": 1.0, + "content": "contribution of our paper.", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 109, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 109, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "(iv) Fuzzy membership values have been shown to be more robust to noise than discrete labels", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 130, + 720, + 198, + 733 + ], + "spans": [ + { + "bbox": [ + 130, + 720, + 198, + 733 + ], + "score": 1.0, + "content": "(Klawonn, 2004).", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 47 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 160 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 104, + 82, + 506, + 160 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 175, + 200, + 186 + ], + "lines": [ + { + "bbox": [ + 106, + 174, + 202, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 202, + 188 + ], + "score": 1.0, + "content": "1.1 RELATED WORK", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 196, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "Means of persistence diagrams. Our work relies on the existence of statistics in the space of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 207, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 250, + 219 + ], + "score": 1.0, + "content": "persistence diagrams. Mileyko et al.", + "type": "text" + }, + { + "bbox": [ + 250, + 207, + 277, + 220 + ], + "score": 0.26, + "content": "\\mathbb { Q } 0 1 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "first showed that means and expectations are well-defined", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "in the space of persistence diagrams. Specifically, they showed that the Fréchet mean, an extension", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "score": 1.0, + "content": "of means onto metric spaces, is well-defined under weak assumptions on the space of persistence", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "diagrams. Turner et al. (2012) then developed an algorithm to compute the Fréchet mean. We adapt", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 251, + 505, + 263 + ], + "score": 1.0, + "content": "the algorithm by Turner et al. to the weighted case, but the combinatoric nature of their algorithm", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 315, + 275 + ], + "score": 1.0, + "content": "makes it computationally intense. Lacombe et al.", + "type": "text" + }, + { + "bbox": [ + 315, + 262, + 343, + 275 + ], + "score": 0.48, + "content": "\\textcircled { 2 0 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "framed the computation of means and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "barycentres in the space of persistence diagram as an optimal transport problem, allowing them to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 284, + 507, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 218, + 297 + ], + "score": 1.0, + "content": "use the Sinkhorn algorithm", + "type": "text" + }, + { + "bbox": [ + 218, + 284, + 318, + 297 + ], + "score": 0.26, + "content": "( \\overline { { \\mathbb { C } \\mathrm { u t u r i } \\ \\& \\ \\mathrm { D o u c e t } } } , \\overline { { 2 0 1 4 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 284, + 507, + 297 + ], + "score": 1.0, + "content": "for fast computation of approximate solutions.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "The vectorisation of the diagram required by the algorithm by Lacombe et al. makes it unsuitable for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "integration into our work, as we remain in the space of persistence diagrams. Techniques to speed up", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 477, + 330 + ], + "score": 1.0, + "content": "the matching problem fundamental to our computation have also been proposed by Vidal et al.", + "type": "text" + }, + { + "bbox": [ + 478, + 317, + 505, + 330 + ], + "score": 0.25, + "content": "\\dot { \\overline { { ( 2 0 2 0 ) } } }", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 327, + 204, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 204, + 341 + ], + "score": 1.0, + "content": "and Kerber et al. (2017).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 196, + 507, + 341 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 345, + 506, + 521 + ], + "lines": [ + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "Learning with persistence-based summaries. Integrating diagrams into machine learning work-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "flows remained challenging even with well-defined means, as the space is non-Hilbertian (Turner &", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 107, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "Spreemann, 2019). As such, efforts have been made to map diagrams into a Hilbert space; primarily", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "either by embedding into finite feature vectors (Kališnik, 2018; Fabio & Ferri, 2015; Chepushtanova", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 401 + ], + "score": 1.0, + "content": "et al., 2015) or functional summaries (Bubenik, 2015; Rieck et al., 2019), or by defining a positive-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 399, + 504, + 412 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 504, + 412 + ], + "score": 1.0, + "content": "definite kernel on diagram space (Reininghaus et al., 2015; Carrière et al., 2017; Le & Yamada, 2018).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "These vectorisations have been integrated into deep learning either by learning parameters for the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "score": 1.0, + "content": "embedding (Hofer et al., 2017; Carrière et al., 2020; Kim et al., 2020; Zhao & Wang, 2019; Zielinski ´", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 432, + 504, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 504, + 446 + ], + "score": 1.0, + "content": "et al., 2019), or as part of a topological loss or regulariser (Chen et al., 2018; Gabrielsson et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "2020; Clough et al., 2020; Moor et al., 2019). However, the embeddings used in these techniques", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 468 + ], + "score": 1.0, + "content": "deform the metric structure of persistence diagram space (Bubenik & Wagner, 2019; Wagner, 2019;", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "score": 1.0, + "content": "Carrière & Bauer, 2019), potentially leading to the loss of important information. Furthermore, these", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "techniques generally require prior knowledge of a ‘correct’ target topology which cannot plausibly", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "be known in most scenarios. In comparison, our algorithm acts in the space of persistence diagrams", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 511 + ], + "score": 1.0, + "content": "so it does not deform the structure of diagram space via embeddings, and is entirely unsupervised,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 508, + 305, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 305, + 523 + ], + "score": 1.0, + "content": "requiring no prior knowledge about the topology.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 28.5, + "bbox_fs": [ + 104, + 344, + 506, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 526, + 505, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 539 + ], + "score": 1.0, + "content": "Hard clustering. Maroulas et al. (2017) gave an algorithm for hard clustering persistence diagrams", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 537, + 504, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 269, + 550 + ], + "score": 1.0, + "content": "based on the algorithm by Turner et al.", + "type": "text" + }, + { + "bbox": [ + 272, + 537, + 336, + 550 + ], + "score": 1.0, + "content": "Lacombe et al.", + "type": "text" + }, + { + "bbox": [ + 336, + 537, + 364, + 550 + ], + "score": 0.26, + "content": "\\textcircled { 2 0 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 537, + 504, + 550 + ], + "score": 1.0, + "content": "gave an alternate implementation", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 548, + 504, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 504, + 560 + ], + "score": 1.0, + "content": "of hard clustering based on their algorithm for barycentre computation, providing a computational", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "speed-up over previous the work by Maroulas et al. The primary advantages of our work over previous", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 569, + 263, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 263, + 582 + ], + "score": 1.0, + "content": "work on hard clustering are as follows.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 525, + 505, + 582 + ] + }, + { + "type": "list", + "bbox": [ + 110, + 593, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 113, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 113, + 594, + 497, + 606 + ], + "score": 1.0, + "content": "(i) The probabilistic membership values allow us to rank datasets in the cluster, enabling top-", + "type": "text" + }, + { + "bbox": [ + 497, + 594, + 504, + 604 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 127, + 603, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 127, + 603, + 505, + 618 + ], + "score": 1.0, + "content": "candidate selection in settings where verifying correctness is expensive. The value provided by", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 127, + 614, + 363, + 628 + ], + "spans": [ + { + "bbox": [ + 127, + 614, + 363, + 628 + ], + "score": 1.0, + "content": "this fuzzy information is demonstrated in the experiments.", + "type": "text" + } + ], + "index": 44, + "is_list_end_line": true + }, + { + "bbox": [ + 110, + 631, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 110, + 631, + 506, + 645 + ], + "score": 1.0, + "content": "(ii) The fuzzy membership values provide information about proximity to all clusters, whereas hard", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 128, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 128, + 643, + 505, + 655 + ], + "score": 1.0, + "content": "labelling loses most of that information. In our experiments we demonstrate that this additional", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 128, + 654, + 285, + 667 + ], + "spans": [ + { + "bbox": [ + 128, + 654, + 285, + 667 + ], + "score": 1.0, + "content": "information can be utilised in practice.", + "type": "text" + } + ], + "index": 47, + "is_list_end_line": true + }, + { + "bbox": [ + 108, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 108, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "(iii) The weighted cost function makes the convergence analysis (which we provide) entirely non-", + "type": "text" + } + ], + "index": 48, + "is_list_start_line": true + }, + { + "bbox": [ + 128, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 128, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "trivial in comparison to the non-fuzzy case. We consider this convergence analysis a primary", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 128, + 693, + 232, + 705 + ], + "spans": [ + { + "bbox": [ + 128, + 693, + 232, + 705 + ], + "score": 1.0, + "content": "contribution of our paper.", + "type": "text" + } + ], + "index": 50, + "is_list_end_line": true + }, + { + "bbox": [ + 109, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 109, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "(iv) Fuzzy membership values have been shown to be more robust to noise than discrete labels", + "type": "text" + } + ], + "index": 51, + "is_list_start_line": true + }, + { + "bbox": [ + 130, + 720, + 198, + 733 + ], + "spans": [ + { + "bbox": [ + 130, + 720, + 198, + 733 + ], + "score": 1.0, + "content": "(Klawonn, 2004).", + "type": "text" + } + ], + "index": 52, + "is_list_end_line": true + } + ], + "index": 47, + "bbox_fs": [ + 108, + 594, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 109, + 82, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 111, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 111, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "(v) Unlike hard clustering, fuzzy clustering is analytically differentiable, allowing integration of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 128, + 93, + 422, + 106 + ], + "spans": [ + { + "bbox": [ + 128, + 93, + 422, + 106 + ], + "score": 1.0, + "content": "the fuzzy clustering step into deep learning methods (Wilder et al., 2019).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 115, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "Geometric equivalents. The most similar unsupervised learning technique to our algorithm is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "Wasserstein Barycentre Clustering (WBC). It clusters datasets of point clouds by the Wasserstein", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "distance between the point clouds, rather than the Wasserstein distance between their persistence", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 433, + 162 + ], + "score": 1.0, + "content": "diagrams. We compare our algorithm experimentally to WBC using ADMM", + "type": "text" + }, + { + "bbox": [ + 434, + 148, + 502, + 161 + ], + "score": 0.34, + "content": "\\pmb { \\mathbb { T e } } \\ \\& \\ \\bot \\pmb { \\mathbb { I } } \\ \\boxed { 2 0 1 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 148, + 505, + 162 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "Bregman ADMM (Ye et al., 2017), Subgradient Descent (Cuturi & Doucet, 2014), Iterative Bregman", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 217, + 183 + ], + "score": 1.0, + "content": "Projection (Benamou et al.,", + "type": "text" + }, + { + "bbox": [ + 218, + 169, + 366, + 183 + ], + "score": 1.0, + "content": "2015), and full linear programming", + "type": "text" + }, + { + "bbox": [ + 366, + 171, + 444, + 183 + ], + "score": 0.28, + "content": "\\mathbb { \\overline { { \\mathbb { L } i \\ \\& \\ \\operatorname { W a n g } } } } , \\mathbb { \\underline { { 2 0 0 8 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 169, + 506, + 183 + ], + "score": 1.0, + "content": ". Each of these", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "algorithms computes or approximates the Wasserstein barycentre in different ways. Theoretically,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 367, + 205 + ], + "score": 1.0, + "content": "fuzzy discrete distribution clustering (d. A. T. de Carvalho et al.,", + "type": "text" + }, + { + "bbox": [ + 368, + 192, + 394, + 205 + ], + "score": 0.37, + "content": "\\textcircled { 2 0 1 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "is similar to our algorithm,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 202, + 441, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 441, + 216 + ], + "score": 1.0, + "content": "but the addition of the diagonal in the persistence diagram makes our work distinct.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 229, + 224, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 226, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 226, + 241 + ], + "score": 1.0, + "content": "1.2 OUR CONTRIBUTIONS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 104, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "1. Our main contribution is an algorithm for Fuzzy c-Means clustering of persistence diagrams,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 439, + 273 + ], + "score": 1.0, + "content": "along with the convergence analysis. Given a collection of persistence diagrams", + "type": "text" + }, + { + "bbox": [ + 439, + 261, + 486, + 272 + ], + "score": 0.92, + "content": "\\mathbb { D } _ { 1 } , \\ldots , \\mathbb { D } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 261, + 505, + 273 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 254, + 285 + ], + "score": 1.0, + "content": "alternatively calculate cluster centres", + "type": "text" + }, + { + "bbox": [ + 254, + 272, + 306, + 283 + ], + "score": 0.93, + "content": "\\mathbb { M } _ { 1 } , \\ldots , \\mathbb { M } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 271, + 402, + 285 + ], + "score": 1.0, + "content": "and membership values", + "type": "text" + }, + { + "bbox": [ + 402, + 272, + 448, + 284 + ], + "score": 0.9, + "content": "r _ { j k } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "which denote", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 283, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 223, + 295 + ], + "score": 1.0, + "content": "the degree to which diagram", + "type": "text" + }, + { + "bbox": [ + 224, + 284, + 236, + 295 + ], + "score": 0.89, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 283, + 340, + 295 + ], + "score": 1.0, + "content": "is associated with cluster", + "type": "text" + }, + { + "bbox": [ + 341, + 283, + 356, + 294 + ], + "score": 0.88, + "content": "\\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 283, + 439, + 295 + ], + "score": 1.0, + "content": ". We prove Theorem", + "type": "text" + }, + { + "bbox": [ + 439, + 283, + 450, + 297 + ], + "score": 0.62, + "content": "^ { 1 , }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "showing that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 294, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 305 + ], + "score": 1.0, + "content": "every convergent subsequence of these alternative update steps tends to a local minimum or saddle", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "point of the cost function. This is the same convergence guarantee provided by traditional FCM", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 149, + 328 + ], + "score": 1.0, + "content": "clustering", + "type": "text" + }, + { + "bbox": [ + 149, + 315, + 235, + 329 + ], + "score": 0.33, + "content": "( \\mathbf { \\mathbb { B } e z d e k \\ e t \\ a l . } ] \\ [ \\mathbf { \\mathbb { 9 } 8 7 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 315, + 505, + 328 + ], + "score": 1.0, + "content": ", but requires additional work as the space of persistence diagrams", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 462, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 462, + 339 + ], + "score": 1.0, + "content": "with the Wasserstein distance has far weaker theoretical properties than Euclidean space.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 505, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "2. Updating the cluster centres requires computing the weighted Fréchet mean. We extend the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 185, + 367 + ], + "score": 1.0, + "content": "algorithm given by", + "type": "text" + }, + { + "bbox": [ + 185, + 354, + 266, + 367 + ], + "score": 0.62, + "content": "\\boxed { \\mathrm { T u r n e r ~ e t ~ a l . } } \\textcircled { 2 0 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "to the weighted case, justifying our addition of weights by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 365, + 379, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 379, + 379 + ], + "score": 1.0, + "content": "extending their proof to show that the updated algorithm converges.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "3. We implement our algorithm in Python, available in the supplementary materials. It works with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "persistence diagrams from commonly used open-source libraries for Topological Data Analysis", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 144, + 417 + ], + "score": 0.26, + "content": "\\mathbf { \\Gamma } _ { \\mathrm { ( T D A ) , } } ^ { \\mathrm { ~ \\tiny ~ 1 ~ } } \\mathfrak { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "o is available for easy integration into current workflows, offering a powerful unsupervised", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 344, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 344, + 427 + ], + "score": 1.0, + "content": "learning algorithm to data science practitioners using TDA.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "4. We demonstrate the application of our algorithm to settings where (i) the properties of persistence", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "diagrams makes clustering them the natural choice over geometric equivalents and (ii) the probabilistic", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 351, + 467 + ], + "score": 1.0, + "content": "membership values can be used to rank candidates for top-", + "type": "text" + }, + { + "bbox": [ + 352, + 455, + 358, + 464 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "selection. Our algorithm classifies", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 464, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 479 + ], + "score": 1.0, + "content": "transformed lattice structures from materials science where geometric equivalents fail, whilst giving", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "probabilistic rankings to help prioritise expensive further investigation. We also cluster the persistence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "diagrams of decision boundaries and labelled datasets, showing that our fuzzy clustering captures", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 499, + 328, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 328, + 510 + ], + "score": 1.0, + "content": "information about model performance on unseen tasks.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 527, + 284, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 286, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 286, + 541 + ], + "score": 1.0, + "content": "2 TOPOLOGICAL PRELIMINARIES", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 506, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "Topological Data Analysis emerged from the study of algebraic topology, providing a toolkit to fully", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 562, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 575 + ], + "score": 1.0, + "content": "describe the topology of a dataset. We offer a quick summary below; for more comprehensive details", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 120, + 585 + ], + "score": 1.0, + "content": "see", + "type": "text" + }, + { + "bbox": [ + 121, + 573, + 211, + 586 + ], + "score": 1.0, + "content": "Edelsbrunner & Harer", + "type": "text" + }, + { + "bbox": [ + 212, + 572, + 238, + 586 + ], + "score": 0.29, + "content": "\\bigoplus \\iiiint \\iiiint \\lVert 0 1 0 \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 573, + 313, + 586 + ], + "score": 1.0, + "content": ". A set of points in", + "type": "text" + }, + { + "bbox": [ + 313, + 573, + 326, + 584 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "are indicative of the shape of the distribution", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 388, + 597 + ], + "score": 1.0, + "content": "they are sampled from. By connecting points that are pairwise within", + "type": "text" + }, + { + "bbox": [ + 388, + 585, + 412, + 595 + ], + "score": 0.9, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "distance of each other,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 595, + 504, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 440, + 608 + ], + "score": 1.0, + "content": "we can create an approximation of the distribution called the Vietoris-Rips complex", + "type": "text" + }, + { + "bbox": [ + 440, + 595, + 503, + 608 + ], + "score": 0.75, + "content": "\\mathrm { ( V i e t o r i s , \\textmu 9 2 7 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 595, + 504, + 608 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 608, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 475, + 619 + ], + "score": 1.0, + "content": "Specifically, we add the convex hull of any collection of points that are pairwise at most", + "type": "text" + }, + { + "bbox": [ + 475, + 609, + 481, + 617 + ], + "score": 0.57, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 608, + 506, + 619 + ], + "score": 1.0, + "content": "apart", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 617, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 131, + 630 + ], + "score": 1.0, + "content": "to the", + "type": "text" + }, + { + "bbox": [ + 132, + 619, + 137, + 628 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 617, + 328, + 630 + ], + "score": 1.0, + "content": "-Vietoris-Rips complex. However, choosing an", + "type": "text" + }, + { + "bbox": [ + 329, + 619, + 335, + 628 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 617, + 506, + 630 + ], + "score": 1.0, + "content": "remains problematic; too low a value and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "key points can remain disconnected, too high a value and the points become fully connected. To", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 436, + 652 + ], + "score": 1.0, + "content": "overcome this we use persistence: we consider the approximation over all values of", + "type": "text" + }, + { + "bbox": [ + 436, + 641, + 442, + 649 + ], + "score": 0.63, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 639, + 506, + 652 + ], + "score": 1.0, + "content": "simultaneously,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 348, + 663 + ], + "score": 1.0, + "content": "and study how the topology of that approximation evolves as", + "type": "text" + }, + { + "bbox": [ + 348, + 653, + 354, + 660 + ], + "score": 0.48, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "grows large. We call the collection of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 662, + 230, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 177, + 673 + ], + "score": 1.0, + "content": "complexes for all", + "type": "text" + }, + { + "bbox": [ + 178, + 663, + 183, + 671 + ], + "score": 0.67, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 662, + 230, + 673 + ], + "score": 1.0, + "content": "a filtration.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 104, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 145, + 691 + ], + "score": 1.0, + "content": "For each", + "type": "text" + }, + { + "bbox": [ + 145, + 681, + 150, + 688 + ], + "score": 0.26, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 677, + 223, + 691 + ], + "score": 1.0, + "content": ", we compute the", + "type": "text" + }, + { + "bbox": [ + 223, + 680, + 229, + 690 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 677, + 441, + 691 + ], + "score": 1.0, + "content": "-homology group. This tells us the topology of the", + "type": "text" + }, + { + "bbox": [ + 441, + 680, + 446, + 688 + ], + "score": 0.72, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "-Vietoris-Rips", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 690, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 690, + 505, + 701 + ], + "score": 1.0, + "content": "complex: the 0-homology counts the number of connected components, the 1-homology counts", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 699, + 503, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 503, + 713 + ], + "score": 1.0, + "content": "the number of holes, the 2-homology counts the number of voids, etc. (Edelsbrunner et al., 2000)", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 120, + 719, + 253, + 734 + ], + "lines": [ + { + "bbox": [ + 121, + 718, + 251, + 736 + ], + "spans": [ + { + "bbox": [ + 121, + 718, + 251, + 736 + ], + "score": 1.0, + "content": "1 Dionysus and Ripser (Bauer, 2019).", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review 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": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 109, + 82, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 111, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 111, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "(v) Unlike hard clustering, fuzzy clustering is analytically differentiable, allowing integration of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 128, + 93, + 422, + 106 + ], + "spans": [ + { + "bbox": [ + 128, + 93, + 422, + 106 + ], + "score": 1.0, + "content": "the fuzzy clustering step into deep learning methods (Wilder et al., 2019).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 111, + 81, + 506, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 115, + 505, + 215 + ], + "lines": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "Geometric equivalents. The most similar unsupervised learning technique to our algorithm is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "Wasserstein Barycentre Clustering (WBC). It clusters datasets of point clouds by the Wasserstein", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "distance between the point clouds, rather than the Wasserstein distance between their persistence", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 433, + 162 + ], + "score": 1.0, + "content": "diagrams. We compare our algorithm experimentally to WBC using ADMM", + "type": "text" + }, + { + "bbox": [ + 434, + 148, + 502, + 161 + ], + "score": 0.34, + "content": "\\pmb { \\mathbb { T e } } \\ \\& \\ \\bot \\pmb { \\mathbb { I } } \\ \\boxed { 2 0 1 4 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 148, + 505, + 162 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "Bregman ADMM (Ye et al., 2017), Subgradient Descent (Cuturi & Doucet, 2014), Iterative Bregman", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 217, + 183 + ], + "score": 1.0, + "content": "Projection (Benamou et al.,", + "type": "text" + }, + { + "bbox": [ + 218, + 169, + 366, + 183 + ], + "score": 1.0, + "content": "2015), and full linear programming", + "type": "text" + }, + { + "bbox": [ + 366, + 171, + 444, + 183 + ], + "score": 0.28, + "content": "\\mathbb { \\overline { { \\mathbb { L } i \\ \\& \\ \\operatorname { W a n g } } } } , \\mathbb { \\underline { { 2 0 0 8 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 169, + 506, + 183 + ], + "score": 1.0, + "content": ". Each of these", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "algorithms computes or approximates the Wasserstein barycentre in different ways. Theoretically,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 367, + 205 + ], + "score": 1.0, + "content": "fuzzy discrete distribution clustering (d. A. T. de Carvalho et al.,", + "type": "text" + }, + { + "bbox": [ + 368, + 192, + 394, + 205 + ], + "score": 0.37, + "content": "\\textcircled { 2 0 1 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "is similar to our algorithm,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 202, + 441, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 441, + 216 + ], + "score": 1.0, + "content": "but the addition of the diagonal in the persistence diagram makes our work distinct.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 115, + 506, + 216 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 229, + 224, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 226, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 226, + 241 + ], + "score": 1.0, + "content": "1.2 OUR CONTRIBUTIONS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 106, + 250, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 104, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "1. Our main contribution is an algorithm for Fuzzy c-Means clustering of persistence diagrams,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 439, + 273 + ], + "score": 1.0, + "content": "along with the convergence analysis. Given a collection of persistence diagrams", + "type": "text" + }, + { + "bbox": [ + 439, + 261, + 486, + 272 + ], + "score": 0.92, + "content": "\\mathbb { D } _ { 1 } , \\ldots , \\mathbb { D } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 261, + 505, + 273 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 254, + 285 + ], + "score": 1.0, + "content": "alternatively calculate cluster centres", + "type": "text" + }, + { + "bbox": [ + 254, + 272, + 306, + 283 + ], + "score": 0.93, + "content": "\\mathbb { M } _ { 1 } , \\ldots , \\mathbb { M } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 271, + 402, + 285 + ], + "score": 1.0, + "content": "and membership values", + "type": "text" + }, + { + "bbox": [ + 402, + 272, + 448, + 284 + ], + "score": 0.9, + "content": "r _ { j k } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "which denote", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 283, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 223, + 295 + ], + "score": 1.0, + "content": "the degree to which diagram", + "type": "text" + }, + { + "bbox": [ + 224, + 284, + 236, + 295 + ], + "score": 0.89, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 283, + 340, + 295 + ], + "score": 1.0, + "content": "is associated with cluster", + "type": "text" + }, + { + "bbox": [ + 341, + 283, + 356, + 294 + ], + "score": 0.88, + "content": "\\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 283, + 439, + 295 + ], + "score": 1.0, + "content": ". We prove Theorem", + "type": "text" + }, + { + "bbox": [ + 439, + 283, + 450, + 297 + ], + "score": 0.62, + "content": "^ { 1 , }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 283, + 505, + 295 + ], + "score": 1.0, + "content": "showing that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 294, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 305 + ], + "score": 1.0, + "content": "every convergent subsequence of these alternative update steps tends to a local minimum or saddle", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 318 + ], + "score": 1.0, + "content": "point of the cost function. This is the same convergence guarantee provided by traditional FCM", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 149, + 328 + ], + "score": 1.0, + "content": "clustering", + "type": "text" + }, + { + "bbox": [ + 149, + 315, + 235, + 329 + ], + "score": 0.33, + "content": "( \\mathbf { \\mathbb { B } e z d e k \\ e t \\ a l . } ] \\ [ \\mathbf { \\mathbb { 9 } 8 7 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 315, + 505, + 328 + ], + "score": 1.0, + "content": ", but requires additional work as the space of persistence diagrams", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 462, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 462, + 339 + ], + "score": 1.0, + "content": "with the Wasserstein distance has far weaker theoretical properties than Euclidean space.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 249, + 506, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 344, + 505, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "2. Updating the cluster centres requires computing the weighted Fréchet mean. We extend the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 354, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 185, + 367 + ], + "score": 1.0, + "content": "algorithm given by", + "type": "text" + }, + { + "bbox": [ + 185, + 354, + 266, + 367 + ], + "score": 0.62, + "content": "\\boxed { \\mathrm { T u r n e r ~ e t ~ a l . } } \\textcircled { 2 0 1 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 354, + 505, + 367 + ], + "score": 1.0, + "content": "to the weighted case, justifying our addition of weights by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 365, + 379, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 379, + 379 + ], + "score": 1.0, + "content": "extending their proof to show that the updated algorithm converges.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 343, + 505, + 379 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 382, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "3. We implement our algorithm in Python, available in the supplementary materials. It works with", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 505, + 406 + ], + "score": 1.0, + "content": "persistence diagrams from commonly used open-source libraries for Topological Data Analysis", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 144, + 417 + ], + "score": 0.26, + "content": "\\mathbf { \\Gamma } _ { \\mathrm { ( T D A ) , } } ^ { \\mathrm { ~ \\tiny ~ 1 ~ } } \\mathfrak { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 405, + 506, + 417 + ], + "score": 1.0, + "content": "o is available for easy integration into current workflows, offering a powerful unsupervised", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 416, + 344, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 344, + 427 + ], + "score": 1.0, + "content": "learning algorithm to data science practitioners using TDA.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 383, + 506, + 427 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 445 + ], + "score": 1.0, + "content": "4. We demonstrate the application of our algorithm to settings where (i) the properties of persistence", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "diagrams makes clustering them the natural choice over geometric equivalents and (ii) the probabilistic", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 351, + 467 + ], + "score": 1.0, + "content": "membership values can be used to rank candidates for top-", + "type": "text" + }, + { + "bbox": [ + 352, + 455, + 358, + 464 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "selection. Our algorithm classifies", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 464, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 479 + ], + "score": 1.0, + "content": "transformed lattice structures from materials science where geometric equivalents fail, whilst giving", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "probabilistic rankings to help prioritise expensive further investigation. We also cluster the persistence", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "diagrams of decision boundaries and labelled datasets, showing that our fuzzy clustering captures", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 499, + 328, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 328, + 510 + ], + "score": 1.0, + "content": "information about model performance on unseen tasks.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 432, + 505, + 510 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 527, + 284, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 286, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 286, + 541 + ], + "score": 1.0, + "content": "2 TOPOLOGICAL PRELIMINARIES", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 506, + 673 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "Topological Data Analysis emerged from the study of algebraic topology, providing a toolkit to fully", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 562, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 506, + 575 + ], + "score": 1.0, + "content": "describe the topology of a dataset. We offer a quick summary below; for more comprehensive details", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 572, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 120, + 585 + ], + "score": 1.0, + "content": "see", + "type": "text" + }, + { + "bbox": [ + 121, + 573, + 211, + 586 + ], + "score": 1.0, + "content": "Edelsbrunner & Harer", + "type": "text" + }, + { + "bbox": [ + 212, + 572, + 238, + 586 + ], + "score": 0.29, + "content": "\\bigoplus \\iiiint \\iiiint \\lVert 0 1 0 \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 573, + 313, + 586 + ], + "score": 1.0, + "content": ". A set of points in", + "type": "text" + }, + { + "bbox": [ + 313, + 573, + 326, + 584 + ], + "score": 0.89, + "content": "\\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "are indicative of the shape of the distribution", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 585, + 388, + 597 + ], + "score": 1.0, + "content": "they are sampled from. By connecting points that are pairwise within", + "type": "text" + }, + { + "bbox": [ + 388, + 585, + 412, + 595 + ], + "score": 0.9, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "distance of each other,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 595, + 504, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 440, + 608 + ], + "score": 1.0, + "content": "we can create an approximation of the distribution called the Vietoris-Rips complex", + "type": "text" + }, + { + "bbox": [ + 440, + 595, + 503, + 608 + ], + "score": 0.75, + "content": "\\mathrm { ( V i e t o r i s , \\textmu 9 2 7 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 595, + 504, + 608 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 608, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 475, + 619 + ], + "score": 1.0, + "content": "Specifically, we add the convex hull of any collection of points that are pairwise at most", + "type": "text" + }, + { + "bbox": [ + 475, + 609, + 481, + 617 + ], + "score": 0.57, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 608, + 506, + 619 + ], + "score": 1.0, + "content": "apart", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 617, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 131, + 630 + ], + "score": 1.0, + "content": "to the", + "type": "text" + }, + { + "bbox": [ + 132, + 619, + 137, + 628 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 617, + 328, + 630 + ], + "score": 1.0, + "content": "-Vietoris-Rips complex. However, choosing an", + "type": "text" + }, + { + "bbox": [ + 329, + 619, + 335, + 628 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 617, + 506, + 630 + ], + "score": 1.0, + "content": "remains problematic; too low a value and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "key points can remain disconnected, too high a value and the points become fully connected. To", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 436, + 652 + ], + "score": 1.0, + "content": "overcome this we use persistence: we consider the approximation over all values of", + "type": "text" + }, + { + "bbox": [ + 436, + 641, + 442, + 649 + ], + "score": 0.63, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 639, + 506, + 652 + ], + "score": 1.0, + "content": "simultaneously,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 348, + 663 + ], + "score": 1.0, + "content": "and study how the topology of that approximation evolves as", + "type": "text" + }, + { + "bbox": [ + 348, + 653, + 354, + 660 + ], + "score": 0.48, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "grows large. We call the collection of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 662, + 230, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 662, + 177, + 673 + ], + "score": 1.0, + "content": "complexes for all", + "type": "text" + }, + { + "bbox": [ + 178, + 663, + 183, + 671 + ], + "score": 0.67, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 662, + 230, + 673 + ], + "score": 1.0, + "content": "a filtration.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 551, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 104, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 145, + 691 + ], + "score": 1.0, + "content": "For each", + "type": "text" + }, + { + "bbox": [ + 145, + 681, + 150, + 688 + ], + "score": 0.26, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 677, + 223, + 691 + ], + "score": 1.0, + "content": ", we compute the", + "type": "text" + }, + { + "bbox": [ + 223, + 680, + 229, + 690 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 677, + 441, + 691 + ], + "score": 1.0, + "content": "-homology group. This tells us the topology of the", + "type": "text" + }, + { + "bbox": [ + 441, + 680, + 446, + 688 + ], + "score": 0.72, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "-Vietoris-Rips", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 107, + 690, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 690, + 505, + 701 + ], + "score": 1.0, + "content": "complex: the 0-homology counts the number of connected components, the 1-homology counts", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 699, + 503, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 503, + 713 + ], + "score": 1.0, + "content": "the number of holes, the 2-homology counts the number of voids, etc. (Edelsbrunner et al., 2000)", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47, + "bbox_fs": [ + 104, + 677, + 505, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 147, + 80, + 499, + 233 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 147, + 80, + 499, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 80, + 499, + 233 + ], + "spans": [ + { + "bbox": [ + 147, + 80, + 499, + 233 + ], + "score": 0.965, + "type": "image", + "image_path": "64ff89fa891ae43bb690e18da28297235b2205ad0db31d4eaf8f06107e1c4f55.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 147, + 80, + 499, + 131.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 147, + 131.0, + 499, + 182.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 147, + 182.0, + 499, + 233.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 243, + 506, + 288 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "Figure 1: An example Vietoris-Rips filtration with its persistence diagram. We only add convex hulls", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 206, + 267 + ], + "score": 1.0, + "content": "when points are pairwise", + "type": "text" + }, + { + "bbox": [ + 207, + 257, + 213, + 264 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 254, + 316, + 267 + ], + "score": 1.0, + "content": "apart, so there is a hole at", + "type": "text" + }, + { + "bbox": [ + 316, + 255, + 340, + 265 + ], + "score": 0.9, + "content": "\\epsilon = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "whilst the diagonals of the square are not", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "close enough to fill the hole in. 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If a topological feature, such as a connected component or hole, persists throughout a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "large range of granularities, then it’s more likely to be a feature of the distribution. If it only persists", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "for a short amount of time, then it’s more likely to be noise (Cohen-Steiner et al., 2007). We can stably", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 370, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 133, + 385 + ], + "score": 1.0, + "content": "map a", + "type": "text" + }, + { + "bbox": [ + 133, + 373, + 140, + 383 + ], + "score": 0.66, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 370, + 506, + 385 + ], + "score": 1.0, + "content": "-PH group into a multiset in the extended plane called a persistence diagram (Chazal et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 381, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 131, + 395 + ], + "score": 0.56, + "content": "\\bar { 2 0 1 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 381, + 506, + 396 + ], + "score": 1.0, + "content": ". Each topological feature has a birth and death: a feature is born when it enters the complex,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "score": 1.0, + "content": "and dies when the complex grows enough to destroy it. For example, in Figure 1(a), a feature is born", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 116, + 416 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 405, + 140, + 415 + ], + "score": 0.9, + "content": "\\epsilon = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 405, + 340, + 416 + ], + "score": 1.0, + "content": "when four lines form a hole. This feature dies at", + "type": "text" + }, + { + "bbox": [ + 340, + 405, + 377, + 415 + ], + "score": 0.88, + "content": "\\epsilon = 1 . 4 2", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "when the hole is filled in. This", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 414, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 369, + 429 + ], + "score": 1.0, + "content": "is shown in the persistence diagram in Figure 1(b) as a point at", + "type": "text" + }, + { + "bbox": [ + 370, + 416, + 404, + 428 + ], + "score": 0.32, + "content": "( 1 , 1 . 4 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 414, + 506, + 429 + ], + "score": 1.0, + "content": "in 1-PH. By computing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "the birth/death points for each topological feature in the filtration, we get a complete picture of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 406, + 449 + ], + "score": 1.0, + "content": "topology of the point cloud at all granularities (Zomorodian & Carlsson,", + "type": "text" + }, + { + "bbox": [ + 406, + 437, + 432, + 448 + ], + "score": 0.36, + "content": "\\overline { { 2 0 0 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 438, + 505, + 450 + ], + "score": 1.0, + "content": ". The persistence", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 394, + 461 + ], + "score": 1.0, + "content": "diagram is the collection of birth/death points, along with the diagonal", + "type": "text" + }, + { + "bbox": [ + 395, + 448, + 483, + 460 + ], + "score": 0.92, + "content": "\\overline { { \\Delta } } \\equiv \\{ ( a , a ) : a \\in \\mathbb { R } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "infinite multiplicity, added in order to make the space of persistence diagrams complete (Mileyko", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 470, + 156, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 156, + 482 + ], + "score": 1.0, + "content": "et al., 2011).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 506, + 243, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 244, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 244, + 520 + ], + "score": 1.0, + "content": "3 ALGORITHMIC DESIGN", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 108, + 536, + 295, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 297, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 297, + 548 + ], + "score": 1.0, + "content": "3.1 CLUSTERING PERSISTENCE DIAGRAMS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 506, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "In order to cluster we need a distance on the space of persistence diagrams. We use the 2-Wasserstein", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 569, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 107, + 571, + 119, + 581 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 569, + 482, + 583 + ], + "score": 1.0, + "content": "metric as it is stable on persistence diagrams of finite point clouds (Chazal et al., 2012).", + "type": "text" + }, + { + "bbox": [ + 482, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "The", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "score": 1.0, + "content": "Wasserstein distance is an optimal transport metric that has found applications across machine", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "learning. In the Euclidean case, it quantifies the smallest distance between optimally matched points.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 603, + 264, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 172, + 615 + ], + "score": 1.0, + "content": "Given diagrams", + "type": "text" + }, + { + "bbox": [ + 172, + 604, + 201, + 615 + ], + "score": 0.92, + "content": "\\mathbb { D } _ { 1 } , \\mathbb { D } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 603, + 264, + 615 + ], + "score": 1.0, + "content": ", the distance is", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 627, + 407, + 664 + ], + "lines": [ + { + "bbox": [ + 203, + 627, + 407, + 664 + ], + "spans": [ + { + "bbox": [ + 203, + 627, + 407, + 664 + ], + "score": 0.94, + "content": "W _ { 2 } ( \\mathbb { D } _ { 1 } , \\mathbb { D } _ { 2 } ) = \\left( \\operatorname* { i n f } _ { \\gamma : \\mathbb { D } _ { 1 } \\to \\mathbb { D } _ { 2 } } \\sum _ { x \\in \\mathbb { D } _ { 1 } } | | x - \\gamma ( x ) | | _ { 2 } ^ { 2 } \\right) ^ { 1 / 2 } ,", + "type": "interline_equation", + "image_path": "044279cb12c1257d323f24efd8909559ccd2be79bc799137572edd50a868f363.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 627, + 407, + 645.5 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 203, + 645.5, + 407, + 664.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 287, + 688 + ], + "score": 1.0, + "content": "where the infimum is taken over all bijections", + "type": "text" + }, + { + "bbox": [ + 288, + 677, + 341, + 689 + ], + "score": 0.91, + "content": "\\gamma : { \\mathbb { D } } _ { 1 } \\to { \\mathbb { D } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 676, + 504, + 688 + ], + "score": 1.0, + "content": ". 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In fact, the closer a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 414, + 723 + ], + "score": 1.0, + "content": "point is to the diagonal, the more likely it is to be noise (Cohen-Steiner et al.,", + "type": "text" + }, + { + "bbox": [ + 415, + 709, + 440, + 722 + ], + "score": 0.35, + "content": "\\boxed { 2 0 0 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 709, + 506, + 723 + ], + "score": 1.0, + "content": ", so this ensures", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 721, + 281, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 281, + 733 + ], + "score": 1.0, + "content": "our distance is not overly affected by noise.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 306, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 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": "image", + "bbox": [ + 147, + 80, + 499, + 233 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 147, + 80, + 499, + 233 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 80, + 499, + 233 + ], + "spans": [ + { + "bbox": [ + 147, + 80, + 499, + 233 + ], + "score": 0.965, + "type": "image", + "image_path": "64ff89fa891ae43bb690e18da28297235b2205ad0db31d4eaf8f06107e1c4f55.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 147, + 80, + 499, + 131.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 147, + 131.0, + 499, + 182.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 147, + 182.0, + 499, + 233.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 243, + 506, + 288 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "Figure 1: An example Vietoris-Rips filtration with its persistence diagram. We only add convex hulls", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 254, + 506, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 206, + 267 + ], + "score": 1.0, + "content": "when points are pairwise", + "type": "text" + }, + { + "bbox": [ + 207, + 257, + 213, + 264 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 254, + 316, + 267 + ], + "score": 1.0, + "content": "apart, so there is a hole at", + "type": "text" + }, + { + "bbox": [ + 316, + 255, + 340, + 265 + ], + "score": 0.9, + "content": "\\epsilon = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 254, + 506, + 267 + ], + "score": 1.0, + "content": "whilst the diagonals of the square are not", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 278 + ], + "score": 1.0, + "content": "close enough to fill the hole in. 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If a topological feature, such as a connected component or hole, persists throughout a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "large range of granularities, then it’s more likely to be a feature of the distribution. If it only persists", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "for a short amount of time, then it’s more likely to be noise (Cohen-Steiner et al., 2007). We can stably", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 370, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 133, + 385 + ], + "score": 1.0, + "content": "map a", + "type": "text" + }, + { + "bbox": [ + 133, + 373, + 140, + 383 + ], + "score": 0.66, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 370, + 506, + 385 + ], + "score": 1.0, + "content": "-PH group into a multiset in the extended plane called a persistence diagram (Chazal et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 381, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 131, + 395 + ], + "score": 0.56, + "content": "\\bar { 2 0 1 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 381, + 506, + 396 + ], + "score": 1.0, + "content": ". Each topological feature has a birth and death: a feature is born when it enters the complex,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 405 + ], + "score": 1.0, + "content": "and dies when the complex grows enough to destroy it. For example, in Figure 1(a), a feature is born", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 116, + 416 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 117, + 405, + 140, + 415 + ], + "score": 0.9, + "content": "\\epsilon = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 405, + 340, + 416 + ], + "score": 1.0, + "content": "when four lines form a hole. This feature dies at", + "type": "text" + }, + { + "bbox": [ + 340, + 405, + 377, + 415 + ], + "score": 0.88, + "content": "\\epsilon = 1 . 4 2", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "when the hole is filled in. This", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 414, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 369, + 429 + ], + "score": 1.0, + "content": "is shown in the persistence diagram in Figure 1(b) as a point at", + "type": "text" + }, + { + "bbox": [ + 370, + 416, + 404, + 428 + ], + "score": 0.32, + "content": "( 1 , 1 . 4 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 414, + 506, + 429 + ], + "score": 1.0, + "content": "in 1-PH. By computing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "the birth/death points for each topological feature in the filtration, we get a complete picture of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 438, + 406, + 449 + ], + "score": 1.0, + "content": "topology of the point cloud at all granularities (Zomorodian & Carlsson,", + "type": "text" + }, + { + "bbox": [ + 406, + 437, + 432, + 448 + ], + "score": 0.36, + "content": "\\overline { { 2 0 0 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 438, + 505, + 450 + ], + "score": 1.0, + "content": ". The persistence", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 447, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 394, + 461 + ], + "score": 1.0, + "content": "diagram is the collection of birth/death points, along with the diagonal", + "type": "text" + }, + { + "bbox": [ + 395, + 448, + 483, + 460 + ], + "score": 0.92, + "content": "\\overline { { \\Delta } } \\equiv \\{ ( a , a ) : a \\in \\mathbb { R } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 447, + 506, + 461 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "infinite multiplicity, added in order to make the space of persistence diagrams complete (Mileyko", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 470, + 156, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 156, + 482 + ], + "score": 1.0, + "content": "et al., 2011).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 317, + 507, + 482 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 506, + 243, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 244, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 244, + 520 + ], + "score": 1.0, + "content": "3 ALGORITHMIC DESIGN", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 108, + 536, + 295, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 297, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 297, + 548 + ], + "score": 1.0, + "content": "3.1 CLUSTERING PERSISTENCE DIAGRAMS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 506, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "In order to cluster we need a distance on the space of persistence diagrams. We use the 2-Wasserstein", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 569, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 107, + 571, + 119, + 581 + ], + "score": 0.86, + "content": "L _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 569, + 482, + 583 + ], + "score": 1.0, + "content": "metric as it is stable on persistence diagrams of finite point clouds (Chazal et al., 2012).", + "type": "text" + }, + { + "bbox": [ + 482, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "The", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 594 + ], + "score": 1.0, + "content": "Wasserstein distance is an optimal transport metric that has found applications across machine", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "learning. In the Euclidean case, it quantifies the smallest distance between optimally matched points.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 603, + 264, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 172, + 615 + ], + "score": 1.0, + "content": "Given diagrams", + "type": "text" + }, + { + "bbox": [ + 172, + 604, + 201, + 615 + ], + "score": 0.92, + "content": "\\mathbb { D } _ { 1 } , \\mathbb { D } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 603, + 264, + 615 + ], + "score": 1.0, + "content": ", the distance is", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 559, + 506, + 615 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 203, + 627, + 407, + 664 + ], + "lines": [ + { + "bbox": [ + 203, + 627, + 407, + 664 + ], + "spans": [ + { + "bbox": [ + 203, + 627, + 407, + 664 + ], + "score": 0.94, + "content": "W _ { 2 } ( \\mathbb { D } _ { 1 } , \\mathbb { D } _ { 2 } ) = \\left( \\operatorname* { i n f } _ { \\gamma : \\mathbb { D } _ { 1 } \\to \\mathbb { D } _ { 2 } } \\sum _ { x \\in \\mathbb { D } _ { 1 } } | | x - \\gamma ( x ) | | _ { 2 } ^ { 2 } \\right) ^ { 1 / 2 } ,", + "type": "interline_equation", + "image_path": "044279cb12c1257d323f24efd8909559ccd2be79bc799137572edd50a868f363.jpg" + } + ] + } + ], + "index": 29.5, + "virtual_lines": [ + { + "bbox": [ + 203, + 627, + 407, + 645.5 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 203, + 645.5, + 407, + 664.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 287, + 688 + ], + "score": 1.0, + "content": "where the infimum is taken over all bijections", + "type": "text" + }, + { + "bbox": [ + 288, + 677, + 341, + 689 + ], + "score": 0.91, + "content": "\\gamma : { \\mathbb { D } } _ { 1 } \\to { \\mathbb { D } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 676, + 504, + 688 + ], + "score": 1.0, + "content": ". Note that as we added the diagonal with", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 687, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 504, + 699 + ], + "score": 1.0, + "content": "infinite multiplicity to each diagram, these bijections exist. If an off-diagonal point is matched to the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 711 + ], + "score": 1.0, + "content": "diagonal the transportation cost is simply the shortest distance to the diagonal. In fact, the closer a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 414, + 723 + ], + "score": 1.0, + "content": "point is to the diagonal, the more likely it is to be noise (Cohen-Steiner et al.,", + "type": "text" + }, + { + "bbox": [ + 415, + 709, + 440, + 722 + ], + "score": 0.35, + "content": "\\boxed { 2 0 0 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 709, + 506, + 723 + ], + "score": 1.0, + "content": ", so this ensures", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 721, + 281, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 281, + 733 + ], + "score": 1.0, + "content": "our distance is not overly affected by noise.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 676, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 505, + 140 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 194, + 95 + ], + "score": 1.0, + "content": "We work in the space", + "type": "text" + }, + { + "bbox": [ + 195, + 81, + 325, + 96 + ], + "score": 0.87, + "content": "\\mathfrak { D } _ { \\mathrm { L } ^ { 2 } } = \\{ \\mathbb { D } : W _ { 2 } ( \\mathbb { D } , \\Delta ) < \\infty \\} , ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "as this leads to a geodesic space with known", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 89, + 508, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 89, + 145, + 109 + ], + "score": 1.0, + "content": "structure", + "type": "text" + }, + { + "bbox": [ + 146, + 93, + 228, + 107 + ], + "score": 0.79, + "content": "\\left\\{ { \\mathrm { I u r n e r ~ e t ~ a l . } } \\right\\} \\left[ { \\mathrm { 2 0 1 2 } } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 89, + 410, + 109 + ], + "score": 1.0, + "content": ". Given a collection of persistence diagrams", + "type": "text" + }, + { + "bbox": [ + 411, + 93, + 478, + 107 + ], + "score": 0.93, + "content": "\\{ \\mathbb { D } _ { j } \\} _ { j = 1 } ^ { n } \\subset \\mathcal { D } _ { \\mathrm { L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 89, + 508, + 109 + ], + "score": 1.0, + "content": "and a", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 104, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 204, + 120 + ], + "score": 1.0, + "content": "fixed number of clusters", + "type": "text" + }, + { + "bbox": [ + 204, + 109, + 210, + 116 + ], + "score": 0.63, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 104, + 335, + 120 + ], + "score": 1.0, + "content": ", we wish to find cluster centres", + "type": "text" + }, + { + "bbox": [ + 335, + 105, + 406, + 119 + ], + "score": 0.92, + "content": "\\{ \\mathbb { M } _ { k } \\} _ { k = 1 } ^ { c } \\subset \\mathcal { D } _ { \\mathrm { L } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 104, + 506, + 120 + ], + "score": 1.0, + "content": ", along with membership", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 135, + 129 + ], + "score": 1.0, + "content": "values", + "type": "text" + }, + { + "bbox": [ + 135, + 117, + 184, + 129 + ], + "score": 0.91, + "content": "r _ { j k } \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 117, + 317, + 129 + ], + "score": 1.0, + "content": "that denote the extent to which", + "type": "text" + }, + { + "bbox": [ + 317, + 117, + 330, + 129 + ], + "score": 0.89, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 117, + 438, + 129 + ], + "score": 1.0, + "content": "is associated with cluster", + "type": "text" + }, + { + "bbox": [ + 438, + 117, + 453, + 128 + ], + "score": 0.89, + "content": "\\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 117, + 505, + 129 + ], + "score": 1.0, + "content": ". We follow", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 353, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 256, + 141 + ], + "score": 1.0, + "content": "probabilistic fuzzy clustering, so that", + "type": "text" + }, + { + "bbox": [ + 257, + 127, + 306, + 141 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { k } r _ { j k } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 127, + 342, + 141 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 342, + 129, + 348, + 139 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 127, + 353, + 141 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 144, + 504, + 167 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 335, + 158 + ], + "score": 1.0, + "content": "We extend the FCM algorithm originally proposed by", + "type": "text" + }, + { + "bbox": [ + 335, + 144, + 398, + 158 + ], + "score": 0.74, + "content": "\\mathbf { B e z d e k } | \\left( \\mathbf { \\overline { { 1 9 8 0 } } } \\right) ", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 144, + 400, + 158 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 405, + 144, + 426, + 158 + ], + "score": 1.0, + "content": "Our", + "type": "text" + }, + { + "bbox": [ + 426, + 147, + 441, + 157 + ], + "score": 0.86, + "content": "r _ { j k }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 144, + 506, + 158 + ], + "score": 1.0, + "content": "is the same as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 407, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 407, + 168 + ], + "score": 1.0, + "content": "traditional FCM clustering, adapted with the Wasserstein distance. That is,", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 172, + 371, + 208 + ], + "lines": [ + { + "bbox": [ + 240, + 172, + 371, + 208 + ], + "spans": [ + { + "bbox": [ + 240, + 172, + 371, + 208 + ], + "score": 0.95, + "content": "r _ { j k } = \\left( \\sum _ { l = 1 } ^ { c } \\frac { W _ { 2 } ( \\mathbb { M } _ { k } , \\mathbb { D } _ { j } ) } { W _ { 2 } ( \\mathbb { M } _ { l } , \\mathbb { D } _ { j } ) } \\right) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "2d89a6075efb8ca2a17e9a94e5ce2dc0a5a92f6c81cce12dbd24305474747ea0.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 172, + 371, + 190.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 240, + 190.0, + 371, + 208.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 219, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 104, + 216, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 216, + 147, + 236 + ], + "score": 1.0, + "content": "To update", + "type": "text" + }, + { + "bbox": [ + 147, + 221, + 163, + 232 + ], + "score": 0.9, + "content": "\\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 216, + 326, + 236 + ], + "score": 1.0, + "content": ", we compute the weighted Fréchet mean", + "type": "text" + }, + { + "bbox": [ + 327, + 219, + 335, + 231 + ], + "score": 0.86, + "content": "\\hat { \\mathbb { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 216, + 447, + 236 + ], + "score": 1.0, + "content": "of the persistence diagrams", + "type": "text" + }, + { + "bbox": [ + 447, + 221, + 483, + 235 + ], + "score": 0.94, + "content": "\\{ \\mathbb { D } _ { j } \\} _ { j = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 216, + 506, + 236 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 232, + 249, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 155, + 248 + ], + "score": 1.0, + "content": "the weights", + "type": "text" + }, + { + "bbox": [ + 155, + 233, + 193, + 248 + ], + "score": 0.93, + "content": "\\{ r _ { j k } ^ { 2 } \\} _ { j = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 232, + 249, + 248 + ], + "score": 1.0, + "content": ". Specifically,", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 252, + 417, + 286 + ], + "lines": [ + { + "bbox": [ + 192, + 252, + 417, + 286 + ], + "spans": [ + { + "bbox": [ + 192, + 252, + 417, + 286 + ], + "score": 0.93, + "content": "\\mathbb { M } _ { k } \\longleftarrow \\operatorname { a r g m i n } _ { \\hat { \\mathbb { D } } } \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( \\hat { \\mathbb { D } } , \\mathbb { D } _ { j } ) ^ { 2 } , \\operatorname { f o r } k = 1 , \\dots , c .", + "type": "interline_equation", + "image_path": "e971481cdc3ef3c9785b66b8e3ba62bb6fb0a41d9f54d8692810ebbf43b1f35c.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 192, + 252, + 417, + 269.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 192, + 269.0, + 417, + 286.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 296, + 506, + 341 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 506, + 309 + ], + "score": 1.0, + "content": "As the weighted Fréchet mean extends weighted centroids to general metric spaces, this gives our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "fuzzy cluster centres. The computation of the weighted Fréchet mean is covered in Section 3.2. By", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "alternatively updating (1) and (2) we get a sequence of iterates. Theorem 1, proven in Appendix A,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 330, + 400, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 400, + 343 + ], + "score": 1.0, + "content": "provides the same convergence guarantees as traditional FCM clustering.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 357 + ], + "score": 1.0, + "content": "Theorem 1. Every convergent subsequence of the sequence of iterates obtained by alternatively", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 318, + 367 + ], + "score": 1.0, + "content": "updating membership values and cluster centres with", + "type": "text" + }, + { + "bbox": [ + 319, + 356, + 331, + 366 + ], + "score": 0.53, + "content": "( l )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "and (2) tends to a local minimum or saddle", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 101, + 360, + 388, + 384 + ], + "spans": [ + { + "bbox": [ + 101, + 360, + 208, + 384 + ], + "score": 1.0, + "content": "point of the cost function", + "type": "text" + }, + { + "bbox": [ + 209, + 366, + 381, + 380 + ], + "score": 0.93, + "content": "\\begin{array} { r } { J ( R , { \\mathbb M } ) = \\sum _ { j = 1 } ^ { n } \\sum _ { k = 1 } ^ { c } r _ { j k } ^ { 2 } W _ { 2 } ( { \\mathbb M } _ { k } , { \\mathbb D } _ { j } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 360, + 388, + 384 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 388, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "Observe that we only guarantee the convergence of subsequences of iterates. This is the same as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "traditional FCM clustering, so we follow the same approach to a stopping condition and run our", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 410, + 475, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 475, + 422 + ], + "score": 1.0, + "content": "algorithm for a fixed number of iterations. The entire algorithm is displayed in Algorithm 1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 209, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 210, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 210, + 447 + ], + "score": 1.0, + "content": "Algorithm 1 FPDCluster", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "table", + "bbox": [ + 109, + 446, + 502, + 572 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 446, + 502, + 572 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 446, + 502, + 572 + ], + "spans": [ + { + "bbox": [ + 109, + 446, + 502, + 572 + ], + "score": 0.728, + "html": "
Input Diagrams D = {Dj}𝑗=1, number of clusters c, maximum iterations MAXITER Output Cluster centres M = {Mk}ε=1, membership values R = {rjk}
1:D = ADDDIAGONALS(D)7: end for
2:M=INITCENTRES(D)8: end for
3: for countin1..MAXITER do9: for k in1..c do
4: for j in1..n do10: M ← WFRECHETMEAN(D,Rk)
5: for k in1..c do11: end for
W2(Mk,D)-1 12: end for
(∑i-1 W) rjk← 6:13: return M, R
", + "type": "table", + "image_path": "727766a169bea9ca3741654e9e120f89d4e770783a4cc91d2f9b6f9acb44b181.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 109, + 446, + 502, + 488.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 109, + 488.0, + 502, + 530.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 109, + 530.0, + 502, + 572.0 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 591, + 324, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 325, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 325, + 604 + ], + "score": 1.0, + "content": "3.2 COMPUTING THE WEIGHTED FRÉCHET MEAN", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 160, + 624 + ], + "score": 1.0, + "content": "Turner et al.", + "type": "text" + }, + { + "bbox": [ + 160, + 611, + 188, + 624 + ], + "score": 0.81, + "content": "\\textcircled { | 2 0 1 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "give an algorithm for the computation of Fréchet means. In this section we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "extend their algorithm and proof of convergence to the weighted case. In Algorithm 1 we add", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 634, + 504, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 504, + 646 + ], + "score": 1.0, + "content": "copies of the diagonal to ensure that each diagram has the same cardinality; denote this cardinality", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 103, + 642, + 508, + 662 + ], + "spans": [ + { + "bbox": [ + 103, + 642, + 118, + 662 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 648, + 128, + 656 + ], + "score": 0.6, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 642, + 370, + 662 + ], + "score": 1.0, + "content": ". To compute the weighted Fréchet mean, we need to find", + "type": "text" + }, + { + "bbox": [ + 371, + 645, + 441, + 658 + ], + "score": 0.93, + "content": "\\mathbb { M } _ { k } = \\bar { \\{ \\boldsymbol { y } ^ { ( i ) } \\} } _ { i = 1 } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 642, + 508, + 662 + ], + "score": 1.0, + "content": "that minimises", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 103, + 653, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 103, + 653, + 423, + 677 + ], + "score": 1.0, + "content": "the Fréchet function in (2). 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That is,", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 144, + 506, + 168 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 172, + 371, + 208 + ], + "lines": [ + { + "bbox": [ + 240, + 172, + 371, + 208 + ], + "spans": [ + { + "bbox": [ + 240, + 172, + 371, + 208 + ], + "score": 0.95, + "content": "r _ { j k } = \\left( \\sum _ { l = 1 } ^ { c } \\frac { W _ { 2 } ( \\mathbb { M } _ { k } , \\mathbb { D } _ { j } ) } { W _ { 2 } ( \\mathbb { M } _ { l } , \\mathbb { D } _ { j } ) } \\right) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "2d89a6075efb8ca2a17e9a94e5ce2dc0a5a92f6c81cce12dbd24305474747ea0.jpg" + } + ] + } + ], + "index": 7.5, + "virtual_lines": [ + { + "bbox": [ + 240, + 172, + 371, + 190.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 240, + 190.0, + 371, + 208.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 104, + 219, + 505, + 247 + ], + "lines": [ + { + "bbox": [ + 104, + 216, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 104, + 216, + 147, + 236 + ], + "score": 1.0, + "content": "To update", + "type": "text" + }, + { + "bbox": [ + 147, + 221, + 163, + 232 + ], + "score": 0.9, + "content": "\\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 216, + 326, + 236 + ], + "score": 1.0, + "content": ", we compute the weighted Fréchet mean", + "type": "text" + }, + { + "bbox": [ + 327, + 219, + 335, + 231 + ], + "score": 0.86, + "content": "\\hat { \\mathbb { D } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 216, + 447, + 236 + ], + "score": 1.0, + "content": "of the persistence diagrams", + "type": "text" + }, + { + "bbox": [ + 447, + 221, + 483, + 235 + ], + "score": 0.94, + "content": "\\{ \\mathbb { D } _ { j } \\} _ { j = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 216, + 506, + 236 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 232, + 249, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 155, + 248 + ], + "score": 1.0, + "content": "the weights", + "type": "text" + }, + { + "bbox": [ + 155, + 233, + 193, + 248 + ], + "score": 0.93, + "content": "\\{ r _ { j k } ^ { 2 } \\} _ { j = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 232, + 249, + 248 + ], + "score": 1.0, + "content": ". 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The computation of the weighted Fréchet mean is covered in Section 3.2. By", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "alternatively updating (1) and (2) we get a sequence of iterates. Theorem 1, proven in Appendix A,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 330, + 400, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 400, + 343 + ], + "score": 1.0, + "content": "provides the same convergence guarantees as traditional FCM clustering.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 296, + 506, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 344, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 357 + ], + "score": 1.0, + "content": "Theorem 1. Every convergent subsequence of the sequence of iterates obtained by alternatively", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 318, + 367 + ], + "score": 1.0, + "content": "updating membership values and cluster centres with", + "type": "text" + }, + { + "bbox": [ + 319, + 356, + 331, + 366 + ], + "score": 0.53, + "content": "( l )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "and (2) tends to a local minimum or saddle", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 101, + 360, + 388, + 384 + ], + "spans": [ + { + "bbox": [ + 101, + 360, + 208, + 384 + ], + "score": 1.0, + "content": "point of the cost function", + "type": "text" + }, + { + "bbox": [ + 209, + 366, + 381, + 380 + ], + "score": 0.93, + "content": "\\begin{array} { r } { J ( R , { \\mathbb M } ) = \\sum _ { j = 1 } ^ { n } \\sum _ { k = 1 } ^ { c } r _ { j k } ^ { 2 } W _ { 2 } ( { \\mathbb M } _ { k } , { \\mathbb D } _ { j } ) ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 360, + 388, + 384 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 101, + 342, + 505, + 384 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 388, + 504, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 388, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 506, + 401 + ], + "score": 1.0, + "content": "Observe that we only guarantee the convergence of subsequences of iterates. This is the same as", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "traditional FCM clustering, so we follow the same approach to a stopping condition and run our", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 410, + 475, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 475, + 422 + ], + "score": 1.0, + "content": "algorithm for a fixed number of iterations. The entire algorithm is displayed in Algorithm 1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 388, + 506, + 422 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 209, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 210, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 210, + 447 + ], + "score": 1.0, + "content": "Algorithm 1 FPDCluster", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "table", + "bbox": [ + 109, + 446, + 502, + 572 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 109, + 446, + 502, + 572 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 446, + 502, + 572 + ], + "spans": [ + { + "bbox": [ + 109, + 446, + 502, + 572 + ], + "score": 0.728, + "html": "
Input Diagrams D = {Dj}𝑗=1, number of clusters c, maximum iterations MAXITER Output Cluster centres M = {Mk}ε=1, membership values R = {rjk}
1:D = ADDDIAGONALS(D)7: end for
2:M=INITCENTRES(D)8: end for
3: for countin1..MAXITER do9: for k in1..c do
4: for j in1..n do10: M ← WFRECHETMEAN(D,Rk)
5: for k in1..c do11: end for
W2(Mk,D)-1 12: end for
(∑i-1 W) rjk← 6:13: return M, R
", + "type": "table", + "image_path": "727766a169bea9ca3741654e9e120f89d4e770783a4cc91d2f9b6f9acb44b181.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 109, + 446, + 502, + 488.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 109, + 488.0, + 502, + 530.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 109, + 530.0, + 502, + 572.0 + ], + "spans": [], + "index": 26 + } + ] + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 591, + 324, + 603 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 325, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 325, + 604 + ], + "score": 1.0, + "content": "3.2 COMPUTING THE WEIGHTED FRÉCHET MEAN", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 611, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 160, + 624 + ], + "score": 1.0, + "content": "Turner et al.", + "type": "text" + }, + { + "bbox": [ + 160, + 611, + 188, + 624 + ], + "score": 0.81, + "content": "\\textcircled { | 2 0 1 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 612, + 505, + 624 + ], + "score": 1.0, + "content": "give an algorithm for the computation of Fréchet means. In this section we", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "extend their algorithm and proof of convergence to the weighted case. In Algorithm 1 we add", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 634, + 504, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 504, + 646 + ], + "score": 1.0, + "content": "copies of the diagonal to ensure that each diagram has the same cardinality; denote this cardinality", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 103, + 642, + 508, + 662 + ], + "spans": [ + { + "bbox": [ + 103, + 642, + 118, + 662 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 648, + 128, + 656 + ], + "score": 0.6, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 642, + 370, + 662 + ], + "score": 1.0, + "content": ". 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Supposing we know these bijections, we can rearrange the Fréchet function into the form", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 312, + 384, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 384, + 329 + ], + "score": 0.73, + "content": "\\begin{array} { r } { F ( \\mathbb { M } _ { k } ) = \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( \\mathbb { M } _ { k } , \\mathbb { D } _ { j } ) ^ { 2 } = \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } | | y ^ { ( i ) } - x _ { j } ^ { ( i ) } | | ^ { 2 } . } \\end{array}", + "type": "inline_equation", + "image_path": "2e0ae1e960ec46332ef7cb3fe29bd3e45c902c0ca5d6e9fdcac8a61e8a28c16f.jpg", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 30, + "bbox_fs": [ + 103, + 611, + 508, + 677 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 122, + 93, + 461, + 228 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 122, + 93, + 461, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 93, + 461, + 228 + ], + "spans": [ + { + "bbox": [ + 122, + 93, + 461, + 228 + ], + "score": 0.946, + "type": "image", + "image_path": "0fb7e8cdd0bd53b2279667088b74f62883e5fb4f3b13a84dfffccc1f17940832.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 122, + 93, + 461, + 138.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 122, + 138.0, + 461, + 183.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 122, + 183.0, + 461, + 228.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 237, + 505, + 271 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "Figure 2: Our algorithm successfully clustered the persistence diagrams, finding cluster centres", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "that represent the topology of the datasets. 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Supposing we know these bijections, we can rearrange the Fréchet function into the form", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 312, + 384, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 384, + 329 + ], + "score": 0.73, + "content": "\\begin{array} { r } { F ( \\mathbb { M } _ { k } ) = \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( \\mathbb { M } _ { k } , \\mathbb { D } _ { j } ) ^ { 2 } = \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } | | y ^ { ( i ) } - x _ { j } ^ { ( i ) } | | ^ { 2 } . } \\end{array}", + "type": "inline_equation", + "image_path": "2e0ae1e960ec46332ef7cb3fe29bd3e45c902c0ca5d6e9fdcac8a61e8a28c16f.jpg" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 506, + 400 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 286, + 348 + ], + "score": 1.0, + "content": "In this form, the summand is minimised for", + "type": "text" + }, + { + "bbox": [ + 287, + 333, + 303, + 347 + ], + "score": 0.91, + "content": "\\boldsymbol y ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "by the weighted Euclidean centroid of the points", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 344, + 509, + 365 + ], + "spans": [ + { + "bbox": [ + 107, + 345, + 147, + 362 + ], + "score": 0.92, + "content": "\\{ x _ { j } ^ { ( i ) } \\} _ { j = 1 } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 344, + 509, + 365 + ], + "score": 1.0, + "content": ". Therefore to compute the weighted Fréchet mean, we need to find the correct bijections.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 360, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 434, + 374 + ], + "score": 1.0, + "content": "We start by using the Hungarian algorithm to find an optimal matching between", + "type": "text" + }, + { + "bbox": [ + 434, + 360, + 450, + 372 + ], + "score": 0.88, + "content": "\\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 360, + 490, + 374 + ], + "score": 1.0, + "content": "and each", + "type": "text" + }, + { + "bbox": [ + 490, + 361, + 503, + 373 + ], + "score": 0.88, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 360, + 505, + 374 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 371, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 141, + 386 + ], + "score": 1.0, + "content": "Given a", + "type": "text" + }, + { + "bbox": [ + 141, + 373, + 153, + 385 + ], + "score": 0.87, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 371, + 217, + 386 + ], + "score": 1.0, + "content": ", for each point", + "type": "text" + }, + { + "bbox": [ + 217, + 372, + 260, + 385 + ], + "score": 0.93, + "content": "\\boldsymbol { y } ^ { ( i ) } \\in \\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 371, + 505, + 386 + ], + "score": 1.0, + "content": ", the Hungarian algorithm will assign an optimally matched", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 385, + 320, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 129, + 401 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 129, + 385, + 169, + 401 + ], + "score": 0.92, + "content": "x _ { j } ^ { ( i ) } \\in \\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 387, + 320, + 401 + ], + "score": 1.0, + "content": ". Specifically, we find matched points", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 414, + 426, + 438 + ], + "lines": [ + { + "bbox": [ + 187, + 414, + 426, + 438 + ], + "spans": [ + { + "bbox": [ + 187, + 414, + 426, + 438 + ], + "score": 0.88, + "content": "\\left[ x _ { j } ^ { ( i ) } \\right] _ { i = 1 } ^ { m } \\longleftarrow \\mathrm { H u n g a r i a n } ( \\mathbb { M } _ { k } , \\mathbb { D } _ { j } ) , \\mathrm { ~ f o r ~ e a c h ~ } j = 1 , \\ldots , n .", + "type": "interline_equation", + "image_path": "813445b60502ff54fa3b6a1d8cbe68cbd361766f4b3ffbff4acd6ad2442e5edc.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 187, + 414, + 426, + 438 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 506, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 167, + 469 + ], + "score": 1.0, + "content": "Now, for each", + "type": "text" + }, + { + "bbox": [ + 168, + 455, + 212, + 468 + ], + "score": 0.93, + "content": "\\boldsymbol { y } ^ { ( i ) } \\in \\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 454, + 464, + 469 + ], + "score": 1.0, + "content": "we need to find the weighted average of the matched poin", + "type": "text" + }, + { + "bbox": [ + 455, + 448, + 505, + 470 + ], + "score": 1.0, + "content": "ts h x(i) in .", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "However, some of these points could be copies of the diagonal, so we need to consider three", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "distinct cases: that each matched point is off-diagonal, that each one is a copy of the diagonal, or", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 493, + 507, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 361, + 507 + ], + "score": 1.0, + "content": "that the points are a mixture of both. We start by partitioning", + "type": "text" + }, + { + "bbox": [ + 361, + 495, + 396, + 505 + ], + "score": 0.92, + "content": "1 , \\ldots , n", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 493, + 507, + 507 + ], + "score": 1.0, + "content": "into the indices of the off-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 102, + 501, + 504, + 527 + ], + "spans": [ + { + "bbox": [ + 102, + 501, + 169, + 527 + ], + "score": 1.0, + "content": "diagonal points", + "type": "text" + }, + { + "bbox": [ + 169, + 505, + 262, + 525 + ], + "score": 0.94, + "content": "\\mathcal { J } _ { \\mathrm { O D } } ^ { ( i ) } = \\left\\{ j : x _ { j } ^ { ( i ) } \\neq \\Delta \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 501, + 412, + 527 + ], + "score": 1.0, + "content": "and the indices of the diagonal points", + "type": "text" + }, + { + "bbox": [ + 412, + 505, + 504, + 525 + ], + "score": 0.96, + "content": "\\mathcal { J } _ { \\mathrm { D } } ^ { ( i ) } = \\left\\{ j : x _ { j } ^ { ( i ) } = \\Delta \\right\\}", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 523, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 144, + 539 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 144, + 525, + 202, + 536 + ], + "score": 0.88, + "content": "i = 1 , \\ldots , m", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 523, + 245, + 539 + ], + "score": 1.0, + "content": ". Now, if", + "type": "text" + }, + { + "bbox": [ + 245, + 525, + 284, + 537 + ], + "score": 0.91, + "content": "{ \\mathfrak { I } } _ { \\mathrm { O D } } = \\ \\varnothing", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 523, + 307, + 539 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 307, + 523, + 323, + 537 + ], + "score": 0.91, + "content": "\\boldsymbol y ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 523, + 482, + 539 + ], + "score": 1.0, + "content": "is a copy of the diagonal. If not, let", + "type": "text" + }, + { + "bbox": [ + 483, + 526, + 505, + 536 + ], + "score": 0.84, + "content": "w =", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 532, + 507, + 565 + ], + "spans": [ + { + "bbox": [ + 107, + 536, + 243, + 558 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\left( \\sum _ { j \\in \\mathcal { J } _ { \\mathrm { o D } } ^ { ( i ) } } r _ { j k } ^ { 2 } \\right) ^ { - 1 } \\sum _ { j \\in \\mathcal { J } _ { \\mathrm { o D } } ^ { ( i ) } } r _ { j k } ^ { 2 } x _ { j } ^ { ( i ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 532, + 253, + 565 + ], + "score": 1.0, + "content": "1 P j2J(i)OD r2jkx(i)j be", + "type": "text" + }, + { + "bbox": [ + 243, + 538, + 448, + 556 + ], + "score": 1.0, + "content": "the weighted mean of the off-diagonal points. If", + "type": "text" + }, + { + "bbox": [ + 448, + 538, + 482, + 554 + ], + "score": 0.93, + "content": "\\mathcal { J } _ { \\mathrm { D } } ^ { ( i ) } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 538, + 507, + 556 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 555, + 467, + 572 + ], + "spans": [ + { + "bbox": [ + 107, + 558, + 143, + 570 + ], + "score": 0.91, + "content": "y ^ { ( i ) } = w", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 555, + 205, + 572 + ], + "score": 1.0, + "content": ". Otherwise, let", + "type": "text" + }, + { + "bbox": [ + 206, + 560, + 221, + 569 + ], + "score": 0.84, + "content": "w _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 555, + 376, + 572 + ], + "score": 1.0, + "content": "be the point on the diagonal closest to", + "type": "text" + }, + { + "bbox": [ + 376, + 560, + 384, + 568 + ], + "score": 0.76, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 555, + 467, + 572 + ], + "score": 1.0, + "content": ". Then our update is", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 583, + 433, + 619 + ], + "lines": [ + { + "bbox": [ + 177, + 583, + 433, + 619 + ], + "spans": [ + { + "bbox": [ + 177, + 583, + 433, + 619 + ], + "score": 0.94, + "content": "y ^ { ( i ) } \\longleftarrow \\frac { \\sum _ { j \\in \\mathcal { J } _ { \\mathrm { 0 D } } ^ { ( i ) } } r _ { j k } ^ { 2 } x _ { j } ^ { ( i ) } + \\sum _ { j \\in \\mathcal { J } _ { \\mathrm { D } } ^ { ( i ) } } r _ { j k } ^ { 2 } w _ { \\Delta } } { \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } } , \\mathrm { ~ f o r ~ } i = 1 , \\dots , m .", + "type": "interline_equation", + "image_path": "c0292e45f5f942bb00f61a98644e345db81e60b2b62814cc75c0e94929e654d0.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 177, + 583, + 433, + 595.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 177, + 595.0, + 433, + 607.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 177, + 607.0, + 433, + 619.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 638, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "We alternate between (3) and (4) until the matching remains the same. Theorem 2, proving that this", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "algorithm converges to a local minimum of the Fréchet function, is proven in Appendix B. Also in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 661, + 404, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 147, + 673 + ], + "score": 1.0, + "content": "Appendix", + "type": "text" + }, + { + "bbox": [ + 148, + 661, + 156, + 671 + ], + "score": 0.31, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 661, + 404, + 673 + ], + "score": 1.0, + "content": "is Algorithm 2, giving an overview of the entire computation.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 681, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 504, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 233, + 695 + ], + "score": 1.0, + "content": "Theorem 2. Given diagrams", + "type": "text" + }, + { + "bbox": [ + 234, + 682, + 246, + 694 + ], + "score": 0.88, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 680, + 336, + 695 + ], + "score": 1.0, + "content": ", membership values", + "type": "text" + }, + { + "bbox": [ + 351, + 680, + 466, + 695 + ], + "score": 1.0, + "content": ", and the Fréchet function", + "type": "text" + }, + { + "bbox": [ + 466, + 680, + 504, + 694 + ], + "score": 0.91, + "content": "F ( { \\hat { \\mathbb { D } } } ) \\mathbf { \\Sigma } =", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 687, + 504, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 196, + 709 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( { \\hat { \\mathbb { D } } } , \\mathbb { D } _ { j } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 687, + 222, + 714 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 223, + 694, + 294, + 708 + ], + "score": 0.92, + "content": "\\mathbb { M } _ { k } \\ = \\ \\{ y ^ { ( i ) } \\} _ { i = 1 } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 687, + 393, + 714 + ], + "score": 1.0, + "content": "is a local minimum of", + "type": "text" + }, + { + "bbox": [ + 393, + 695, + 402, + 705 + ], + "score": 0.76, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 687, + 497, + 714 + ], + "score": 1.0, + "content": "if and only if there is", + "type": "text" + }, + { + "bbox": [ + 497, + 697, + 504, + 705 + ], + "score": 0.58, + "content": "a", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 708, + 439, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 221, + 722 + ], + "score": 1.0, + "content": "unique optimal pairing from", + "type": "text" + }, + { + "bbox": [ + 222, + 709, + 237, + 720 + ], + "score": 0.89, + "content": "\\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 708, + 294, + 722 + ], + "score": 1.0, + "content": "to each of the", + "type": "text" + }, + { + "bbox": [ + 295, + 709, + 307, + 722 + ], + "score": 0.89, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 708, + 347, + 722 + ], + "score": 1.0, + "content": "and each", + "type": "text" + }, + { + "bbox": [ + 348, + 708, + 363, + 721 + ], + "score": 0.91, + "content": "\\boldsymbol y ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 708, + 439, + 722 + ], + "score": 1.0, + "content": "is updated via (4).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 307, + 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 2021", + "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": "image", + "bbox": [ + 122, + 93, + 461, + 228 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 122, + 93, + 461, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 93, + 461, + 228 + ], + "spans": [ + { + "bbox": [ + 122, + 93, + 461, + 228 + ], + "score": 0.946, + "type": "image", + "image_path": "0fb7e8cdd0bd53b2279667088b74f62883e5fb4f3b13a84dfffccc1f17940832.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 122, + 93, + 461, + 138.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 122, + 138.0, + 461, + 183.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 122, + 183.0, + 461, + 228.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 237, + 505, + 271 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 106, + 237, + 505, + 249 + ], + "score": 1.0, + "content": "Figure 2: Our algorithm successfully clustered the persistence diagrams, finding cluster centres", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "score": 1.0, + "content": "that represent the topology of the datasets. 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Therefore to compute the weighted Fréchet mean, we need to find the correct bijections.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 360, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 434, + 374 + ], + "score": 1.0, + "content": "We start by using the Hungarian algorithm to find an optimal matching between", + "type": "text" + }, + { + "bbox": [ + 434, + 360, + 450, + 372 + ], + "score": 0.88, + "content": "\\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 360, + 490, + 374 + ], + "score": 1.0, + "content": "and each", + "type": "text" + }, + { + "bbox": [ + 490, + 361, + 503, + 373 + ], + "score": 0.88, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 360, + 505, + 374 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 371, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 141, + 386 + ], + "score": 1.0, + "content": "Given a", + "type": "text" + }, + { + "bbox": [ + 141, + 373, + 153, + 385 + ], + "score": 0.87, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 371, + 217, + 386 + ], + "score": 1.0, + "content": ", for each point", + "type": "text" + }, + { + "bbox": [ + 217, + 372, + 260, + 385 + ], + "score": 0.93, + "content": "\\boldsymbol { y } ^ { ( i ) } \\in \\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 371, + 505, + 386 + ], + "score": 1.0, + "content": ", the Hungarian algorithm will assign an optimally matched", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 385, + 320, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 129, + 401 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 129, + 385, + 169, + 401 + ], + "score": 0.92, + "content": "x _ { j } ^ { ( i ) } \\in \\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 387, + 320, + 401 + ], + "score": 1.0, + "content": ". Specifically, we find matched points", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 104, + 333, + 509, + 401 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 187, + 414, + 426, + 438 + ], + "lines": [ + { + "bbox": [ + 187, + 414, + 426, + 438 + ], + "spans": [ + { + "bbox": [ + 187, + 414, + 426, + 438 + ], + "score": 0.88, + "content": "\\left[ x _ { j } ^ { ( i ) } \\right] _ { i = 1 } ^ { m } \\longleftarrow \\mathrm { H u n g a r i a n } ( \\mathbb { M } _ { k } , \\mathbb { D } _ { j } ) , \\mathrm { ~ f o r ~ e a c h ~ } j = 1 , \\ldots , n .", + "type": "interline_equation", + "image_path": "813445b60502ff54fa3b6a1d8cbe68cbd361766f4b3ffbff4acd6ad2442e5edc.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 187, + 414, + 426, + 438 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 506, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 167, + 469 + ], + "score": 1.0, + "content": "Now, for each", + "type": "text" + }, + { + "bbox": [ + 168, + 455, + 212, + 468 + ], + "score": 0.93, + "content": "\\boldsymbol { y } ^ { ( i ) } \\in \\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 454, + 464, + 469 + ], + "score": 1.0, + "content": "we need to find the weighted average of the matched poin", + "type": "text" + }, + { + "bbox": [ + 455, + 448, + 505, + 470 + ], + "score": 1.0, + "content": "ts h x(i) in .", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 506, + 484 + ], + "score": 1.0, + "content": "However, some of these points could be copies of the diagonal, so we need to consider three", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 506, + 495 + ], + "score": 1.0, + "content": "distinct cases: that each matched point is off-diagonal, that each one is a copy of the diagonal, or", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 493, + 507, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 361, + 507 + ], + "score": 1.0, + "content": "that the points are a mixture of both. 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If", + "type": "text" + }, + { + "bbox": [ + 448, + 538, + 482, + 554 + ], + "score": 0.93, + "content": "\\mathcal { J } _ { \\mathrm { D } } ^ { ( i ) } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 538, + 507, + 556 + ], + "score": 1.0, + "content": ", then", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 555, + 467, + 572 + ], + "spans": [ + { + "bbox": [ + 107, + 558, + 143, + 570 + ], + "score": 0.91, + "content": "y ^ { ( i ) } = w", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 555, + 205, + 572 + ], + "score": 1.0, + "content": ". Otherwise, let", + "type": "text" + }, + { + "bbox": [ + 206, + 560, + 221, + 569 + ], + "score": 0.84, + "content": "w _ { \\Delta }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 555, + 376, + 572 + ], + "score": 1.0, + "content": "be the point on the diagonal closest to", + "type": "text" + }, + { + "bbox": [ + 376, + 560, + 384, + 568 + ], + "score": 0.76, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 555, + 467, + 572 + ], + "score": 1.0, + "content": ". Then our update is", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 102, + 448, + 507, + 572 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 583, + 433, + 619 + ], + "lines": [ + { + "bbox": [ + 177, + 583, + 433, + 619 + ], + "spans": [ + { + "bbox": [ + 177, + 583, + 433, + 619 + ], + "score": 0.94, + "content": "y ^ { ( i ) } \\longleftarrow \\frac { \\sum _ { j \\in \\mathcal { J } _ { \\mathrm { 0 D } } ^ { ( i ) } } r _ { j k } ^ { 2 } x _ { j } ^ { ( i ) } + \\sum _ { j \\in \\mathcal { J } _ { \\mathrm { D } } ^ { ( i ) } } r _ { j k } ^ { 2 } w _ { \\Delta } } { \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } } , \\mathrm { ~ f o r ~ } i = 1 , \\dots , m .", + "type": "interline_equation", + "image_path": "c0292e45f5f942bb00f61a98644e345db81e60b2b62814cc75c0e94929e654d0.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 177, + 583, + 433, + 595.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 177, + 595.0, + 433, + 607.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 177, + 607.0, + 433, + 619.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 638, + 505, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 651 + ], + "score": 1.0, + "content": "We alternate between (3) and (4) until the matching remains the same. Theorem 2, proving that this", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "algorithm converges to a local minimum of the Fréchet function, is proven in Appendix B. Also in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 661, + 404, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 147, + 673 + ], + "score": 1.0, + "content": "Appendix", + "type": "text" + }, + { + "bbox": [ + 148, + 661, + 156, + 671 + ], + "score": 0.31, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 661, + 404, + 673 + ], + "score": 1.0, + "content": "is Algorithm 2, giving an overview of the entire computation.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 639, + 505, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 681, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 680, + 504, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 233, + 695 + ], + "score": 1.0, + "content": "Theorem 2. Given diagrams", + "type": "text" + }, + { + "bbox": [ + 234, + 682, + 246, + 694 + ], + "score": 0.88, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 680, + 336, + 695 + ], + "score": 1.0, + "content": ", membership values", + "type": "text" + }, + { + "bbox": [ + 351, + 680, + 466, + 695 + ], + "score": 1.0, + "content": ", and the Fréchet function", + "type": "text" + }, + { + "bbox": [ + 466, + 680, + 504, + 694 + ], + "score": 0.91, + "content": "F ( { \\hat { \\mathbb { D } } } ) \\mathbf { \\Sigma } =", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 687, + 504, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 196, + 709 + ], + "score": 0.93, + "content": "\\textstyle \\sum _ { j = 1 } ^ { n } r _ { j k } ^ { 2 } W _ { 2 } ( { \\hat { \\mathbb { D } } } , \\mathbb { D } _ { j } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 687, + 222, + 714 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 223, + 694, + 294, + 708 + ], + "score": 0.92, + "content": "\\mathbb { M } _ { k } \\ = \\ \\{ y ^ { ( i ) } \\} _ { i = 1 } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 687, + 393, + 714 + ], + "score": 1.0, + "content": "is a local minimum of", + "type": "text" + }, + { + "bbox": [ + 393, + 695, + 402, + 705 + ], + "score": 0.76, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 687, + 497, + 714 + ], + "score": 1.0, + "content": "if and only if there is", + "type": "text" + }, + { + "bbox": [ + 497, + 697, + 504, + 705 + ], + "score": 0.58, + "content": "a", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 708, + 439, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 221, + 722 + ], + "score": 1.0, + "content": "unique optimal pairing from", + "type": "text" + }, + { + "bbox": [ + 222, + 709, + 237, + 720 + ], + "score": 0.89, + "content": "\\mathbb { M } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 708, + 294, + 722 + ], + "score": 1.0, + "content": "to each of the", + "type": "text" + }, + { + "bbox": [ + 295, + 709, + 307, + 722 + ], + "score": 0.89, + "content": "\\mathbb { D } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 708, + 347, + 722 + ], + "score": 1.0, + "content": "and each", + "type": "text" + }, + { + "bbox": [ + 348, + 708, + 363, + 721 + ], + "score": 0.91, + "content": "\\boldsymbol y ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 708, + 439, + 722 + ], + "score": 1.0, + "content": "is updated via (4).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 680, + 504, + 722 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 122, + 82, + 488, + 228 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 122, + 82, + 488, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 82, + 488, + 228 + ], + "spans": [ + { + "bbox": [ + 122, + 82, + 488, + 228 + ], + "score": 0.421, + "type": "image", + "image_path": "3445a13ce1545f247cd45442b94aace5714885dd37d873b1a7c6954ed5ad08ac.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 122, + 82, + 488, + 130.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 122, + 130.66666666666666, + 488, + 179.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 122, + 179.33333333333331, + 488, + 227.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 250, + 505, + 295 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "Figure 3: Heatmaps showing average number of iterations for fuzzy clustering of persistence diagrams", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "(left) and the weighted Fréchet mean computation (right) to converge. Convergence of the FCM", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 366, + 286 + ], + "score": 1.0, + "content": "algorithm is determined when the cost function is stable to within", + "type": "text" + }, + { + "bbox": [ + 367, + 273, + 397, + 284 + ], + "score": 0.9, + "content": "\\pm 0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 272, + 506, + 286 + ], + "score": 1.0, + "content": ". Convergence experiments", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 284, + 429, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 429, + 296 + ], + "score": 1.0, + "content": "were carried out on randomly generated persistence diagrams with three repeats.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 325, + 200, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 202, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 202, + 339 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 356, + 205, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 207, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 207, + 369 + ], + "score": 1.0, + "content": "4.1 SYNTHETIC DATA", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "Exemplar clustering. We start by demonstrating our algorithm on a simple synthetic dataset designed", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "to highlight its ability to cluster based on the topology of the underlying datasets. We produce three", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "datasets of noise, three datasets of a ring, and three datasets of figure-of-eights, all shown in Figure", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 413, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 427 + ], + "score": 1.0, + "content": "2(a). In Figure 2(b) we show the corresponding 1-PH persistence diagrams. Note that the persistence", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "diagrams have either zero, one, or two significant off-diagonal points, corresponding to zero, one, or", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "two holes in the datasets. We then use our algorithm to cluster the nine persistence diagrams into", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "three clusters. Having only been given the list of diagrams, the number of clusters, and the maximum", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "score": 1.0, + "content": "number of iterations, our algorithm successfully clusters the diagrams based on their topology. Figure", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "2(c) shows that the cluster centres have zero, one, or two off-diagonal points: our algorithm has", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "found cluster centres that reflect the topological features of the datasets. 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Details", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 511, + 309, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 309, + 525 + ], + "score": 1.0, + "content": "of these timing experiments are in Appendix C.1.3", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 506, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 530, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 504, + 541 + ], + "score": 1.0, + "content": "Empirical behaviour. Figure 4 shows the results of experiments run to determine the empirical", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "score": 1.0, + "content": "performance of our algorithm. We give theoretical guarantees that every convergent subsequence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 550, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 565 + ], + "score": 1.0, + "content": "will tend to a local minimum, but in practice it remains important that our algorithm will converge,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 562, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 575 + ], + "score": 1.0, + "content": "and within a reasonable timeframe. To this end we ran experiments on a total of 825 randomly", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "generated persistence diagrams, recording the number of iterations and cost functions for both the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "FCM clustering and the weighted Fréchet mean (WFM) computation. We considered the FCM to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 595, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 336, + 607 + ], + "score": 1.0, + "content": "have converged when the cost function remained within", + "type": "text" + }, + { + "bbox": [ + 337, + 595, + 367, + 606 + ], + "score": 0.9, + "content": "\\pm 0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 596, + 504, + 607 + ], + "score": 1.0, + "content": ". As explained in Section 3.2, the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 604, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 620 + ], + "score": 1.0, + "content": "WFM converges when the matching stays the same. Our experiments showed that the FCM clustering", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 230, + 630 + ], + "score": 1.0, + "content": "consistently converges within", + "type": "text" + }, + { + "bbox": [ + 230, + 618, + 243, + 627 + ], + "score": 0.68, + "content": "{ < } 5", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 616, + 506, + 630 + ], + "score": 1.0, + "content": "iterations, regardless of the number of diagrams and points per", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "score": 1.0, + "content": "diagram (note that the time per iteration increases as the number of points/diagrams increases, even", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 639, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 651 + ], + "score": 1.0, + "content": "if the number of iterations remains stable). We had no experiments in which the algorithm did not", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "converge. The WFM computation requires more iterations as both number of diagrams and number", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "of points per diagram increases, but we once again experienced no failures to converge in each of our", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "experiments. In general, running the algorithm offered no difficulties on a laptop, and we believe the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 683, + 373, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 373, + 695 + ], + "score": 1.0, + "content": "algorithm is ready for use by practitioners in the TDA community.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 29 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 721, + 348, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 348, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 348, + 733 + ], + "score": 1.0, + "content": "3 Further details of all experiments are available in Appendix C.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 122, + 82, + 488, + 228 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 122, + 82, + 488, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 82, + 488, + 228 + ], + "spans": [ + { + "bbox": [ + 122, + 82, + 488, + 228 + ], + "score": 0.421, + "type": "image", + "image_path": "3445a13ce1545f247cd45442b94aace5714885dd37d873b1a7c6954ed5ad08ac.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 122, + 82, + 488, + 130.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 122, + 130.66666666666666, + 488, + 179.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 122, + 179.33333333333331, + 488, + 227.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 250, + 505, + 295 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 505, + 264 + ], + "score": 1.0, + "content": "Figure 3: Heatmaps showing average number of iterations for fuzzy clustering of persistence diagrams", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 506, + 275 + ], + "score": 1.0, + "content": "(left) and the weighted Fréchet mean computation (right) to converge. Convergence of the FCM", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 272, + 506, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 366, + 286 + ], + "score": 1.0, + "content": "algorithm is determined when the cost function is stable to within", + "type": "text" + }, + { + "bbox": [ + 367, + 273, + 397, + 284 + ], + "score": 0.9, + "content": "\\pm 0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 272, + 506, + 286 + ], + "score": 1.0, + "content": ". Convergence experiments", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 284, + 429, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 429, + 296 + ], + "score": 1.0, + "content": "were carried out on randomly generated persistence diagrams with three repeats.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "title", + "bbox": [ + 108, + 325, + 200, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 202, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 202, + 339 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 356, + 205, + 367 + ], + "lines": [ + { + "bbox": [ + 105, + 355, + 207, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 207, + 369 + ], + "score": 1.0, + "content": "4.1 SYNTHETIC DATA", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 505, + 393 + ], + "score": 1.0, + "content": "Exemplar clustering. We start by demonstrating our algorithm on a simple synthetic dataset designed", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "to highlight its ability to cluster based on the topology of the underlying datasets. We produce three", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "datasets of noise, three datasets of a ring, and three datasets of figure-of-eights, all shown in Figure", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 413, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 427 + ], + "score": 1.0, + "content": "2(a). In Figure 2(b) we show the corresponding 1-PH persistence diagrams. Note that the persistence", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 438 + ], + "score": 1.0, + "content": "diagrams have either zero, one, or two significant off-diagonal points, corresponding to zero, one, or", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "two holes in the datasets. We then use our algorithm to cluster the nine persistence diagrams into", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "three clusters. Having only been given the list of diagrams, the number of clusters, and the maximum", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "score": 1.0, + "content": "number of iterations, our algorithm successfully clusters the diagrams based on their topology. Figure", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 468, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 481 + ], + "score": 1.0, + "content": "2(c) shows that the cluster centres have zero, one, or two off-diagonal points: our algorithm has", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 492 + ], + "score": 1.0, + "content": "found cluster centres that reflect the topological features of the datasets. Because we are reducing the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "cardinality and dimensionality of datasets by mapping into persistence diagrams, we also demonstrate", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "a speed-up of at least an order of magnitude over Wasserstein barycentre clustering methods. Details", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 511, + 309, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 309, + 525 + ], + "score": 1.0, + "content": "of these timing experiments are in Appendix C.1.3", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 381, + 506, + 525 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 529, + 506, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 530, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 504, + 541 + ], + "score": 1.0, + "content": "Empirical behaviour. Figure 4 shows the results of experiments run to determine the empirical", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "score": 1.0, + "content": "performance of our algorithm. We give theoretical guarantees that every convergent subsequence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 550, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 565 + ], + "score": 1.0, + "content": "will tend to a local minimum, but in practice it remains important that our algorithm will converge,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 562, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 575 + ], + "score": 1.0, + "content": "and within a reasonable timeframe. To this end we ran experiments on a total of 825 randomly", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "generated persistence diagrams, recording the number of iterations and cost functions for both the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "FCM clustering and the weighted Fréchet mean (WFM) computation. We considered the FCM to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 595, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 336, + 607 + ], + "score": 1.0, + "content": "have converged when the cost function remained within", + "type": "text" + }, + { + "bbox": [ + 337, + 595, + 367, + 606 + ], + "score": 0.9, + "content": "\\pm 0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 596, + 504, + 607 + ], + "score": 1.0, + "content": ". As explained in Section 3.2, the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 604, + 506, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 620 + ], + "score": 1.0, + "content": "WFM converges when the matching stays the same. Our experiments showed that the FCM clustering", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 616, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 230, + 630 + ], + "score": 1.0, + "content": "consistently converges within", + "type": "text" + }, + { + "bbox": [ + 230, + 618, + 243, + 627 + ], + "score": 0.68, + "content": "{ < } 5", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 616, + 506, + 630 + ], + "score": 1.0, + "content": "iterations, regardless of the number of diagrams and points per", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "score": 1.0, + "content": "diagram (note that the time per iteration increases as the number of points/diagrams increases, even", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 639, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 651 + ], + "score": 1.0, + "content": "if the number of iterations remains stable). We had no experiments in which the algorithm did not", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "converge. The WFM computation requires more iterations as both number of diagrams and number", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "of points per diagram increases, but we once again experienced no failures to converge in each of our", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 672, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 684 + ], + "score": 1.0, + "content": "experiments. In general, running the algorithm offered no difficulties on a laptop, and we believe the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 683, + 373, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 373, + 695 + ], + "score": 1.0, + "content": "algorithm is ready for use by practitioners in the TDA community.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 530, + 506, + 695 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 142, + 74, + 471, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 142, + 74, + 471, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 142, + 74, + 471, + 248 + ], + "spans": [ + { + "bbox": [ + 142, + 74, + 471, + 248 + ], + "score": 0.908, + "type": "image", + "image_path": "1af76cbcab2ee1ea6421350136a2706e44ee85ab360b8d60afaf3f80eda85385.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 142, + 74, + 471, + 132.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 142, + 132.0, + 471, + 190.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 142, + 190.0, + 471, + 248.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 257, + 505, + 280 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "Figure 4: Cubic structures (top) and carbon allotropes (bottom). Our algorithm can cluster transformed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 268, + 356, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 356, + 280 + ], + "score": 1.0, + "content": "lattice structures where comparable geometric algorithms fail.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 108, + 319, + 227, + 329 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 227, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 227, + 330 + ], + "score": 1.0, + "content": "4.2 LATTICE STRUCTURES", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "A key property for machine learning in materials science has been identified as “invariance to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "basis symmetries of physics [...] rotation, reflection, translation” (Schmidt et al., 2019). Removing the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "need for a standardised coordinate system allows machine learning methods to be applied to a broader", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "range of existing coordinate datasets generated by experimental methods (e.g., x-ray diffraction and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "scanning electron microscope imaging) and computational methods (e.g., density functional theory).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "Persistence diagrams, which capture affine transformation-invariant properties of datasets, are ideally", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "suited for application in this domain. Our algorithm enables successful unsupervised learning on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 452, + 436 + ], + "score": 1.0, + "content": "these datasets for the first time. Additionally, the fuzzy membership values allow top-", + "type": "text" + }, + { + "bbox": [ + 453, + 424, + 460, + 434 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "ranking of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "candidates suggested by our algorithm. This is particularly important in materials science, where", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 445, + 338, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 338, + 459 + ], + "score": 1.0, + "content": "further investigation of materials can be extremely costly.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "The large majority of solids are comprised of lattices: regularly repeating unit cells of atoms. This", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 356, + 486 + ], + "score": 1.0, + "content": "lattice structure directly determines the properties of a material", + "type": "text" + }, + { + "bbox": [ + 356, + 473, + 443, + 486 + ], + "score": 0.65, + "content": "\\dot { ( \\mathbb { H } \\mathrm { u l l ~ } \\& \\mathrm { ~ B a c o n } ) } \\dot { \\mathbb { Z } } ^ { 0 1 1 } \\mathrm { 1 } \\mathrm { 1 } \\mathrm { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "and it has been", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "predicted that machine learning will reveal presently unknown links between structure and property", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 273, + 508 + ], + "score": 1.0, + "content": "by identifying new trends across materials", + "type": "text" + }, + { + "bbox": [ + 274, + 495, + 406, + 508 + ], + "score": 0.3, + "content": "( \\dot { \\overline { { { \\mathrm { M e r e d i g } } } } } , \\dot { \\underline { { 2 0 1 9 } } } , \\dot { \\overline { { { \\mathrm { W e i ~ e t ~ a l . } } } } } , \\dot { \\underline { { 2 0 1 9 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 495, + 505, + 508 + ], + "score": 1.0, + "content": ". We apply our algorithm", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "to two examples of lattice structures from materials science: cubic structures and carbon allotropes.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 518, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 529 + ], + "score": 1.0, + "content": "Cubic structures are important because they are ubiquitous. The most common lattice structures,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "particularly amongst pure metals, are face-centred cubic (FCC) structures and body-centred cubic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 175, + 552 + ], + "score": 1.0, + "content": "(BCC) structures", + "type": "text" + }, + { + "bbox": [ + 175, + 539, + 231, + 552 + ], + "score": 0.9, + "content": "\\bar { ( \\mathrm { P u t n i s } , \\mathrm { } \\textmu { 9 9 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 539, + 505, + 552 + ], + "score": 1.0, + "content": ", shown in Figures 4(a) and 4(b). Carbon allotropes, such as graphene", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 550, + 503, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 503, + 562 + ], + "score": 1.0, + "content": "and diamond, are widely anticipated to revolutionise electronics and optoelectronics (Wang et al.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 130, + 573 + ], + "score": 0.32, + "content": "\\boxed { 2 0 1 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 560, + 505, + 574 + ], + "score": 1.0, + "content": ". We focus on the carbon allotropes diamond and cis-hinged polydiacetylene, shown in Figures", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 572, + 163, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 163, + 585 + ], + "score": 1.0, + "content": "4(c) and 4(d).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "We use atomic positions for the unit-cells of iron mp-150 and iron mp-13 from the Materials Project", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 107, + 600, + 177, + 613 + ], + "score": 0.72, + "content": "\\underline { { \\mathrm { ( J a i n ~ e t ~ a l . } } } , \\overline { { 2 0 1 3 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 599, + 506, + 613 + ], + "score": 1.0, + "content": ", representing BCC and FCC structures respectively, for our first experiment. For", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "our second experiment we use diamond and cis-hinged polydiacetylene unit-cell atomic positions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 282, + 634 + ], + "score": 1.0, + "content": "from the Samara Carbon Allotrope Database", + "type": "text" + }, + { + "bbox": [ + 282, + 622, + 374, + 635 + ], + "score": 0.32, + "content": "( \\overline { { { \\mathrm { H o f f m a n n ~ e t ~ a l . } } } } ) , \\overline { { { 2 0 1 6 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 621, + 505, + 634 + ], + "score": 1.0, + "content": ". We simulate distinct collections", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "of lattices by transforming the atomic coordinates, with no information about bonds given to the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "algorithms. The properties of persistence diagrams mean that we can successfully cluster the atomic", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 655, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 504, + 666 + ], + "score": 1.0, + "content": "coordinates derived from the same base unit-cell regardless of the transformations applied to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "coordinate system, fulfilling the key property identified above (we consider a clustering successful", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "when all datasets from the same lattice structure have their highest membership values for the same", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "cluster). In comparison, we run Wasserstein barycentre clustering on the same datasets using several", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "score": 1.0, + "content": "state-of-the-art algorithms for barycentre computation and approximation. Each can only successfully", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "cluster the cubic structures after reflection, and none of them successfully cluster the carbon allotropes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 444, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 444, + 733 + ], + "score": 1.0, + "content": "after any transformations. 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Our algorithm can cluster transformed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 268, + 356, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 356, + 280 + ], + "score": 1.0, + "content": "lattice structures where comparable geometric algorithms fail.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 108, + 319, + 227, + 329 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 227, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 227, + 330 + ], + "score": 1.0, + "content": "4.2 LATTICE STRUCTURES", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "A key property for machine learning in materials science has been identified as “invariance to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "basis symmetries of physics [...] rotation, reflection, translation” (Schmidt et al., 2019). Removing the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 382 + ], + "score": 1.0, + "content": "need for a standardised coordinate system allows machine learning methods to be applied to a broader", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 506, + 393 + ], + "score": 1.0, + "content": "range of existing coordinate datasets generated by experimental methods (e.g., x-ray diffraction and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "scanning electron microscope imaging) and computational methods (e.g., density functional theory).", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "Persistence diagrams, which capture affine transformation-invariant properties of datasets, are ideally", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "suited for application in this domain. Our algorithm enables successful unsupervised learning on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 452, + 436 + ], + "score": 1.0, + "content": "these datasets for the first time. Additionally, the fuzzy membership values allow top-", + "type": "text" + }, + { + "bbox": [ + 453, + 424, + 460, + 434 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "ranking of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "candidates suggested by our algorithm. This is particularly important in materials science, where", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 445, + 338, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 338, + 459 + ], + "score": 1.0, + "content": "further investigation of materials can be extremely costly.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 347, + 506, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "The large majority of solids are comprised of lattices: regularly repeating unit cells of atoms. This", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 473, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 356, + 486 + ], + "score": 1.0, + "content": "lattice structure directly determines the properties of a material", + "type": "text" + }, + { + "bbox": [ + 356, + 473, + 443, + 486 + ], + "score": 0.65, + "content": "\\dot { ( \\mathbb { H } \\mathrm { u l l ~ } \\& \\mathrm { ~ B a c o n } ) } \\dot { \\mathbb { Z } } ^ { 0 1 1 } \\mathrm { 1 } \\mathrm { 1 } \\mathrm { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 473, + 505, + 486 + ], + "score": 1.0, + "content": "and it has been", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "predicted that machine learning will reveal presently unknown links between structure and property", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 495, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 273, + 508 + ], + "score": 1.0, + "content": "by identifying new trends across materials", + "type": "text" + }, + { + "bbox": [ + 274, + 495, + 406, + 508 + ], + "score": 0.3, + "content": "( \\dot { \\overline { { { \\mathrm { M e r e d i g } } } } } , \\dot { \\underline { { 2 0 1 9 } } } , \\dot { \\overline { { { \\mathrm { W e i ~ e t ~ a l . } } } } } , \\dot { \\underline { { 2 0 1 9 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 495, + 505, + 508 + ], + "score": 1.0, + "content": ". We apply our algorithm", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "to two examples of lattice structures from materials science: cubic structures and carbon allotropes.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 518, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 529 + ], + "score": 1.0, + "content": "Cubic structures are important because they are ubiquitous. The most common lattice structures,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "particularly amongst pure metals, are face-centred cubic (FCC) structures and body-centred cubic", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 175, + 552 + ], + "score": 1.0, + "content": "(BCC) structures", + "type": "text" + }, + { + "bbox": [ + 175, + 539, + 231, + 552 + ], + "score": 0.9, + "content": "\\bar { ( \\mathrm { P u t n i s } , \\mathrm { } \\textmu { 9 9 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 539, + 505, + 552 + ], + "score": 1.0, + "content": ", shown in Figures 4(a) and 4(b). Carbon allotropes, such as graphene", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 550, + 503, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 503, + 562 + ], + "score": 1.0, + "content": "and diamond, are widely anticipated to revolutionise electronics and optoelectronics (Wang et al.,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 130, + 573 + ], + "score": 0.32, + "content": "\\boxed { 2 0 1 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 560, + 505, + 574 + ], + "score": 1.0, + "content": ". We focus on the carbon allotropes diamond and cis-hinged polydiacetylene, shown in Figures", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 572, + 163, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 163, + 585 + ], + "score": 1.0, + "content": "4(c) and 4(d).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 462, + 506, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "We use atomic positions for the unit-cells of iron mp-150 and iron mp-13 from the Materials Project", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 107, + 600, + 177, + 613 + ], + "score": 0.72, + "content": "\\underline { { \\mathrm { ( J a i n ~ e t ~ a l . } } } , \\overline { { 2 0 1 3 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 599, + 506, + 613 + ], + "score": 1.0, + "content": ", representing BCC and FCC structures respectively, for our first experiment. For", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "our second experiment we use diamond and cis-hinged polydiacetylene unit-cell atomic positions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 282, + 634 + ], + "score": 1.0, + "content": "from the Samara Carbon Allotrope Database", + "type": "text" + }, + { + "bbox": [ + 282, + 622, + 374, + 635 + ], + "score": 0.32, + "content": "( \\overline { { { \\mathrm { H o f f m a n n ~ e t ~ a l . } } } } ) , \\overline { { { 2 0 1 6 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 621, + 505, + 634 + ], + "score": 1.0, + "content": ". We simulate distinct collections", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "of lattices by transforming the atomic coordinates, with no information about bonds given to the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "algorithms. The properties of persistence diagrams mean that we can successfully cluster the atomic", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 655, + 504, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 504, + 666 + ], + "score": 1.0, + "content": "coordinates derived from the same base unit-cell regardless of the transformations applied to the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "coordinate system, fulfilling the key property identified above (we consider a clustering successful", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "when all datasets from the same lattice structure have their highest membership values for the same", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "cluster). In comparison, we run Wasserstein barycentre clustering on the same datasets using several", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 711 + ], + "score": 1.0, + "content": "state-of-the-art algorithms for barycentre computation and approximation. Each can only successfully", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "cluster the cubic structures after reflection, and none of them successfully cluster the carbon allotropes", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 720, + 444, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 444, + 733 + ], + "score": 1.0, + "content": "after any transformations. We give specific values for these results in Appendix C.2.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 588, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 233, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 234, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 234, + 95 + ], + "score": 1.0, + "content": "4.3 DECISION BOUNDARIES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 107, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "score": 1.0, + "content": "Learnt models have been shown to perform better on datasets which have a similar persistence", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 118, + 506, + 132 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 273, + 132 + ], + "score": 1.0, + "content": "diagram to the model’s decision boundary", + "type": "text" + }, + { + "bbox": [ + 274, + 119, + 378, + 131 + ], + "score": 0.58, + "content": "( \\mathrm { \\mathbb { R a m a m u r t h y ~ e t ~ a l . } } , \\mathrm { \\mathbb { 2 0 1 9 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 118, + 506, + 132 + ], + "score": 1.0, + "content": ". 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We compute the 1-persistence diagrams of the tasks’ and models’ decision", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 218, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 228 + ], + "score": 1.0, + "content": "boundaries and cluster them to obtain membership values and cluster centres. To view task and model", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "score": 1.0, + "content": "proximity through our clustering, we find the cluster centre with the highest membership value for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 238, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 252 + ], + "score": 1.0, + "content": "each task, and consider the models closest to that cluster centre. Note that in general you cannot do", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "this with hard clustering: most of the time a path will not exist from task to cluster centre to model,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 507, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 507, + 275 + ], + "score": 1.0, + "content": "because each task/model is only associated with a single cluster. This contrasts with fuzzy clustering,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 272, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 506, + 284 + ], + "score": 1.0, + "content": "where you have information about how close each model/task is to each cluster centre. We further", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 282, + 381, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 381, + 295 + ], + "score": 1.0, + "content": "discuss why this does not work for hard clustering in Appendix C.3.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 312 + ], + "score": 1.0, + "content": "To assess the ability of our model/task clustering, we performed the above experiment on three", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 271, + 324 + ], + "score": 1.0, + "content": "different datasets: MNIST (LeCun et al.,", + "type": "text" + }, + { + "bbox": [ + 271, + 311, + 297, + 323 + ], + "score": 0.31, + "content": "\\textcircled { 2 0 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 310, + 366, + 324 + ], + "score": 1.0, + "content": ", FashionMNIST", + "type": "text" + }, + { + "bbox": [ + 367, + 311, + 441, + 323 + ], + "score": 0.52, + "content": "( \\overline { { \\mathrm { B i a o ~ e t ~ a l . } } } , \\overline { { 2 0 1 7 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 310, + 506, + 324 + ], + "score": 1.0, + "content": ", and Kuzushiji-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 321, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 210, + 335 + ], + "score": 1.0, + "content": "MNIST (Clanuwat et al.,", + "type": "text" + }, + { + "bbox": [ + 210, + 322, + 236, + 334 + ], + "score": 0.4, + "content": "\\boxed { 2 0 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 321, + 506, + 335 + ], + "score": 1.0, + "content": ". We repeat each experiment three times. Our goal is to evaluate", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "whether or not the clustering is capturing information about model performance on tasks, so as a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "baseline we use the average performance of all models on a fixed task, averaged over all tasks. Recall", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "that we are just using the topology of the decision boundary: no information about model performance", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "on the task was used until we assessed the ability of the clustering to capture information about the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "tasks. We start by verifying what happens if we use the model closest to the cluster centre associated", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "with the task (i.e., top-1). We see a significant increase in performance, indicating that the topological", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "fuzzy clustering has selected the model trained on the task: this means the clustering is working.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 408, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 423 + ], + "score": 1.0, + "content": "We also compute the top-3 and top-2 performance change over average. We still see a statistically", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "significant increase in performance over average performance, indicating that our membership values", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "are capturing information about model performance on unseen tasks. These results are shown in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "Table 1. By successfully matching models to tasks using fuzzy clustering, we offer a new technique", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 452, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 467 + ], + "score": 1.0, + "content": "for future model marketplaces, demonstrating the value of topological clustering when analysing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 464, + 191, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 191, + 477 + ], + "score": 1.0, + "content": "decision boundaries.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 25.5 + }, + { + "type": "table", + "bbox": [ + 150, + 537, + 460, + 610 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 494, + 506, + 528 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 494, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 505, + 508 + ], + "score": 1.0, + "content": "Table 1: Performance increase/decrease over average task performance when using learnt fuzzy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "membership values for model selection. 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Performance change vs random model selection (%)
Top-3Top-2Top-1
MNIST+6.17±2.18+10.81±1.88+20.88±4.08
FashionMNIST+16.46±4.00+21.94±4.73+23.30±8.72
Kuzushiji+6.61±1.78+11.18±2.45+21.89±5.54
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On each of these tasks, we train a model. 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We compute the 1-persistence diagrams of the tasks’ and models’ decision", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 218, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 505, + 228 + ], + "score": 1.0, + "content": "boundaries and cluster them to obtain membership values and cluster centres. To view task and model", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "score": 1.0, + "content": "proximity through our clustering, we find the cluster centre with the highest membership value for", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 238, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 506, + 252 + ], + "score": 1.0, + "content": "each task, and consider the models closest to that cluster centre. Note that in general you cannot do", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 506, + 262 + ], + "score": 1.0, + "content": "this with hard clustering: most of the time a path will not exist from task to cluster centre to model,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 507, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 507, + 275 + ], + "score": 1.0, + "content": "because each task/model is only associated with a single cluster. This contrasts with fuzzy clustering,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 272, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 506, + 284 + ], + "score": 1.0, + "content": "where you have information about how close each model/task is to each cluster centre. We further", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 282, + 381, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 381, + 295 + ], + "score": 1.0, + "content": "discuss why this does not work for hard clustering in Appendix C.3.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 106, + 507, + 295 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 506, + 312 + ], + "score": 1.0, + "content": "To assess the ability of our model/task clustering, we performed the above experiment on three", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 271, + 324 + ], + "score": 1.0, + "content": "different datasets: MNIST (LeCun et al.,", + "type": "text" + }, + { + "bbox": [ + 271, + 311, + 297, + 323 + ], + "score": 0.31, + "content": "\\textcircled { 2 0 1 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 310, + 366, + 324 + ], + "score": 1.0, + "content": ", FashionMNIST", + "type": "text" + }, + { + "bbox": [ + 367, + 311, + 441, + 323 + ], + "score": 0.52, + "content": "( \\overline { { \\mathrm { B i a o ~ e t ~ a l . } } } , \\overline { { 2 0 1 7 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 310, + 506, + 324 + ], + "score": 1.0, + "content": ", and Kuzushiji-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 321, + 506, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 210, + 335 + ], + "score": 1.0, + "content": "MNIST (Clanuwat et al.,", + "type": "text" + }, + { + "bbox": [ + 210, + 322, + 236, + 334 + ], + "score": 0.4, + "content": "\\boxed { 2 0 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 321, + 506, + 335 + ], + "score": 1.0, + "content": ". We repeat each experiment three times. Our goal is to evaluate", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "whether or not the clustering is capturing information about model performance on tasks, so as a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 505, + 356 + ], + "score": 1.0, + "content": "baseline we use the average performance of all models on a fixed task, averaged over all tasks. Recall", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "that we are just using the topology of the decision boundary: no information about model performance", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "on the task was used until we assessed the ability of the clustering to capture information about the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 389 + ], + "score": 1.0, + "content": "tasks. We start by verifying what happens if we use the model closest to the cluster centre associated", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "with the task (i.e., top-1). We see a significant increase in performance, indicating that the topological", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "fuzzy clustering has selected the model trained on the task: this means the clustering is working.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 408, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 423 + ], + "score": 1.0, + "content": "We also compute the top-3 and top-2 performance change over average. We still see a statistically", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "significant increase in performance over average performance, indicating that our membership values", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "are capturing information about model performance on unseen tasks. These results are shown in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "Table 1. By successfully matching models to tasks using fuzzy clustering, we offer a new technique", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 452, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 467 + ], + "score": 1.0, + "content": "for future model marketplaces, demonstrating the value of topological clustering when analysing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 464, + 191, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 191, + 477 + ], + "score": 1.0, + "content": "decision boundaries.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 300, + 506, + 477 + ] + }, + { + "type": "table", + "bbox": [ + 150, + 537, + 460, + 610 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 494, + 506, + 528 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 494, + 505, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 505, + 508 + ], + "score": 1.0, + "content": "Table 1: Performance increase/decrease over average task performance when using learnt fuzzy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 518 + ], + "score": 1.0, + "content": "membership values for model selection. The increase in performance demonstrates that our fuzzy", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 517, + 400, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 400, + 529 + ], + "score": 1.0, + "content": "clustering automatically clusters models near tasks they perform well on.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "table_body", + "bbox": [ + 150, + 537, + 460, + 610 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 150, + 537, + 460, + 610 + ], + "spans": [ + { + "bbox": [ + 150, + 537, + 460, + 610 + ], + "score": 0.982, + "html": "
Performance change vs random model selection (%)
Top-3Top-2Top-1
MNIST+6.17±2.18+10.81±1.88+20.88±4.08
FashionMNIST+16.46±4.00+21.94±4.73+23.30±8.72
Kuzushiji+6.61±1.78+11.18±2.45+21.89±5.54
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Our topological fuzzy clustering", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "paves the way for applications in materials science and pre-trained model marketplaces, where we", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 721, + 493, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 493, + 732 + ], + "score": 1.0, + "content": "envisage our algorithm being used as part of automated materials discovery and model selection.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42.5, + "bbox_fs": [ + 104, + 687, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 176, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 95 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 504, + 123 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 506, + 113 + ], + "score": 1.0, + "content": "Ulrich Bauer. 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Input Diagrams D = {Dj}𝑗=1, number of clusters c, maximum iterations MAXITER Output Cluster centres M = {Mk}ε=1, membership values R = {rjk}
1:D = ADDDIAGONALS(D)7: end for
2:M=INITCENTRES(D)8: end for
3: for countin1..MAXITER do9: for k in1..c do
4: for j in1..n do10: M ← WFRECHETMEAN(D,Rk)
5: for k in1..c do11: end for
W2(Mk,D)-1 12: end for
(∑i-1 W) rjk← 6:13: return M, R
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Performance change vs random model selection (%)
Top-3Top-2Top-1
MNIST+6.17±2.18+10.81±1.88+20.88±4.08
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0000000000000000000000000000000000000000..8c11d04b80dfc3847e15101f16e637d991152029 --- /dev/null +++ b/parse/train/S1dIzvclg/S1dIzvclg.md @@ -0,0 +1,299 @@ +# A RECURRENT NEURAL NETWORK WITHOUT CHAOS + +# Thomas Laurent + +# James von Brecht + +Department of Mathematics Loyola Marymount University Los Angeles, CA 90045, USA tlaurent@lmu.edu + +Department of Mathematics California State University, Long Beach Long Beach, CA 90840, USA james.vonbrecht@csulb.edu + +# ABSTRACT + +We introduce an exceptionally simple gated recurrent neural network (RNN) that achieves performance comparable to well-known gated architectures, such as LSTMs and GRUs, on the word-level language modeling task. We prove that our model has simple, predicable and non-chaotic dynamics. This stands in stark contrast to more standard gated architectures, whose underlying dynamical systems exhibit chaotic behavior. + +# 1 INTRODUCTION + +Gated recurrent neural networks, such as the Long Short Term Memory network (LSTM) introduced by Hochreiter & Schmidhuber (1997) and the Gated Recurrent Unit (GRU) proposed by Cho et al. (2014), prove highly effective for machine learning tasks that involve sequential data. We propose an exceptionally simple variant of these gated architectures. The basic model takes the form + +$$ +h _ { t } = \theta _ { t } \odot \operatorname { t a n h } ( h _ { t - 1 } ) + \eta _ { t } \odot \operatorname { t a n h } ( W x _ { t } ) , +$$ + +where $\odot$ stands for the Hadamard product. The horizontal/forget gate (i.e. $\theta _ { t }$ ) and the vertical/input gate (i.e. $\eta _ { t }$ ) take the usual form used in most gated RNN architectures. Specifically + +$$ +\begin{array} { r } { \theta _ { t } : = \sigma \left( U _ { \theta } h _ { t - 1 } + V _ { \theta } x _ { t } + b _ { \theta } \right) \quad \mathrm { a n d } \quad \eta _ { t } : = \sigma \left( U _ { \eta } h _ { t - 1 } + V _ { \eta } x _ { t } + b _ { \eta } \right) } \end{array} +$$ + +where $\sigma ( x ) : = ( 1 + \mathrm { e } ^ { - x } ) ^ { - 1 }$ denotes the logistic sigmoid function. The network (1)–(2) has quite intuitive dynamics. Suppose the data $x _ { t }$ present the model with a sequence + +$$ +( W x _ { t } ) ( i ) = { \left\{ \begin{array} { l l } { 1 0 } & { { \mathrm { i f ~ } } t = T } \\ { 0 } & { { \mathrm { o t h e r w i s e } } , } \end{array} \right. } +$$ + +where $( W x _ { t } ) ( i )$ stands for the $i ^ { \mathrm { t h } }$ component of the vector $W x _ { t }$ . In other words we consider an input sequence $x _ { t }$ for which the learned $i ^ { \mathrm { t h } }$ feature $( W x _ { t } ) ( i )$ remains off except at time $T$ . When initialized from $h _ { 0 } = 0$ , the corresponding response of the network to this “impulse” in the $i ^ { \mathrm { t h } }$ feature is + +$$ +h _ { t } ( i ) \approx { \left\{ \begin{array} { l l } { 0 } & { { \mathrm { i f ~ } } t < T } \\ { \eta _ { T } } & { { \mathrm { i f ~ } } t = T } \\ { \alpha _ { t } } & { { \mathrm { i f ~ } } t > T } \end{array} \right. } +$$ + +with $\alpha _ { t }$ a sequence that relaxes toward zero. The forget gate $\theta _ { t }$ control the rate of this relaxation. Thus $h _ { t } ( i )$ activates when presented with a strong $i ^ { \mathrm { t h } }$ feature, and then relaxes toward zero until the data present the network once again with strong $i ^ { \mathrm { t h } }$ feature. Overall this leads to a dynamically simple model, in which the activation patterns in the hidden states of the network have a clear cause and predictable subsequent behavior. + +Dynamics of this sort do not occur in other RNN models. Instead, the three most popular recurrent neural network architectures, namely the vanilla RNN, the LSTM and the GRU, have complex, irregular, and unpredictable dynamics. Even in the absence of input data, these networks can give rise to chaotic dynamical systems. In other words, when presented with null input data the activation patterns in their hidden states do not necessarily follow a predictable path. The proposed network (1)–(2) has rather dull and minimalist dynamics in comparison; its only attractor is the zero state, and so it stands at the polar-opposite end of the spectrum from chaotic systems. Perhaps surprisingly, at least in the light of this comparison, the proposed network (1) performs as well as LSTMs and GRUs on the word level language modeling task. We therefore conclude that the ability of an RNN to form chaotic temporal dynamics, in the sense we describe in Section 2, cannot explain its success on word-level language modeling tasks. + +In the next section, we review the phenomenon of chaos in RNNs via both synthetic examples and trained models. We also prove a precise, quantified description of the dynamical picture (3)–(4) for the proposed network. In particular, we show that the dynamical system induced by the proposed network is never chaotic, and for this reason we refer to it as a Chaos-Free Network (CFN). The final section provides a series of experiments that demonstrate that CFN achieve results comparable to LSTM on the word-level language modeling task. All together, these observations show that an architecture as simple as (1)–(2) can achieve performance comparable to the more dynamically complex LSTM. + +# 2 CHAOS IN RECURRENT NEURAL NETWORKS + +The study of RNNs from a dynamical systems point-of-view has brought fruitful insights into generic features of RNNs (Sussillo & Barak, 2013; Pascanu et al., 2013). We shall pursue a brief investigation of CFN, LSTM and GRU networks using this formalism, as it allows us to identify key distinctions between them. Recall that for a given mapping $\Phi : \mathbb { R } ^ { d } \mapsto \mathbb { R } ^ { d }$ , a given initial time $t _ { 0 } \in \mathbb { N }$ and a given initial state $\mathfrak { u } _ { 0 } \in \mathbb { R } ^ { d }$ , a simple repeated iteration of the mapping $\Phi$ + +$$ +\begin{array} { c c } { \mathfrak { u } _ { t + 1 } = \Phi ( \mathfrak { u } _ { t } ) } & { t > t _ { 0 } , } \\ { \mathfrak { u } _ { t _ { 0 } } = \mathfrak { u } _ { 0 } } & { t = t _ { 0 } , } \end{array} +$$ + +defines a discrete-time dynamical system. The index $t \in \mathbb N$ represents the current time, while the point $\mathfrak { u } _ { t } \in \mathbb { R } ^ { d }$ represents the current state of the system. The set of all visited states $\mathcal { O } ^ { + } ( \mathfrak { u } _ { 0 } ) : =$ $\left\{ \mathfrak { u } _ { t _ { 0 } } , \mathfrak { u } _ { t _ { 0 } + 1 } , \dotsc , \mathfrak { u } _ { t _ { 0 } + n } , \dotsc \right\}$ defines the forward trajectory or forward orbit through $\mathfrak { u } _ { 0 }$ . An attractor for the dynamical system is a set that is invariant (any trajectory that starts in the set remains in the set) and that attracts all trajectories that start sufficiently close to it. The attractors of chaotic dynamical systems are often fractal sets, and for this reason they are referred to as strange attractors. + +Most RNNs generically take the functional form + +$$ +\mathsf { u } _ { t } = \Psi ( \mathfrak { u } _ { t - 1 } , W _ { 1 } x _ { t } , W _ { 2 } x _ { t } , \ldots , W _ { k } x _ { t } ) , +$$ + +where $x _ { t }$ denotes the $t ^ { \mathrm { t h } }$ input data point. For example, in the case of the CFN (1)–(2), we have $W _ { 1 } = W$ , $W _ { 2 } = V _ { \theta }$ and $W _ { 3 } = V _ { \eta }$ . To gain insight into the underlying design of the architecture of an RNN, it proves usefull to consider how trajectories behave when they are not influenced by any external input. This lead us to consider the dynamical system + +$$ +\mathfrak { u } _ { t } = \Phi ( \mathfrak { u } _ { t - 1 } ) \qquad \Phi ( \mathfrak { u } ) : = \Psi ( \mathfrak { u } , 0 , 0 , \dots , 0 ) , +$$ + +which we refer to as the dynamical system induced by the recurrent neural network. The timeinvariant system (6) is much more tractable than (5), and it offers a mean to investigate the inner working of a given architecture; it separates the influence of input data $x _ { t }$ , which can produce essentially any possible response, from the model itself. Studying trajectories that are not influenced by external data will give us an indication on the ability of a given RNN to generate complex and sophisticated trajectories by its own. As we shall see shortly, the dynamical system induced by a CFN has excessively simple and predictable trajectories: all of them converge to the zero state. In other words, its only attractor is the zero state. This is in sharp contrast with the dynamical systems induced by LSTM or GRU, who can exhibit chaotic behaviors and have strange attractors. + +The learned parameters $W _ { j }$ in (5) describe how data influence the evolution of hidden states at each time step. From a modeling perspective, (6) would occur in the scenario where a trained RNN has learned a weak coupling between a specific data point $x _ { t _ { 0 } }$ and the hidden state at that time, in the sense that the data influence is small and so all $W _ { j } x _ { t _ { 0 } } \approx 0$ nearly vanish. The hidden state then transitions according to $\mathfrak { u } _ { t _ { 0 } } \approx \Psi ( \mathfrak { u } _ { t _ { 0 } - 1 } , 0 , 0 , \dots , 0 ) = \Phi ( \mathfrak { u } _ { t _ { 0 } - 1 } )$ . + +We refer to Bertschinger & Natschlager (2004) for a study of the chaotic behavior of a simplified ¨ vanilla RNN with a specific statistical model, namely an i.i.d. Bernoulli process, for the input data as well as a specific statistical model, namely i.i.d. Gaussian, for the weights of the recurrence matrix. + +![](images/4ab3671f85507ce9fae8efa2bef387ae843514737e6c1f8f9ec26fbe10f6ea41.jpg) +Figure 1: Strange attractor of a 2-unit LSTM. Successive zooms (from left to right) reveal the selfrepeating, fractal nature of the attractor. Colored boxes depict zooming regions. + +# 2.1 CHAOTIC BEHAVIOR OF LSTM AND GRU IN THE ABSENCE OF INPUT DATA + +In this subsection we briefly show that LSTM and GRU, in the absence of input data, can lead to dynamical systems $\mathfrak { u } _ { t } = \Phi ( \mathfrak { u } _ { t - 1 } )$ that are chaotic in the classical sense of the term (Strogatz, 2014). Figure 1 depicts the strange attractor of the dynamical system: + +$$ +\begin{array} { r l } & { \mathbf { u } _ { t } = \Big [ _ { c _ { t } } ^ { h _ { t } } \Big ] \quad \mathbf { u } \mapsto { \boldsymbol { \Phi } } ( \mathbf { u } ) = \Big [ ^ { o \odot \operatorname { t a n h } \big ( f \odot c + i \odot g \big ) } \Big ] } \\ & { { \small i : = \sigma ( W _ { i } h + b _ { i } ) \quad f : = \sigma ( W _ { f } h + b _ { f } ) \quad o : = \sigma ( W _ { o } h + b _ { o } ) \quad g : = \operatorname { t a n h } ( W _ { g } h + b _ { g } ) } } \end{array} +$$ + +induced by a two-unit LSTM with weight matrices + +$$ +W _ { i } = \left[ - 1 \quad - 4 \right] \quad W _ { o } = \left[ { \begin{array} { r r } { 4 } & { 1 } \\ { - 9 } & { - 7 } \end{array} } \right] \quad W _ { f } = \left[ { \begin{array} { r r } { - 2 } & { 6 } \\ { 0 } & { - 6 } \end{array} } \right] \quad W _ { g } = \left[ { \begin{array} { r r } { - 1 } & { - 6 } \\ { 6 } & { - 9 } \end{array} } \right] +$$ + +and zero bias for the model parameters. These weights were randomly generated from a normal distribution with standard deviation 5 and then rounded to the nearest integer. Figure 1(a) was obtained by choosing an initial state ${ \mathfrak { u } } _ { 0 } ~ = ~ ( h _ { 0 } , c _ { 0 } )$ uniformly at random in $[ 0 , 1 ] ^ { 2 } \times [ 0 , \dot { 1 } ] ^ { 2 }$ and plotting the h-component of the iterates ${ \mathfrak { u } } _ { t } ~ = ~ ( h _ { t } , c _ { t } )$ for $t$ between $1 0 ^ { 3 }$ and $1 0 ^ { \bar { 5 } }$ (so this figure should be regarded as a two dimensional projection of a four dimensional attractor, which explain its tangled appearance). Most trajectories starting in $[ 0 , 1 ] ^ { 2 } \times [ 0 , 1 ] ^ { 2 }$ converge toward the depicted attractor. The resemblance between this attractor and classical strange attractors such as the Henon ´ attractor is striking (see Figure 5 in the appendix for a depiction of the Henon attractor). Successive ´ zooms on the branch of the LSTM attractor from Figure 1(a) reveal its fractal nature. Figure 1(b) is an enlargement of the red box in Figure 1(a), and Figure 1(c) is an enlargement of the magenta box in Figure 1(b). We see that the structure repeats itself as we zoom in. + +The most practical consequence of chaos is that the long-term behavior of their forward orbits can exhibit a high degree of sensitivity to the initial states $\mathfrak { u } _ { 0 }$ . Figure 2 provides an example of such behavior for the dynamical system (7)–(9). An initial condition $\mathfrak { u } _ { 0 }$ was drawn uniformly at random in $[ 0 , 1 ] ^ { 2 } \times [ 0 , 1 ] ^ { 2 }$ . We then computed 100, 000 small amplitude perturbations $\hat { \mathfrak { u } } _ { 0 }$ of $\mathfrak { u } _ { 0 }$ by adding a small random number drawn uniformly from $[ - 1 0 ^ { - 7 } , 1 0 ^ { - 7 } ]$ to each component. We then iterated (7)–(9) for 200 steps and plotted the $\mathbf { h }$ -component of the final state $\hat { \textmd u } _ { 2 0 0 }$ for each of the 100, 000 trials on Figure 2(a). The collection of these 100, 000 final states essentially fills out the entire attractor, despite the fact that their initial conditions are highly localized (i.e. at distance of no more than $1 0 ^ { - 7 }$ ) around a fixed point. In other words, the time $t = 2 0 0$ map of the dynamical system will map a small neighborhood around a fixed initial condition $\mathfrak { u } _ { 0 }$ to the entire attractor. Figure 2(b) additionally illustrates this sensitivity to initial conditions for points on the attractor itself. We take an initial condition $\mathfrak { u } _ { 0 }$ on the attractor and perturb it by $1 0 ^ { - 7 }$ to a nearby initial condition $\hat { \mathfrak { u } } _ { 0 }$ . We then plot the distance $\left\| \hat { \mathbf { u } } _ { t } - \mathbf { u } _ { t } \right\|$ between the two corresponding trajectories for the first 200 time steps. After an initial phase of agreement, the trajectories strongly diverge. + +The synthetic example (7)–(9) illustrates the potentially chaotic nature of the LSTM architecture. We now show that chaotic behavior occurs for trained models as well, and not just for synthetically generated instances. We take the parameter values of an LSTM with 228 hidden units trained on the + +![](images/08703ee2db3c758c4fb6bd9c2358d1258e01e76cbaa7406e8cd80b1c7aee6c0d.jpg) +Figure 2: (a): A small neighborhood around a fixed initial condition $\mathfrak { u } _ { 0 }$ , after 200 iterations, is mapped to the entire attractor. (b): Two trajectories starting starting within $1 0 ^ { - 7 }$ of one another strongly diverge after 50 steps. + +![](images/5d4ac4bc4740c0bee2b2e32fef8a96f54fb4405a873f08e76cd9d14db5cc0d2c.jpg) +Figure 3: 228-unit LSTM trained on Penn Treebank. (a): In the absence of input data, the system is chaotic and nearby trajectories diverge. (b): In the presence of input data, the system is mostly driven by the external input. Trajectories starting far apart converge. + +Penn Treebank corpus without dropout (c.f. the experimental section for the precise procedure). We then set all data inputs $x _ { t }$ to zero and run the corresponding induced dynamical system. Two trajectories starting from nearby initial conditions $\mathfrak { u } _ { 0 }$ and $\hat { \mathfrak { u } } _ { 0 }$ were computed (as before $\hat { \mathfrak { u } } _ { 0 }$ was obtained by adding to each components of $\mathfrak { u } _ { 0 }$ a small random number drawn uniformly from $[ - 1 0 ^ { - 7 } , 1 0 ^ { - 7 } ] )$ . Figure 3(a) plots the first component h(1) of the hidden state for both trajectories over the first 1600 time steps. After an initial phase of agreement, the forward trajectories $\mathcal { O } ^ { + } ( \mathfrak { u } _ { 0 } )$ and $\mathcal { O } ^ { + } ( \hat { \mathfrak { u } } _ { 0 } )$ strongly diverge. We also see that both trajectories exhibit the typical aperiodic behavior that characterizes chaotic systems. If the inputs $x _ { t }$ do not vanish, but come from actual word-level data, then the behavior is very different. The LSTM is now no longer an autonomous system whose dynamics are driven by its hidden states, but a time dependent system whose dynamics are mostly driven by the external inputs. Figure 3(b) shows the first component $h ( 1 )$ of the hidden states of two trajectories that start with initial conditions $\mathfrak { u } _ { 0 }$ and $\hat { \mathfrak { u } } _ { 0 }$ that are far apart. The sensitivity to initial condition disappears, and instead the trajectories converge toward each other after about 70 steps. The memory of this initial difference is lost. Overall these experiments indicate that a trained LSTM, when it is not driven by external inputs, can be chaotic. In the presence of input data, the LSTM becomes a forced system whose dynamics are dominated by external forcing. + +Like LSTM networks, GRU can also lead to dynamical systems that are chaotic and they can also have strange attractors. The depiction of such an attractor, in the case of a two-unit GRU, is provided in Figure 6 of the appendix. + +# 2.2 CHAOS-FREE BEHAVIOR OF THE CFN + +The dynamical behavior of the CFN is dramatically different from that of the LSTM. In this subsection we start by showing that the hidden states of the CFN activate and relax toward zero in a predictable fashion in response to input data. On one hand, this shows that the CFN cannot produce non-trivial dynamics without some influence from data. On the other, this leads to an interpretable model; any non-trivial activations in the hidden states of a CFN have a clear cause emanating from data-driven activation. This follows from a precise, quantified description of the intuitive picture (3)–(4) sketched in the introduction. + +We begin with the following simple estimate that sheds light on how the hidden states of the CFN activate and then relax toward the origin. + +Lemma 1. For any $T , k > 0$ we have + +$$ +\left| h _ { T + k } ( i ) \right| \leq \Theta ^ { k } \left| h _ { T } ( i ) \right| + \frac { H } { 1 - \Theta } \left( \operatorname* { m a x } _ { T \leq t \leq T + k } \left| ( W x _ { t } ) ( i ) \right| \right) +$$ + +where $\Theta$ and $H$ are the maximum values of the $i ^ { \mathrm { t h } }$ components of the $\theta$ and $\eta$ gate in the time interval $[ T , T + k ]$ , that is: + +$$ +\Theta = \operatorname* { m a x } _ { T \leq t \leq T + k } \theta _ { t } ( i ) a n d H = \operatorname* { m a x } _ { T \leq t \leq T + k } \eta _ { t } ( i ) . +$$ + +This estimate shows that if during a time interval $[ T _ { 1 } , T _ { 2 } ]$ one of (i) the embedded inputs $W x _ { t }$ have weak $i ^ { \mathrm { t h } }$ feature (i.e. $\operatorname* { m a x } _ { T \leq t \leq T + k } | ( W x _ { t } ) ( i ) |$ is small), (ii) or the input gates $\eta _ { t }$ have their $i ^ { \mathrm { t h } }$ component close to zero (i.e. $H$ is small), + +occurs then the $i ^ { \mathrm { t h } }$ component of the hidden state $h _ { t }$ will relaxes toward zero at a rate that depends on the value of the $i ^ { \mathrm { t h } }$ component the the forget gate. Overall this leads to the following simple picture: $h _ { t } ( i )$ activates when presented with an embedded input $W x _ { t }$ with strong $i ^ { \mathrm { t h } }$ feature, and then relaxes toward zero until the data present the network once again with strong $i ^ { \mathrm { t h } }$ feature. The strength of the activation and the decay rate are controlled by the $i ^ { \mathrm { t h } }$ component of the input and forget gates. The proof of Lemma 1 is elementary — + +Proof of Lemma $^ { l }$ . Using the non-expansivity of the hyperbolic tangent, i.e. $| \operatorname { t a n h } ( x ) | \leq | x |$ , and the triangle inequality, we obtain from (1) + +$$ +| h _ { t } ( i ) | \leq \Theta | h _ { t - 1 } ( i ) | + H \operatorname* { m a x } _ { T \leq t \leq T + k } | ( W x _ { t } ) ( i ) | +$$ + +whenever $t$ is in the interval $[ T , T + k ]$ . Iterating this inequality and summing the geometric series then gives + +$$ +| h _ { T + k } ( i ) | \leq \Theta ^ { k } | h _ { T } ( i ) | + \left( \frac { 1 - \Theta ^ { k } } { 1 - \Theta } \right) \ H \operatorname* { m a x } _ { T \leq t \leq T + k } | ( W x _ { t } ) ( i ) | +$$ + +from which we easily conclude. + +We now turn toward the analysis of the long-term behavior of the the dynamical system + +$$ +\mathfrak { u } _ { t } = h _ { t } , \qquad \mathfrak { u } \mapsto \Phi ( \mathfrak { u } ) : = \sigma \left( U _ { \theta } \mathfrak { u } + b _ { \theta } \right) \odot \operatorname { t a n h } ( \mathfrak { u } ) . +$$ + +induced by a CFN. The following lemma shows that the only attractor of this dynamical system is the zero state. + +Lemma 2. Starting from any initial state $\mathfrak { u } _ { 0 }$ , the trajectory $\mathcal { O } ^ { + } ( \mathfrak { u } _ { 0 } )$ will eventually converge to the zero state. That is, $\begin{array} { r } { \operatorname* { l i m } _ { t + \infty } \mathfrak { u } _ { t } = 0 } \end{array}$ regardless of the the initial state $\mathfrak { u } _ { 0 }$ . + +Proof. From the definition of $\Phi$ we clearly have that the sequence defined by $\mathfrak { u } _ { t + 1 } = \Phi ( \mathfrak { u } _ { t } )$ satisfies $- 1 < \mathfrak { u } _ { t } ( i ) < 1$ for all $t$ and all $i$ . Since the sequence ${ \mathfrak { u } } _ { t }$ is bounded, so is the sequence $\mathbf { v } _ { t } : =$ $U _ { \theta } \mathfrak { u } _ { t } + b _ { \theta }$ . That is there exists a finite $C > 0$ such that $( U _ { \theta } \mathfrak { u } _ { t } ) ( i ) + b _ { \theta } ( i ) < C$ for all $t$ and $i$ . Using the non-expansivity of the hyperbolic tangent, we then obtain that $| \mathfrak { u } _ { t } ( i ) | \leq \sigma ( C ) | \mathfrak { u } _ { t - 1 } ( i ) |$ , for all $t$ and all $i$ . We conclude by noting that $0 < \sigma ( C ) < 1$ . □ + +Lemma 2 remains true for a multi-layer CFN, that is, a CFN in which the first layer is defined by (1) and the subsequent layers $2 \le \ell \le L$ are defined by: + +$$ +\begin{array} { r } { h _ { t } ^ { ( \ell ) } = \theta _ { t } ^ { ( \ell ) } \odot \operatorname { t a n h } ( h _ { t - 1 } ^ { ( \ell ) } ) + \eta _ { t } ^ { ( \ell ) } \odot \operatorname { t a n h } ( W ^ { ( \ell ) } h _ { t } ^ { ( \ell - 1 ) } ) . } \end{array} +$$ + +Assume that $W x _ { t } = 0$ for all $t > T$ , then an extension of the arguments contained in the proof of the two previous lemmas shows that + +$$ +| h _ { T + k } ^ { ( \ell ) } | \leq C ( 1 + k ) ^ { ( \ell - 1 ) } \Theta ^ { k } +$$ + +where $0 < \Theta < 1$ is the maximal values for the input gates involved in layer 1 to $\ell$ of the network, and $C > 0$ is some constant depending only on the norms $\| W ^ { ( j ) } \| _ { \infty }$ of the matrices and the sizes $| h _ { T } ^ { ( j ) } |$ of the initial conditions at all previous $1 \le j \le \ell$ levels. Estimate (11) shows that Lemma 2 remains true for multi-layer architectures. + +![](images/4b9915eb4051f66eba1c7295abb31996ba44df45291410f4220266ec7b001ad0.jpg) +Figure 4: A 2-layer, 224-unit CFN trained on Penn Treebank. All inputs $x _ { t }$ are zero after $t = 1 0 0 0$ , i.e. the time-point indicated by the dashed line. At left: plot of the 10 “slowest” units of the first layer. At right: plot of the 10 slowest units of the second layer. The second layer retains information much longer than the first layer. + +Inequality (11) shows that higher levels (i.e. larger $\ell$ ) decay more slowly, and remain non-trivial, while earlier levels (i.e. smaller $\ell$ ) decay more quickly. We illustrate this behavior computationally with a simple experiment. We take a 2-layer, 224-unit CFN network trained on Penn Treebank and feed it the following input data: The first 1000 inputs $x _ { t }$ are the first 1000 words of the test set of Penn Treebank; All subsequent inputs are zero. In other words, $x _ { t } = 0$ if $t > 1 0 0 0$ . For each of the two layers we then select the 10 units that decay the slowest after $t > 1 0 0 0$ and plot them on Figure 4. The figure illustrates that the second layer retains information for much longer than the first layer. To quantify this observation we define the relaxation time (or half-life) of the $\bar { i } ^ { \mathrm { t h } }$ unit as the smallest $T$ such that + +$$ +| h _ { 1 0 0 0 + T } ( i ) | < 0 . 5 | h _ { 1 0 0 0 } ( i ) | . +$$ + +Using this definition yields average relaxation times of 2.2 time steps for the first layer and 23.2 time steps for the second layer. The first layer has a standard deviations of approximately 5 steps while the second layer has a standard deviation of approximately 75 time steps. A more fine-grained analysis reveals that some units in the second layer have relaxation times of several hundred steps. For instance, if instead of averaging the relaxation times over the whole layer we average them over the top quartile (i.e. the $2 5 \%$ units that decay the most slowly) we get 4.8 time steps and 85.6 time steps for the first and second layers, respectively. In other words, by restricting attention to long-term units the difference between the first and second layers becomes much more striking. + +Overall, this experiment conforms with the analysis (11), and indicates that adding a third or fourth layer would potentially allow a multi-layer CFN architecture to retain information for even longer. + +# 3 EXPERIMENTS + +In this section we show that despite its simplicity, the CFN network achieves performance comparable to the much more complex LSTM network on the word level language modeling task. We use two datasets for these experiments, namely the Penn Treebank corpus (Marcus et al., 1993) and the Text8 corpus (Mikolov et al., 2014). We consider both one-layer and two-layer CFNs and LSTMs for our experiments. We train both CFN and LSTM networks in a similar fashion and always compare models that use the same number of parameters. We compare their performance with and without dropout, and show that in both cases they obtain similar results. We also provide results published in Mikolov et al. (2014), Jozefowicz et al. (2015) and Sukhbaatar et al. (2015) for the sake of comparison. + +Table 1: Experiments on Penn Treebank without dropout. + +
ModelSizeTrainingVal. perp.Test perp.
VanillaRNN5M parametersJozefowicz et al. (2015)1122.9
GRU5M parametersJozefowicz et al. (2015)=108.2
LSTM5M parametersJozefowicz et al. (2015)-109.7
LSTM (1 layer)5M parametersTrained by us108.4105.1
CFN (2 layers)5M parametersTrained by us109.3106.3
+ +Table 2: Experiments on Text8 without dropout + +
ModelSizeTrainingPerp.on development set
Vanilla RNN500 hidden unitsMikolov et al. (2014)184
SCRN500 hidden unitsMikolov et al. (2014)161
LSTM500 hidden unitsMikolov et al. (2014)156
MemN2N500 hidden unitsSukhbaatar et al. (2015)147
LSTM(2 layers)46.4M parametersTrained by us139.9
CFN (2 layers)46.4M parametersTrained by us142.0
+ +For concreteness, the exact implementation for the two-layer architecture of our model is + +$$ +\begin{array} { r l } & { h _ { t } ^ { ( 0 ) } = W ^ { ( 0 ) } x _ { t } } \\ & { \hat { h } _ { t } ^ { ( 0 ) } = \mathrm { D r o p } ( h _ { t } ^ { ( 0 ) } , p ) } \\ & { h _ { t } ^ { ( 1 ) } = \theta _ { t } ^ { ( 1 ) } \odot \mathrm { t a n h } ( h _ { t - 1 } ^ { ( 1 ) } ) + \eta _ { t } ^ { ( 1 ) } \odot \mathrm { t a n h } ( W ^ { ( 1 ) } \hat { h } _ { t } ^ { ( 0 ) } ) } \\ & { \hat { h } _ { t } ^ { ( 1 ) } = \mathrm { D r o p } ( h _ { t } ^ { ( 1 ) } , p ) } \\ & { h _ { t } ^ { ( 2 ) } = \theta _ { t } ^ { ( 2 ) } \odot \mathrm { t a n h } ( h _ { t - 1 } ^ { ( 2 ) } ) + \eta _ { t } ^ { ( 2 ) } \odot \mathrm { t a n h } ( W ^ { ( 2 ) } \hat { h } _ { t } ^ { ( 1 ) } ) } \\ & { \hat { h } _ { t } ^ { ( 2 ) } = \mathrm { D r o p } ( h _ { t } ^ { ( 2 ) } , p ) } \\ & { y _ { t } = \mathrm { L o g s o f u n a x } ( W ^ { ( 3 ) } \hat { h } _ { t } ^ { ( 2 ) } + b ) } \end{array} +$$ + +where $\mathrm { D r o p } ( z , p )$ denotes the dropout operator with a probability $p$ of setting components in $z$ to zero. We compute the gates according to + +$$ +\begin{array} { r l } & { \theta _ { t } ^ { ( \ell ) } : = \sigma \left( U _ { \theta } ^ { ( \ell ) } \widetilde { h } _ { t - 1 } ^ { ( \ell ) } + V _ { \theta } ^ { ( \ell ) } \widetilde { h } _ { t } ^ { ( \ell - 1 ) } + b _ { \theta } \right) \mathrm { a n d } \eta _ { t } ^ { ( \ell ) } : = \sigma \left( U _ { \eta } ^ { ( \ell ) } \widetilde { h } _ { t - 1 } ^ { ( \ell ) } + V _ { \eta } ^ { ( \ell ) } \widetilde { h } _ { t } ^ { ( \ell - 1 ) } + b _ { \eta } \right) } \\ & { \mathrm { w h e r e } \quad \widetilde { h } _ { t - 1 } ^ { ( \ell ) } = \mathrm { D r o p } ( h _ { t - 1 } ^ { ( \ell ) } , q ) \quad \mathrm { a n d } \quad \widetilde { h } _ { t } ^ { ( \ell - 1 ) } = \mathrm { D r o p } ( h _ { t } ^ { ( \ell - 1 ) } , q ) , } \end{array} +$$ + +and thus the model has two dropout hyperparameters. The parameter $p$ controls the amount of dropout between layers; the parameter $q$ controls the amount of dropout inside each gate. We use a similar dropout strategy for the LSTM, in that all sigmoid gates $f , o$ and $i$ receive the same amount $q$ of dropout. + +To train the CFN and LSTM networks, we use a simple online steepest descent algorithm. We update the weights $w$ via + +$$ +\begin{array} { r } { \boldsymbol { w } ^ { ( k + 1 ) } = \boldsymbol { w } ^ { ( k ) } - \mathrm { l r } \cdot \vec { p } \quad \mathrm { w h e r e } \quad \vec { p } = \frac { \nabla _ { w } L } { \| \nabla _ { w } L \| _ { 2 } } , } \end{array} +$$ + +where lr is the learning rate and $\nabla _ { w } L$ denotes the approximate gradient of the loss with respect to the weights as estimated from a certain number of presented examples. We use the usual backpropagation through time approximation when estimating the gradient: we unroll the net $T$ steps in the past and neglect longer dependencies. In all experiments, the CFN and LSTM networks are unrolled for $T = 3 5$ steps and we take minibatches of size 20. As all search directions $\vec { p }$ have Euclidean norm $\| \vec { p } \| _ { 2 } = 1$ , we perform no gradient clipping during training. + +We initialize all the weights in the CFN, except for the bias of the gates, uniformly at random in $[ - 0 . 0 7 , 0 . 0 7 ]$ . We initialize the bias $b _ { \theta }$ and $b _ { \eta }$ of the gates to 1 and $- 1$ , respectively, so that at the beginning of the training $\theta _ { t } \approx \sigma ( 1 ) \approx 0 . 7 3$ and $\eta _ { t } \approx \sigma ( - 1 ) \approx 0 . 2 3$ . We initialize the weights of the LSTM in exactly the same way; the bias for the forget and input gate are initialized to 1 and $- 1$ , and all the other weights are initialized uniformly in $[ - 0 . 0 7 , 0 . 0 7 ]$ . This initialization scheme favors the flow of information in the horizontal direction. The importance of a careful initialization of the forget gate was pointed out in Gers et al. (2000) and Jozefowicz et al. (2015). Finally, we initialize all hidden states to zero for both models. + +Table 3: Experiments on Penn Treebank with dropout. + +
ModelSizeTrainingVal. perp.Test perp.
Vanilla RNN GRU LSTM20M parametersJozefowicz et al. (2015)103.097.7
20M parametersJozefowicz et al. (2015)95.591.7
20M parametersJozefowicz et al. (2015)83.378.8
LSTM (2 layers) CFN (2 layers)20M parametersTrained by us78.474.3
20M parametersTrained by us79.774.9
50M parametersTrained by us75.971.8
LSTM(2 layers) CFN (2 layers)50M parametersTrained by us77.072.2
+ +Dataset Construction. The Penn Treebank Corpus has 1 million words and a vocabulary size of 10,000. We used the code from Zaremba et al. (2014) to construct and split the dataset into a training set (929K words), a validation set (73K words) and a test set (82K words). The Text8 corpus has 100 million characters and a vocabulary size of 44,000. We used the script from Mikolov et al. (2014) to construct and split the dataset into a training set (first 99M characters) and a development set (last 1M characters). + +Experiments without Dropout. Tables 1 and 2 provide a comparison of various recurrent network architectures without dropout evaluated on the Penn Treebank corpus and the Text8 corpus. The last two rows of each table provide results for LSTM and CFN networks trained and initialized in the manner described above. We have tried both one and two layer architectures, and reported only the best result. The learning rate schedules used for each network are described in the appendix. + +We also report results published in Jozefowicz et al. (2015) were a vanilla RNN, a GRU and an LSTM network were trained on Penn Treebank, each of them having 5 million parameters (only the test perplexity was reported). Finally we report results published in Mikolov et al. (2014) and Sukhbaatar et al. (2015) where various networks are trained on Text8. Of these four networks, only the LSTM network from Mikolov et al. (2014) has the same number of parameters than the CFN and LSTM networks we trained (46.4M parameters). The vanilla RNN, Structurally Constrained Recurrent Network (SCRN) and End-To-End Memory Network (MemN2N) all have 500 units, but less than $4 6 . 4 \mathbf { M }$ parameters. We nonetheless indicate their performance in Table 2 to provide some context. + +Experiments with Dropout. Table 3 provides a comparison of various recurrent network architectures with dropout evaluated on the Penn Treebank corpus. The first three rows report results published in (Jozefowicz et al., 2015) and the last four rows provide results for LSTM and CFN networks trained and initialized with the strategy previously described. The dropout rate $p$ and $q$ are chosen as follows: For the experiments with 20M parameters, we set $p = 5 5 \%$ and $q = 4 5 \%$ for the CFN and $p = 6 0 \%$ and $q = 4 0 \%$ for the LSTM; For the experiments with 50M parameters, we set $p = 6 5 \%$ and $q = 5 5 \%$ for the CFN and $p = 7 0 \%$ and $q = \hat { 5 } 0 \%$ for the LSTM. + +# 4 CONCLUSION + +Despite its simple dynamics, the CFN obtains results that compare well against LSTM networks and GRUs on word-level language modeling. This indicates that it might be possible, in general, to build RNNs that perform well while avoiding the intricate, uninterpretable and potentially chaotic dynamics that can occur in LSTMs and GRUs. Of course, it remains to be seen if dynamically simple RNNs such as the proposed CFN can perform well on a wide variety of tasks, potentially requiring longer term dependencies than the one needed for word level language modeling. The experiments presented in Section 2 indicate a plausible path forward — activations in the higher layers of a multi-layer CFN decay at a slower rate than the activations in the lower layers. In theory, complexity and long-term dependencies can therefore be captured using a more “feed-forward” approach (i.e. stacking layers) rather than relying on the intricate and hard to interpret dynamics of an LSTM or a GRU. + +Overall, the CFN is a simple model and it therefore has the potential of being mathematically wellunderstood. In particular, Section 2 reveals that the dynamics of its hidden states are inherently more interpretable than those of an LSTM. The mathematical analysis here provides a few key insights into the network, in both the presence and absence of input data, but obviously more work is needed before a complete picture can emerge. We hope that this investigation opens up new avenues of inquiry, and that such an understanding will drive subsequent improvements. + +# REFERENCES + +Nils Bertschinger and Thomas Natschlager. Real-time computation at the edge of chaos in recurrent ¨ neural networks. Neural computation, 16(7):1413–1436, 2004. + +Kyunghyun Cho, Bart Van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Hol- ¨ ger Schwenk, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv preprint arXiv:1406.1078, 2014. + +Felix A Gers, Jurgen Schmidhuber, and Fred Cummins. Learning to forget: Continual prediction¨ with lstm. Neural computation, 12(10):2451–2471, 2000. + +Michel Henon. A two-dimensional mapping with a strange attractor. ´ Communications in Mathematical Physics, 50(1):69–77, 1976. + +Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8): 1735–1780, 1997. + +Rafal Jozefowicz, Wojciech Zaremba, and Ilya Sutskever. An empirical exploration of recurrent network architectures. In Proceedings of the 32nd International Conference on Machine Learning, 2015. + +Mitchell P Marcus, Mary Ann Marcinkiewicz, and Beatrice Santorini. Building a large annotated corpus of english: The penn treebank. Computational linguistics, 19(2):313–330, 1993. + +Tomas Mikolov, Armand Joulin, Sumit Chopra, Michael Mathieu, and Marc’Aurelio Ranzato. Learning longer memory in recurrent neural networks. arXiv preprint arXiv:1412.7753, 2014. + +Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio. On the difficulty of training recurrent neural networks. ICML (3), 28:1310–1318, 2013. + +Steven H Strogatz. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview press, 2014. + +Sainbayar Sukhbaatar, Jason Weston, Rob Fergus, et al. End-to-end memory networks. In Advances in neural information processing systems, pp. 2440–2448, 2015. + +David Sussillo and Omri Barak. Opening the black box: low-dimensional dynamics in highdimensional recurrent neural networks. Neural computation, 25(3):626–649, 2013. + +Wojciech Zaremba, Ilya Sutskever, and Oriol Vinyals. Recurrent neural network regularization. arXiv preprint arXiv:1409.2329, 2014. + +# APPENDIX + +Strange attractor of the Henon map. ´ For the sake of comparison, we provide in Figure 5 a depiction of a well-known strange attractor (the Henon attractor) arising from a discrete-time dynamical ´ system. We generate these pictures by reproducing the numerical experiments from Henon (1976). ´ The discrete dynamical system considered here is the two dimensional map + +$$ +x _ { t + 1 } = y _ { t } + 1 - a x _ { t } ^ { 2 } , \quad y _ { t + 1 } = b x _ { t } , +$$ + +with parameters set to $a = 1 . 4$ and $b = 0 . 3$ . We obtain Figure 5(a) by choosing the initial state $( x _ { 0 } , \overset { \cdot } { y } _ { 0 } ) = ( 0 , 0 )$ and plotting the iterates $( x _ { t } , y _ { t } )$ for $t$ between $1 0 ^ { 3 }$ and $\mathrm { i 0 ^ { 5 } }$ . All trajectories starting close to the origin at time $t = 0$ converge toward the depicted attractor. Successive zooms on the branch of the attractor reveal its fractal nature. The structure repeats in a fashion remarkably similar to the 2-unit LSTM in Section 2. + +![](images/af8103f1056a9d902f5bc2f706b76e1ba0c9d4bcba580680c4fe764aa5c50e7b.jpg) +Figure 5: Strange attractor of the Henon map. From left to right: The H ´ enon attractor, enlargement ´ of the red box, enlargement of the magenta box. + +Strange attractor of a 2-unit GRU. As with LSTMs, the GRU gated architecture can induce a chaotic dynamical system. Figure 6 depicts the strange attractor of the dynamical system + +$$ +\begin{array} { r l } { \mathfrak { u } _ { t } = h _ { t } , \quad } & { \mathfrak { u } \mapsto \Phi ( \mathfrak { u } ) : = ( 1 - z ) \odot \mathfrak { u } + z \odot \operatorname { t a n h } \left( U ( r \odot \mathfrak { u } ) \right) } \\ & { z : = \sigma \left( W _ { z } \mathfrak { u } + b _ { z } \right) \quad r : = \sigma \left( W _ { r } \mathfrak { u } + b _ { r } \right) , } \end{array} +$$ + +induced by a two-dimensional GRU, with weight matrices + +$$ +W _ { z } = { \left[ \begin{array} { l l } { 0 } & { 1 } \\ { 1 } & { 1 } \end{array} \right] } \quad W _ { r } = { \left[ \begin{array} { l l } { 0 } & { 1 } \\ { 1 } & { 0 } \end{array} \right] } \quad U = { \left[ \begin{array} { l l } { - 5 } & { - 8 } \\ { 8 } & { 5 } \end{array} \right] } +$$ + +and zero bias for the model parameters. Here also successive zooms on the branch of the attractor reveal its fractal nature. As in the LSTM, the forward trajectories of this dynamical system exhibit a high degree of sensitivity to initial states. + +![](images/dc8aae1d5143b80ef97bf8a2f7a5473f59db5a7ce47cf69e861690d77e9875dc.jpg) +Figure 6: Strange attractor of a two-unit GRU. Successive zooms reveal the fractal nature of the attractor. + +Network sizes and learning rate schedules used in the experiments. In the Penn Treebank experiment without dropout (Table 1), the CFN network has two hidden layers of 224 units each for a total of 5 million parameters. The LSTM has one hidden layer with 228 units for a total of 5 million parameters as well. We also tried a two-layer LSTM with 5 million parameters but the result was worse (test perplexity of 110.6) and we did not report it in the table. For the Text8 experiments (Table 2), the LSTM has two hidden layers with 481 hidden units for a total 46.4 million parameters. We also tried a one-layer LSTM with 46.4 million parameters but the result was worse (perplexity of 140.8). The CFN has two hidden layers with 495 units each, for a total of 46.4 million parameters as well. + +For both experiments without dropout (Table 1 and 2), we used a simple and aggressive learning rate schedule: at each epoch, lr is divided by 3. For the CFN the initial learning rate was chosen to be $\mathrm { { I r } _ { 0 } = 5 . 5 }$ for PTB and $\mathrm { { l r } _ { 0 } = 5 }$ for Text8. For the LSTM we chose $\mathrm { { l r } _ { 0 } } = 7$ for PTB and $\mathrm { { l r } _ { 0 } = 5 }$ for Text8. + +In the Penn Treebank experiment with dropout (Table 3), the CFN with 20M parameters has two hidden layers of 731 units each and the LSTM with 20M parameters trained by us has two hidden layers of 655 units each. We also tried a one-layer LSTM with 20M parameters and it led to similar but slightly worse results than the two-layer architecture. For both network, the learning rate was divided by 1.1 each time the validation perplexity did not decrease by at least $1 \%$ . The initial learning rate were chosen to be $\mathrm { { l r } _ { 0 } } = 7$ for the CFN and $\mathrm { { I r } _ { 0 } = 5 }$ for the LSTM. \ No newline at end of file diff --git a/parse/train/S1dIzvclg/S1dIzvclg_content_list.json b/parse/train/S1dIzvclg/S1dIzvclg_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..ffe1f782eb47b0c621a604a0164e3d879e2677db --- /dev/null +++ b/parse/train/S1dIzvclg/S1dIzvclg_content_list.json @@ -0,0 +1,1474 @@ +[ + { + "type": "text", + "text": "A RECURRENT NEURAL NETWORK WITHOUT CHAOS ", + "text_level": 1, + "bbox": [ + 173, + 101, + 797, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Thomas Laurent ", + "text_level": 1, + "bbox": [ + 184, + 146, + 302, + 159 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "James von Brecht ", + "text_level": 1, + "bbox": [ + 477, + 145, + 601, + 159 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Mathematics Loyola Marymount University Los Angeles, CA 90045, USA tlaurent@lmu.edu ", + "bbox": [ + 183, + 160, + 385, + 214 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Mathematics California State University, Long Beach Long Beach, CA 90840, USA james.vonbrecht@csulb.edu ", + "bbox": [ + 477, + 160, + 740, + 215 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 252, + 544, + 267 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We introduce an exceptionally simple gated recurrent neural network (RNN) that achieves performance comparable to well-known gated architectures, such as LSTMs and GRUs, on the word-level language modeling task. We prove that our model has simple, predicable and non-chaotic dynamics. This stands in stark contrast to more standard gated architectures, whose underlying dynamical systems exhibit chaotic behavior. ", + "bbox": [ + 232, + 282, + 766, + 367 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 393, + 336, + 410 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Gated recurrent neural networks, such as the Long Short Term Memory network (LSTM) introduced by Hochreiter & Schmidhuber (1997) and the Gated Recurrent Unit (GRU) proposed by Cho et al. (2014), prove highly effective for machine learning tasks that involve sequential data. We propose an exceptionally simple variant of these gated architectures. The basic model takes the form ", + "bbox": [ + 173, + 425, + 825, + 482 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/8ece9b9c8380423888284d0e6573a2fed4a7cf4d49de8243483d8839189028c2.jpg", + "text": "$$\nh _ { t } = \\theta _ { t } \\odot \\operatorname { t a n h } ( h _ { t - 1 } ) + \\eta _ { t } \\odot \\operatorname { t a n h } ( W x _ { t } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 354, + 488, + 642, + 506 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "where $\\odot$ stands for the Hadamard product. The horizontal/forget gate (i.e. $\\theta _ { t }$ ) and the vertical/input gate (i.e. $\\eta _ { t }$ ) take the usual form used in most gated RNN architectures. Specifically ", + "bbox": [ + 173, + 512, + 828, + 541 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/deb18df6ab3078acfbd0240a01f73698e9eb6fc26fddfeb70550d2da13bf4486.jpg", + "text": "$$\n\\begin{array} { r } { \\theta _ { t } : = \\sigma \\left( U _ { \\theta } h _ { t - 1 } + V _ { \\theta } x _ { t } + b _ { \\theta } \\right) \\quad \\mathrm { a n d } \\quad \\eta _ { t } : = \\sigma \\left( U _ { \\eta } h _ { t - 1 } + V _ { \\eta } x _ { t } + b _ { \\eta } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 547, + 733, + 565 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "where $\\sigma ( x ) : = ( 1 + \\mathrm { e } ^ { - x } ) ^ { - 1 }$ denotes the logistic sigmoid function. The network (1)–(2) has quite intuitive dynamics. Suppose the data $x _ { t }$ present the model with a sequence ", + "bbox": [ + 174, + 573, + 826, + 603 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/b2165571585a3bd8b81d33312d9846f5e2be6432656dfe8d42c4a6abb18fa2ae.jpg", + "text": "$$\n( W x _ { t } ) ( i ) = { \\left\\{ \\begin{array} { l l } { 1 0 } & { { \\mathrm { i f ~ } } t = T } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e } } , } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 397, + 609, + 598, + 645 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "where $( W x _ { t } ) ( i )$ stands for the $i ^ { \\mathrm { t h } }$ component of the vector $W x _ { t }$ . In other words we consider an input sequence $x _ { t }$ for which the learned $i ^ { \\mathrm { t h } }$ feature $( W x _ { t } ) ( i )$ remains off except at time $T$ . When initialized from $h _ { 0 } = 0$ , the corresponding response of the network to this “impulse” in the $i ^ { \\mathrm { t h } }$ feature is ", + "bbox": [ + 173, + 654, + 825, + 709 + ], + "page_idx": 0 + }, + { + "type": "equation", + "img_path": "images/0f4905068be5407b7b5c3691ad2547029cf43c5193ade93a5dd8fd3b135882f1.jpg", + "text": "$$\nh _ { t } ( i ) \\approx { \\left\\{ \\begin{array} { l l } { 0 } & { { \\mathrm { i f ~ } } t < T } \\\\ { \\eta _ { T } } & { { \\mathrm { i f ~ } } t = T } \\\\ { \\alpha _ { t } } & { { \\mathrm { i f ~ } } t > T } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 416, + 708, + 583, + 760 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "with $\\alpha _ { t }$ a sequence that relaxes toward zero. The forget gate $\\theta _ { t }$ control the rate of this relaxation. Thus $h _ { t } ( i )$ activates when presented with a strong $i ^ { \\mathrm { t h } }$ feature, and then relaxes toward zero until the data present the network once again with strong $i ^ { \\mathrm { t h } }$ feature. Overall this leads to a dynamically simple model, in which the activation patterns in the hidden states of the network have a clear cause and predictable subsequent behavior. ", + "bbox": [ + 174, + 762, + 825, + 833 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Dynamics of this sort do not occur in other RNN models. Instead, the three most popular recurrent neural network architectures, namely the vanilla RNN, the LSTM and the GRU, have complex, irregular, and unpredictable dynamics. Even in the absence of input data, these networks can give rise to chaotic dynamical systems. In other words, when presented with null input data the activation patterns in their hidden states do not necessarily follow a predictable path. The proposed network (1)–(2) has rather dull and minimalist dynamics in comparison; its only attractor is the zero state, and so it stands at the polar-opposite end of the spectrum from chaotic systems. Perhaps surprisingly, at least in the light of this comparison, the proposed network (1) performs as well as LSTMs and GRUs on the word level language modeling task. We therefore conclude that the ability of an RNN to form chaotic temporal dynamics, in the sense we describe in Section 2, cannot explain its success on word-level language modeling tasks. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the next section, we review the phenomenon of chaos in RNNs via both synthetic examples and trained models. We also prove a precise, quantified description of the dynamical picture (3)–(4) for the proposed network. In particular, we show that the dynamical system induced by the proposed network is never chaotic, and for this reason we refer to it as a Chaos-Free Network (CFN). The final section provides a series of experiments that demonstrate that CFN achieve results comparable to LSTM on the word-level language modeling task. All together, these observations show that an architecture as simple as (1)–(2) can achieve performance comparable to the more dynamically complex LSTM. ", + "bbox": [ + 173, + 180, + 825, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 CHAOS IN RECURRENT NEURAL NETWORKS", + "text_level": 1, + "bbox": [ + 174, + 311, + 578, + 328 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The study of RNNs from a dynamical systems point-of-view has brought fruitful insights into generic features of RNNs (Sussillo & Barak, 2013; Pascanu et al., 2013). We shall pursue a brief investigation of CFN, LSTM and GRU networks using this formalism, as it allows us to identify key distinctions between them. Recall that for a given mapping $\\Phi : \\mathbb { R } ^ { d } \\mapsto \\mathbb { R } ^ { d }$ , a given initial time $t _ { 0 } \\in \\mathbb { N }$ and a given initial state $\\mathfrak { u } _ { 0 } \\in \\mathbb { R } ^ { d }$ , a simple repeated iteration of the mapping $\\Phi$ ", + "bbox": [ + 174, + 342, + 825, + 412 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/0405c648e882191d3bc5229f2801828a1787e0dbe5c91b086aeab56b19c0eea0.jpg", + "text": "$$\n\\begin{array} { c c } { \\mathfrak { u } _ { t + 1 } = \\Phi ( \\mathfrak { u } _ { t } ) } & { t > t _ { 0 } , } \\\\ { \\mathfrak { u } _ { t _ { 0 } } = \\mathfrak { u } _ { 0 } } & { t = t _ { 0 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 419, + 417, + 576, + 454 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "defines a discrete-time dynamical system. The index $t \\in \\mathbb N$ represents the current time, while the point $\\mathfrak { u } _ { t } \\in \\mathbb { R } ^ { d }$ represents the current state of the system. The set of all visited states $\\mathcal { O } ^ { + } ( \\mathfrak { u } _ { 0 } ) : =$ $\\left\\{ \\mathfrak { u } _ { t _ { 0 } } , \\mathfrak { u } _ { t _ { 0 } + 1 } , \\dotsc , \\mathfrak { u } _ { t _ { 0 } + n } , \\dotsc \\right\\}$ defines the forward trajectory or forward orbit through $\\mathfrak { u } _ { 0 }$ . An attractor for the dynamical system is a set that is invariant (any trajectory that starts in the set remains in the set) and that attracts all trajectories that start sufficiently close to it. The attractors of chaotic dynamical systems are often fractal sets, and for this reason they are referred to as strange attractors. ", + "bbox": [ + 174, + 457, + 825, + 541 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Most RNNs generically take the functional form ", + "bbox": [ + 174, + 547, + 490, + 563 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/8eaff2b9199aa07be2091afb170ede552f1c313f822ae4b66602b216e31fc836.jpg", + "text": "$$\n\\mathsf { u } _ { t } = \\Psi ( \\mathfrak { u } _ { t - 1 } , W _ { 1 } x _ { t } , W _ { 2 } x _ { t } , \\ldots , W _ { k } x _ { t } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 366, + 568, + 630, + 585 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $x _ { t }$ denotes the $t ^ { \\mathrm { t h } }$ input data point. For example, in the case of the CFN (1)–(2), we have $W _ { 1 } = W$ , $W _ { 2 } = V _ { \\theta }$ and $W _ { 3 } = V _ { \\eta }$ . To gain insight into the underlying design of the architecture of an RNN, it proves usefull to consider how trajectories behave when they are not influenced by any external input. This lead us to consider the dynamical system ", + "bbox": [ + 173, + 589, + 825, + 646 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/1fe29f71d6e7969485e8484385448b70a4c656e453cb661ec0f67cc824b4ebc1.jpg", + "text": "$$\n\\mathfrak { u } _ { t } = \\Phi ( \\mathfrak { u } _ { t - 1 } ) \\qquad \\Phi ( \\mathfrak { u } ) : = \\Psi ( \\mathfrak { u } , 0 , 0 , \\dots , 0 ) ,\n$$", + "text_format": "latex", + "bbox": [ + 348, + 650, + 648, + 667 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "which we refer to as the dynamical system induced by the recurrent neural network. The timeinvariant system (6) is much more tractable than (5), and it offers a mean to investigate the inner working of a given architecture; it separates the influence of input data $x _ { t }$ , which can produce essentially any possible response, from the model itself. Studying trajectories that are not influenced by external data will give us an indication on the ability of a given RNN to generate complex and sophisticated trajectories by its own. As we shall see shortly, the dynamical system induced by a CFN has excessively simple and predictable trajectories: all of them converge to the zero state. In other words, its only attractor is the zero state. This is in sharp contrast with the dynamical systems induced by LSTM or GRU, who can exhibit chaotic behaviors and have strange attractors. ", + "bbox": [ + 173, + 672, + 825, + 799 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The learned parameters $W _ { j }$ in (5) describe how data influence the evolution of hidden states at each time step. From a modeling perspective, (6) would occur in the scenario where a trained RNN has learned a weak coupling between a specific data point $x _ { t _ { 0 } }$ and the hidden state at that time, in the sense that the data influence is small and so all $W _ { j } x _ { t _ { 0 } } \\approx 0$ nearly vanish. The hidden state then transitions according to $\\mathfrak { u } _ { t _ { 0 } } \\approx \\Psi ( \\mathfrak { u } _ { t _ { 0 } - 1 } , 0 , 0 , \\dots , 0 ) = \\Phi ( \\mathfrak { u } _ { t _ { 0 } - 1 } )$ . ", + "bbox": [ + 174, + 804, + 825, + 876 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We refer to Bertschinger & Natschlager (2004) for a study of the chaotic behavior of a simplified ¨ vanilla RNN with a specific statistical model, namely an i.i.d. Bernoulli process, for the input data as well as a specific statistical model, namely i.i.d. Gaussian, for the weights of the recurrence matrix. ", + "bbox": [ + 176, + 881, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/4ab3671f85507ce9fae8efa2bef387ae843514737e6c1f8f9ec26fbe10f6ea41.jpg", + "image_caption": [ + "Figure 1: Strange attractor of a 2-unit LSTM. Successive zooms (from left to right) reveal the selfrepeating, fractal nature of the attractor. Colored boxes depict zooming regions. " + ], + "image_footnote": [], + "bbox": [ + 186, + 109, + 812, + 257 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 CHAOTIC BEHAVIOR OF LSTM AND GRU IN THE ABSENCE OF INPUT DATA ", + "text_level": 1, + "bbox": [ + 176, + 327, + 732, + 343 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this subsection we briefly show that LSTM and GRU, in the absence of input data, can lead to dynamical systems $\\mathfrak { u } _ { t } = \\Phi ( \\mathfrak { u } _ { t - 1 } )$ that are chaotic in the classical sense of the term (Strogatz, 2014). Figure 1 depicts the strange attractor of the dynamical system: ", + "bbox": [ + 174, + 353, + 823, + 396 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c42eee4b570fc26ea71facc0ab66ff5308346a2a12d667f65f253478912be5e6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { u } _ { t } = \\Big [ _ { c _ { t } } ^ { h _ { t } } \\Big ] \\quad \\mathbf { u } \\mapsto { \\boldsymbol { \\Phi } } ( \\mathbf { u } ) = \\Big [ ^ { o \\odot \\operatorname { t a n h } \\big ( f \\odot c + i \\odot g \\big ) } \\Big ] } \\\\ & { { \\small i : = \\sigma ( W _ { i } h + b _ { i } ) \\quad f : = \\sigma ( W _ { f } h + b _ { f } ) \\quad o : = \\sigma ( W _ { o } h + b _ { o } ) \\quad g : = \\operatorname { t a n h } ( W _ { g } h + b _ { g } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 401, + 781, + 457 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "induced by a two-unit LSTM with weight matrices ", + "bbox": [ + 173, + 460, + 508, + 474 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b907cdb69264bb9c09eb86b0b7a273aa9dc1241b0395aa3bd91fe4b7dbd55c96.jpg", + "text": "$$\nW _ { i } = \\left[ - 1 \\quad - 4 \\right] \\quad W _ { o } = \\left[ { \\begin{array} { r r } { 4 } & { 1 } \\\\ { - 9 } & { - 7 } \\end{array} } \\right] \\quad W _ { f } = \\left[ { \\begin{array} { r r } { - 2 } & { 6 } \\\\ { 0 } & { - 6 } \\end{array} } \\right] \\quad W _ { g } = \\left[ { \\begin{array} { r r } { - 1 } & { - 6 } \\\\ { 6 } & { - 9 } \\end{array} } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 228, + 479, + 767, + 516 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "and zero bias for the model parameters. These weights were randomly generated from a normal distribution with standard deviation 5 and then rounded to the nearest integer. Figure 1(a) was obtained by choosing an initial state ${ \\mathfrak { u } } _ { 0 } ~ = ~ ( h _ { 0 } , c _ { 0 } )$ uniformly at random in $[ 0 , 1 ] ^ { 2 } \\times [ 0 , \\dot { 1 } ] ^ { 2 }$ and plotting the h-component of the iterates ${ \\mathfrak { u } } _ { t } ~ = ~ ( h _ { t } , c _ { t } )$ for $t$ between $1 0 ^ { 3 }$ and $1 0 ^ { \\bar { 5 } }$ (so this figure should be regarded as a two dimensional projection of a four dimensional attractor, which explain its tangled appearance). Most trajectories starting in $[ 0 , 1 ] ^ { 2 } \\times [ 0 , 1 ] ^ { 2 }$ converge toward the depicted attractor. The resemblance between this attractor and classical strange attractors such as the Henon ´ attractor is striking (see Figure 5 in the appendix for a depiction of the Henon attractor). Successive ´ zooms on the branch of the LSTM attractor from Figure 1(a) reveal its fractal nature. Figure 1(b) is an enlargement of the red box in Figure 1(a), and Figure 1(c) is an enlargement of the magenta box in Figure 1(b). We see that the structure repeats itself as we zoom in. ", + "bbox": [ + 173, + 520, + 825, + 674 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The most practical consequence of chaos is that the long-term behavior of their forward orbits can exhibit a high degree of sensitivity to the initial states $\\mathfrak { u } _ { 0 }$ . Figure 2 provides an example of such behavior for the dynamical system (7)–(9). An initial condition $\\mathfrak { u } _ { 0 }$ was drawn uniformly at random in $[ 0 , 1 ] ^ { 2 } \\times [ 0 , 1 ] ^ { 2 }$ . We then computed 100, 000 small amplitude perturbations $\\hat { \\mathfrak { u } } _ { 0 }$ of $\\mathfrak { u } _ { 0 }$ by adding a small random number drawn uniformly from $[ - 1 0 ^ { - 7 } , 1 0 ^ { - 7 } ]$ to each component. We then iterated (7)–(9) for 200 steps and plotted the $\\mathbf { h }$ -component of the final state $\\hat { \\textmd u } _ { 2 0 0 }$ for each of the 100, 000 trials on Figure 2(a). The collection of these 100, 000 final states essentially fills out the entire attractor, despite the fact that their initial conditions are highly localized (i.e. at distance of no more than $1 0 ^ { - 7 }$ ) around a fixed point. In other words, the time $t = 2 0 0$ map of the dynamical system will map a small neighborhood around a fixed initial condition $\\mathfrak { u } _ { 0 }$ to the entire attractor. Figure 2(b) additionally illustrates this sensitivity to initial conditions for points on the attractor itself. We take an initial condition $\\mathfrak { u } _ { 0 }$ on the attractor and perturb it by $1 0 ^ { - 7 }$ to a nearby initial condition $\\hat { \\mathfrak { u } } _ { 0 }$ . We then plot the distance $\\left\\| \\hat { \\mathbf { u } } _ { t } - \\mathbf { u } _ { t } \\right\\|$ between the two corresponding trajectories for the first 200 time steps. After an initial phase of agreement, the trajectories strongly diverge. ", + "bbox": [ + 174, + 680, + 825, + 875 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The synthetic example (7)–(9) illustrates the potentially chaotic nature of the LSTM architecture. We now show that chaotic behavior occurs for trained models as well, and not just for synthetically generated instances. We take the parameter values of an LSTM with 228 hidden units trained on the ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/08703ee2db3c758c4fb6bd9c2358d1258e01e76cbaa7406e8cd80b1c7aee6c0d.jpg", + "image_caption": [ + "Figure 2: (a): A small neighborhood around a fixed initial condition $\\mathfrak { u } _ { 0 }$ , after 200 iterations, is mapped to the entire attractor. (b): Two trajectories starting starting within $1 0 ^ { - 7 }$ of one another strongly diverge after 50 steps. " + ], + "image_footnote": [], + "bbox": [ + 223, + 112, + 761, + 262 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/5d4ac4bc4740c0bee2b2e32fef8a96f54fb4405a873f08e76cd9d14db5cc0d2c.jpg", + "image_caption": [ + "Figure 3: 228-unit LSTM trained on Penn Treebank. (a): In the absence of input data, the system is chaotic and nearby trajectories diverge. (b): In the presence of input data, the system is mostly driven by the external input. Trajectories starting far apart converge. " + ], + "image_footnote": [], + "bbox": [ + 178, + 343, + 797, + 453 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Penn Treebank corpus without dropout (c.f. the experimental section for the precise procedure). We then set all data inputs $x _ { t }$ to zero and run the corresponding induced dynamical system. Two trajectories starting from nearby initial conditions $\\mathfrak { u } _ { 0 }$ and $\\hat { \\mathfrak { u } } _ { 0 }$ were computed (as before $\\hat { \\mathfrak { u } } _ { 0 }$ was obtained by adding to each components of $\\mathfrak { u } _ { 0 }$ a small random number drawn uniformly from $[ - 1 0 ^ { - 7 } , 1 0 ^ { - 7 } ] )$ . Figure 3(a) plots the first component h(1) of the hidden state for both trajectories over the first 1600 time steps. After an initial phase of agreement, the forward trajectories $\\mathcal { O } ^ { + } ( \\mathfrak { u } _ { 0 } )$ and $\\mathcal { O } ^ { + } ( \\hat { \\mathfrak { u } } _ { 0 } )$ strongly diverge. We also see that both trajectories exhibit the typical aperiodic behavior that characterizes chaotic systems. If the inputs $x _ { t }$ do not vanish, but come from actual word-level data, then the behavior is very different. The LSTM is now no longer an autonomous system whose dynamics are driven by its hidden states, but a time dependent system whose dynamics are mostly driven by the external inputs. Figure 3(b) shows the first component $h ( 1 )$ of the hidden states of two trajectories that start with initial conditions $\\mathfrak { u } _ { 0 }$ and $\\hat { \\mathfrak { u } } _ { 0 }$ that are far apart. The sensitivity to initial condition disappears, and instead the trajectories converge toward each other after about 70 steps. The memory of this initial difference is lost. Overall these experiments indicate that a trained LSTM, when it is not driven by external inputs, can be chaotic. In the presence of input data, the LSTM becomes a forced system whose dynamics are dominated by external forcing. ", + "bbox": [ + 174, + 537, + 825, + 760 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Like LSTM networks, GRU can also lead to dynamical systems that are chaotic and they can also have strange attractors. The depiction of such an attractor, in the case of a two-unit GRU, is provided in Figure 6 of the appendix. ", + "bbox": [ + 176, + 767, + 823, + 809 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 CHAOS-FREE BEHAVIOR OF THE CFN ", + "text_level": 1, + "bbox": [ + 176, + 828, + 472, + 842 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The dynamical behavior of the CFN is dramatically different from that of the LSTM. In this subsection we start by showing that the hidden states of the CFN activate and relax toward zero in a predictable fashion in response to input data. On one hand, this shows that the CFN cannot produce non-trivial dynamics without some influence from data. On the other, this leads to an interpretable model; any non-trivial activations in the hidden states of a CFN have a clear cause emanating from data-driven activation. This follows from a precise, quantified description of the intuitive picture (3)–(4) sketched in the introduction. ", + "bbox": [ + 174, + 853, + 823, + 922 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We begin with the following simple estimate that sheds light on how the hidden states of the CFN activate and then relax toward the origin. ", + "bbox": [ + 174, + 138, + 823, + 167 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Lemma 1. For any $T , k > 0$ we have ", + "bbox": [ + 173, + 171, + 423, + 185 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d4e742ca30a93b5ab05e8a4b0d56958bb41e9b338436f08f253454dbeac0a044.jpg", + "text": "$$\n\\left| h _ { T + k } ( i ) \\right| \\leq \\Theta ^ { k } \\left| h _ { T } ( i ) \\right| + \\frac { H } { 1 - \\Theta } \\left( \\operatorname* { m a x } _ { T \\leq t \\leq T + k } \\left| ( W x _ { t } ) ( i ) \\right| \\right)\n$$", + "text_format": "latex", + "bbox": [ + 303, + 189, + 694, + 224 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\Theta$ and $H$ are the maximum values of the $i ^ { \\mathrm { t h } }$ components of the $\\theta$ and $\\eta$ gate in the time interval $[ T , T + k ]$ , that is: ", + "bbox": [ + 174, + 229, + 821, + 260 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/1fef17c985d3e193982441a8ad679415c5927f1da57dc54926e0c315d9c82d6e.jpg", + "text": "$$\n\\Theta = \\operatorname* { m a x } _ { T \\leq t \\leq T + k } \\theta _ { t } ( i ) a n d H = \\operatorname* { m a x } _ { T \\leq t \\leq T + k } \\eta _ { t } ( i ) .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 263, + 663, + 287 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This estimate shows that if during a time interval $[ T _ { 1 } , T _ { 2 } ]$ one of (i) the embedded inputs $W x _ { t }$ have weak $i ^ { \\mathrm { t h } }$ feature (i.e. $\\operatorname* { m a x } _ { T \\leq t \\leq T + k } | ( W x _ { t } ) ( i ) |$ is small), (ii) or the input gates $\\eta _ { t }$ have their $i ^ { \\mathrm { t h } }$ component close to zero (i.e. $H$ is small), ", + "bbox": [ + 174, + 300, + 596, + 316 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 204, + 325, + 818, + 362 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "occurs then the $i ^ { \\mathrm { t h } }$ component of the hidden state $h _ { t }$ will relaxes toward zero at a rate that depends on the value of the $i ^ { \\mathrm { t h } }$ component the the forget gate. Overall this leads to the following simple picture: $h _ { t } ( i )$ activates when presented with an embedded input $W x _ { t }$ with strong $i ^ { \\mathrm { t h } }$ feature, and then relaxes toward zero until the data present the network once again with strong $i ^ { \\mathrm { t h } }$ feature. The strength of the activation and the decay rate are controlled by the $i ^ { \\mathrm { t h } }$ component of the input and forget gates. The proof of Lemma 1 is elementary — ", + "bbox": [ + 173, + 372, + 825, + 457 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof of Lemma $^ { l }$ . Using the non-expansivity of the hyperbolic tangent, i.e. $| \\operatorname { t a n h } ( x ) | \\leq | x |$ , and the triangle inequality, we obtain from (1) ", + "bbox": [ + 174, + 460, + 823, + 489 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4025c9f5991f725da7cef6eb16c9ccf303e66840580cdd19daea6eac91005de3.jpg", + "text": "$$\n| h _ { t } ( i ) | \\leq \\Theta | h _ { t - 1 } ( i ) | + H \\operatorname* { m a x } _ { T \\leq t \\leq T + k } | ( W x _ { t } ) ( i ) |\n$$", + "text_format": "latex", + "bbox": [ + 336, + 493, + 661, + 518 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "whenever $t$ is in the interval $[ T , T + k ]$ . Iterating this inequality and summing the geometric series then gives ", + "bbox": [ + 174, + 523, + 825, + 551 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/671ef617f269e8fcb86cf3420c3b8733f20e3386c12059eecf1c691f35285404.jpg", + "text": "$$\n| h _ { T + k } ( i ) | \\leq \\Theta ^ { k } | h _ { T } ( i ) | + \\left( \\frac { 1 - \\Theta ^ { k } } { 1 - \\Theta } \\right) \\ H \\operatorname* { m a x } _ { T \\leq t \\leq T + k } | ( W x _ { t } ) ( i ) |\n$$", + "text_format": "latex", + "bbox": [ + 287, + 550, + 710, + 587 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "from which we easily conclude. ", + "bbox": [ + 174, + 587, + 383, + 602 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now turn toward the analysis of the long-term behavior of the the dynamical system ", + "bbox": [ + 173, + 616, + 750, + 632 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/49b2dc2b20f33d3389fd05b9ab29c46e9b653d915e01350b47e0d8be0ed3fc46.jpg", + "text": "$$\n\\mathfrak { u } _ { t } = h _ { t } , \\qquad \\mathfrak { u } \\mapsto \\Phi ( \\mathfrak { u } ) : = \\sigma \\left( U _ { \\theta } \\mathfrak { u } + b _ { \\theta } \\right) \\odot \\operatorname { t a n h } ( \\mathfrak { u } ) .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 636, + 673, + 654 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "induced by a CFN. The following lemma shows that the only attractor of this dynamical system is the zero state. ", + "bbox": [ + 173, + 659, + 826, + 686 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Lemma 2. Starting from any initial state $\\mathfrak { u } _ { 0 }$ , the trajectory $\\mathcal { O } ^ { + } ( \\mathfrak { u } _ { 0 } )$ will eventually converge to the zero state. That is, $\\begin{array} { r } { \\operatorname* { l i m } _ { t + \\infty } \\mathfrak { u } _ { t } = 0 } \\end{array}$ regardless of the the initial state $\\mathfrak { u } _ { 0 }$ . ", + "bbox": [ + 171, + 689, + 823, + 719 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proof. From the definition of $\\Phi$ we clearly have that the sequence defined by $\\mathfrak { u } _ { t + 1 } = \\Phi ( \\mathfrak { u } _ { t } )$ satisfies $- 1 < \\mathfrak { u } _ { t } ( i ) < 1$ for all $t$ and all $i$ . Since the sequence ${ \\mathfrak { u } } _ { t }$ is bounded, so is the sequence $\\mathbf { v } _ { t } : =$ $U _ { \\theta } \\mathfrak { u } _ { t } + b _ { \\theta }$ . That is there exists a finite $C > 0$ such that $( U _ { \\theta } \\mathfrak { u } _ { t } ) ( i ) + b _ { \\theta } ( i ) < C$ for all $t$ and $i$ . Using the non-expansivity of the hyperbolic tangent, we then obtain that $| \\mathfrak { u } _ { t } ( i ) | \\leq \\sigma ( C ) | \\mathfrak { u } _ { t - 1 } ( i ) |$ , for all $t$ and all $i$ . We conclude by noting that $0 < \\sigma ( C ) < 1$ . □ ", + "bbox": [ + 173, + 727, + 825, + 797 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Lemma 2 remains true for a multi-layer CFN, that is, a CFN in which the first layer is defined by (1) and the subsequent layers $2 \\le \\ell \\le L$ are defined by: ", + "bbox": [ + 173, + 811, + 820, + 842 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/46273fa3fbb15bd25def44caf75adf177f0dcee21b22cbdf8fbdff07a14b380a.jpg", + "text": "$$\n\\begin{array} { r } { h _ { t } ^ { ( \\ell ) } = \\theta _ { t } ^ { ( \\ell ) } \\odot \\operatorname { t a n h } ( h _ { t - 1 } ^ { ( \\ell ) } ) + \\eta _ { t } ^ { ( \\ell ) } \\odot \\operatorname { t a n h } ( W ^ { ( \\ell ) } h _ { t } ^ { ( \\ell - 1 ) } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 316, + 845, + 679, + 867 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assume that $W x _ { t } = 0$ for all $t > T$ , then an extension of the arguments contained in the proof of the two previous lemmas shows that ", + "bbox": [ + 173, + 871, + 825, + 900 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d1bfa2fe06e723a98455c86df8864837d70afd6add0a43e8fb8ac07787f8178c.jpg", + "text": "$$\n| h _ { T + k } ^ { ( \\ell ) } | \\leq C ( 1 + k ) ^ { ( \\ell - 1 ) } \\Theta ^ { k }\n$$", + "text_format": "latex", + "bbox": [ + 405, + 904, + 593, + 926 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $0 < \\Theta < 1$ is the maximal values for the input gates involved in layer 1 to $\\ell$ of the network, and $C > 0$ is some constant depending only on the norms $\\| W ^ { ( j ) } \\| _ { \\infty }$ of the matrices and the sizes $| h _ { T } ^ { ( j ) } |$ of the initial conditions at all previous $1 \\le j \\le \\ell$ levels. Estimate (11) shows that Lemma 2 remains true for multi-layer architectures. ", + "bbox": [ + 174, + 103, + 825, + 165 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/4b9915eb4051f66eba1c7295abb31996ba44df45291410f4220266ec7b001ad0.jpg", + "image_caption": [ + "Figure 4: A 2-layer, 224-unit CFN trained on Penn Treebank. All inputs $x _ { t }$ are zero after $t = 1 0 0 0$ , i.e. the time-point indicated by the dashed line. At left: plot of the 10 “slowest” units of the first layer. At right: plot of the 10 slowest units of the second layer. The second layer retains information much longer than the first layer. " + ], + "image_footnote": [], + "bbox": [ + 184, + 191, + 799, + 375 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Inequality (11) shows that higher levels (i.e. larger $\\ell$ ) decay more slowly, and remain non-trivial, while earlier levels (i.e. smaller $\\ell$ ) decay more quickly. We illustrate this behavior computationally with a simple experiment. We take a 2-layer, 224-unit CFN network trained on Penn Treebank and feed it the following input data: The first 1000 inputs $x _ { t }$ are the first 1000 words of the test set of Penn Treebank; All subsequent inputs are zero. In other words, $x _ { t } = 0$ if $t > 1 0 0 0$ . For each of the two layers we then select the 10 units that decay the slowest after $t > 1 0 0 0$ and plot them on Figure 4. The figure illustrates that the second layer retains information for much longer than the first layer. To quantify this observation we define the relaxation time (or half-life) of the $\\bar { i } ^ { \\mathrm { t h } }$ unit as the smallest $T$ such that ", + "bbox": [ + 173, + 459, + 825, + 584 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/87d6ad34cf0f111bcd0c08770d92bd3556f2ef20ee2aaed2af55ebc835643e05.jpg", + "text": "$$\n| h _ { 1 0 0 0 + T } ( i ) | < 0 . 5 | h _ { 1 0 0 0 } ( i ) | .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 583, + 596, + 601 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Using this definition yields average relaxation times of 2.2 time steps for the first layer and 23.2 time steps for the second layer. The first layer has a standard deviations of approximately 5 steps while the second layer has a standard deviation of approximately 75 time steps. A more fine-grained analysis reveals that some units in the second layer have relaxation times of several hundred steps. For instance, if instead of averaging the relaxation times over the whole layer we average them over the top quartile (i.e. the $2 5 \\%$ units that decay the most slowly) we get 4.8 time steps and 85.6 time steps for the first and second layers, respectively. In other words, by restricting attention to long-term units the difference between the first and second layers becomes much more striking. ", + "bbox": [ + 173, + 602, + 825, + 713 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Overall, this experiment conforms with the analysis (11), and indicates that adding a third or fourth layer would potentially allow a multi-layer CFN architecture to retain information for even longer. ", + "bbox": [ + 174, + 719, + 821, + 747 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 767, + 326, + 784 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section we show that despite its simplicity, the CFN network achieves performance comparable to the much more complex LSTM network on the word level language modeling task. We use two datasets for these experiments, namely the Penn Treebank corpus (Marcus et al., 1993) and the Text8 corpus (Mikolov et al., 2014). We consider both one-layer and two-layer CFNs and LSTMs for our experiments. We train both CFN and LSTM networks in a similar fashion and always compare models that use the same number of parameters. We compare their performance with and without dropout, and show that in both cases they obtain similar results. We also provide results published in Mikolov et al. (2014), Jozefowicz et al. (2015) and Sukhbaatar et al. (2015) for the sake of comparison. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/95ca597f29224d4d9e38b9fae75593cdfd19132a1ce524e7089a79db6d47c458.jpg", + "table_caption": [ + "Table 1: Experiments on Penn Treebank without dropout. " + ], + "table_footnote": [], + "table_body": "
ModelSizeTrainingVal. perp.Test perp.
VanillaRNN5M parametersJozefowicz et al. (2015)1122.9
GRU5M parametersJozefowicz et al. (2015)=108.2
LSTM5M parametersJozefowicz et al. (2015)-109.7
LSTM (1 layer)5M parametersTrained by us108.4105.1
CFN (2 layers)5M parametersTrained by us109.3106.3
", + "bbox": [ + 228, + 136, + 766, + 215 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/13010b8a1cde63f588820e2fe3757fd6b547cd477d251c9e09d762daa33754ec.jpg", + "table_caption": [ + "Table 2: Experiments on Text8 without dropout " + ], + "table_footnote": [], + "table_body": "
ModelSizeTrainingPerp.on development set
Vanilla RNN500 hidden unitsMikolov et al. (2014)184
SCRN500 hidden unitsMikolov et al. (2014)161
LSTM500 hidden unitsMikolov et al. (2014)156
MemN2N500 hidden unitsSukhbaatar et al. (2015)147
LSTM(2 layers)46.4M parametersTrained by us139.9
CFN (2 layers)46.4M parametersTrained by us142.0
", + "bbox": [ + 209, + 262, + 784, + 356 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For concreteness, the exact implementation for the two-layer architecture of our model is ", + "bbox": [ + 171, + 380, + 756, + 395 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/f61fb71b07a4ec6f80cb442a8e6aaf4d105ed19ca5d5ead4748b5d0ffeb0fe41.jpg", + "text": "$$\n\\begin{array} { r l } & { h _ { t } ^ { ( 0 ) } = W ^ { ( 0 ) } x _ { t } } \\\\ & { \\hat { h } _ { t } ^ { ( 0 ) } = \\mathrm { D r o p } ( h _ { t } ^ { ( 0 ) } , p ) } \\\\ & { h _ { t } ^ { ( 1 ) } = \\theta _ { t } ^ { ( 1 ) } \\odot \\mathrm { t a n h } ( h _ { t - 1 } ^ { ( 1 ) } ) + \\eta _ { t } ^ { ( 1 ) } \\odot \\mathrm { t a n h } ( W ^ { ( 1 ) } \\hat { h } _ { t } ^ { ( 0 ) } ) } \\\\ & { \\hat { h } _ { t } ^ { ( 1 ) } = \\mathrm { D r o p } ( h _ { t } ^ { ( 1 ) } , p ) } \\\\ & { h _ { t } ^ { ( 2 ) } = \\theta _ { t } ^ { ( 2 ) } \\odot \\mathrm { t a n h } ( h _ { t - 1 } ^ { ( 2 ) } ) + \\eta _ { t } ^ { ( 2 ) } \\odot \\mathrm { t a n h } ( W ^ { ( 2 ) } \\hat { h } _ { t } ^ { ( 1 ) } ) } \\\\ & { \\hat { h } _ { t } ^ { ( 2 ) } = \\mathrm { D r o p } ( h _ { t } ^ { ( 2 ) } , p ) } \\\\ & { y _ { t } = \\mathrm { L o g s o f u n a x } ( W ^ { ( 3 ) } \\hat { h } _ { t } ^ { ( 2 ) } + b ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 323, + 400, + 673, + 559 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\mathrm { D r o p } ( z , p )$ denotes the dropout operator with a probability $p$ of setting components in $z$ to zero. We compute the gates according to ", + "bbox": [ + 173, + 561, + 823, + 590 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/202900b9cbe4c8e5fbb3c2ed2cb33efe9da62e6d76f8ac4e7361a5ef37670eff.jpg", + "text": "$$\n\\begin{array} { r l } & { \\theta _ { t } ^ { ( \\ell ) } : = \\sigma \\left( U _ { \\theta } ^ { ( \\ell ) } \\widetilde { h } _ { t - 1 } ^ { ( \\ell ) } + V _ { \\theta } ^ { ( \\ell ) } \\widetilde { h } _ { t } ^ { ( \\ell - 1 ) } + b _ { \\theta } \\right) \\mathrm { a n d } \\eta _ { t } ^ { ( \\ell ) } : = \\sigma \\left( U _ { \\eta } ^ { ( \\ell ) } \\widetilde { h } _ { t - 1 } ^ { ( \\ell ) } + V _ { \\eta } ^ { ( \\ell ) } \\widetilde { h } _ { t } ^ { ( \\ell - 1 ) } + b _ { \\eta } \\right) } \\\\ & { \\mathrm { w h e r e } \\quad \\widetilde { h } _ { t - 1 } ^ { ( \\ell ) } = \\mathrm { D r o p } ( h _ { t - 1 } ^ { ( \\ell ) } , q ) \\quad \\mathrm { a n d } \\quad \\widetilde { h } _ { t } ^ { ( \\ell - 1 ) } = \\mathrm { D r o p } ( h _ { t } ^ { ( \\ell - 1 ) } , q ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 594, + 794, + 647 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "and thus the model has two dropout hyperparameters. The parameter $p$ controls the amount of dropout between layers; the parameter $q$ controls the amount of dropout inside each gate. We use a similar dropout strategy for the LSTM, in that all sigmoid gates $f , o$ and $i$ receive the same amount $q$ of dropout. ", + "bbox": [ + 174, + 650, + 825, + 707 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To train the CFN and LSTM networks, we use a simple online steepest descent algorithm. We update the weights $w$ via ", + "bbox": [ + 176, + 713, + 823, + 739 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/801723b5343efdf98541939ed59ebd201335281dd3056eda7bff52a499e3ee5a.jpg", + "text": "$$\n\\begin{array} { r } { \\boldsymbol { w } ^ { ( k + 1 ) } = \\boldsymbol { w } ^ { ( k ) } - \\mathrm { l r } \\cdot \\vec { p } \\quad \\mathrm { w h e r e } \\quad \\vec { p } = \\frac { \\nabla _ { w } L } { \\| \\nabla _ { w } L \\| _ { 2 } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 739, + 658, + 761 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where lr is the learning rate and $\\nabla _ { w } L$ denotes the approximate gradient of the loss with respect to the weights as estimated from a certain number of presented examples. We use the usual backpropagation through time approximation when estimating the gradient: we unroll the net $T$ steps in the past and neglect longer dependencies. In all experiments, the CFN and LSTM networks are unrolled for $T = 3 5$ steps and we take minibatches of size 20. As all search directions $\\vec { p }$ have Euclidean norm $\\| \\vec { p } \\| _ { 2 } = 1$ , we perform no gradient clipping during training. ", + "bbox": [ + 173, + 762, + 825, + 848 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We initialize all the weights in the CFN, except for the bias of the gates, uniformly at random in $[ - 0 . 0 7 , 0 . 0 7 ]$ . We initialize the bias $b _ { \\theta }$ and $b _ { \\eta }$ of the gates to 1 and $- 1$ , respectively, so that at the beginning of the training $\\theta _ { t } \\approx \\sigma ( 1 ) \\approx 0 . 7 3$ and $\\eta _ { t } \\approx \\sigma ( - 1 ) \\approx 0 . 2 3$ . We initialize the weights of the LSTM in exactly the same way; the bias for the forget and input gate are initialized to 1 and $- 1$ , and all the other weights are initialized uniformly in $[ - 0 . 0 7 , 0 . 0 7 ]$ . This initialization scheme favors the flow of information in the horizontal direction. The importance of a careful initialization of the forget gate was pointed out in Gers et al. (2000) and Jozefowicz et al. (2015). Finally, we initialize all hidden states to zero for both models. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/febe66354cf1c25d6909834c20d0efa1f3419549ef9aa09f98a8bd5110c40637.jpg", + "table_caption": [ + "Table 3: Experiments on Penn Treebank with dropout. " + ], + "table_footnote": [], + "table_body": "
ModelSizeTrainingVal. perp.Test perp.
Vanilla RNN GRU LSTM20M parametersJozefowicz et al. (2015)103.097.7
20M parametersJozefowicz et al. (2015)95.591.7
20M parametersJozefowicz et al. (2015)83.378.8
LSTM (2 layers) CFN (2 layers)20M parametersTrained by us78.474.3
20M parametersTrained by us79.774.9
50M parametersTrained by us75.971.8
LSTM(2 layers) CFN (2 layers)50M parametersTrained by us77.072.2
", + "bbox": [ + 222, + 135, + 772, + 242 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 268, + 823, + 310 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Dataset Construction. The Penn Treebank Corpus has 1 million words and a vocabulary size of 10,000. We used the code from Zaremba et al. (2014) to construct and split the dataset into a training set (929K words), a validation set (73K words) and a test set (82K words). The Text8 corpus has 100 million characters and a vocabulary size of 44,000. We used the script from Mikolov et al. (2014) to construct and split the dataset into a training set (first 99M characters) and a development set (last 1M characters). ", + "bbox": [ + 174, + 318, + 825, + 401 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Experiments without Dropout. Tables 1 and 2 provide a comparison of various recurrent network architectures without dropout evaluated on the Penn Treebank corpus and the Text8 corpus. The last two rows of each table provide results for LSTM and CFN networks trained and initialized in the manner described above. We have tried both one and two layer architectures, and reported only the best result. The learning rate schedules used for each network are described in the appendix. ", + "bbox": [ + 174, + 409, + 825, + 478 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also report results published in Jozefowicz et al. (2015) were a vanilla RNN, a GRU and an LSTM network were trained on Penn Treebank, each of them having 5 million parameters (only the test perplexity was reported). Finally we report results published in Mikolov et al. (2014) and Sukhbaatar et al. (2015) where various networks are trained on Text8. Of these four networks, only the LSTM network from Mikolov et al. (2014) has the same number of parameters than the CFN and LSTM networks we trained (46.4M parameters). The vanilla RNN, Structurally Constrained Recurrent Network (SCRN) and End-To-End Memory Network (MemN2N) all have 500 units, but less than $4 6 . 4 \\mathbf { M }$ parameters. We nonetheless indicate their performance in Table 2 to provide some context. ", + "bbox": [ + 174, + 484, + 825, + 611 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Experiments with Dropout. Table 3 provides a comparison of various recurrent network architectures with dropout evaluated on the Penn Treebank corpus. The first three rows report results published in (Jozefowicz et al., 2015) and the last four rows provide results for LSTM and CFN networks trained and initialized with the strategy previously described. The dropout rate $p$ and $q$ are chosen as follows: For the experiments with 20M parameters, we set $p = 5 5 \\%$ and $q = 4 5 \\%$ for the CFN and $p = 6 0 \\%$ and $q = 4 0 \\%$ for the LSTM; For the experiments with 50M parameters, we set $p = 6 5 \\%$ and $q = 5 5 \\%$ for the CFN and $p = 7 0 \\%$ and $q = \\hat { 5 } 0 \\%$ for the LSTM. ", + "bbox": [ + 174, + 617, + 825, + 715 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 738, + 318, + 753 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Despite its simple dynamics, the CFN obtains results that compare well against LSTM networks and GRUs on word-level language modeling. This indicates that it might be possible, in general, to build RNNs that perform well while avoiding the intricate, uninterpretable and potentially chaotic dynamics that can occur in LSTMs and GRUs. Of course, it remains to be seen if dynamically simple RNNs such as the proposed CFN can perform well on a wide variety of tasks, potentially requiring longer term dependencies than the one needed for word level language modeling. The experiments presented in Section 2 indicate a plausible path forward — activations in the higher layers of a multi-layer CFN decay at a slower rate than the activations in the lower layers. In theory, complexity and long-term dependencies can therefore be captured using a more “feed-forward” approach (i.e. stacking layers) rather than relying on the intricate and hard to interpret dynamics of an LSTM or a GRU. ", + "bbox": [ + 174, + 770, + 825, + 922 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Overall, the CFN is a simple model and it therefore has the potential of being mathematically wellunderstood. In particular, Section 2 reveals that the dynamics of its hidden states are inherently more interpretable than those of an LSTM. The mathematical analysis here provides a few key insights into the network, in both the presence and absence of input data, but obviously more work is needed before a complete picture can emerge. We hope that this investigation opens up new avenues of inquiry, and that such an understanding will drive subsequent improvements. ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 209, + 285, + 223 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Nils Bertschinger and Thomas Natschlager. Real-time computation at the edge of chaos in recurrent ¨ neural networks. Neural computation, 16(7):1413–1436, 2004. ", + "bbox": [ + 173, + 232, + 823, + 260 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Kyunghyun Cho, Bart Van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Hol- ¨ ger Schwenk, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv preprint arXiv:1406.1078, 2014. ", + "bbox": [ + 174, + 268, + 821, + 311 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Felix A Gers, Jurgen Schmidhuber, and Fred Cummins. Learning to forget: Continual prediction¨ with lstm. Neural computation, 12(10):2451–2471, 2000. ", + "bbox": [ + 171, + 320, + 820, + 349 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Michel Henon. A two-dimensional mapping with a strange attractor. ´ Communications in Mathematical Physics, 50(1):69–77, 1976. ", + "bbox": [ + 174, + 358, + 821, + 387 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8): 1735–1780, 1997. ", + "bbox": [ + 173, + 396, + 821, + 425 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Rafal Jozefowicz, Wojciech Zaremba, and Ilya Sutskever. An empirical exploration of recurrent network architectures. In Proceedings of the 32nd International Conference on Machine Learning, 2015. ", + "bbox": [ + 174, + 433, + 823, + 477 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Mitchell P Marcus, Mary Ann Marcinkiewicz, and Beatrice Santorini. Building a large annotated corpus of english: The penn treebank. Computational linguistics, 19(2):313–330, 1993. ", + "bbox": [ + 173, + 484, + 821, + 515 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Tomas Mikolov, Armand Joulin, Sumit Chopra, Michael Mathieu, and Marc’Aurelio Ranzato. Learning longer memory in recurrent neural networks. arXiv preprint arXiv:1412.7753, 2014. ", + "bbox": [ + 174, + 522, + 820, + 553 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio. On the difficulty of training recurrent neural networks. ICML (3), 28:1310–1318, 2013. ", + "bbox": [ + 174, + 560, + 821, + 590 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Steven H Strogatz. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview press, 2014. ", + "bbox": [ + 173, + 598, + 821, + 627 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Sainbayar Sukhbaatar, Jason Weston, Rob Fergus, et al. End-to-end memory networks. In Advances in neural information processing systems, pp. 2440–2448, 2015. ", + "bbox": [ + 174, + 636, + 821, + 666 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "David Sussillo and Omri Barak. Opening the black box: low-dimensional dynamics in highdimensional recurrent neural networks. Neural computation, 25(3):626–649, 2013. ", + "bbox": [ + 173, + 674, + 821, + 703 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Wojciech Zaremba, Ilya Sutskever, and Oriol Vinyals. Recurrent neural network regularization. arXiv preprint arXiv:1409.2329, 2014. ", + "bbox": [ + 173, + 712, + 823, + 741 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 264, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Strange attractor of the Henon map. ´ For the sake of comparison, we provide in Figure 5 a depiction of a well-known strange attractor (the Henon attractor) arising from a discrete-time dynamical ´ system. We generate these pictures by reproducing the numerical experiments from Henon (1976). ´ The discrete dynamical system considered here is the two dimensional map ", + "bbox": [ + 173, + 133, + 825, + 189 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/b8685b044468dbf9b9868051363615a3fb6115e2a1e0b6dff75d2f78243d437b.jpg", + "text": "$$\nx _ { t + 1 } = y _ { t } + 1 - a x _ { t } ^ { 2 } , \\quad y _ { t + 1 } = b x _ { t } ,\n$$", + "text_format": "latex", + "bbox": [ + 375, + 195, + 620, + 213 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "with parameters set to $a = 1 . 4$ and $b = 0 . 3$ . We obtain Figure 5(a) by choosing the initial state $( x _ { 0 } , \\overset { \\cdot } { y } _ { 0 } ) = ( 0 , 0 )$ and plotting the iterates $( x _ { t } , y _ { t } )$ for $t$ between $1 0 ^ { 3 }$ and $\\mathrm { i 0 ^ { 5 } }$ . All trajectories starting close to the origin at time $t = 0$ converge toward the depicted attractor. Successive zooms on the branch of the attractor reveal its fractal nature. The structure repeats in a fashion remarkably similar to the 2-unit LSTM in Section 2. ", + "bbox": [ + 173, + 219, + 825, + 290 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/af8103f1056a9d902f5bc2f706b76e1ba0c9d4bcba580680c4fe764aa5c50e7b.jpg", + "image_caption": [ + "Figure 5: Strange attractor of the Henon map. From left to right: The H ´ enon attractor, enlargement ´ of the red box, enlargement of the magenta box. " + ], + "image_footnote": [], + "bbox": [ + 187, + 309, + 807, + 462 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Strange attractor of a 2-unit GRU. As with LSTMs, the GRU gated architecture can induce a chaotic dynamical system. Figure 6 depicts the strange attractor of the dynamical system ", + "bbox": [ + 171, + 530, + 823, + 559 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/6c3fa4611a1f9ada92ecf89d64065c2f10fedb30a151ae4a50a3fb49530381d3.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathfrak { u } _ { t } = h _ { t } , \\quad } & { \\mathfrak { u } \\mapsto \\Phi ( \\mathfrak { u } ) : = ( 1 - z ) \\odot \\mathfrak { u } + z \\odot \\operatorname { t a n h } \\left( U ( r \\odot \\mathfrak { u } ) \\right) } \\\\ & { z : = \\sigma \\left( W _ { z } \\mathfrak { u } + b _ { z } \\right) \\quad r : = \\sigma \\left( W _ { r } \\mathfrak { u } + b _ { r } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 564, + 709, + 603 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "induced by a two-dimensional GRU, with weight matrices ", + "bbox": [ + 174, + 607, + 557, + 622 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/e98a47012e0c67c5b2329f69c7632c2443512181ab921bba32457e735b9dca70.jpg", + "text": "$$\nW _ { z } = { \\left[ \\begin{array} { l l } { 0 } & { 1 } \\\\ { 1 } & { 1 } \\end{array} \\right] } \\quad W _ { r } = { \\left[ \\begin{array} { l l } { 0 } & { 1 } \\\\ { 1 } & { 0 } \\end{array} \\right] } \\quad U = { \\left[ \\begin{array} { l l } { - 5 } & { - 8 } \\\\ { 8 } & { 5 } \\end{array} \\right] }\n$$", + "text_format": "latex", + "bbox": [ + 328, + 627, + 669, + 662 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "and zero bias for the model parameters. Here also successive zooms on the branch of the attractor reveal its fractal nature. As in the LSTM, the forward trajectories of this dynamical system exhibit a high degree of sensitivity to initial states. ", + "bbox": [ + 173, + 667, + 825, + 710 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/dc8aae1d5143b80ef97bf8a2f7a5473f59db5a7ce47cf69e861690d77e9875dc.jpg", + "image_caption": [ + "Figure 6: Strange attractor of a two-unit GRU. Successive zooms reveal the fractal nature of the attractor. " + ], + "image_footnote": [], + "bbox": [ + 173, + 731, + 808, + 871 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Network sizes and learning rate schedules used in the experiments. In the Penn Treebank experiment without dropout (Table 1), the CFN network has two hidden layers of 224 units each for a total of 5 million parameters. The LSTM has one hidden layer with 228 units for a total of 5 million parameters as well. We also tried a two-layer LSTM with 5 million parameters but the result was worse (test perplexity of 110.6) and we did not report it in the table. For the Text8 experiments (Table 2), the LSTM has two hidden layers with 481 hidden units for a total 46.4 million parameters. We also tried a one-layer LSTM with 46.4 million parameters but the result was worse (perplexity of 140.8). The CFN has two hidden layers with 495 units each, for a total of 46.4 million parameters as well. ", + "bbox": [ + 173, + 103, + 825, + 228 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "For both experiments without dropout (Table 1 and 2), we used a simple and aggressive learning rate schedule: at each epoch, lr is divided by 3. For the CFN the initial learning rate was chosen to be $\\mathrm { { I r } _ { 0 } = 5 . 5 }$ for PTB and $\\mathrm { { l r } _ { 0 } = 5 }$ for Text8. For the LSTM we chose $\\mathrm { { l r } _ { 0 } } = 7$ for PTB and $\\mathrm { { l r } _ { 0 } = 5 }$ for Text8. ", + "bbox": [ + 174, + 236, + 825, + 291 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "In the Penn Treebank experiment with dropout (Table 3), the CFN with 20M parameters has two hidden layers of 731 units each and the LSTM with 20M parameters trained by us has two hidden layers of 655 units each. We also tried a one-layer LSTM with 20M parameters and it led to similar but slightly worse results than the two-layer architecture. For both network, the learning rate was divided by 1.1 each time the validation perplexity did not decrease by at least $1 \\%$ . The initial learning rate were chosen to be $\\mathrm { { l r } _ { 0 } } = 7$ for the CFN and $\\mathrm { { I r } _ { 0 } = 5 }$ for the LSTM. 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We prove that our", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 258, + 470, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 470, + 270 + ], + "score": 1.0, + "content": "model has simple, predicable and non-chaotic dynamics. This stands in stark con-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 269, + 469, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 269, + 469, + 281 + ], + "score": 1.0, + "content": "trast to more standard gated architectures, whose underlying dynamical systems", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 280, + 242, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 242, + 291 + ], + "score": 1.0, + "content": "exhibit chaotic behavior.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 108, + 312, + 206, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 208, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 208, + 328 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 337, + 505, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "score": 1.0, + "content": "Gated recurrent neural networks, such as the Long Short Term Memory network (LSTM) introduced", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "by Hochreiter & Schmidhuber (1997) and the Gated Recurrent Unit (GRU) proposed by Cho et al.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 359, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 373 + ], + "score": 1.0, + "content": "(2014), prove highly effective for machine learning tasks that involve sequential data. We propose", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 371, + 476, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 476, + 383 + ], + "score": 1.0, + "content": "an exceptionally simple variant of these gated architectures. 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Overall this leads to a dynamically", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "simple model, in which the activation patterns in the hidden states of the network have a clear cause", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 256, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 256, + 662 + ], + "score": 1.0, + "content": "and predictable subsequent behavior.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "score": 1.0, + "content": "Dynamics of this sort do not occur in other RNN models. 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We prove that our", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 258, + 470, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 258, + 470, + 270 + ], + "score": 1.0, + "content": "model has simple, predicable and non-chaotic dynamics. This stands in stark con-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 269, + 469, + 281 + ], + "spans": [ + { + "bbox": [ + 141, + 269, + 469, + 281 + ], + "score": 1.0, + "content": "trast to more standard gated architectures, whose underlying dynamical systems", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 280, + 242, + 291 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 242, + 291 + ], + "score": 1.0, + "content": "exhibit chaotic behavior.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 141, + 225, + 470, + 291 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 312, + 206, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 208, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 208, + 328 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 337, + 505, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 350 + ], + "score": 1.0, + "content": "Gated recurrent neural networks, such as the Long Short Term Memory network (LSTM) introduced", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "by Hochreiter & Schmidhuber (1997) and the Gated Recurrent Unit (GRU) proposed by Cho et al.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 359, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 373 + ], + "score": 1.0, + "content": "(2014), prove highly effective for machine learning tasks that involve sequential data. We propose", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 371, + 476, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 476, + 383 + ], + "score": 1.0, + "content": "an exceptionally simple variant of these gated architectures. The basic model takes the form", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 337, + 505, + 383 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 217, + 387, + 393, + 401 + ], + "lines": [ + { + "bbox": [ + 217, + 387, + 393, + 401 + ], + "spans": [ + { + "bbox": [ + 217, + 387, + 393, + 401 + ], + "score": 0.91, + "content": "h _ { t } = \\theta _ { t } \\odot \\operatorname { t a n h } ( h _ { t - 1 } ) + \\eta _ { t } \\odot \\operatorname { t a n h } ( W x _ { t } ) ,", + "type": "interline_equation", + "image_path": "8ece9b9c8380423888284d0e6573a2fed4a7cf4d49de8243483d8839189028c2.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 217, + 387, + 393, + 401 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 507, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 133, + 419 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 408, + 142, + 417 + ], + "score": 0.84, + "content": "\\odot", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 406, + 404, + 419 + ], + "score": 1.0, + "content": "stands for the Hadamard product. 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Overall this leads to a dynamically", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "simple model, in which the activation patterns in the hidden states of the network have a clear cause", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 256, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 256, + 662 + ], + "score": 1.0, + "content": "and predictable subsequent behavior.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 605, + 506, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 677 + ], + "score": 1.0, + "content": "Dynamics of this sort do not occur in other RNN models. Instead, the three most popular recurrent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "neural network architectures, namely the vanilla RNN, the LSTM and the GRU, have complex,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "irregular, and unpredictable dynamics. Even in the absence of input data, these networks can give", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "rise to chaotic dynamical systems. In other words, when presented with null input data the activation", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "patterns in their hidden states do not necessarily follow a predictable path. The proposed network", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "(1)–(2) has rather dull and minimalist dynamics in comparison; its only attractor is the zero state,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "and so it stands at the polar-opposite end of the spectrum from chaotic systems. Perhaps surprisingly,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "at least in the light of this comparison, the proposed network (1) performs as well as LSTMs and", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "GRUs on the word level language modeling task. We therefore conclude that the ability of an RNN", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "to form chaotic temporal dynamics, in the sense we describe in Section 2, cannot explain its success", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 266, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 266, + 139 + ], + "score": 1.0, + "content": "on word-level language modeling tasks.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 666, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "and so it stands at the polar-opposite end of the spectrum from chaotic systems. Perhaps surprisingly,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "at least in the light of this comparison, the proposed network (1) performs as well as LSTMs and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "GRUs on the word level language modeling task. We therefore conclude that the ability of an RNN", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "to form chaotic temporal dynamics, in the sense we describe in Section 2, cannot explain its success", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 266, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 266, + 139 + ], + "score": 1.0, + "content": "on word-level language modeling tasks.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "In the next section, we review the phenomenon of chaos in RNNs via both synthetic examples and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "trained models. We also prove a precise, quantified description of the dynamical picture (3)–(4) for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "the proposed network. In particular, we show that the dynamical system induced by the proposed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "network is never chaotic, and for this reason we refer to it as a Chaos-Free Network (CFN). The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "final section provides a series of experiments that demonstrate that CFN achieve results comparable", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "to LSTM on the word-level language modeling task. All together, these observations show that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 504, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 504, + 222 + ], + "score": 1.0, + "content": "an architecture as simple as (1)–(2) can achieve performance comparable to the more dynamically", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 175, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 175, + 232 + ], + "score": 1.0, + "content": "complex LSTM.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 107, + 247, + 354, + 260 + ], + "lines": [ + { + "bbox": [ + 104, + 245, + 354, + 262 + ], + "spans": [ + { + "bbox": [ + 104, + 245, + 354, + 262 + ], + "score": 1.0, + "content": "2 CHAOS IN RECURRENT NEURAL NETWORKS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "The study of RNNs from a dynamical systems point-of-view has brought fruitful insights into", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 283, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 295 + ], + "score": 1.0, + "content": "generic features of RNNs (Sussillo & Barak, 2013; Pascanu et al., 2013). We shall pursue a brief", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 293, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 307 + ], + "score": 1.0, + "content": "investigation of CFN, LSTM and GRU networks using this formalism, as it allows us to identify", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 507, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 364, + 319 + ], + "score": 1.0, + "content": "key distinctions between them. Recall that for a given mapping", + "type": "text" + }, + { + "bbox": [ + 365, + 304, + 423, + 315 + ], + "score": 0.92, + "content": "\\Phi : \\mathbb { R } ^ { d } \\mapsto \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 303, + 507, + 319 + ], + "score": 1.0, + "content": ", a given initial time", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 449, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 136, + 327 + ], + "score": 0.91, + "content": "t _ { 0 } \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 315, + 230, + 330 + ], + "score": 1.0, + "content": "and a given initial state", + "type": "text" + }, + { + "bbox": [ + 231, + 315, + 266, + 327 + ], + "score": 0.92, + "content": "\\mathfrak { u } _ { 0 } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 315, + 441, + 330 + ], + "score": 1.0, + "content": ", a simple repeated iteration of the mapping", + "type": "text" + }, + { + "bbox": [ + 441, + 316, + 449, + 326 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 331, + 353, + 360 + ], + "lines": [ + { + "bbox": [ + 257, + 331, + 353, + 360 + ], + "spans": [ + { + "bbox": [ + 257, + 331, + 353, + 360 + ], + "score": 0.9, + "content": "\\begin{array} { c c } { \\mathfrak { u } _ { t + 1 } = \\Phi ( \\mathfrak { u } _ { t } ) } & { t > t _ { 0 } , } \\\\ { \\mathfrak { u } _ { t _ { 0 } } = \\mathfrak { u } _ { 0 } } & { t = t _ { 0 } , } \\end{array}", + "type": "interline_equation", + "image_path": "0405c648e882191d3bc5229f2801828a1787e0dbe5c91b086aeab56b19c0eea0.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 257, + 331, + 353, + 345.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 257, + 345.5, + 353, + 360.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 322, + 375 + ], + "score": 1.0, + "content": "defines a discrete-time dynamical system. The index", + "type": "text" + }, + { + "bbox": [ + 322, + 363, + 350, + 373 + ], + "score": 0.9, + "content": "t \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "represents the current time, while the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 130, + 387 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 130, + 373, + 167, + 385 + ], + "score": 0.9, + "content": "\\mathfrak { u } _ { t } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 372, + 456, + 387 + ], + "score": 1.0, + "content": "represents the current state of the system. The set of all visited states", + "type": "text" + }, + { + "bbox": [ + 456, + 374, + 505, + 386 + ], + "score": 0.91, + "content": "\\mathcal { O } ^ { + } ( \\mathfrak { u } _ { 0 } ) : =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 107, + 385, + 219, + 397 + ], + "score": 0.91, + "content": "\\left\\{ \\mathfrak { u } _ { t _ { 0 } } , \\mathfrak { u } _ { t _ { 0 } + 1 } , \\dotsc , \\mathfrak { u } _ { t _ { 0 } + n } , \\dotsc \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 384, + 437, + 398 + ], + "score": 1.0, + "content": "defines the forward trajectory or forward orbit through", + "type": "text" + }, + { + "bbox": [ + 438, + 386, + 449, + 396 + ], + "score": 0.85, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 384, + 506, + 398 + ], + "score": 1.0, + "content": ". An attractor", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "for the dynamical system is a set that is invariant (any trajectory that starts in the set remains in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 405, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 420 + ], + "score": 1.0, + "content": "the set) and that attracts all trajectories that start sufficiently close to it. The attractors of chaotic", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "dynamical systems are often fractal sets, and for this reason they are referred to as strange attractors.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 300, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 302, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 302, + 447 + ], + "score": 1.0, + "content": "Most RNNs generically take the functional form", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 450, + 386, + 464 + ], + "lines": [ + { + "bbox": [ + 224, + 450, + 386, + 464 + ], + "spans": [ + { + "bbox": [ + 224, + 450, + 386, + 464 + ], + "score": 0.92, + "content": "\\mathsf { u } _ { t } = \\Psi ( \\mathfrak { u } _ { t - 1 } , W _ { 1 } x _ { t } , W _ { 2 } x _ { t } , \\ldots , W _ { k } x _ { t } ) ,", + "type": "interline_equation", + "image_path": "8eaff2b9199aa07be2091afb170ede552f1c313f822ae4b66602b216e31fc836.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 224, + 450, + 386, + 464 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 133, + 480 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 469, + 145, + 478 + ], + "score": 0.82, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 466, + 195, + 480 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 196, + 466, + 208, + 477 + ], + "score": 0.88, + "content": "t ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "input data point. For example, in the case of the CFN (1)–(2), we have", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 146, + 489 + ], + "score": 0.87, + "content": "W _ { 1 } = W", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 478, + 149, + 491 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 150, + 479, + 188, + 490 + ], + "score": 0.88, + "content": "W _ { 2 } = V _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 478, + 207, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 207, + 479, + 245, + 491 + ], + "score": 0.92, + "content": "W _ { 3 } = V _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 478, + 506, + 491 + ], + "score": 1.0, + "content": ". To gain insight into the underlying design of the architecture of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 488, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 503 + ], + "score": 1.0, + "content": "an RNN, it proves usefull to consider how trajectories behave when they are not influenced by any", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 500, + 353, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 353, + 513 + ], + "score": 1.0, + "content": "external input. This lead us to consider the dynamical system", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 515, + 397, + 529 + ], + "lines": [ + { + "bbox": [ + 213, + 515, + 397, + 529 + ], + "spans": [ + { + "bbox": [ + 213, + 515, + 397, + 529 + ], + "score": 0.91, + "content": "\\mathfrak { u } _ { t } = \\Phi ( \\mathfrak { u } _ { t - 1 } ) \\qquad \\Phi ( \\mathfrak { u } ) : = \\Psi ( \\mathfrak { u } , 0 , 0 , \\dots , 0 ) ,", + "type": "interline_equation", + "image_path": "1fe29f71d6e7969485e8484385448b70a4c656e453cb661ec0f67cc824b4ebc1.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 213, + 515, + 397, + 529 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 504, + 545 + ], + "score": 1.0, + "content": "which we refer to as the dynamical system induced by the recurrent neural network. The time-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "invariant system (6) is much more tractable than (5), and it offers a mean to investigate the inner", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 408, + 568 + ], + "score": 1.0, + "content": "working of a given architecture; it separates the influence of input data", + "type": "text" + }, + { + "bbox": [ + 408, + 557, + 419, + 566 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 556, + 505, + 568 + ], + "score": 1.0, + "content": ", which can produce", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "essentially any possible response, from the model itself. Studying trajectories that are not influenced", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "by external data will give us an indication on the ability of a given RNN to generate complex and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "sophisticated trajectories by its own. As we shall see shortly, the dynamical system induced by a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "CFN has excessively simple and predictable trajectories: all of them converge to the zero state. In", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "other words, its only attractor is the zero state. This is in sharp contrast with the dynamical systems", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 621, + 471, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 471, + 634 + ], + "score": 1.0, + "content": "induced by LSTM or GRU, who can exhibit chaotic behaviors and have strange attractors.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 202, + 650 + ], + "score": 1.0, + "content": "The learned parameters", + "type": "text" + }, + { + "bbox": [ + 203, + 638, + 217, + 650 + ], + "score": 0.89, + "content": "W _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "in (5) describe how data influence the evolution of hidden states at each", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "time step. From a modeling perspective, (6) would occur in the scenario where a trained RNN has", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 328, + 673 + ], + "score": 1.0, + "content": "learned a weak coupling between a specific data point", + "type": "text" + }, + { + "bbox": [ + 328, + 661, + 342, + 671 + ], + "score": 0.86, + "content": "x _ { t _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "and the hidden state at that time, in the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 302, + 684 + ], + "score": 1.0, + "content": "sense that the data influence is small and so all", + "type": "text" + }, + { + "bbox": [ + 302, + 671, + 351, + 682 + ], + "score": 0.88, + "content": "W _ { j } x _ { t _ { 0 } } \\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "nearly vanish. The hidden state then", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 681, + 369, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 202, + 695 + ], + "score": 1.0, + "content": "transitions according to", + "type": "text" + }, + { + "bbox": [ + 203, + 682, + 364, + 694 + ], + "score": 0.91, + "content": "\\mathfrak { u } _ { t _ { 0 } } \\approx \\Psi ( \\mathfrak { u } _ { t _ { 0 } - 1 } , 0 , 0 , \\dots , 0 ) = \\Phi ( \\mathfrak { u } _ { t _ { 0 } - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 681, + 369, + 695 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "We refer to Bertschinger & Natschlager (2004) for a study of the chaotic behavior of a simplified ¨", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "vanilla RNN with a specific statistical model, namely an i.i.d. Bernoulli process, for the input data as", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "well as a specific statistical model, namely i.i.d. Gaussian, for the weights of the recurrence matrix.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49 + } + ], + "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 2017", + "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, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 81, + 505, + 139 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 143, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "In the next section, we review the phenomenon of chaos in RNNs via both synthetic examples and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "trained models. We also prove a precise, quantified description of the dynamical picture (3)–(4) for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "the proposed network. In particular, we show that the dynamical system induced by the proposed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "network is never chaotic, and for this reason we refer to it as a Chaos-Free Network (CFN). The", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "final section provides a series of experiments that demonstrate that CFN achieve results comparable", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "to LSTM on the word-level language modeling task. All together, these observations show that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 504, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 504, + 222 + ], + "score": 1.0, + "content": "an architecture as simple as (1)–(2) can achieve performance comparable to the more dynamically", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 175, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 175, + 232 + ], + "score": 1.0, + "content": "complex LSTM.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 144, + 506, + 232 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 247, + 354, + 260 + ], + "lines": [ + { + "bbox": [ + 104, + 245, + 354, + 262 + ], + "spans": [ + { + "bbox": [ + 104, + 245, + 354, + 262 + ], + "score": 1.0, + "content": "2 CHAOS IN RECURRENT NEURAL NETWORKS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "The study of RNNs from a dynamical systems point-of-view has brought fruitful insights into", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 283, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 295 + ], + "score": 1.0, + "content": "generic features of RNNs (Sussillo & Barak, 2013; Pascanu et al., 2013). We shall pursue a brief", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 293, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 307 + ], + "score": 1.0, + "content": "investigation of CFN, LSTM and GRU networks using this formalism, as it allows us to identify", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 507, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 364, + 319 + ], + "score": 1.0, + "content": "key distinctions between them. Recall that for a given mapping", + "type": "text" + }, + { + "bbox": [ + 365, + 304, + 423, + 315 + ], + "score": 0.92, + "content": "\\Phi : \\mathbb { R } ^ { d } \\mapsto \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 303, + 507, + 319 + ], + "score": 1.0, + "content": ", a given initial time", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 315, + 449, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 136, + 327 + ], + "score": 0.91, + "content": "t _ { 0 } \\in \\mathbb { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 315, + 230, + 330 + ], + "score": 1.0, + "content": "and a given initial state", + "type": "text" + }, + { + "bbox": [ + 231, + 315, + 266, + 327 + ], + "score": 0.92, + "content": "\\mathfrak { u } _ { 0 } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 315, + 441, + 330 + ], + "score": 1.0, + "content": ", a simple repeated iteration of the mapping", + "type": "text" + }, + { + "bbox": [ + 441, + 316, + 449, + 326 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 272, + 507, + 330 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 331, + 353, + 360 + ], + "lines": [ + { + "bbox": [ + 257, + 331, + 353, + 360 + ], + "spans": [ + { + "bbox": [ + 257, + 331, + 353, + 360 + ], + "score": 0.9, + "content": "\\begin{array} { c c } { \\mathfrak { u } _ { t + 1 } = \\Phi ( \\mathfrak { u } _ { t } ) } & { t > t _ { 0 } , } \\\\ { \\mathfrak { u } _ { t _ { 0 } } = \\mathfrak { u } _ { 0 } } & { t = t _ { 0 } , } \\end{array}", + "type": "interline_equation", + "image_path": "0405c648e882191d3bc5229f2801828a1787e0dbe5c91b086aeab56b19c0eea0.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 257, + 331, + 353, + 345.5 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 257, + 345.5, + 353, + 360.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 322, + 375 + ], + "score": 1.0, + "content": "defines a discrete-time dynamical system. The index", + "type": "text" + }, + { + "bbox": [ + 322, + 363, + 350, + 373 + ], + "score": 0.9, + "content": "t \\in \\mathbb N", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "represents the current time, while the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 130, + 387 + ], + "score": 1.0, + "content": "point", + "type": "text" + }, + { + "bbox": [ + 130, + 373, + 167, + 385 + ], + "score": 0.9, + "content": "\\mathfrak { u } _ { t } \\in \\mathbb { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 372, + 456, + 387 + ], + "score": 1.0, + "content": "represents the current state of the system. The set of all visited states", + "type": "text" + }, + { + "bbox": [ + 456, + 374, + 505, + 386 + ], + "score": 0.91, + "content": "\\mathcal { O } ^ { + } ( \\mathfrak { u } _ { 0 } ) : =", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 107, + 385, + 219, + 397 + ], + "score": 0.91, + "content": "\\left\\{ \\mathfrak { u } _ { t _ { 0 } } , \\mathfrak { u } _ { t _ { 0 } + 1 } , \\dotsc , \\mathfrak { u } _ { t _ { 0 } + n } , \\dotsc \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 384, + 437, + 398 + ], + "score": 1.0, + "content": "defines the forward trajectory or forward orbit through", + "type": "text" + }, + { + "bbox": [ + 438, + 386, + 449, + 396 + ], + "score": 0.85, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 384, + 506, + 398 + ], + "score": 1.0, + "content": ". An attractor", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "for the dynamical system is a set that is invariant (any trajectory that starts in the set remains in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 405, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 420 + ], + "score": 1.0, + "content": "the set) and that attracts all trajectories that start sufficiently close to it. The attractors of chaotic", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 431 + ], + "score": 1.0, + "content": "dynamical systems are often fractal sets, and for this reason they are referred to as strange attractors.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 363, + 506, + 431 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 300, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 302, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 302, + 447 + ], + "score": 1.0, + "content": "Most RNNs generically take the functional form", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 434, + 302, + 447 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 450, + 386, + 464 + ], + "lines": [ + { + "bbox": [ + 224, + 450, + 386, + 464 + ], + "spans": [ + { + "bbox": [ + 224, + 450, + 386, + 464 + ], + "score": 0.92, + "content": "\\mathsf { u } _ { t } = \\Psi ( \\mathfrak { u } _ { t - 1 } , W _ { 1 } x _ { t } , W _ { 2 } x _ { t } , \\ldots , W _ { k } x _ { t } ) ,", + "type": "interline_equation", + "image_path": "8eaff2b9199aa07be2091afb170ede552f1c313f822ae4b66602b216e31fc836.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 224, + 450, + 386, + 464 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 467, + 505, + 512 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 133, + 480 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 469, + 145, + 478 + ], + "score": 0.82, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 466, + 195, + 480 + ], + "score": 1.0, + "content": "denotes the", + "type": "text" + }, + { + "bbox": [ + 196, + 466, + 208, + 477 + ], + "score": 0.88, + "content": "t ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "input data point. For example, in the case of the CFN (1)–(2), we have", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 146, + 489 + ], + "score": 0.87, + "content": "W _ { 1 } = W", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 478, + 149, + 491 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 150, + 479, + 188, + 490 + ], + "score": 0.88, + "content": "W _ { 2 } = V _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 478, + 207, + 491 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 207, + 479, + 245, + 491 + ], + "score": 0.92, + "content": "W _ { 3 } = V _ { \\eta }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 478, + 506, + 491 + ], + "score": 1.0, + "content": ". To gain insight into the underlying design of the architecture of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 488, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 503 + ], + "score": 1.0, + "content": "an RNN, it proves usefull to consider how trajectories behave when they are not influenced by any", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 500, + 353, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 353, + 513 + ], + "score": 1.0, + "content": "external input. This lead us to consider the dynamical system", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 466, + 506, + 513 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 515, + 397, + 529 + ], + "lines": [ + { + "bbox": [ + 213, + 515, + 397, + 529 + ], + "spans": [ + { + "bbox": [ + 213, + 515, + 397, + 529 + ], + "score": 0.91, + "content": "\\mathfrak { u } _ { t } = \\Phi ( \\mathfrak { u } _ { t - 1 } ) \\qquad \\Phi ( \\mathfrak { u } ) : = \\Psi ( \\mathfrak { u } , 0 , 0 , \\dots , 0 ) ,", + "type": "interline_equation", + "image_path": "1fe29f71d6e7969485e8484385448b70a4c656e453cb661ec0f67cc824b4ebc1.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 213, + 515, + 397, + 529 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 534, + 504, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 504, + 545 + ], + "score": 1.0, + "content": "which we refer to as the dynamical system induced by the recurrent neural network. The time-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 557 + ], + "score": 1.0, + "content": "invariant system (6) is much more tractable than (5), and it offers a mean to investigate the inner", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 556, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 408, + 568 + ], + "score": 1.0, + "content": "working of a given architecture; it separates the influence of input data", + "type": "text" + }, + { + "bbox": [ + 408, + 557, + 419, + 566 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 556, + 505, + 568 + ], + "score": 1.0, + "content": ", which can produce", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "essentially any possible response, from the model itself. Studying trajectories that are not influenced", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 591 + ], + "score": 1.0, + "content": "by external data will give us an indication on the ability of a given RNN to generate complex and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "score": 1.0, + "content": "sophisticated trajectories by its own. As we shall see shortly, the dynamical system induced by a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 612 + ], + "score": 1.0, + "content": "CFN has excessively simple and predictable trajectories: all of them converge to the zero state. In", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "other words, its only attractor is the zero state. This is in sharp contrast with the dynamical systems", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 621, + 471, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 471, + 634 + ], + "score": 1.0, + "content": "induced by LSTM or GRU, who can exhibit chaotic behaviors and have strange attractors.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 534, + 506, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 202, + 650 + ], + "score": 1.0, + "content": "The learned parameters", + "type": "text" + }, + { + "bbox": [ + 203, + 638, + 217, + 650 + ], + "score": 0.89, + "content": "W _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "in (5) describe how data influence the evolution of hidden states at each", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "time step. From a modeling perspective, (6) would occur in the scenario where a trained RNN has", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 328, + 673 + ], + "score": 1.0, + "content": "learned a weak coupling between a specific data point", + "type": "text" + }, + { + "bbox": [ + 328, + 661, + 342, + 671 + ], + "score": 0.86, + "content": "x _ { t _ { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 660, + 505, + 673 + ], + "score": 1.0, + "content": "and the hidden state at that time, in the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 670, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 302, + 684 + ], + "score": 1.0, + "content": "sense that the data influence is small and so all", + "type": "text" + }, + { + "bbox": [ + 302, + 671, + 351, + 682 + ], + "score": 0.88, + "content": "W _ { j } x _ { t _ { 0 } } \\approx 0", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 670, + 506, + 684 + ], + "score": 1.0, + "content": "nearly vanish. The hidden state then", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 681, + 369, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 202, + 695 + ], + "score": 1.0, + "content": "transitions according to", + "type": "text" + }, + { + "bbox": [ + 203, + 682, + 364, + 694 + ], + "score": 0.91, + "content": "\\mathfrak { u } _ { t _ { 0 } } \\approx \\Psi ( \\mathfrak { u } _ { t _ { 0 } - 1 } , 0 , 0 , \\dots , 0 ) = \\Phi ( \\mathfrak { u } _ { t _ { 0 } - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 681, + 369, + 695 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 638, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "We refer to Bertschinger & Natschlager (2004) for a study of the chaotic behavior of a simplified ¨", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "vanilla RNN with a specific statistical model, namely an i.i.d. Bernoulli process, for the input data as", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "well as a specific statistical model, namely i.i.d. Gaussian, for the weights of the recurrence matrix.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49, + "bbox_fs": [ + 106, + 698, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 114, + 87, + 497, + 204 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 87, + 497, + 204 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 87, + 497, + 204 + ], + "spans": [ + { + "bbox": [ + 114, + 87, + 497, + 204 + ], + "score": 0.955, + "type": "image", + "image_path": "4ab3671f85507ce9fae8efa2bef387ae843514737e6c1f8f9ec26fbe10f6ea41.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 114, + 87, + 497, + 126.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 114, + 126.0, + 497, + 165.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 114, + 165.0, + 497, + 204.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 217, + 504, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 217, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 504, + 230 + ], + "score": 1.0, + "content": "Figure 1: Strange attractor of a 2-unit LSTM. Successive zooms (from left to right) reveal the self-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 227, + 426, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 426, + 242 + ], + "score": 1.0, + "content": "repeating, fractal nature of the attractor. Colored boxes depict zooming regions.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 108, + 259, + 448, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 260, + 450, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 450, + 273 + ], + "score": 1.0, + "content": "2.1 CHAOTIC BEHAVIOR OF LSTM AND GRU IN THE ABSENCE OF INPUT DATA", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 504, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "In this subsection we briefly show that LSTM and GRU, in the absence of input data, can lead to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 184, + 304 + ], + "score": 1.0, + "content": "dynamical systems", + "type": "text" + }, + { + "bbox": [ + 184, + 291, + 241, + 304 + ], + "score": 0.93, + "content": "\\mathfrak { u } _ { t } = \\Phi ( \\mathfrak { u } _ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "that are chaotic in the classical sense of the term (Strogatz, 2014).", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 302, + 358, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 358, + 316 + ], + "score": 1.0, + "content": "Figure 1 depicts the strange attractor of the dynamical system:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 318, + 478, + 362 + ], + "lines": [ + { + "bbox": [ + 131, + 318, + 478, + 362 + ], + "spans": [ + { + "bbox": [ + 131, + 318, + 478, + 362 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbf { u } _ { t } = \\Big [ _ { c _ { t } } ^ { h _ { t } } \\Big ] \\quad \\mathbf { u } \\mapsto { \\boldsymbol { \\Phi } } ( \\mathbf { u } ) = \\Big [ ^ { o \\odot \\operatorname { t a n h } \\big ( f \\odot c + i \\odot g \\big ) } \\Big ] } \\\\ & { { \\small i : = \\sigma ( W _ { i } h + b _ { i } ) \\quad f : = \\sigma ( W _ { f } h + b _ { f } ) \\quad o : = \\sigma ( W _ { o } h + b _ { o } ) \\quad g : = \\operatorname { t a n h } ( W _ { g } h + b _ { g } ) } } \\end{array}", + "type": "interline_equation", + "image_path": "c42eee4b570fc26ea71facc0ab66ff5308346a2a12d667f65f253478912be5e6.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 131, + 318, + 478, + 332.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 131, + 332.6666666666667, + 478, + 347.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 131, + 347.33333333333337, + 478, + 362.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 311, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 311, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 311, + 379 + ], + "score": 1.0, + "content": "induced by a two-unit LSTM with weight matrices", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 380, + 470, + 409 + ], + "lines": [ + { + "bbox": [ + 140, + 380, + 470, + 409 + ], + "spans": [ + { + "bbox": [ + 140, + 380, + 470, + 409 + ], + "score": 0.93, + "content": "W _ { i } = \\left[ - 1 \\quad - 4 \\right] \\quad W _ { o } = \\left[ { \\begin{array} { r r } { 4 } & { 1 } \\\\ { - 9 } & { - 7 } \\end{array} } \\right] \\quad W _ { f } = \\left[ { \\begin{array} { r r } { - 2 } & { 6 } \\\\ { 0 } & { - 6 } \\end{array} } \\right] \\quad W _ { g } = \\left[ { \\begin{array} { r r } { - 1 } & { - 6 } \\\\ { 6 } & { - 9 } \\end{array} } \\right]", + "type": "interline_equation", + "image_path": "b907cdb69264bb9c09eb86b0b7a273aa9dc1241b0395aa3bd91fe4b7dbd55c96.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 140, + 380, + 470, + 389.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 140, + 389.6666666666667, + 470, + 399.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 140, + 399.33333333333337, + 470, + 409.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "and zero bias for the model parameters. These weights were randomly generated from a normal", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "distribution with standard deviation 5 and then rounded to the nearest integer. Figure 1(a) was", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 259, + 447 + ], + "score": 1.0, + "content": "obtained by choosing an initial state", + "type": "text" + }, + { + "bbox": [ + 259, + 434, + 319, + 446 + ], + "score": 0.92, + "content": "{ \\mathfrak { u } } _ { 0 } ~ = ~ ( h _ { 0 } , c _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 434, + 422, + 447 + ], + "score": 1.0, + "content": "uniformly at random in", + "type": "text" + }, + { + "bbox": [ + 423, + 434, + 486, + 447 + ], + "score": 0.91, + "content": "[ 0 , 1 ] ^ { 2 } \\times [ 0 , \\dot { 1 } ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 273, + 459 + ], + "score": 1.0, + "content": "plotting the h-component of the iterates", + "type": "text" + }, + { + "bbox": [ + 273, + 446, + 330, + 457 + ], + "score": 0.92, + "content": "{ \\mathfrak { u } } _ { t } ~ = ~ ( h _ { t } , c _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 445, + 347, + 459 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 347, + 447, + 353, + 456 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 445, + 391, + 459 + ], + "score": 1.0, + "content": "between", + "type": "text" + }, + { + "bbox": [ + 391, + 445, + 407, + 456 + ], + "score": 0.88, + "content": "1 0 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 445, + 427, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 428, + 445, + 443, + 456 + ], + "score": 0.82, + "content": "1 0 ^ { \\bar { 5 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "(so this figure", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "should be regarded as a two dimensional projection of a four dimensional attractor, which explain", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 320, + 480 + ], + "score": 1.0, + "content": "its tangled appearance). Most trajectories starting in", + "type": "text" + }, + { + "bbox": [ + 321, + 467, + 383, + 479 + ], + "score": 0.93, + "content": "[ 0 , 1 ] ^ { 2 } \\times [ 0 , 1 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "converge toward the depicted", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "score": 1.0, + "content": "attractor. The resemblance between this attractor and classical strange attractors such as the Henon ´", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "attractor is striking (see Figure 5 in the appendix for a depiction of the Henon attractor). Successive ´", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "zooms on the branch of the LSTM attractor from Figure 1(a) reveal its fractal nature. Figure 1(b) is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "an enlargement of the red box in Figure 1(a), and Figure 1(c) is an enlargement of the magenta box", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 523, + 383, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 383, + 535 + ], + "score": 1.0, + "content": "in Figure 1(b). We see that the structure repeats itself as we zoom in.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "The most practical consequence of chaos is that the long-term behavior of their forward orbits can", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 329, + 563 + ], + "score": 1.0, + "content": "exhibit a high degree of sensitivity to the initial states", + "type": "text" + }, + { + "bbox": [ + 330, + 552, + 341, + 561 + ], + "score": 0.84, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 550, + 506, + 563 + ], + "score": 1.0, + "content": ". Figure 2 provides an example of such", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 362, + 574 + ], + "score": 1.0, + "content": "behavior for the dynamical system (7)–(9). An initial condition", + "type": "text" + }, + { + "bbox": [ + 362, + 563, + 373, + 573 + ], + "score": 0.84, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "was drawn uniformly at random", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 117, + 585 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 118, + 572, + 179, + 584 + ], + "score": 0.92, + "content": "[ 0 , 1 ] ^ { 2 } \\times [ 0 , 1 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 571, + 420, + 585 + ], + "score": 1.0, + "content": ". We then computed 100, 000 small amplitude perturbations", + "type": "text" + }, + { + "bbox": [ + 420, + 572, + 431, + 583 + ], + "score": 0.88, + "content": "\\hat { \\mathfrak { u } } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 571, + 443, + 585 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 443, + 573, + 454, + 583 + ], + "score": 0.85, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "by adding a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 291, + 596 + ], + "score": 1.0, + "content": "small random number drawn uniformly from", + "type": "text" + }, + { + "bbox": [ + 291, + 582, + 351, + 595 + ], + "score": 0.93, + "content": "[ - 1 0 ^ { - 7 } , 1 0 ^ { - 7 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "to each component. We then iterated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 258, + 606 + ], + "score": 1.0, + "content": "(7)–(9) for 200 steps and plotted the", + "type": "text" + }, + { + "bbox": [ + 258, + 595, + 265, + 604 + ], + "score": 0.38, + "content": "\\mathbf { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 594, + 383, + 606 + ], + "score": 1.0, + "content": "-component of the final state", + "type": "text" + }, + { + "bbox": [ + 384, + 594, + 403, + 605 + ], + "score": 0.88, + "content": "\\hat { \\textmd u } _ { 2 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "for each of the 100, 000", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "trials on Figure 2(a). The collection of these 100, 000 final states essentially fills out the entire", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "attractor, despite the fact that their initial conditions are highly localized (i.e. at distance of no more", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 127, + 640 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 127, + 626, + 149, + 638 + ], + "score": 0.88, + "content": "1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 626, + 345, + 640 + ], + "score": 1.0, + "content": ") around a fixed point. In other words, the time", + "type": "text" + }, + { + "bbox": [ + 345, + 627, + 381, + 637 + ], + "score": 0.84, + "content": "t = 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "map of the dynamical system", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 357, + 650 + ], + "score": 1.0, + "content": "will map a small neighborhood around a fixed initial condition", + "type": "text" + }, + { + "bbox": [ + 357, + 640, + 368, + 649 + ], + "score": 0.84, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "to the entire attractor. Figure 2(b)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "additionally illustrates this sensitivity to initial conditions for points on the attractor itself. We take", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 185, + 673 + ], + "score": 1.0, + "content": "an initial condition", + "type": "text" + }, + { + "bbox": [ + 186, + 661, + 197, + 671 + ], + "score": 0.84, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 659, + 334, + 673 + ], + "score": 1.0, + "content": "on the attractor and perturb it by", + "type": "text" + }, + { + "bbox": [ + 334, + 659, + 356, + 671 + ], + "score": 0.9, + "content": "1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 659, + 472, + 673 + ], + "score": 1.0, + "content": "to a nearby initial condition", + "type": "text" + }, + { + "bbox": [ + 472, + 660, + 483, + 671 + ], + "score": 0.82, + "content": "\\hat { \\mathfrak { u } } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 197, + 684 + ], + "score": 1.0, + "content": "then plot the distance", + "type": "text" + }, + { + "bbox": [ + 197, + 671, + 239, + 683 + ], + "score": 0.93, + "content": "\\left\\| \\hat { \\mathbf { u } } _ { t } - \\mathbf { u } _ { t } \\right\\|", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "between the two corresponding trajectories for the first 200 time", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 681, + 407, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 407, + 695 + ], + "score": 1.0, + "content": "steps. After an initial phase of agreement, the trajectories strongly diverge.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "The synthetic example (7)–(9) illustrates the potentially chaotic nature of the LSTM architecture.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "We now show that chaotic behavior occurs for trained models as well, and not just for synthetically", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "generated instances. We take the parameter values of an LSTM with 228 hidden units trained on the", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "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 2017", + "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": [ + 114, + 87, + 497, + 204 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 114, + 87, + 497, + 204 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 114, + 87, + 497, + 204 + ], + "spans": [ + { + "bbox": [ + 114, + 87, + 497, + 204 + ], + "score": 0.955, + "type": "image", + "image_path": "4ab3671f85507ce9fae8efa2bef387ae843514737e6c1f8f9ec26fbe10f6ea41.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 114, + 87, + 497, + 126.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 114, + 126.0, + 497, + 165.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 114, + 165.0, + 497, + 204.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 217, + 504, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 217, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 504, + 230 + ], + "score": 1.0, + "content": "Figure 1: Strange attractor of a 2-unit LSTM. Successive zooms (from left to right) reveal the self-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 227, + 426, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 426, + 242 + ], + "score": 1.0, + "content": "repeating, fractal nature of the attractor. Colored boxes depict zooming regions.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 108, + 259, + 448, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 260, + 450, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 450, + 273 + ], + "score": 1.0, + "content": "2.1 CHAOTIC BEHAVIOR OF LSTM AND GRU IN THE ABSENCE OF INPUT DATA", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 504, + 314 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "In this subsection we briefly show that LSTM and GRU, in the absence of input data, can lead to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 184, + 304 + ], + "score": 1.0, + "content": "dynamical systems", + "type": "text" + }, + { + "bbox": [ + 184, + 291, + 241, + 304 + ], + "score": 0.93, + "content": "\\mathfrak { u } _ { t } = \\Phi ( \\mathfrak { u } _ { t - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 292, + 505, + 304 + ], + "score": 1.0, + "content": "that are chaotic in the classical sense of the term (Strogatz, 2014).", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 302, + 358, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 358, + 316 + ], + "score": 1.0, + "content": "Figure 1 depicts the strange attractor of the dynamical system:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 280, + 505, + 316 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 131, + 318, + 478, + 362 + ], + "lines": [ + { + "bbox": [ + 131, + 318, + 478, + 362 + ], + "spans": [ + { + "bbox": [ + 131, + 318, + 478, + 362 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { \\mathbf { u } _ { t } = \\Big [ _ { c _ { t } } ^ { h _ { t } } \\Big ] \\quad \\mathbf { u } \\mapsto { \\boldsymbol { \\Phi } } ( \\mathbf { u } ) = \\Big [ ^ { o \\odot \\operatorname { t a n h } \\big ( f \\odot c + i \\odot g \\big ) } \\Big ] } \\\\ & { { \\small i : = \\sigma ( W _ { i } h + b _ { i } ) \\quad f : = \\sigma ( W _ { f } h + b _ { f } ) \\quad o : = \\sigma ( W _ { o } h + b _ { o } ) \\quad g : = \\operatorname { t a n h } ( W _ { g } h + b _ { g } ) } } \\end{array}", + "type": "interline_equation", + "image_path": "c42eee4b570fc26ea71facc0ab66ff5308346a2a12d667f65f253478912be5e6.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 131, + 318, + 478, + 332.6666666666667 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 131, + 332.6666666666667, + 478, + 347.33333333333337 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 131, + 347.33333333333337, + 478, + 362.00000000000006 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 365, + 311, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 311, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 311, + 379 + ], + "score": 1.0, + "content": "induced by a two-unit LSTM with weight matrices", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 363, + 311, + 379 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 380, + 470, + 409 + ], + "lines": [ + { + "bbox": [ + 140, + 380, + 470, + 409 + ], + "spans": [ + { + "bbox": [ + 140, + 380, + 470, + 409 + ], + "score": 0.93, + "content": "W _ { i } = \\left[ - 1 \\quad - 4 \\right] \\quad W _ { o } = \\left[ { \\begin{array} { r r } { 4 } & { 1 } \\\\ { - 9 } & { - 7 } \\end{array} } \\right] \\quad W _ { f } = \\left[ { \\begin{array} { r r } { - 2 } & { 6 } \\\\ { 0 } & { - 6 } \\end{array} } \\right] \\quad W _ { g } = \\left[ { \\begin{array} { r r } { - 1 } & { - 6 } \\\\ { 6 } & { - 9 } \\end{array} } \\right]", + "type": "interline_equation", + "image_path": "b907cdb69264bb9c09eb86b0b7a273aa9dc1241b0395aa3bd91fe4b7dbd55c96.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 140, + 380, + 470, + 389.6666666666667 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 140, + 389.6666666666667, + 470, + 399.33333333333337 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 140, + 399.33333333333337, + 470, + 409.00000000000006 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "and zero bias for the model parameters. These weights were randomly generated from a normal", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 436 + ], + "score": 1.0, + "content": "distribution with standard deviation 5 and then rounded to the nearest integer. Figure 1(a) was", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 259, + 447 + ], + "score": 1.0, + "content": "obtained by choosing an initial state", + "type": "text" + }, + { + "bbox": [ + 259, + 434, + 319, + 446 + ], + "score": 0.92, + "content": "{ \\mathfrak { u } } _ { 0 } ~ = ~ ( h _ { 0 } , c _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 434, + 422, + 447 + ], + "score": 1.0, + "content": "uniformly at random in", + "type": "text" + }, + { + "bbox": [ + 423, + 434, + 486, + 447 + ], + "score": 0.91, + "content": "[ 0 , 1 ] ^ { 2 } \\times [ 0 , \\dot { 1 } ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 445, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 273, + 459 + ], + "score": 1.0, + "content": "plotting the h-component of the iterates", + "type": "text" + }, + { + "bbox": [ + 273, + 446, + 330, + 457 + ], + "score": 0.92, + "content": "{ \\mathfrak { u } } _ { t } ~ = ~ ( h _ { t } , c _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 445, + 347, + 459 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 347, + 447, + 353, + 456 + ], + "score": 0.75, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 445, + 391, + 459 + ], + "score": 1.0, + "content": "between", + "type": "text" + }, + { + "bbox": [ + 391, + 445, + 407, + 456 + ], + "score": 0.88, + "content": "1 0 ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 445, + 427, + 459 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 428, + 445, + 443, + 456 + ], + "score": 0.82, + "content": "1 0 ^ { \\bar { 5 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 445, + 506, + 459 + ], + "score": 1.0, + "content": "(so this figure", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "should be regarded as a two dimensional projection of a four dimensional attractor, which explain", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 320, + 480 + ], + "score": 1.0, + "content": "its tangled appearance). Most trajectories starting in", + "type": "text" + }, + { + "bbox": [ + 321, + 467, + 383, + 479 + ], + "score": 0.93, + "content": "[ 0 , 1 ] ^ { 2 } \\times [ 0 , 1 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "converge toward the depicted", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 490 + ], + "score": 1.0, + "content": "attractor. The resemblance between this attractor and classical strange attractors such as the Henon ´", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "attractor is striking (see Figure 5 in the appendix for a depiction of the Henon attractor). Successive ´", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "zooms on the branch of the LSTM attractor from Figure 1(a) reveal its fractal nature. Figure 1(b) is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "an enlargement of the red box in Figure 1(a), and Figure 1(c) is an enlargement of the magenta box", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 523, + 383, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 383, + 535 + ], + "score": 1.0, + "content": "in Figure 1(b). We see that the structure repeats itself as we zoom in.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 412, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "The most practical consequence of chaos is that the long-term behavior of their forward orbits can", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 329, + 563 + ], + "score": 1.0, + "content": "exhibit a high degree of sensitivity to the initial states", + "type": "text" + }, + { + "bbox": [ + 330, + 552, + 341, + 561 + ], + "score": 0.84, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 550, + 506, + 563 + ], + "score": 1.0, + "content": ". Figure 2 provides an example of such", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 362, + 574 + ], + "score": 1.0, + "content": "behavior for the dynamical system (7)–(9). An initial condition", + "type": "text" + }, + { + "bbox": [ + 362, + 563, + 373, + 573 + ], + "score": 0.84, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "was drawn uniformly at random", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 117, + 585 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 118, + 572, + 179, + 584 + ], + "score": 0.92, + "content": "[ 0 , 1 ] ^ { 2 } \\times [ 0 , 1 ] ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 571, + 420, + 585 + ], + "score": 1.0, + "content": ". We then computed 100, 000 small amplitude perturbations", + "type": "text" + }, + { + "bbox": [ + 420, + 572, + 431, + 583 + ], + "score": 0.88, + "content": "\\hat { \\mathfrak { u } } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 571, + 443, + 585 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 443, + 573, + 454, + 583 + ], + "score": 0.85, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "by adding a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 291, + 596 + ], + "score": 1.0, + "content": "small random number drawn uniformly from", + "type": "text" + }, + { + "bbox": [ + 291, + 582, + 351, + 595 + ], + "score": 0.93, + "content": "[ - 1 0 ^ { - 7 } , 1 0 ^ { - 7 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "to each component. We then iterated", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 258, + 606 + ], + "score": 1.0, + "content": "(7)–(9) for 200 steps and plotted the", + "type": "text" + }, + { + "bbox": [ + 258, + 595, + 265, + 604 + ], + "score": 0.38, + "content": "\\mathbf { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 594, + 383, + 606 + ], + "score": 1.0, + "content": "-component of the final state", + "type": "text" + }, + { + "bbox": [ + 384, + 594, + 403, + 605 + ], + "score": 0.88, + "content": "\\hat { \\textmd u } _ { 2 0 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "for each of the 100, 000", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "trials on Figure 2(a). The collection of these 100, 000 final states essentially fills out the entire", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "attractor, despite the fact that their initial conditions are highly localized (i.e. at distance of no more", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 127, + 640 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 127, + 626, + 149, + 638 + ], + "score": 0.88, + "content": "1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 626, + 345, + 640 + ], + "score": 1.0, + "content": ") around a fixed point. In other words, the time", + "type": "text" + }, + { + "bbox": [ + 345, + 627, + 381, + 637 + ], + "score": 0.84, + "content": "t = 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "map of the dynamical system", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 357, + 650 + ], + "score": 1.0, + "content": "will map a small neighborhood around a fixed initial condition", + "type": "text" + }, + { + "bbox": [ + 357, + 640, + 368, + 649 + ], + "score": 0.84, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "to the entire attractor. Figure 2(b)", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "additionally illustrates this sensitivity to initial conditions for points on the attractor itself. We take", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 185, + 673 + ], + "score": 1.0, + "content": "an initial condition", + "type": "text" + }, + { + "bbox": [ + 186, + 661, + 197, + 671 + ], + "score": 0.84, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 659, + 334, + 673 + ], + "score": 1.0, + "content": "on the attractor and perturb it by", + "type": "text" + }, + { + "bbox": [ + 334, + 659, + 356, + 671 + ], + "score": 0.9, + "content": "1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 659, + 472, + 673 + ], + "score": 1.0, + "content": "to a nearby initial condition", + "type": "text" + }, + { + "bbox": [ + 472, + 660, + 483, + 671 + ], + "score": 0.82, + "content": "\\hat { \\mathfrak { u } } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 659, + 506, + 673 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 671, + 506, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 197, + 684 + ], + "score": 1.0, + "content": "then plot the distance", + "type": "text" + }, + { + "bbox": [ + 197, + 671, + 239, + 683 + ], + "score": 0.93, + "content": "\\left\\| \\hat { \\mathbf { u } } _ { t } - \\mathbf { u } _ { t } \\right\\|", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 671, + 506, + 684 + ], + "score": 1.0, + "content": "between the two corresponding trajectories for the first 200 time", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 681, + 407, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 407, + 695 + ], + "score": 1.0, + "content": "steps. After an initial phase of agreement, the trajectories strongly diverge.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 539, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "The synthetic example (7)–(9) illustrates the potentially chaotic nature of the LSTM architecture.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "We now show that chaotic behavior occurs for trained models as well, and not just for synthetically", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "generated instances. We take the parameter values of an LSTM with 228 hidden units trained on the", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42, + "bbox_fs": [ + 106, + 698, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 137, + 89, + 466, + 208 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 137, + 89, + 466, + 208 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 137, + 89, + 466, + 208 + ], + "spans": [ + { + "bbox": [ + 137, + 89, + 466, + 208 + ], + "score": 0.951, + "type": "image", + "image_path": "08703ee2db3c758c4fb6bd9c2358d1258e01e76cbaa7406e8cd80b1c7aee6c0d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 137, + 89, + 466, + 128.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 137, + 128.66666666666666, + 466, + 168.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 137, + 168.33333333333331, + 466, + 207.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 221, + 505, + 255 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 221, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 395, + 234 + ], + "score": 1.0, + "content": "Figure 2: (a): A small neighborhood around a fixed initial condition", + "type": "text" + }, + { + "bbox": [ + 395, + 223, + 406, + 232 + ], + "score": 0.85, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 221, + 506, + 234 + ], + "score": 1.0, + "content": ", after 200 iterations, is", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 232, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 419, + 244 + ], + "score": 1.0, + "content": "mapped to the entire attractor. (b): Two trajectories starting starting within", + "type": "text" + }, + { + "bbox": [ + 419, + 232, + 441, + 243 + ], + "score": 0.9, + "content": "1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 232, + 506, + 244 + ], + "score": 1.0, + "content": "of one another", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 242, + 232, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 232, + 257 + ], + "score": 1.0, + "content": "strongly diverge after 50 steps.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "image", + "bbox": [ + 109, + 272, + 488, + 359 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 272, + 488, + 359 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 109, + 272, + 488, + 359 + ], + "spans": [ + { + "bbox": [ + 109, + 272, + 488, + 359 + ], + "score": 0.964, + "type": "image", + "image_path": "5d4ac4bc4740c0bee2b2e32fef8a96f54fb4405a873f08e76cd9d14db5cc0d2c.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 109, + 272, + 488, + 301.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 109, + 301.0, + 488, + 330.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 109, + 330.0, + 488, + 359.0 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 371, + 505, + 405 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 505, + 384 + ], + "score": 1.0, + "content": "Figure 3: 228-unit LSTM trained on Penn Treebank. (a): In the absence of input data, the system", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 382, + 504, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 504, + 395 + ], + "score": 1.0, + "content": "is chaotic and nearby trajectories diverge. (b): In the presence of input data, the system is mostly", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 393, + 380, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 380, + 407 + ], + "score": 1.0, + "content": "driven by the external input. Trajectories starting far apart converge.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "Penn Treebank corpus without dropout (c.f. the experimental section for the precise procedure). We", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 198, + 451 + ], + "score": 1.0, + "content": "then set all data inputs", + "type": "text" + }, + { + "bbox": [ + 198, + 439, + 208, + 448 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 437, + 505, + 451 + ], + "score": 1.0, + "content": "to zero and run the corresponding induced dynamical system. Two trajec-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 448, + 505, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 286, + 461 + ], + "score": 1.0, + "content": "tories starting from nearby initial conditions", + "type": "text" + }, + { + "bbox": [ + 287, + 450, + 298, + 460 + ], + "score": 0.83, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 448, + 317, + 461 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 448, + 328, + 460 + ], + "score": 0.87, + "content": "\\hat { \\mathfrak { u } } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 448, + 437, + 461 + ], + "score": 1.0, + "content": "were computed (as before", + "type": "text" + }, + { + "bbox": [ + 437, + 448, + 448, + 459 + ], + "score": 0.86, + "content": "\\hat { \\mathfrak { u } } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 448, + 505, + 461 + ], + "score": 1.0, + "content": "was obtained", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 239, + 471 + ], + "score": 1.0, + "content": "by adding to each components of", + "type": "text" + }, + { + "bbox": [ + 239, + 461, + 250, + 471 + ], + "score": 0.86, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 459, + 438, + 471 + ], + "score": 1.0, + "content": "a small random number drawn uniformly from", + "type": "text" + }, + { + "bbox": [ + 438, + 459, + 500, + 471 + ], + "score": 0.91, + "content": "[ - 1 0 ^ { - 7 } , 1 0 ^ { - 7 } ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 459, + 505, + 471 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 471, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 482 + ], + "score": 1.0, + "content": "Figure 3(a) plots the first component h(1) of the hidden state for both trajectories over the first", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 479, + 504, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 418, + 495 + ], + "score": 1.0, + "content": "1600 time steps. After an initial phase of agreement, the forward trajectories", + "type": "text" + }, + { + "bbox": [ + 418, + 481, + 451, + 493 + ], + "score": 0.92, + "content": "\\mathcal { O } ^ { + } ( \\mathfrak { u } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 479, + 470, + 495 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 481, + 504, + 493 + ], + "score": 0.93, + "content": "\\mathcal { O } ^ { + } ( \\hat { \\mathfrak { u } } _ { 0 } )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 493, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 506, + 505 + ], + "score": 1.0, + "content": "strongly diverge. We also see that both trajectories exhibit the typical aperiodic behavior that char-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 262, + 516 + ], + "score": 1.0, + "content": "acterizes chaotic systems. If the inputs", + "type": "text" + }, + { + "bbox": [ + 262, + 505, + 273, + 514 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "do not vanish, but come from actual word-level data, then", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "the behavior is very different. The LSTM is now no longer an autonomous system whose dynamics", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "are driven by its hidden states, but a time dependent system whose dynamics are mostly driven by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 340, + 549 + ], + "score": 1.0, + "content": "the external inputs. Figure 3(b) shows the first component", + "type": "text" + }, + { + "bbox": [ + 340, + 536, + 360, + 548 + ], + "score": 0.91, + "content": "h ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "of the hidden states of two trajecto-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 251, + 560 + ], + "score": 1.0, + "content": "ries that start with initial conditions", + "type": "text" + }, + { + "bbox": [ + 252, + 549, + 263, + 558 + ], + "score": 0.83, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 546, + 281, + 560 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 547, + 293, + 558 + ], + "score": 0.88, + "content": "\\hat { \\mathfrak { u } } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "that are far apart. The sensitivity to initial condition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "disappears, and instead the trajectories converge toward each other after about 70 steps. The mem-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 569, + 507, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 507, + 581 + ], + "score": 1.0, + "content": "ory of this initial difference is lost. Overall these experiments indicate that a trained LSTM, when it", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "is not driven by external inputs, can be chaotic. In the presence of input data, the LSTM becomes a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 590, + 373, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 373, + 605 + ], + "score": 1.0, + "content": "forced system whose dynamics are dominated by external forcing.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 108, + 608, + 504, + 641 + ], + "lines": [ + { + "bbox": [ + 104, + 606, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 505, + 621 + ], + "score": 1.0, + "content": "Like LSTM networks, GRU can also lead to dynamical systems that are chaotic and they can also", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "have strange attractors. The depiction of such an attractor, in the case of a two-unit GRU, is provided", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 630, + 220, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 220, + 642 + ], + "score": 1.0, + "content": "in Figure 6 of the appendix.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 656, + 289, + 667 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 291, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 291, + 668 + ], + "score": 1.0, + "content": "2.2 CHAOS-FREE BEHAVIOR OF THE CFN", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 107, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "The dynamical behavior of the CFN is dramatically different from that of the LSTM. 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(a): In the absence of input data, the system", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 382, + 504, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 504, + 395 + ], + "score": 1.0, + "content": "is chaotic and nearby trajectories diverge. (b): In the presence of input data, the system is mostly", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 393, + 380, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 380, + 407 + ], + "score": 1.0, + "content": "driven by the external input. Trajectories starting far apart converge.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 426, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 440 + ], + "score": 1.0, + "content": "Penn Treebank corpus without dropout (c.f. the experimental section for the precise procedure). We", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 198, + 451 + ], + "score": 1.0, + "content": "then set all data inputs", + "type": "text" + }, + { + "bbox": [ + 198, + 439, + 208, + 448 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 437, + 505, + 451 + ], + "score": 1.0, + "content": "to zero and run the corresponding induced dynamical system. 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We also see that both trajectories exhibit the typical aperiodic behavior that char-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 262, + 516 + ], + "score": 1.0, + "content": "acterizes chaotic systems. If the inputs", + "type": "text" + }, + { + "bbox": [ + 262, + 505, + 273, + 514 + ], + "score": 0.84, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "do not vanish, but come from actual word-level data, then", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 527 + ], + "score": 1.0, + "content": "the behavior is very different. The LSTM is now no longer an autonomous system whose dynamics", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "are driven by its hidden states, but a time dependent system whose dynamics are mostly driven by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 340, + 549 + ], + "score": 1.0, + "content": "the external inputs. Figure 3(b) shows the first component", + "type": "text" + }, + { + "bbox": [ + 340, + 536, + 360, + 548 + ], + "score": 0.91, + "content": "h ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "of the hidden states of two trajecto-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 251, + 560 + ], + "score": 1.0, + "content": "ries that start with initial conditions", + "type": "text" + }, + { + "bbox": [ + 252, + 549, + 263, + 558 + ], + "score": 0.83, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 546, + 281, + 560 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 547, + 293, + 558 + ], + "score": 0.88, + "content": "\\hat { \\mathfrak { u } } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "that are far apart. The sensitivity to initial condition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "disappears, and instead the trajectories converge toward each other after about 70 steps. The mem-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 569, + 507, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 507, + 581 + ], + "score": 1.0, + "content": "ory of this initial difference is lost. Overall these experiments indicate that a trained LSTM, when it", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "is not driven by external inputs, can be chaotic. 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In this sub-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "section we start by showing that the hidden states of the CFN activate and relax toward zero in a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "predictable fashion in response to input data. On one hand, this shows that the CFN cannot produce", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "non-trivial dynamics without some influence from data. 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This follows from a precise, quantified description of the intuitive picture", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 253, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 253, + 106 + ], + "score": 1.0, + "content": "(3)–(4) sketched in the introduction.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 677, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "data-driven activation. This follows from a precise, quantified description of the intuitive picture", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 253, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 253, + 106 + ], + "score": 1.0, + "content": "(3)–(4) sketched in the introduction.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 504, + 133 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "We begin with the following simple estimate that sheds light on how the hidden states of the CFN", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 272, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 272, + 135 + ], + "score": 1.0, + "content": "activate and then relax toward the origin.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 136, + 259, + 147 + ], + "lines": [ + { + "bbox": [ + 106, + 135, + 259, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 187, + 148 + ], + "score": 1.0, + "content": "Lemma 1. For any", + "type": "text" + }, + { + "bbox": [ + 187, + 136, + 223, + 147 + ], + "score": 0.91, + "content": "T , k > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 135, + 259, + 148 + ], + "score": 1.0, + "content": "we have", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 150, + 425, + 178 + ], + "lines": [ + { + "bbox": [ + 186, + 150, + 425, + 178 + ], + "spans": [ + { + "bbox": [ + 186, + 150, + 425, + 178 + ], + "score": 0.92, + "content": "\\left| h _ { T + k } ( i ) \\right| \\leq \\Theta ^ { k } \\left| h _ { T } ( i ) \\right| + \\frac { H } { 1 - \\Theta } \\left( \\operatorname* { m a x } _ { T \\leq t \\leq T + k } \\left| ( W x _ { t } ) ( i ) \\right| \\right)", + "type": "interline_equation", + "image_path": "d4e742ca30a93b5ab05e8a4b0d56958bb41e9b338436f08f253454dbeac0a044.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 186, + 150, + 425, + 178 + ], + 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The proof of Lemma 1 is elementary —", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 173, + 378 + ], + "score": 1.0, + "content": "Proof of Lemma", + "type": "text" + }, + { + "bbox": [ + 174, + 366, + 179, + 375 + ], + "score": 0.35, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 365, + 416, + 378 + ], + "score": 1.0, + "content": ". 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Iterating this inequality and summing the geometric series", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 426, + 150, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 150, + 440 + ], + "score": 1.0, + "content": "then gives", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 436, + 435, + 465 + ], + "lines": [ + { + "bbox": [ + 176, + 436, + 435, + 465 + ], + "spans": [ + { + "bbox": [ + 176, + 436, + 435, + 465 + ], + "score": 0.94, + "content": "| h _ { T + k } ( i ) | \\leq \\Theta ^ { k } | h _ { T } ( i ) | + \\left( \\frac { 1 - \\Theta ^ { k } } { 1 - \\Theta } \\right) \\ H \\operatorname* { m a x } _ { T \\leq t \\leq T + k } | ( W x _ { t } ) ( i ) |", + "type": "interline_equation", + "image_path": "671ef617f269e8fcb86cf3420c3b8733f20e3386c12059eecf1c691f35285404.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 176, + 436, + 435, + 445.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 176, + 445.6666666666667, + 435, + 455.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 176, + 455.33333333333337, + 435, + 465.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 235, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 236, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 236, + 478 + ], + "score": 1.0, + "content": "from which we easily conclude.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 459, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 459, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 459, + 503 + ], + "score": 1.0, + "content": "We now turn toward the analysis of the long-term behavior of the the dynamical system", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 504, + 412, + 518 + ], + "lines": [ + { + "bbox": [ + 198, + 504, + 412, + 518 + ], + "spans": [ + { + "bbox": [ + 198, + 504, + 412, + 518 + ], + "score": 0.89, + "content": "\\mathfrak { u } _ { t } = h _ { t } , \\qquad \\mathfrak { u } \\mapsto \\Phi ( \\mathfrak { u } ) : = \\sigma \\left( U _ { \\theta } \\mathfrak { u } + b _ { \\theta } \\right) \\odot \\operatorname { t a n h } ( \\mathfrak { u } ) .", + "type": "interline_equation", + "image_path": "49b2dc2b20f33d3389fd05b9ab29c46e9b653d915e01350b47e0d8be0ed3fc46.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 198, + 504, + 412, + 518 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 506, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "induced by a CFN. The following lemma shows that the only attractor of this dynamical system is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 532, + 164, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 164, + 545 + ], + "score": 1.0, + "content": "the zero state.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 105, + 546, + 504, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 274, + 560 + ], + "score": 1.0, + "content": "Lemma 2. Starting from any initial state", + "type": "text" + }, + { + "bbox": [ + 275, + 549, + 285, + 558 + ], + "score": 0.8, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 546, + 346, + 560 + ], + "score": 1.0, + "content": ", the trajectory", + "type": "text" + }, + { + "bbox": [ + 346, + 547, + 379, + 559 + ], + "score": 0.92, + "content": "\\mathcal { O } ^ { + } ( \\mathfrak { u } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "will eventually converge to the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 557, + 397, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 182, + 572 + ], + "score": 1.0, + "content": "zero state. That is,", + "type": "text" + }, + { + "bbox": [ + 182, + 559, + 252, + 570 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { t + \\infty } \\mathfrak { u } _ { t } = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 557, + 383, + 572 + ], + "score": 1.0, + "content": "regardless of the the initial state", + "type": "text" + }, + { + "bbox": [ + 383, + 560, + 394, + 569 + ], + "score": 0.8, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 557, + 397, + 572 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 225, + 590 + ], + "score": 1.0, + "content": "Proof. From the definition of", + "type": "text" + }, + { + "bbox": [ + 226, + 577, + 234, + 587 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 576, + 413, + 590 + ], + "score": 1.0, + "content": "we clearly have that the sequence defined by", + "type": "text" + }, + { + "bbox": [ + 413, + 577, + 470, + 588 + ], + "score": 0.94, + "content": "\\mathfrak { u } _ { t + 1 } = \\Phi ( \\mathfrak { u } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 586, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 178, + 599 + ], + "score": 0.91, + "content": "- 1 < \\mathfrak { u } _ { t } ( i ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 586, + 208, + 601 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 208, + 588, + 213, + 597 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 586, + 246, + 601 + ], + "score": 1.0, + "content": "and all", + "type": "text" + }, + { + "bbox": [ + 246, + 588, + 251, + 597 + ], + "score": 0.61, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 586, + 338, + 601 + ], + "score": 1.0, + "content": ". Since the sequence", + "type": "text" + }, + { + "bbox": [ + 338, + 589, + 348, + 599 + ], + "score": 0.85, + "content": "{ \\mathfrak { u } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 586, + 477, + 601 + ], + "score": 1.0, + "content": "is bounded, so is the sequence", + "type": "text" + }, + { + "bbox": [ + 478, + 589, + 505, + 599 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { t } : =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 597, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 148, + 610 + ], + "score": 0.9, + "content": "U _ { \\theta } \\mathfrak { u } _ { t } + b _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 597, + 259, + 612 + ], + "score": 1.0, + "content": ". That is there exists a finite", + "type": "text" + }, + { + "bbox": [ + 259, + 599, + 287, + 609 + ], + "score": 0.91, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 597, + 326, + 612 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 327, + 599, + 420, + 610 + ], + "score": 0.91, + "content": "( U _ { \\theta } \\mathfrak { u } _ { t } ) ( i ) + b _ { \\theta } ( i ) < C", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 597, + 447, + 612 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 447, + 600, + 452, + 609 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 597, + 469, + 612 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 600, + 474, + 609 + ], + "score": 0.67, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 597, + 506, + 612 + ], + "score": 1.0, + "content": ". Using", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 504, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 371, + 622 + ], + "score": 1.0, + "content": "the non-expansivity of the hyperbolic tangent, we then obtain that", + "type": "text" + }, + { + "bbox": [ + 372, + 610, + 468, + 622 + ], + "score": 0.92, + "content": "| \\mathfrak { u } _ { t } ( i ) | \\leq \\sigma ( C ) | \\mathfrak { u } _ { t - 1 } ( i ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 609, + 498, + 622 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 499, + 611, + 504, + 619 + ], + "score": 0.79, + "content": "t", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 620, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 136, + 632 + ], + "score": 1.0, + "content": "and all", + "type": "text" + }, + { + "bbox": [ + 136, + 621, + 141, + 630 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 621, + 257, + 632 + ], + "score": 1.0, + "content": ". 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Overall this leads to the following simple", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 315, + 507, + 331 + ], + "spans": [ + { + "bbox": [ + 104, + 315, + 141, + 331 + ], + "score": 1.0, + "content": "picture:", + "type": "text" + }, + { + "bbox": [ + 141, + 317, + 163, + 330 + ], + "score": 0.92, + "content": "h _ { t } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 315, + 368, + 331 + ], + "score": 1.0, + "content": "activates when presented with an embedded input", + "type": "text" + }, + { + "bbox": [ + 369, + 318, + 390, + 329 + ], + "score": 0.9, + "content": "W x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 315, + 440, + 331 + ], + "score": 1.0, + "content": "with strong", + "type": "text" + }, + { + "bbox": [ + 440, + 317, + 453, + 327 + ], + "score": 0.88, + "content": "i ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 315, + 507, + 331 + ], + "score": 1.0, + "content": "feature, and", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 326, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 104, + 326, + 438, + 341 + ], + "score": 1.0, + "content": "then relaxes toward zero until the data present the network once again with strong", + "type": "text" + }, + { + "bbox": [ + 439, + 328, + 451, + 339 + ], + "score": 0.86, + "content": "i ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 326, + 506, + 341 + ], + "score": 1.0, + "content": "feature. The", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 375, + 352 + ], + "score": 1.0, + "content": "strength of the activation and the decay rate are controlled by the", + "type": "text" + }, + { + "bbox": [ + 375, + 339, + 388, + 349 + ], + "score": 0.88, + "content": "i ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "component of the input and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 350, + 320, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 320, + 363 + ], + "score": 1.0, + "content": "forget gates. The proof of Lemma 1 is elementary —", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 293, + 507, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 365, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 173, + 378 + ], + "score": 1.0, + "content": "Proof of Lemma", + "type": "text" + }, + { + "bbox": [ + 174, + 366, + 179, + 375 + ], + "score": 0.35, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 365, + 416, + 378 + ], + "score": 1.0, + "content": ". Using the non-expansivity of the hyperbolic tangent, i.e.", + "type": "text" + }, + { + "bbox": [ + 417, + 365, + 484, + 378 + ], + "score": 0.92, + "content": "| \\operatorname { t a n h } ( x ) | \\leq | x |", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 365, + 505, + 378 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 376, + 276, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 276, + 390 + ], + "score": 1.0, + "content": "the triangle inequality, we obtain from (1)", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 365, + 505, + 390 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 391, + 405, + 411 + ], + "lines": [ + { + "bbox": [ + 206, + 391, + 405, + 411 + ], + "spans": [ + { + "bbox": [ + 206, + 391, + 405, + 411 + ], + "score": 0.91, + "content": "| h _ { t } ( i ) | \\leq \\Theta | h _ { t - 1 } ( i ) | + H \\operatorname* { m a x } _ { T \\leq t \\leq T + k } | ( W x _ { t } ) ( i ) |", + "type": "interline_equation", + "image_path": "4025c9f5991f725da7cef6eb16c9ccf303e66840580cdd19daea6eac91005de3.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 206, + 391, + 405, + 411 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 415, + 505, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 147, + 429 + ], + "score": 1.0, + "content": "whenever", + "type": "text" + }, + { + "bbox": [ + 147, + 417, + 152, + 425 + ], + "score": 0.8, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 415, + 222, + 429 + ], + "score": 1.0, + "content": "is in the interval", + "type": "text" + }, + { + "bbox": [ + 222, + 415, + 264, + 428 + ], + "score": 0.93, + "content": "[ T , T + k ]", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 415, + 505, + 429 + ], + "score": 1.0, + "content": ". Iterating this inequality and summing the geometric series", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 426, + 150, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 150, + 440 + ], + "score": 1.0, + "content": "then gives", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 415, + 505, + 440 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 176, + 436, + 435, + 465 + ], + "lines": [ + { + "bbox": [ + 176, + 436, + 435, + 465 + ], + "spans": [ + { + "bbox": [ + 176, + 436, + 435, + 465 + ], + "score": 0.94, + "content": "| h _ { T + k } ( i ) | \\leq \\Theta ^ { k } | h _ { T } ( i ) | + \\left( \\frac { 1 - \\Theta ^ { k } } { 1 - \\Theta } \\right) \\ H \\operatorname* { m a x } _ { T \\leq t \\leq T + k } | ( W x _ { t } ) ( i ) |", + "type": "interline_equation", + "image_path": "671ef617f269e8fcb86cf3420c3b8733f20e3386c12059eecf1c691f35285404.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 176, + 436, + 435, + 445.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 176, + 445.6666666666667, + 435, + 455.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 176, + 455.33333333333337, + 435, + 465.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 235, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 463, + 236, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 236, + 478 + ], + "score": 1.0, + "content": "from which we easily conclude.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 463, + 236, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 488, + 459, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 459, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 459, + 503 + ], + "score": 1.0, + "content": "We now turn toward the analysis of the long-term behavior of the the dynamical system", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 487, + 459, + 503 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 504, + 412, + 518 + ], + "lines": [ + { + "bbox": [ + 198, + 504, + 412, + 518 + ], + "spans": [ + { + "bbox": [ + 198, + 504, + 412, + 518 + ], + "score": 0.89, + "content": "\\mathfrak { u } _ { t } = h _ { t } , \\qquad \\mathfrak { u } \\mapsto \\Phi ( \\mathfrak { u } ) : = \\sigma \\left( U _ { \\theta } \\mathfrak { u } + b _ { \\theta } \\right) \\odot \\operatorname { t a n h } ( \\mathfrak { u } ) .", + "type": "interline_equation", + "image_path": "49b2dc2b20f33d3389fd05b9ab29c46e9b653d915e01350b47e0d8be0ed3fc46.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 198, + 504, + 412, + 518 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 506, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "induced by a CFN. The following lemma shows that the only attractor of this dynamical system is", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 532, + 164, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 164, + 545 + ], + "score": 1.0, + "content": "the zero state.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 520, + 505, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 546, + 504, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 274, + 560 + ], + "score": 1.0, + "content": "Lemma 2. Starting from any initial state", + "type": "text" + }, + { + "bbox": [ + 275, + 549, + 285, + 558 + ], + "score": 0.8, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 546, + 346, + 560 + ], + "score": 1.0, + "content": ", the trajectory", + "type": "text" + }, + { + "bbox": [ + 346, + 547, + 379, + 559 + ], + "score": 0.92, + "content": "\\mathcal { O } ^ { + } ( \\mathfrak { u } _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "will eventually converge to the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 557, + 397, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 182, + 572 + ], + "score": 1.0, + "content": "zero state. That is,", + "type": "text" + }, + { + "bbox": [ + 182, + 559, + 252, + 570 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { t + \\infty } \\mathfrak { u } _ { t } = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 557, + 383, + 572 + ], + "score": 1.0, + "content": "regardless of the the initial state", + "type": "text" + }, + { + "bbox": [ + 383, + 560, + 394, + 569 + ], + "score": 0.8, + "content": "\\mathfrak { u } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 557, + 397, + 572 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 546, + 505, + 572 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 576, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 225, + 590 + ], + "score": 1.0, + "content": "Proof. From the definition of", + "type": "text" + }, + { + "bbox": [ + 226, + 577, + 234, + 587 + ], + "score": 0.84, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 576, + 413, + 590 + ], + "score": 1.0, + "content": "we clearly have that the sequence defined by", + "type": "text" + }, + { + "bbox": [ + 413, + 577, + 470, + 588 + ], + "score": 0.94, + "content": "\\mathfrak { u } _ { t + 1 } = \\Phi ( \\mathfrak { u } _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 576, + 506, + 590 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 586, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 178, + 599 + ], + "score": 0.91, + "content": "- 1 < \\mathfrak { u } _ { t } ( i ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 586, + 208, + 601 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 208, + 588, + 213, + 597 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 586, + 246, + 601 + ], + "score": 1.0, + "content": "and all", + "type": "text" + }, + { + "bbox": [ + 246, + 588, + 251, + 597 + ], + "score": 0.61, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 586, + 338, + 601 + ], + "score": 1.0, + "content": ". Since the sequence", + "type": "text" + }, + { + "bbox": [ + 338, + 589, + 348, + 599 + ], + "score": 0.85, + "content": "{ \\mathfrak { u } } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 586, + 477, + 601 + ], + "score": 1.0, + "content": "is bounded, so is the sequence", + "type": "text" + }, + { + "bbox": [ + 478, + 589, + 505, + 599 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { t } : =", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 597, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 148, + 610 + ], + "score": 0.9, + "content": "U _ { \\theta } \\mathfrak { u } _ { t } + b _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 597, + 259, + 612 + ], + "score": 1.0, + "content": ". That is there exists a finite", + "type": "text" + }, + { + "bbox": [ + 259, + 599, + 287, + 609 + ], + "score": 0.91, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 597, + 326, + 612 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 327, + 599, + 420, + 610 + ], + "score": 0.91, + "content": "( U _ { \\theta } \\mathfrak { u } _ { t } ) ( i ) + b _ { \\theta } ( i ) < C", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 597, + 447, + 612 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 447, + 600, + 452, + 609 + ], + "score": 0.78, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 597, + 469, + 612 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 600, + 474, + 609 + ], + "score": 0.67, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 597, + 506, + 612 + ], + "score": 1.0, + "content": ". Using", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 504, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 371, + 622 + ], + "score": 1.0, + "content": "the non-expansivity of the hyperbolic tangent, we then obtain that", + "type": "text" + }, + { + "bbox": [ + 372, + 610, + 468, + 622 + ], + "score": 0.92, + "content": "| \\mathfrak { u } _ { t } ( i ) | \\leq \\sigma ( C ) | \\mathfrak { u } _ { t - 1 } ( i ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 609, + 498, + 622 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 499, + 611, + 504, + 619 + ], + "score": 0.79, + "content": "t", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 620, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 136, + 632 + ], + "score": 1.0, + "content": "and all", + "type": "text" + }, + { + "bbox": [ + 136, + 621, + 141, + 630 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 621, + 257, + 632 + ], + "score": 1.0, + "content": ". We conclude by noting that", + "type": "text" + }, + { + "bbox": [ + 257, + 620, + 316, + 632 + ], + "score": 0.93, + "content": "0 < \\sigma ( C ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 621, + 320, + 632 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 621, + 506, + 632 + ], + "score": 0.99, + "content": "□", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 576, + 506, + 632 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 643, + 502, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 642, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 658 + ], + "score": 1.0, + "content": "Lemma 2 remains true for a multi-layer CFN, that is, a CFN in which the first layer is defined by (1)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 655, + 318, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 211, + 668 + ], + "score": 1.0, + "content": "and the subsequent layers", + "type": "text" + }, + { + "bbox": [ + 211, + 655, + 255, + 666 + ], + "score": 0.92, + "content": "2 \\le \\ell \\le L", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 655, + 318, + 668 + ], + "score": 1.0, + "content": "are defined by:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 642, + 505, + 668 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 670, + 416, + 687 + ], + "lines": [ + { + "bbox": [ + 194, + 670, + 416, + 687 + ], + "spans": [ + { + "bbox": [ + 194, + 670, + 416, + 687 + ], + "score": 0.9, + "content": "\\begin{array} { r } { h _ { t } ^ { ( \\ell ) } = \\theta _ { t } ^ { ( \\ell ) } \\odot \\operatorname { t a n h } ( h _ { t - 1 } ^ { ( \\ell ) } ) + \\eta _ { t } ^ { ( \\ell ) } \\odot \\operatorname { t a n h } ( W ^ { ( \\ell ) } h _ { t } ^ { ( \\ell - 1 ) } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "46273fa3fbb15bd25def44caf75adf177f0dcee21b22cbdf8fbdff07a14b380a.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 194, + 670, + 416, + 687 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 690, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 159, + 703 + ], + "score": 1.0, + "content": "Assume that", + "type": "text" + }, + { + "bbox": [ + 159, + 691, + 200, + 702 + ], + "score": 0.92, + "content": "W x _ { t } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 690, + 228, + 703 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 229, + 691, + 255, + 701 + ], + "score": 0.89, + "content": "t > T", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 690, + 506, + 703 + ], + "score": 1.0, + "content": ", then an extension of the arguments contained in the proof of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 702, + 254, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 254, + 713 + ], + "score": 1.0, + "content": "the two previous lemmas shows that", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41.5, + "bbox_fs": [ + 106, + 690, + 506, + 713 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 248, + 716, + 363, + 734 + ], + "lines": [ + { + "bbox": [ + 248, + 716, + 363, + 734 + ], + "spans": [ + { + "bbox": [ + 248, + 716, + 363, + 734 + ], + "score": 0.93, + "content": "| h _ { T + k } ^ { ( \\ell ) } | \\leq C ( 1 + k ) ^ { ( \\ell - 1 ) } \\Theta ^ { k }", + "type": "interline_equation", + "image_path": "d1bfa2fe06e723a98455c86df8864837d70afd6add0a43e8fb8ac07787f8178c.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 248, + 716, + 363, + 734 + ], + "spans": [], + "index": 43 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 131 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 504, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 133, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 83, + 181, + 93 + ], + "score": 0.91, + "content": "0 < \\Theta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 82, + 434, + 95 + ], + "score": 1.0, + "content": "is the maximal values for the input gates involved in layer 1 to", + "type": "text" + }, + { + "bbox": [ + 435, + 83, + 441, + 92 + ], + "score": 0.75, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 82, + 504, + 95 + ], + "score": 1.0, + "content": "of the network,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 123, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 94, + 153, + 105 + ], + "score": 0.9, + "content": "C > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 93, + 345, + 108 + ], + "score": 1.0, + "content": "is some constant depending only on the norms", + "type": "text" + }, + { + "bbox": [ + 345, + 93, + 386, + 107 + ], + "score": 0.94, + "content": "\\| W ^ { ( j ) } \\| _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 93, + 506, + 108 + ], + "score": 1.0, + "content": "of the matrices and the sizes", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 102, + 509, + 125 + ], + "spans": [ + { + "bbox": [ + 107, + 106, + 129, + 121 + ], + "score": 0.92, + "content": "| h _ { T } ^ { ( j ) } |", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 102, + 287, + 125 + ], + "score": 1.0, + "content": "of the initial conditions at all previous", + "type": "text" + }, + { + "bbox": [ + 287, + 108, + 331, + 120 + ], + "score": 0.92, + "content": "1 \\le j \\le \\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 102, + 509, + 125 + ], + "score": 1.0, + "content": "levels. Estimate (11) shows that Lemma 2", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 276, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 276, + 132 + ], + "score": 1.0, + "content": "remains true for multi-layer architectures.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "image", + "bbox": [ + 113, + 152, + 489, + 297 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 152, + 489, + 297 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 152, + 489, + 297 + ], + "spans": [ + { + "bbox": [ + 113, + 152, + 489, + 297 + ], + "score": 0.97, + "type": "image", + "image_path": "4b9915eb4051f66eba1c7295abb31996ba44df45291410f4220266ec7b001ad0.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 113, + 152, + 489, + 200.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 113, + 200.33333333333334, + 489, + 248.66666666666669 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 113, + 248.66666666666669, + 489, + 297.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 309, + 505, + 354 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 310, + 504, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 396, + 322 + ], + "score": 1.0, + "content": "Figure 4: A 2-layer, 224-unit CFN trained on Penn Treebank. All inputs", + "type": "text" + }, + { + "bbox": [ + 396, + 312, + 407, + 321 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 310, + 462, + 322 + ], + "score": 1.0, + "content": "are zero after", + "type": "text" + }, + { + "bbox": [ + 463, + 310, + 501, + 321 + ], + "score": 0.87, + "content": "t = 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 310, + 504, + 322 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "i.e. the time-point indicated by the dashed line. At left: plot of the 10 “slowest” units of the first", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "layer. At right: plot of the 10 slowest units of the second layer. The second layer retains information", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 341, + 236, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 236, + 356 + ], + "score": 1.0, + "content": "much longer than the first layer.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + } + ], + "index": 6.75 + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 317, + 376 + ], + "score": 1.0, + "content": "Inequality (11) shows that higher levels (i.e. larger", + "type": "text" + }, + { + "bbox": [ + 317, + 366, + 323, + 375 + ], + "score": 0.54, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 365, + 504, + 376 + ], + "score": 1.0, + "content": ") decay more slowly, and remain non-trivial,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 235, + 388 + ], + "score": 1.0, + "content": "while earlier levels (i.e. smaller", + "type": "text" + }, + { + "bbox": [ + 236, + 376, + 241, + 386 + ], + "score": 0.44, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 376, + 505, + 388 + ], + "score": 1.0, + "content": ") decay more quickly. We illustrate this behavior computationally", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 387, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 398 + ], + "score": 1.0, + "content": "with a simple experiment. We take a 2-layer, 224-unit CFN network trained on Penn Treebank and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 398, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 325, + 409 + ], + "score": 1.0, + "content": "feed it the following input data: The first 1000 inputs", + "type": "text" + }, + { + "bbox": [ + 326, + 399, + 336, + 408 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 398, + 506, + 409 + ], + "score": 1.0, + "content": "are the first 1000 words of the test set of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 361, + 421 + ], + "score": 1.0, + "content": "Penn Treebank; All subsequent inputs are zero. In other words,", + "type": "text" + }, + { + "bbox": [ + 361, + 409, + 390, + 420 + ], + "score": 0.91, + "content": "x _ { t } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 408, + 399, + 421 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 400, + 408, + 438, + 419 + ], + "score": 0.89, + "content": "t > 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 408, + 506, + 421 + ], + "score": 1.0, + "content": ". For each of the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 367, + 432 + ], + "score": 1.0, + "content": "two layers we then select the 10 units that decay the slowest after", + "type": "text" + }, + { + "bbox": [ + 368, + 420, + 406, + 430 + ], + "score": 0.89, + "content": "t > 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "and plot them on Figure", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "4. The figure illustrates that the second layer retains information for much longer than the first layer.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 104, + 440, + 413, + 454 + ], + "score": 1.0, + "content": "To quantify this observation we define the relaxation time (or half-life) of the", + "type": "text" + }, + { + "bbox": [ + 413, + 441, + 426, + 451 + ], + "score": 0.88, + "content": "\\bar { i } ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 440, + 506, + 454 + ], + "score": 1.0, + "content": "unit as the smallest", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 452, + 155, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 115, + 462 + ], + "score": 0.75, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 452, + 155, + 464 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 462, + 365, + 476 + ], + "lines": [ + { + "bbox": [ + 245, + 462, + 365, + 476 + ], + "spans": [ + { + "bbox": [ + 245, + 462, + 365, + 476 + ], + "score": 0.9, + "content": "| h _ { 1 0 0 0 + T } ( i ) | < 0 . 5 | h _ { 1 0 0 0 } ( i ) | .", + "type": "interline_equation", + "image_path": "87d6ad34cf0f111bcd0c08770d92bd3556f2ef20ee2aaed2af55ebc835643e05.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 245, + 462, + 365, + 476 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 477, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "Using this definition yields average relaxation times of 2.2 time steps for the first layer and 23.2", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "time steps for the second layer. The first layer has a standard deviations of approximately 5 steps", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "while the second layer has a standard deviation of approximately 75 time steps. A more fine-grained", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "analysis reveals that some units in the second layer have relaxation times of several hundred steps.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "For instance, if instead of averaging the relaxation times over the whole layer we average them over", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 204, + 544 + ], + "score": 1.0, + "content": "the top quartile (i.e. the", + "type": "text" + }, + { + "bbox": [ + 204, + 532, + 224, + 542 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "units that decay the most slowly) we get 4.8 time steps and 85.6 time", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "steps for the first and second layers, respectively. In other words, by restricting attention to long-term", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 552, + 447, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 447, + 566 + ], + "score": 1.0, + "content": "units the difference between the first and second layers becomes much more striking.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 503, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "Overall, this experiment conforms with the analysis (11), and indicates that adding a third or fourth", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 580, + 501, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 501, + 595 + ], + "score": 1.0, + "content": "layer would potentially allow a multi-layer CFN architecture to retain information for even longer.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 108, + 608, + 200, + 621 + ], + "lines": [ + { + "bbox": [ + 104, + 606, + 202, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 202, + 623 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "In this section we show that despite its simplicity, the CFN network achieves performance compa-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "rable to the much more complex LSTM network on the word level language modeling task. We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "use two datasets for these experiments, namely the Penn Treebank corpus (Marcus et al., 1993)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 666, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 678 + ], + "score": 1.0, + "content": "and the Text8 corpus (Mikolov et al., 2014). We consider both one-layer and two-layer CFNs and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "LSTMs for our experiments. We train both CFN and LSTM networks in a similar fashion and al-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "ways compare models that use the same number of parameters. We compare their performance with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 699, + 504, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 710 + ], + "score": 1.0, + "content": "and without dropout, and show that in both cases they obtain similar results. We also provide results", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "published in Mikolov et al. (2014), Jozefowicz et al. (2015) and Sukhbaatar et al. 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Estimate (11) shows that Lemma 2", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 119, + 276, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 276, + 132 + ], + "score": 1.0, + "content": "remains true for multi-layer architectures.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 104, + 82, + 509, + 132 + ] + }, + { + "type": "image", + "bbox": [ + 113, + 152, + 489, + 297 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 113, + 152, + 489, + 297 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 113, + 152, + 489, + 297 + ], + "spans": [ + { + "bbox": [ + 113, + 152, + 489, + 297 + ], + "score": 0.97, + "type": "image", + "image_path": "4b9915eb4051f66eba1c7295abb31996ba44df45291410f4220266ec7b001ad0.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 113, + 152, + 489, + 200.33333333333334 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 113, + 200.33333333333334, + 489, + 248.66666666666669 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 113, + 248.66666666666669, + 489, + 297.0 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 309, + 505, + 354 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 310, + 504, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 396, + 322 + ], + "score": 1.0, + "content": "Figure 4: A 2-layer, 224-unit CFN trained on Penn Treebank. All inputs", + "type": "text" + }, + { + "bbox": [ + 396, + 312, + 407, + 321 + ], + "score": 0.85, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 310, + 462, + 322 + ], + "score": 1.0, + "content": "are zero after", + "type": "text" + }, + { + "bbox": [ + 463, + 310, + 501, + 321 + ], + "score": 0.87, + "content": "t = 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 310, + 504, + 322 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 333 + ], + "score": 1.0, + "content": "i.e. the time-point indicated by the dashed line. At left: plot of the 10 “slowest” units of the first", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "layer. At right: plot of the 10 slowest units of the second layer. The second layer retains information", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 341, + 236, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 236, + 356 + ], + "score": 1.0, + "content": "much longer than the first layer.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + } + ], + "index": 6.75 + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 505, + 463 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 317, + 376 + ], + "score": 1.0, + "content": "Inequality (11) shows that higher levels (i.e. larger", + "type": "text" + }, + { + "bbox": [ + 317, + 366, + 323, + 375 + ], + "score": 0.54, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 365, + 504, + 376 + ], + "score": 1.0, + "content": ") decay more slowly, and remain non-trivial,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 235, + 388 + ], + "score": 1.0, + "content": "while earlier levels (i.e. smaller", + "type": "text" + }, + { + "bbox": [ + 236, + 376, + 241, + 386 + ], + "score": 0.44, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 376, + 505, + 388 + ], + "score": 1.0, + "content": ") decay more quickly. We illustrate this behavior computationally", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 387, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 505, + 398 + ], + "score": 1.0, + "content": "with a simple experiment. We take a 2-layer, 224-unit CFN network trained on Penn Treebank and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 398, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 325, + 409 + ], + "score": 1.0, + "content": "feed it the following input data: The first 1000 inputs", + "type": "text" + }, + { + "bbox": [ + 326, + 399, + 336, + 408 + ], + "score": 0.86, + "content": "x _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 398, + 506, + 409 + ], + "score": 1.0, + "content": "are the first 1000 words of the test set of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 361, + 421 + ], + "score": 1.0, + "content": "Penn Treebank; All subsequent inputs are zero. In other words,", + "type": "text" + }, + { + "bbox": [ + 361, + 409, + 390, + 420 + ], + "score": 0.91, + "content": "x _ { t } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 408, + 399, + 421 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 400, + 408, + 438, + 419 + ], + "score": 0.89, + "content": "t > 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 408, + 506, + 421 + ], + "score": 1.0, + "content": ". For each of the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 367, + 432 + ], + "score": 1.0, + "content": "two layers we then select the 10 units that decay the slowest after", + "type": "text" + }, + { + "bbox": [ + 368, + 420, + 406, + 430 + ], + "score": 0.89, + "content": "t > 1 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "and plot them on Figure", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 443 + ], + "score": 1.0, + "content": "4. The figure illustrates that the second layer retains information for much longer than the first layer.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 440, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 104, + 440, + 413, + 454 + ], + "score": 1.0, + "content": "To quantify this observation we define the relaxation time (or half-life) of the", + "type": "text" + }, + { + "bbox": [ + 413, + 441, + 426, + 451 + ], + "score": 0.88, + "content": "\\bar { i } ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 440, + 506, + 454 + ], + "score": 1.0, + "content": "unit as the smallest", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 452, + 155, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 115, + 462 + ], + "score": 0.75, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 452, + 155, + 464 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 365, + 506, + 464 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 462, + 365, + 476 + ], + "lines": [ + { + "bbox": [ + 245, + 462, + 365, + 476 + ], + "spans": [ + { + "bbox": [ + 245, + 462, + 365, + 476 + ], + "score": 0.9, + "content": "| h _ { 1 0 0 0 + T } ( i ) | < 0 . 5 | h _ { 1 0 0 0 } ( i ) | .", + "type": "interline_equation", + "image_path": "87d6ad34cf0f111bcd0c08770d92bd3556f2ef20ee2aaed2af55ebc835643e05.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 245, + 462, + 365, + 476 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 477, + 505, + 565 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "Using this definition yields average relaxation times of 2.2 time steps for the first layer and 23.2", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "time steps for the second layer. The first layer has a standard deviations of approximately 5 steps", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "while the second layer has a standard deviation of approximately 75 time steps. A more fine-grained", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 506, + 523 + ], + "score": 1.0, + "content": "analysis reveals that some units in the second layer have relaxation times of several hundred steps.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "For instance, if instead of averaging the relaxation times over the whole layer we average them over", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 204, + 544 + ], + "score": 1.0, + "content": "the top quartile (i.e. the", + "type": "text" + }, + { + "bbox": [ + 204, + 532, + 224, + 542 + ], + "score": 0.87, + "content": "2 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "units that decay the most slowly) we get 4.8 time steps and 85.6 time", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "steps for the first and second layers, respectively. In other words, by restricting attention to long-term", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 552, + 447, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 447, + 566 + ], + "score": 1.0, + "content": "units the difference between the first and second layers becomes much more striking.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 477, + 506, + 566 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 570, + 503, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "Overall, this experiment conforms with the analysis (11), and indicates that adding a third or fourth", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 580, + 501, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 501, + 595 + ], + "score": 1.0, + "content": "layer would potentially allow a multi-layer CFN architecture to retain information for even longer.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 570, + 505, + 595 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 608, + 200, + 621 + ], + "lines": [ + { + "bbox": [ + 104, + 606, + 202, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 202, + 623 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "In this section we show that despite its simplicity, the CFN network achieves performance compa-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "rable to the much more complex LSTM network on the word level language modeling task. We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "use two datasets for these experiments, namely the Penn Treebank corpus (Marcus et al., 1993)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 666, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 678 + ], + "score": 1.0, + "content": "and the Text8 corpus (Mikolov et al., 2014). We consider both one-layer and two-layer CFNs and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 504, + 689 + ], + "score": 1.0, + "content": "LSTMs for our experiments. We train both CFN and LSTM networks in a similar fashion and al-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "ways compare models that use the same number of parameters. We compare their performance with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 699, + 504, + 710 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 504, + 710 + ], + "score": 1.0, + "content": "and without dropout, and show that in both cases they obtain similar results. We also provide results", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "published in Mikolov et al. (2014), Jozefowicz et al. (2015) and Sukhbaatar et al. (2015) for the sake", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 721, + 168, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 168, + 733 + ], + "score": 1.0, + "content": "of comparison.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 632, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 140, + 108, + 469, + 171 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 189, + 89, + 421, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 189, + 88, + 421, + 102 + ], + "spans": [ + { + "bbox": [ + 189, + 88, + 421, + 102 + ], + "score": 1.0, + "content": "Table 1: Experiments on Penn Treebank without dropout.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 140, + 108, + 469, + 171 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 108, + 469, + 171 + ], + "spans": [ + { + "bbox": [ + 140, + 108, + 469, + 171 + ], + "score": 0.983, + "html": "
ModelSizeTrainingVal. perp.Test perp.
VanillaRNN5M parametersJozefowicz et al. (2015)1122.9
GRU5M parametersJozefowicz et al. (2015)=108.2
LSTM5M parametersJozefowicz et al. (2015)-109.7
LSTM (1 layer)5M parametersTrained by us108.4105.1
CFN (2 layers)5M parametersTrained by us109.3106.3
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ModelSizeTrainingPerp.on development set
Vanilla RNN500 hidden unitsMikolov et al. (2014)184
SCRN500 hidden unitsMikolov et al. (2014)161
LSTM500 hidden unitsMikolov et al. (2014)156
MemN2N500 hidden unitsSukhbaatar et al. (2015)147
LSTM(2 layers)46.4M parametersTrained by us139.9
CFN (2 layers)46.4M parametersTrained by us142.0
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This initialization scheme favors", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "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 2017", + "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": [ + 140, + 108, + 469, + 171 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 189, + 89, + 421, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 189, + 88, + 421, + 102 + ], + "spans": [ + { + "bbox": [ + 189, + 88, + 421, + 102 + ], + "score": 1.0, + "content": "Table 1: Experiments on Penn Treebank without dropout.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 140, + 108, + 469, + 171 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 140, + 108, + 469, + 171 + ], + "spans": [ + { + "bbox": [ + 140, + 108, + 469, + 171 + ], + "score": 0.983, + "html": "
ModelSizeTrainingVal. perp.Test perp.
VanillaRNN5M parametersJozefowicz et al. (2015)1122.9
GRU5M parametersJozefowicz et al. (2015)=108.2
LSTM5M parametersJozefowicz et al. (2015)-109.7
LSTM (1 layer)5M parametersTrained by us108.4105.1
CFN (2 layers)5M parametersTrained by us109.3106.3
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ModelSizeTrainingPerp.on development set
Vanilla RNN500 hidden unitsMikolov et al. (2014)184
SCRN500 hidden unitsMikolov et al. (2014)161
LSTM500 hidden unitsMikolov et al. (2014)156
MemN2N500 hidden unitsSukhbaatar et al. (2015)147
LSTM(2 layers)46.4M parametersTrained by us139.9
CFN (2 layers)46.4M parametersTrained by us142.0
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We initialize the weights of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 487, + 722 + ], + "score": 1.0, + "content": "the LSTM in exactly the same way; the bias for the forget and input gate are initialized to 1 and", + "type": "text" + }, + { + "bbox": [ + 488, + 710, + 501, + 720 + ], + "score": 0.76, + "content": "- 1", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 719, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 315, + 732 + ], + "score": 1.0, + "content": "and all the other weights are initialized uniformly in", + "type": "text" + }, + { + "bbox": [ + 315, + 721, + 369, + 732 + ], + "score": 0.73, + "content": "[ - 0 . 0 7 , 0 . 0 7 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 719, + 505, + 732 + ], + "score": 1.0, + "content": ". This initialization scheme favors", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 213, + 505, + 225 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 505, + 225 + ], + "score": 1.0, + "content": "the flow of information in the horizontal direction. The importance of a careful initialization of the", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "forget gate was pointed out in Gers et al. (2000) and Jozefowicz et al. (2015). Finally, we initialize", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 236, + 271, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 271, + 247 + ], + "score": 1.0, + "content": "all hidden states to zero for both models.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 676, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 136, + 107, + 473, + 192 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 196, + 89, + 414, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 195, + 88, + 415, + 102 + ], + "spans": [ + { + "bbox": [ + 195, + 88, + 415, + 102 + ], + "score": 1.0, + "content": "Table 3: Experiments on Penn Treebank with dropout.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 136, + 107, + 473, + 192 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 136, + 107, + 473, + 192 + ], + "spans": [ + { + "bbox": [ + 136, + 107, + 473, + 192 + ], + "score": 0.986, + "html": "
ModelSizeTrainingVal. perp.Test perp.
Vanilla RNN GRU LSTM20M parametersJozefowicz et al. (2015)103.097.7
20M parametersJozefowicz et al. (2015)95.591.7
20M parametersJozefowicz et al. (2015)83.378.8
LSTM (2 layers) CFN (2 layers)20M parametersTrained by us78.474.3
20M parametersTrained by us79.774.9
50M parametersTrained by us75.971.8
LSTM(2 layers) CFN (2 layers)50M parametersTrained by us77.072.2
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The importance of a careful initialization of the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "forget gate was pointed out in Gers et al. (2000) and Jozefowicz et al. (2015). Finally, we initialize", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 236, + 271, + 247 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 271, + 247 + ], + "score": 1.0, + "content": "all hidden states to zero for both models.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 252, + 505, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 264 + ], + "score": 1.0, + "content": "Dataset Construction. 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Tables 1 and 2 provide a comparison of various recurrent network", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "architectures without dropout evaluated on the Penn Treebank corpus and the Text8 corpus. The last", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "two rows of each table provide results for LSTM and CFN networks trained and initialized in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "manner described above. We have tried both one and two layer architectures, and reported only the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 369, + 477, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 477, + 380 + ], + "score": 1.0, + "content": "best result. The learning rate schedules used for each network are described in the appendix.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "We also report results published in Jozefowicz et al. (2015) were a vanilla RNN, a GRU and an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "score": 1.0, + "content": "LSTM network were trained on Penn Treebank, each of them having 5 million parameters (only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "the test perplexity was reported). Finally we report results published in Mikolov et al. (2014) and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 416, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 431 + ], + "score": 1.0, + "content": "Sukhbaatar et al. (2015) where various networks are trained on Text8. Of these four networks, only", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "the LSTM network from Mikolov et al. (2014) has the same number of parameters than the CFN", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "and LSTM networks we trained (46.4M parameters). The vanilla RNN, Structurally Constrained", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "Recurrent Network (SCRN) and End-To-End Memory Network (MemN2N) all have 500 units, but", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 144, + 474 + ], + "score": 1.0, + "content": "less than", + "type": "text" + }, + { + "bbox": [ + 144, + 462, + 172, + 472 + ], + "score": 0.34, + "content": "4 6 . 4 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "parameters. We nonetheless indicate their performance in Table 2 to provide some", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 140, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 140, + 484 + ], + "score": 1.0, + "content": "context.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "Experiments with Dropout. Table 3 provides a comparison of various recurrent network archi-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "score": 1.0, + "content": "tectures with dropout evaluated on the Penn Treebank corpus. The first three rows report results", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "published in (Jozefowicz et al., 2015) and the last four rows provide results for LSTM and CFN", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 459, + 535 + ], + "score": 1.0, + "content": "networks trained and initialized with the strategy previously described. 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This indicates that it might be possible, in general, to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "build RNNs that perform well while avoiding the intricate, uninterpretable and potentially chaotic", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "dynamics that can occur in LSTMs and GRUs. Of course, it remains to be seen if dynamically", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "simple RNNs such as the proposed CFN can perform well on a wide variety of tasks, potentially", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "requiring longer term dependencies than the one needed for word level language modeling. The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "experiments presented in Section 2 indicate a plausible path forward — activations in the higher", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "layers of a multi-layer CFN decay at a slower rate than the activations in the lower layers. 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ModelSizeTrainingVal. perp.Test perp.
Vanilla RNN GRU LSTM20M parametersJozefowicz et al. (2015)103.097.7
20M parametersJozefowicz et al. (2015)95.591.7
20M parametersJozefowicz et al. (2015)83.378.8
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20M parametersTrained by us79.774.9
50M parametersTrained by us75.971.8
LSTM(2 layers) CFN (2 layers)50M parametersTrained by us77.072.2
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Tables 1 and 2 provide a comparison of various recurrent network", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "architectures without dropout evaluated on the Penn Treebank corpus and the Text8 corpus. The last", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "two rows of each table provide results for LSTM and CFN networks trained and initialized in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "manner described above. We have tried both one and two layer architectures, and reported only the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 369, + 477, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 477, + 380 + ], + "score": 1.0, + "content": "best result. The learning rate schedules used for each network are described in the appendix.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 324, + 505, + 380 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 484 + ], + "lines": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "We also report results published in Jozefowicz et al. (2015) were a vanilla RNN, a GRU and an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "score": 1.0, + "content": "LSTM network were trained on Penn Treebank, each of them having 5 million parameters (only", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "the test perplexity was reported). Finally we report results published in Mikolov et al. (2014) and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 416, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 431 + ], + "score": 1.0, + "content": "Sukhbaatar et al. (2015) where various networks are trained on Text8. Of these four networks, only", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 505, + 442 + ], + "score": 1.0, + "content": "the LSTM network from Mikolov et al. (2014) has the same number of parameters than the CFN", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "and LSTM networks we trained (46.4M parameters). The vanilla RNN, Structurally Constrained", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "Recurrent Network (SCRN) and End-To-End Memory Network (MemN2N) all have 500 units, but", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 144, + 474 + ], + "score": 1.0, + "content": "less than", + "type": "text" + }, + { + "bbox": [ + 144, + 462, + 172, + 472 + ], + "score": 0.34, + "content": "4 6 . 4 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "parameters. We nonetheless indicate their performance in Table 2 to provide some", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 140, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 140, + 484 + ], + "score": 1.0, + "content": "context.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 385, + 506, + 484 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "Experiments with Dropout. Table 3 provides a comparison of various recurrent network archi-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 504, + 512 + ], + "score": 1.0, + "content": "tectures with dropout evaluated on the Penn Treebank corpus. The first three rows report results", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "published in (Jozefowicz et al., 2015) and the last four rows provide results for LSTM and CFN", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 459, + 535 + ], + "score": 1.0, + "content": "networks trained and initialized with the strategy previously described. The dropout rate", + "type": "text" + }, + { + "bbox": [ + 459, + 524, + 466, + 533 + ], + "score": 0.78, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 522, + 483, + 535 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 483, + 524, + 489, + 534 + ], + "score": 0.77, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 381, + 546 + ], + "score": 1.0, + "content": "chosen as follows: For the experiments with 20M parameters, we set", + "type": "text" + }, + { + "bbox": [ + 382, + 533, + 420, + 545 + ], + "score": 0.92, + "content": "p = 5 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 532, + 437, + 546 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 438, + 533, + 475, + 545 + ], + "score": 0.92, + "content": "q = 4 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 532, + 505, + 546 + ], + "score": 1.0, + "content": "for the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 544, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 145, + 557 + ], + "score": 1.0, + "content": "CFN and", + "type": "text" + }, + { + "bbox": [ + 146, + 544, + 184, + 555 + ], + "score": 0.91, + "content": "p = 6 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 544, + 203, + 557 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 203, + 544, + 241, + 556 + ], + "score": 0.92, + "content": "q = 4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 544, + 506, + 557 + ], + "score": 1.0, + "content": "for the LSTM; For the experiments with 50M parameters, we set", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 555, + 424, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 145, + 567 + ], + "score": 0.91, + "content": "p = 6 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 555, + 162, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 163, + 555, + 200, + 567 + ], + "score": 0.91, + "content": "q = 5 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 555, + 268, + 568 + ], + "score": 1.0, + "content": "for the CFN and", + "type": "text" + }, + { + "bbox": [ + 269, + 555, + 307, + 567 + ], + "score": 0.92, + "content": "p = 7 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 555, + 324, + 568 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 325, + 555, + 362, + 567 + ], + "score": 0.92, + "content": "q = \\hat { 5 } 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 555, + 424, + 568 + ], + "score": 1.0, + "content": "for the LSTM.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 489, + 506, + 568 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 585, + 195, + 597 + ], + "lines": [ + { + "bbox": [ + 104, + 583, + 197, + 600 + ], + "spans": [ + { + "bbox": [ + 104, + 583, + 197, + 600 + ], + "score": 1.0, + "content": "4 CONCLUSION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 610, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "Despite its simple dynamics, the CFN obtains results that compare well against LSTM networks", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "and GRUs on word-level language modeling. This indicates that it might be possible, in general, to", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "build RNNs that perform well while avoiding the intricate, uninterpretable and potentially chaotic", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "dynamics that can occur in LSTMs and GRUs. Of course, it remains to be seen if dynamically", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "simple RNNs such as the proposed CFN can perform well on a wide variety of tasks, potentially", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "requiring longer term dependencies than the one needed for word level language modeling. The", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "experiments presented in Section 2 indicate a plausible path forward — activations in the higher", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "layers of a multi-layer CFN decay at a slower rate than the activations in the lower layers. In theory,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "complexity and long-term dependencies can therefore be captured using a more “feed-forward”", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "approach (i.e. stacking layers) rather than relying on the intricate and hard to interpret dynamics of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 191, + 731 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 191, + 731 + ], + "score": 1.0, + "content": "an LSTM or a GRU.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 611, + 506, + 731 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 504, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 504, + 94 + ], + "score": 1.0, + "content": "Overall, the CFN is a simple model and it therefore has the potential of being mathematically well-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "understood. In particular, Section 2 reveals that the dynamics of its hidden states are inherently more", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "interpretable than those of an LSTM. The mathematical analysis here provides a few key insights", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "into the network, in both the presence and absence of input data, but obviously more work is needed", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "before a complete picture can emerge. 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On the difficulty of training recurrent neural", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 116, + 456, + 289, + 467 + ], + "spans": [ + { + "bbox": [ + 116, + 456, + 289, + 467 + ], + "score": 1.0, + "content": "networks. ICML (3), 28:1310–1318, 2013.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 443, + 505, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 503, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 505, + 489 + ], + "score": 1.0, + "content": "Steven H Strogatz. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 115, + 486, + 279, + 498 + ], + "spans": [ + { + "bbox": [ + 115, + 486, + 279, + 498 + ], + "score": 1.0, + "content": "and engineering. 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All trajectories starting", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 213, + 209 + ], + "score": 1.0, + "content": "close to the origin at time", + "type": "text" + }, + { + "bbox": [ + 214, + 197, + 239, + 207 + ], + "score": 0.9, + "content": "t = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 196, + 506, + 209 + ], + "score": 1.0, + "content": "converge toward the depicted attractor. Successive zooms on the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 208, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 505, + 219 + ], + "score": 1.0, + "content": "branch of the attractor reveal its fractal nature. 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We generate these pictures by reproducing the numerical experiments from Henon (1976). ´", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 410, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 410, + 153 + ], + "score": 1.0, + "content": "The discrete dynamical system considered here is the two dimensional map", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 106, + 506, + 153 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 155, + 380, + 169 + ], + "lines": [ + { + "bbox": [ + 230, + 155, + 380, + 169 + ], + "spans": [ + { + "bbox": [ + 230, + 155, + 380, + 169 + ], + "score": 0.91, + "content": "x _ { t + 1 } = y _ { t } + 1 - a x _ { t } ^ { 2 } , \\quad y _ { t + 1 } = b x _ { t } ,", + "type": "interline_equation", + "image_path": "b8685b044468dbf9b9868051363615a3fb6115e2a1e0b6dff75d2f78243d437b.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 230, + 155, + 380, + 169 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 174, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 173, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 200, + 187 + ], + "score": 1.0, + "content": "with parameters set to", + "type": "text" + }, + { + "bbox": [ + 200, + 175, + 236, + 185 + ], + "score": 0.89, + "content": "a = 1 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 173, + 255, + 187 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 255, + 174, + 290, + 185 + ], + "score": 0.89, + "content": "b = 0 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 173, + 506, + 187 + ], + "score": 1.0, + "content": ". 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We also tried a one-layer LSTM with 20M parameters and it led to similar", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "score": 1.0, + "content": "but slightly worse results than the two-layer architecture. For both network, the learning rate was", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 436, + 293 + ], + "score": 1.0, + "content": "divided by 1.1 each time the validation perplexity did not decrease by at least", + "type": "text" + }, + { + "bbox": [ + 437, + 281, + 451, + 291 + ], + "score": 0.88, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 280, + 505, + 293 + ], + "score": 1.0, + "content": ". The initial", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 424, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 232, + 303 + ], + "score": 1.0, + "content": "learning rate were chosen to be", + "type": "text" + }, + { + "bbox": [ + 233, + 292, + 264, + 303 + ], + "score": 0.92, + "content": "\\mathrm { { l r } _ { 0 } } = 7", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 292, + 331, + 303 + ], + "score": 1.0, + "content": "for the CFN and", + "type": "text" + }, + { + "bbox": [ + 332, + 292, + 362, + 303 + ], + "score": 0.91, + "content": "\\mathrm { { I r } _ { 0 } = 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 292, + 424, + 303 + ], + "score": 1.0, + "content": "for the LSTM.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 236, + 506, + 303 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git 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ModelSizeTrainingPerp.on development set
Vanilla RNN500 hidden unitsMikolov et al. (2014)184
SCRN500 hidden unitsMikolov et al. (2014)161
LSTM500 hidden unitsMikolov et al. (2014)156
MemN2N500 hidden unitsSukhbaatar et al. (2015)147
LSTM(2 layers)46.4M parametersTrained by us139.9
CFN (2 layers)46.4M parametersTrained by us142.0
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ModelSizeTrainingVal. perp.Test perp.
VanillaRNN5M parametersJozefowicz et al. (2015)1122.9
GRU5M parametersJozefowicz et al. (2015)=108.2
LSTM5M parametersJozefowicz et al. (2015)-109.7
LSTM (1 layer)5M parametersTrained by us108.4105.1
CFN (2 layers)5M parametersTrained by us109.3106.3
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ModelSizeTrainingVal. perp.Test perp.
Vanilla RNN GRU LSTM20M parametersJozefowicz et al. (2015)103.097.7
20M parametersJozefowicz et al. (2015)95.591.7
20M parametersJozefowicz et al. (2015)83.378.8
LSTM (2 layers) CFN (2 layers)20M parametersTrained by us78.474.3
20M parametersTrained by us79.774.9
50M parametersTrained by us75.971.8
LSTM(2 layers) CFN (2 layers)50M parametersTrained by us77.072.2
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Seward1,2 Günter Klambauer1 + +Martin Heusel1 + +Hubert Ramsauer1 + +Sepp Hochreiter1 + +1LIT AI Lab & Institute of Bioinformatics, Johannes Kepler University Linz, Austria 2Zalando Research, Mühlenstraße 25, 10243 Berlin, Germany {unterthiner,nessler,seward,klambauer,mhe,ramsauer,hochreit}@bioinf.jku.at + +# ABSTRACT + +Generative adversarial networks (GANs) evolved into one of the most successful unsupervised techniques for generating realistic images. Even though it has recently been shown that GAN training converges, GAN models often end up in local Nash equilibria that are associated with mode collapse or otherwise fail to model the target distribution. We introduce Coulomb GANs, which pose the GAN learning problem as a potential field, where generated samples are attracted to training set samples but repel each other. The discriminator learns a potential field while the generator decreases the energy by moving its samples along the vector (force) field determined by the gradient of the potential field. Through decreasing the energy, the GAN model learns to generate samples according to the whole target distribution and does not only cover some of its modes. We prove that Coulomb GANs possess only one Nash equilibrium which is optimal in the sense that the model distribution equals the target distribution. We show the efficacy of Coulomb GANs on LSUN bedrooms, CelebA faces, CIFAR-10 and the Google Billion Word text generation. + +# 1 INTRODUCTION + +Generative adversarial networks (GANs) (Goodfellow et al., 2014) excel at constructing realistic images (Radford et al., 2016; Ledig et al., 2016; Isola et al., 2017; Arjovsky et al., 2017; Berthelot et al., 2017) and text (Gulrajani et al., 2017). In GAN learning, a discriminator network guides the learning of another, generative network. This procedure can be considered as a game between the generator which constructs synthetic data and the discriminator which separates synthetic data from training set data (Goodfellow, 2017). The generator’s goal is to construct data which the discriminator cannot tell apart from training set data. GAN convergence points are local Nash equilibria. At these local Nash equilibria neither the discriminator nor the generator can locally improve its objective. + +Despite their recent successes, GANs have several problems. First (I), until recently it was not clear if in general gradient-based GAN learning could converge to one of the local Nash equilibria (Salimans et al., 2016; Goodfellow, 2014; Goodfellow et al., 2014). It is even possible to construct counterexamples (Goodfellow, 2017). Second (II), GANs suffer from “mode collapsing”, where the model generates samples only in certain regions which are called modes. While these modes contain realistic samples, the variety is low and only a few prototypes are generated. Mode collapsing is less likely if the generator is trained with batch normalization, since the network is bound to create a certain variance among its generated samples within one batch (Radford et al., 2016; Chintala et al., 2016). However batch normalization introduces fluctuations of normalizing constants which can be harmful (Klambauer et al., 2017; Goodfellow, 2017). To avoid mode collapsing without batch normalization, several methods have been proposed (Che et al., 2017; Metz et al., 2016; Salimans et al., + +2016). Third (III), GANs cannot assure that the density of training samples is correctly modeled by the generator. The discriminator only tells the generator whether a region is more likely to contain samples from the training set or synthetic samples. Therefore the discriminator can only distinguish the support of the model distribution from the support of the target distribution. Beyond matching the support of distributions, GANs with proper objectives may learn to locally align model and target densities via averaging over many training examples. On a global scale, however, GANs fail to equalize model and target densities. The discriminator does not inform the generator globally where probability mass is missing. Consequently, standard GANs are not assured to capture the global sample density and are prone to neglect large parts of the target distribution. The next paragraph gives an example of this. Fourth (IV), the discriminator of GANs may forget previous modeling errors of the generator which then may reappear, a property that leads to oscillatory behavior instead of convergence (Goodfellow, 2017). + +Recently, problem (I) was solved by proving that GAN learning does indeed converge when discriminator and generator are learned using a two time-scale learning rule (Heusel et al., 2017). Convergence means that the expected SGD-gradient of both the discriminator objective and the generator objective are zero. Thus, neither the generator nor the discriminator can locally improve, i.e., learning has reached a local Nash equilibrium. However, convergence alone does not guarantee good generative performance. It is possible to converge to sub-optimal solutions which are local Nash equilibria. Mode collapse is a special case of a local Nash equilibrium associated with suboptimal generative performance. For example, assume a two mode real world distribution where one mode contains too few and the other mode too many generator samples. If no real world samples are between these two distinct modes, then the discriminator penalizes to move generated samples outside the modes. Therefore the generated samples cannot be correctly distributed over the modes. Thus, standard GANs cannot capture the global sample density such that the resulting generators are prone to neglect large parts of the real world distribution. A more detailed example is listed in the Appendix in Section A.1. + +In this paper, we introduce a novel GAN model, the Coulomb GAN, which has only one Nash equilibrium. We are later going to show that this Nash equilibrium is optimal, i.e., the model distribution matches the target distribution. We propose Coulomb GANs to avoid the GAN shortcoming (II) to (IV) by using a potential field created by point charges analogously to the electric field in physics. The next section will introduce the idea of learning in a potential field and prove that its only solution is optimal. We will then show how learning the discriminator and generator works in a Coulomb GAN and discuss the assumptions needed for our optimality proof. In Section 3 we will then see that the Coulomb GAN does indeed work well in practice and that the samples it produces have very large variability and appear to capture the original distribution very well. + +Related Work. Several GAN approaches have been suggested for bringing the target and model distributions in alignment using not just local discriminator information: Geometric GANs combine samples via a linear support vector machine which uses the discriminator outputs as samples, therefore they are much more robust to mode collapsing (Lim & Ye, 2017). Energy-Based GANs (Zhao et al., 2017) and their later improvement BEGANs (Berthelot et al., 2017) optimize an energy landscape based on auto-encoders. McGANs match mean and covariance of synthetic and target data, therefore are more suited than standard GANs to approximate the target distribution (Mroueh et al., 2017). In a similar fashion, Generative Moment Matching Networks (Li et al., 2015) and MMD nets (Dziugaite et al., 2015) directly optimize a generator network to match a training distribution by using a loss function based on the maximum mean discrepancy (MMD) criterion (Gretton et al., 2012). These approaches were later expanded to include an MMD criterion with learnable kernels and discriminators (Li et al., 2017). The MMD criterion that these later approaches optimize has a form similar to the energy function that Coulomb GANs optimize (cf. Eq. (33)). However, all MMD approaches end up using either Gaussian or Laplace kernels, which are not guaranteed to find the optimal solution where the model distribution matches the target distribution. In contrast, the Plummer kernel which is employed in this work has been shown to lead to the optimal solution (Hochreiter & Obermayer, 2005). We show that even a simplified version of the Plummer kernel, the low-dimensional Plummer kernel, ensures that gradient descent convergences to the optimal solution as stated by Theorem 1. Furthermore, most MMD GAN approaches use the MMD directly as loss function though the number of possible samples in a mini-batch is limited. Therefore MMD approaches face a sampling problem in high-dimensional spaces. The Coulomb GAN instead learns a discriminator network that gradually improves its approximation of the potential field via learning on many mini-batches. The discriminator network also tracks the slowly changing generator distribution during learning. Most importantly however, our approach is, to the best of our knowledge, the first one for which optimality, i.e., ability to perfectly learn a target distribution, can be proved. + +![](images/1dd35099253bbe238ca744ba58c3e5789a9a64144bbf15b96950a65d14400fb8.jpg) +Figure 1: The vector field of a Coulomb GAN. The basic idea behind the Coulomb GAN: true samples (blue) and generated samples (red) create a potential field (scalar field). Blue samples act as sinks that attract the red samples, which repel each other. The superimposed vector field shows the forces acting on the generator samples to equalize potential differences, and the background color shows the potential at each position. Best viewed in color. + +The use of the Coulomb potential for learning is not new. Coulomb Potential Learning was proposed to store arbitrary many patterns in a potential field with perfect recall and without spurious patterns (Perrone & Cooper, 1995). Another related work is the Potential Support Vector Machine (PSVM), which minimizes Coulomb potential differences (Hochreiter & Mozer, 2001; Hochreiter et al., 2003). Hochreiter & Obermayer (2005) also used a potential function based on Plummer kernels for optimal unsupervised learning, on which we base our work on Coulomb GANs. + +# 2 COULOMB GANS + +# 2.1 GENERAL CONSIDERATIONS ON GANS + +We assume data samples $\pmb { a } \in \mathbb { R } ^ { m }$ for a model density $p _ { x } ( . )$ and a target density $p _ { y } ( . )$ . The goal of GAN learning is to modify the model in a way to obtain $\dot { p } _ { x } ( . ) = p _ { y } \bar { ( . ) }$ . We define the difference of densities $\rho ( { \pmb a } ) = p _ { y } ( { \pmb a } ) - p _ { x } ( { \pmb a } )$ which should be pushed toward zero for all $\pmb { a } \in \mathbb { R } ^ { m }$ during learning. In the GAN setting, the discriminator $D ( { \pmb a } )$ is a function $D : \mathbb { R } ^ { m } \mathbb { R }$ that learns to discriminate between generated and target samples and predicts how likely it is that $\textbf { \em a }$ is sampled from the target distribution. In conventional GANs, $D ( { \pmb a } )$ is usually optimized to approximate the probability of seeing a target sample, or $\rho ( { \pmb a } )$ or some similar function. The generator $G ( z )$ is a continuous function $G : \mathbb { R } ^ { n } \mathbb { R } ^ { m }$ which maps some $n$ -dimensional random variable $_ z$ into the space of target samples. $_ z$ is typically sampled from a multivariate Gaussian or Uniform distribution. + +In order to improve the generator, a GAN uses the gradient of the discriminator $\nabla _ { a } D ( { \pmb a } )$ with respect to the discriminator input $\begin{array} { r } { \pmb { a } = G ( \pmb { z } ) } \end{array}$ for learning. The objective of the generator is a scalar function $D ( G ( z ) )$ , therefore the gradient of the objective function is just a scaled version of the gradient $\nabla _ { a } D ( { \pmb a } )$ which would then propagate further to the parameters of $G$ . This gradient $\nabla _ { a } D ( { \pmb a } )$ tells the generator in which direction $\rho ( { \pmb a } )$ becomes larger, i.e., in which direction the ratio of target examples increases. The generator changes slightly so that $_ { z }$ is now mapped to a new $\pmb { a } ^ { \prime } = G ^ { \prime } ( \pmb { z } )$ , moving the sample generated by $_ { z }$ a little bit towards the direction where $\rho ( { \pmb a } )$ was larger, i.e., where target examples were more likely. However, $\rho ( { \pmb a } )$ and its derivative only take into account the local neighborhood of $\textbf { \em a }$ , since regions of the sample space that are distant from $\textbf { \em a }$ do not have much influence on $\rho ( { \pmb a } )$ . Regions of data space that have strong support in $p _ { y }$ but not in $p _ { x }$ will not be noticed by the generator via discriminator gradients. The restriction to local environments hampers GAN learning significantly (Arjovsky & Bottou, 2017; Arjovsky et al., 2017). + +The theoretical analysis of GAN learning can be done at three different levels: (1) in the space of distributions $p _ { x }$ and $p _ { y }$ regardless of the fact that $p _ { x }$ is realized by $G$ and $p _ { z }$ , (2) in the space of functions $G$ and $D$ regardless of the fact that $G$ and $D$ are typically realized by a parametric form, i.e., as neural networks, or (3) in the space of the parameters of $G$ and $D$ . Goodfellow et al. (2014) use (1) to prove convergence of GAN learning in their Proposition 2 in a hypothetical scenario where the learning algorithm operates by making small, local moves in $p _ { x }$ space. In order to see that level (1) and (2) should both be understood as hypothetical scenarios, remember that in all practical implementations, $p _ { x }$ can only be altered implicitly by making small changes to the generator function G, which in turn can only be changed implicitly by small steps in its parameters. Even if we assume that the mapping from a distribution $p _ { x }$ to the generator $G$ that induced it exists and is unique, this mapping from $p _ { x }$ to the space of $G$ is not continuous. To see this, consider changing a distribution $p 1 _ { x }$ to a new distribution $p 2 _ { x }$ by moving a small amount $\epsilon$ of its density to an isolated region in space where $p 1 _ { x }$ has no support. Let’s further assume this region has distance $d$ to any other regions of support of $p 1 _ { x }$ . By letting $\epsilon 0$ , the distance between $p 1 _ { x }$ and $p 2 _ { x }$ becomes smaller, yet the distance between the inducing generator functions $G _ { 1 }$ and $G _ { 2 }$ (e.g. using the supremum norm on bounded functions) will not tend to zero because for at least one function input $_ { z }$ we have: $| G _ { 1 } ( z ) - G _ { 2 } ( z ) | \geqslant d$ . Because of this, we need to go further than the distribution space when analyzing GAN learning. In practice, when learning GANs, we are restricted to small steps in parameter space, which in turn lead to small steps in function space and finally to small steps in distribution space. But not all small steps in distribution space can be realized this way as shown in the example above. This causes local Nash equilibria in the function space, because even though in distribution space it would be easy to escape by making small steps, such a step would require very large changes in function space and is thus not realizable. In this paper we show that Coulomb GANs do not exhibit any local Nash equilibria in the space of the functions $G$ and $D$ . To the best of our knowledge, this is the first formulation of GAN learning that can guarantee this property. Of course, Coulomb GANs are learned as parametrized neural networks, and as we will discuss in Subsection 2.4.2, Coulomb GANs are not immune to the usual issues that arise from parameter learning, such as over- and underfitting, which can cause local Nash Equilibria due to a bad choice of parameters. + +# 2.2 FROM CONVENTIONAL GANS TO POTENTIALS + +If the density $p _ { x } ( . )$ or $p _ { y } ( . )$ approaches a Dirac delta-distribution, gradients vanish since the density approaches zero except for the exact location of data points. Similarly, electric point charges are often represented by Dirac delta-distributions, however the electric potential created by a point charge has influence everywhere in the space, not just locally. The electric potential (Coulomb potential) created by the point charge $Q$ is $\begin{array} { r } { \Phi _ { C } = \frac { 1 } { 4 \pi \varepsilon _ { 0 } } \frac { Q } { r } } \end{array}$ , where $r$ is the distance to the location of $Q$ and $\varepsilon _ { \mathrm { 0 } }$ is the dielectric constant. Motivated by this electric potential, we introduce a similar concept for GAN learning: Instead of the difference of densities $\rho ( { \pmb a } )$ , we rather consider a potential function $\Phi ( a )$ defined as + +$$ +\Phi ( { \pmb a } ) = \int \rho ( { \pmb b } ) \ k ( { \pmb a } , { \pmb b } ) \ \mathrm { d } { \pmb b } , +$$ + +with some kernel $k \left( a , b \right)$ which defines the influence of a point at $^ { b }$ onto a point at $^ { a }$ . The crucial advantage of potentials $\Phi ( a )$ is that each point can influence each other point in space if $k$ is chosen properly. If we minimize this potential $\Phi ( a )$ we are at the same time minimizing the difference of densities $\rho ( { \pmb a } )$ : For all kernels $k$ it holds that if $\rho ( \pmb { b } ) = 0$ for all $^ { b }$ then $\Phi ( { \pmb a } ) = 0$ for all $^ { a }$ . We must still show that (i) $\Phi ( { \pmb a } ) = 0$ for all $\textbf { \em a }$ then $\rho ( \pmb { b } ) = 0$ for all $^ { b }$ , and even more importantly, (ii) whether a gradient optimization of $\Phi ( a )$ leads to $\Phi ( { \pmb a } ) = 0$ for all $\textbf { \em a }$ . This is not the case for every kernel. Indeed only for particular kernels $k$ gradient optimization of $\Phi ( a )$ leads to $\rho ( \pmb { b } ) = 0$ for all $^ { b }$ , that is, $p _ { x } ( b ) \stackrel { \cdot } { = } p _ { y } ( \bar { b } )$ for all $^ { b }$ (Hochreiter & Obermayer, 2005) (see also Theorem 1 below). An example for such a kernel $k$ is the one leading to the Coulomb potential $\Phi _ { C }$ from above, where $\begin{array} { r } { k \left( \mathbf { { a } } , \mathbf { { b } } \right) \ { \stackrel { * } { = } } \ { \frac { 1 } { \left\| \mathbf { { a } } - \mathbf { { b } } \right\| } } } \end{array}$ for $m = 3$ . As we will see in the following, the ability to have samples that influence each other over long distances, like charges in a Coulomb potential, will lead to GANs with a single, optimal Nash equilibrium. + +# 2.3 GANS AS ELECTRICAL FIELDS + +For Coulomb GANs, the generator objective is derived from electrical field dynamics: real and generated samples generate a potential field, where samples of the same class (real vs. generated) repel each other, but attract samples of the opposite class. However, real data points are fixed in space, so the only samples that can move are the generated ones. In turn, the gradient of the potential with respect to the input samples creates a vector field in the space of samples. The generator can move its samples along the forces generated by this field. Such a field is depicted in Figure 1. The discriminator learns to predict the potential function, in order to approximate the current potential landscape of all samples, not just the ones in the current mini-batch. Meanwhile, the generator learns to distribute its samples across the whole field in such a way that the energy is minimized, thus naturally avoids mode collapse and covering the whole region of support of the data. The energy is minimal and equal to zero only if all potential differences are zero and the model distribution is equal to the target distribution. + +Within an electrostatic field, the strength of the force on one particle depends on its distance to other particles and their charges. If left to move freely, the particles will organize themselves into a constellation where all forces equal out and no potential differences are present. For continuous charge distributions, the potential field is constant without potential differences if charges no longer move since forces are equaled out. If the potential field is constant, then the difference of densities $\rho$ is constant, too. Otherwise the potential field would have local bumps. The same behavior is modeled within our Coulomb GAN, except that real and generated samples replace the positive and negative particles, respectively, and that the real data points remain fixed. Only the generated samples are allowed to move freely, in order to minimize $\rho$ . The generated samples are attracted by real samples, so they move towards them. At the same time, generated samples should repel each other, so they do not clump together, which would lead to mode collapsing. + +Analogously to electrostatics, the potential $\Phi ( a )$ from Eq. (1) gives rise to a field $\begin{array} { l l } { \mathbf { E } ( \pmb { a } ) } & { = } \end{array}$ $- \nabla _ { a } \bar { \Phi } ( a )$ . and to an energy function $\begin{array} { r } { F \left( \rho \right) = \frac { 1 } { 2 } \int \rho ( \mathbf { { a } } ) \Phi ( \mathbf { { a } } ) \mathrm { { d } } \mathbf { { a } } } \end{array}$ . The field ${ \bf E } ( a )$ applies a force on charges at $^ { a }$ which pushes the charges toward lower energy constellations. Ultimately, the Coulomb GAN aims to make the potential $\Phi$ zero everywhere via the field ${ \bf E } ( a )$ , which is the negative gradient of $\Phi$ . For proper kernels $k$ , it can be shown that (i) $\Phi$ can be pushed to zero via its negative gradient given by the field and (ii) that $\Phi ( { \pmb a } ) = 0$ for all $\textbf { \em a }$ implies $\bar { { \rho } } ( { \pmb a } ) = 0$ for all $^ { a }$ , therefore, $\bar { p } _ { x } ( { \pmb a } ) = \bar { p } _ { y } ( { \pmb a } )$ for all $^ { a }$ (Hochreiter & Obermayer, 2005) (see also Theorem 1 below). + +# 2.3.1 LEARNING PROCESS + +During learning we do not change $\Phi$ or $\rho$ directly. Instead, the location $\begin{array} { r } { \pmb { a } = G ( \pmb { z } ) } \end{array}$ to which the random variable $_ z$ is mapped changes to a new location $\pmb { a } ^ { \prime } = G ^ { \prime } ( \pmb { z } )$ . For the GAN optimization dynamics, we assume that generator samples $\pmb { a } = G ( \pmb { z } )$ can move freely, which is ensured by a sufficiently complex generator. Importantly, generator samples originating from random variables $_ { z }$ do neither disappear nor are they newly created but are conserved. This conservation is expressed by the continuity equation (Schwartz, 1972) that describes how the difference between distributions $\rho ( { \pmb a } )$ changes as the particles are moving along the field, i.e., how moving samples during the learning process changes our densities: + +$$ +{ \dot { \rho } } ( { \mathbf { a } } ) \ = \ - \nabla \cdot ( \rho ( { \mathbf { a } } ) \ v ( { \mathbf { a } } ) ) +$$ + +for sample density difference $\rho$ and unit charges that move with “velocity” ${ \pmb v } ( { \pmb a } ) = \mathrm { s i g n } ( \rho ( { \pmb a } ) ) { \bf E } ( { \pmb a } )$ . The continuity equation is crucial as it establishes the connection between moving samples and changing the generator density and thereby $\rho$ . The sign function of the velocity indicates whether positive or negative charges are present at $\textbf { \em a }$ . The divergence operator $\mathbf { \omega } ^ { 6 6 } \nabla \cdot \mathbf { \vec { \mu } } ^ { 5 }$ determines whether samples move toward or outward of $\textbf { \em a }$ for a given field. Basically, the continuity equation says that if the generator density increases, then generator samples must flow into the region and if the generator density decreases, they flow outwards. We assume that differently charged particles cancel each other. If generator samples are moved away from a location $\textbf { \em a }$ then $\rho ( { \pmb a } )$ is increasing while $\rho ( { \pmb a } )$ is decreasing when generator samples are moved toward $\textbf { \em a }$ . The continuity equation is also obtained as a first order ODE to move particles in a potential field (Dembo & Zeitouni, 1988), therefore describes the dynamics how the densities are changing. We obtain + +$$ +\dot { \rho } ( a ) = - \mathrm { s i g n } ( \rho ( a ) ) \nabla \cdot \left( \rho ( a ) { \bf E } ( a ) \right) = - \nabla \cdot \left( | \rho ( a ) | { \bf E } ( a ) \right) . +$$ + +The density difference $\rho ( { \pmb a } )$ indicates how many samples are locally available for being moved. At each local minimum and local maximum $\textbf { \em a }$ of $\rho$ we obtain $\pmb { \nabla } _ { a } \rho ( \pmb { a } ) = 0$ . Using the product rule for the divergence operator, at points $^ { a }$ that are minima or maxima, Eq. (3) reduces to + +$$ +\dot { \rho } ( { \pmb a } ) = - \operatorname { s i g n } ( \rho ( { \pmb a } ) ) \rho ( { \pmb a } ) { \pmb \nabla } \cdot { \bf E } ( { \pmb a } ) . +$$ + +In order to ensure that $\rho$ converges to zero, it is necessary and sufficient that $\mathrm { s i g n } ( \pmb { \nabla } \cdot \mathbf { E } ( \pmb { a } ) ) =$ $\mathrm { s i g n } ( \rho ( a ) )$ , where $\div a \rho ( { \pmb a } ) = 0$ , as this condition ensures the uniform decrease of the maximal absolute density differences $\left| \rho ( \pmb { a } _ { \mathrm { m a x } } ) \right|$ . + +# 2.3.2 CHOICE OF KERNEL + +As discussed before, the choice of kernel is crucial for Coulomb GANs. The $m$ -dimensional Coulomb kernel and the $m$ -dimensional Plummer kernel lead to (i) $\Phi$ that is pushed to zero via the field it creates and (ii) that $\Phi ( { \pmb a } ) = 0$ for all $^ { a }$ implies $\rho ( { \pmb a } ) = 0$ for all $\textbf { \em a }$ , therefore, $p _ { x } ( { \pmb a } ) = p _ { y } ( { \pmb a } )$ for all $\textbf { \em a }$ (Hochreiter & Obermayer, 2005). Thus, gradient learning with these kernels has been proved to converge to an optimal solution. However, both the $m$ -dimensional Coulomb and the $m$ - dimensional Plummer kernel lead to numerical instabilities if $m$ is large. Therefore the Coulomb potential $\Phi ( a )$ for the Coulomb GAN was constructed by a low-dimensional Plummer kernel $k$ with parameters $d \leqslant m - 2$ and $\epsilon$ : + +$$ +\Phi ( { \pmb a } ) = \int \rho ( { \pmb b } ) ~ k ~ ( { \pmb a } , { \pmb b } ) ~ \mathrm { d } { \pmb b } , \quad k ( { \pmb a } , { \pmb b } ) = \frac { 1 } { ( \sqrt { \| { \pmb a } - { \pmb b } \| ^ { 2 } + \epsilon ^ { 2 } } ) ^ { d } } . +$$ + +The original Plummer kernel is obtained with $d = m - 2$ . The resulting field and potential energy is + +$$ +\begin{array} { l } { \displaystyle \mathbf { E } ( \pmb { a } ) = - \int \rho ( \pmb { b } ) \nabla _ { a } k ( \pmb { a } , \pmb { b } ) \mathrm { d } \pmb { b } = - \nabla _ { a } \Phi ( \pmb { a } ) \ , } \\ { \displaystyle F ( \rho ) = \frac { 1 } { 2 } \int \rho ( \pmb { a } ) \Phi ( \pmb { a } ) \mathrm { d } \pmb { a } = \frac { 1 } { 2 } \int \displaystyle \int \rho ( \pmb { a } ) \rho ( \pmb { b } ) k ( \pmb { a } , \pmb { b } ) \mathrm { d } \pmb { b } \mathrm { d } \pmb { a } . } \end{array} +$$ + +The next theorem states that for freely moving generated samples, $\rho$ converges to zero, that is, $p _ { x } ( . ) = p _ { y } ( . )$ , when using this potential function $\Phi ( a )$ . + +Theorem 1 (Convergence with low-dimensional Plummer kernel). For $\pmb { a } , \pmb { b } \in \mathbb { R } ^ { m }$ , $d \leqslant m - 2$ , and $\epsilon > 0$ the densities $p _ { x } ( . )$ and $p _ { y } ( . )$ equalize over time when minimizing energy $F$ with the low-dimensional Plummer kernel by gradient descent. The convergence is faster for larger $d$ . + +Proof. See Section A.2. + +# 2.4 DEFINITION OF THE COULOMB GAN + +The Coulomb GAN minimizes the electric potential energy from Eq. (6) using a stochastic gradient descent based approach using mini-batches. Appendix Section A.4 contains the equations for the Coulomb potential, field, and energy in this case. Generator samples are obtained by drawing $N _ { x }$ random numbers $z _ { i }$ and transforming them into outputs $\pmb { x } _ { i } = G ( z _ { i } )$ . Each mini-batch also includes $N _ { y }$ real world samples $\mathbf { \nabla } _ { \mathbf { \boldsymbol { y } } _ { i } }$ . This gives rise to a mini-batch specific potential, where in Eq. (5) we use $\rho ( { \mathbf { \bar { a } } } ) = p _ { y } ( { \mathbf { \bar { a } } } ) - \bar { p _ { x } } ( { \mathbf { \bar { a } } } )$ and replace the expectations by empirical means using the drawn samples: + +$$ +\hat { \Phi } ( { \pmb a } ) = \frac { 1 } { N _ { y } } \sum _ { i = 1 } ^ { N _ { y } } k \left( { \pmb a } , { \pmb y } _ { i } \right) - \frac { 1 } { N _ { x } } \sum _ { i = 1 } ^ { N _ { x } } k \left( { \pmb a } , { \pmb x } _ { i } \right) . +$$ + +It is tempting to have a generator network that directly minimizes this potential $\hat { \Phi }$ between generated and training set points. In fact, we show that $\hat { \Phi }$ is an unbiased estimate for $\Phi$ in Appendix Section A.4. However, the estimate has very high variance: for example, if a mini-batch fails to sample training data from an existing mode, the field would drive all generated samples that have been generated at this mode to move elsewhere. The high variance has to be counteracted by extremely low learning rates, which makes learning infeasible in practice, as confirmed by initial experiments. Our solution to this problem is to have a network that generalizes over the mini-batch specific potentials: each mini-batch contains different generator samples $\mathcal { X } = \pmb { x } _ { i }$ for $i = 1 , \ldots , N _ { x }$ and real world samples $\mathbf { \boldsymbol { y } } = \mathbf { \boldsymbol { y } } _ { i }$ for $i = 1 , \ldots , N _ { y }$ , they create a batch-specific potential $\hat { \Phi }$ . The goal of the discriminator is to learn $\operatorname { E } _ { \mathcal { X } , \mathcal { Y } } ( \hat { \Phi } ( \pmb { a } ) ) = \Phi ( \pmb { a } )$ , i.e., the potential averaged over many mini-batches. Thus the discriminator function $D$ fulfills a similar role as other typical GAN discriminator functions, i.e., it discriminates between real and generated data such that for any point in space $^ { a }$ , $D ( { \pmb a } )$ should be greater than zero if the $p _ { y } ( { \pmb a } ) > p _ { x } ( { \pmb a } )$ and smaller than zero otherwise. In particular $D ( { \pmb a } )$ also indicates, via its gradient and its potential properties, directions toward regions where training set samples are predominant and where generator samples are predominant. + +The generator in turn tries to move all of its samples according to the vector field into areas where generator samples are missing and training set samples are predominant. The generator minimizes the approximated energy $F$ as predicted by the discriminator. The loss $\mathcal { L } _ { D }$ for the discriminator and $\mathcal { L } _ { G }$ for the generator are given by: + +$$ +\begin{array} { l } { { \displaystyle { \mathcal L } _ { D } ( D ; G ) = \frac { 1 } { 2 } \mathrm { E } _ { p _ { a } } \left( ( D ( { \pmb a } ) - \frac { \hat { \Phi } ( { \pmb a } ) ) ^ { 2 } } { 2 } \right) } } \\ { { \displaystyle { \mathcal L } _ { G } ( G ; D ) = - \frac { 1 } { 2 } \mathrm { E } _ { p _ { z } } \left( D ( G ( { \pmb z } ) ) \right) . } } \end{array} +$$ + +Where $\begin{array} { r } { p ( { \pmb a } ) = 1 / 2 \int { \mathcal N } ( { \pmb a } ; G ( z ) , \epsilon I ) p _ { z } ( z ) \mathrm { d } z + 1 / 2 \int { \mathcal N } ( { \pmb a } ; { \pmb y } , \epsilon I ) p _ { y } ( { \pmb y } ) \mathrm { d } { \pmb y } , } \end{array}$ , i.e., a distribution where each point of support both of the generator and the real world distribution is surrounded with a Gaussian ball of width $\epsilon I$ similar to Bishop et al. (1998), in order to overcome the problem that the generator distribution is only a sub-manifold of $\mathbb { R } ^ { m }$ . These loss functions cause the approximated potential values $D ( { \pmb a } )$ that are negative are pushed toward zero. Finally, the Coulomb GAN, like all other GANs, consists of two parts: a generator to generate model samples, and a discriminator that provides its learning signal. Without a discriminator, our would be very similar to GMMNs (Li et al., 2015), as can be seen in Eq. (33), but with an optimal Kernel specifically tailored to the problem of estimating differences between probability distributions. + +We use each mini-batch only for one update of the discriminator and the generator. It is important to note that the discriminator uses each sample in the mini batch twice: once as a point to generate the mini-batch specific potential $\hat { \Phi }$ , and once as a point in space for the evaluation of the potential $\hat { \Phi }$ and its approximation $D$ . Using each sample twice is done for performance reasons, but not strictly necessary: the discriminator could learn the potential field by sampling points that lie between generator and real samples as in Gulrajani et al. (2017), but we are mainly interested in correct predictions in the vicinity of generator samples. Pseudocode for the learning algorithm is detailed in Algorithm 1 in the appendix. + +# 2.4.1 OPTIMALITY OF THE SOLUTION + +Convergence of the GAN learning process was proved for a two time-scales update rule by Heusel et al. (2017). A local Nash equilibrium is a pair of generator and discriminator $( D ^ { * } , G ^ { * } )$ that fulfills the two conditions + +$$ +D ^ { * } = \arg \operatorname* { m i n } _ { D \in U ( D ^ { * } ) } \mathcal { L } _ { D } ( D ; G ^ { * } ) \quad \mathrm { a n d } \quad G ^ { * } = \arg \operatorname* { m i n } _ { G \in U ( G ^ { * } ) } \mathcal { L } _ { G } ( G ; D ^ { * } ) . +$$ + +for some neighborhoods $U ( D ^ { * } )$ and $U ( G ^ { * } )$ . We show in the following Theorem 2 that for Coulomb GANs every local Nash equilibrium necessarily is identical to the unique global Nash equilibrium. In other words, any equilibrium point of the Coulomb GAN that is found to be local optimal has to be the one global Nash equilibrium as the minimization of the energy $F ( \rho )$ in Eq. (33) leads to a single, global optimum at $p _ { y } = p _ { x }$ . + +Theorem 2 (Optimal Solution). If the pair $( D ^ { * } , G ^ { * } )$ is a local Nash equilibrium for the Coulomb GAN objectives, then it is the global Nash equilibrium, and no other local Nash equilibria exist, and G∗ has output distribution px = py. + +Proof. See Appendix Section A.3. + +# 2.4.2 COULOMB GANS IN PRACTICE + +To implement GANs in practice, we need learnable models for $G$ and $D$ . We assume that our models for $G$ and $D$ are continuously differentiable with respect to their parameters and inputs. Toward this end, GANs are typically implemented as neural networks optimized by (some variant of) gradient descent. Thus we may not find the optimal $G ^ { * }$ or $D ^ { * }$ , since neural networks may suffer from capacity or optimization issues. Recent research indicates that the effect of local minima in deep learning vanishes with increasing depth (Dauphin et al., 2014; Choromanska et al., 2015; Kawaguchi, 2016), such that this limitation becomes less restrictive as our ability to train deep networks grows thanks to hardware and optimization improvements. + +The main problem with learning Coulomb GANs is to approximate the potential function $\Phi$ , which is a complex function in a high-dimensional space, since the potential can be very non-linear and non-smooth. When learning the discriminator, we must ensure that enough data is sampled and averaged over. We already lessened the non-linear function problem by using a low-dimensional Plummer kernel. But still, this kernel can introduce large non-linearities if samples are close to each other. It is crucial that the discriminator learns slow enough to accurately estimate the potential function which is induced by the current generator. The generator, in turn, must be even slower since it must be tracked by the discriminator. These approximation problems are supposed to be tackled by the research community in near future, which would enable optimal GAN learning. + +The formulation of GAN learning as a potential field naturally solves the mode collapsing issue: the example described in Section A.1, where a normal GAN cannot get out of a local Nash equilibria is not a converged solution for the Coulomb GAN: If all probability mass of the generator lies in one of the modes, then both attracting forces from real-world samples located at the other mode as well as repelling forces from the over-represented generator mode will act upon the generator until it generates samples at the other mode as well. + +# 3 EXPERIMENTS + +In all of our experiments, we used a low-dimensional Plummer Kernel of dimensionality $d = 3$ . This kernel both gave best computational performance and has low risk of running into numerical issues. We used a batch size of 128. To evaluate the quality of a GAN, the FID metric as proposed by Heusel et al. (2017) was calculated by using $5 0 \mathrm { k }$ samples drawn from the generator, while the training set statistics were calculated using the whole training set. We compare to BEGAN (Berthelot et al., 2017), DCGAN (Radford et al., 2016) and WGAN-GP (Gulrajani et al., 2017) both in their original version as well as when using the two-timescale update-rule (TTUR), using the settings from Heusel et al. (2017). We additionally compare to MMD-GAN (Li et al., 2017), which is conceptually very similar to the Coulomb GAN, but uses a Gaussian Kernel instead of the Plummer Kernel. We use the dataset-specific settings recommended in (Li et al., 2017) and report the highest FID score over the course of training. All images shown in this paper were produced with a random seed and not cherry picked. The implementation used for these experiments is available online1. The appendix Section A.5 contains an additional toy example demonstrating that Coulomb GANs do not suffer from mode collapse when fitting a simple Gaussian Mixture of 25 components. + +# 3.1 IMAGE DATASETS + +To demonstrate the ability of the Coulomb GAN to learn distributions in high dimensional spaces, we trained a Coulomb GAN on several popular image data sets: The cropped and centered images of celebrities from the Large-scale CelebFaces Attributes (“CelebA”) data set (Liu et al., 2015), the LSUN bedrooms data set consists of over 3 million 64x64 pixel images of the bedrooms category of the large scale image database LSUN (Yu et al., 2015) as well as the CIFAR-10 data set. For these experiments, we used the DCGAN architecture (Radford et al., 2016) with a few modifications: our convolutional kernels all have a kernel size of 5x5, our random seed that serves as input to the generator has fewer dimensions: 32 for CelebA and LSUN bedrooms, and 16 for CIFAR-10. Furthermore, the discriminator uses twice as many feature channels in each layer as in the DCGAN architecture. For the Plummer kernel, $\epsilon$ was set to 1. We used the Adam optimizer with a learning rate of $1 0 ^ { - 4 }$ for the generator and $5 \cdot 1 0 ^ { - 5 }$ for the discriminator. To improve convergence performance, we used the tanh output activation function (LeCun et al., 1998). For regularization we used an L2 weight decay term with a weighting factor of $1 0 ^ { - 7 }$ . Learning was stopped by monitoring the FID metric (Heusel et al., 2017). Once learning plateaus, we scaled the learning rate down by a factor of 10 and let it continue once more until the FID plateaus. The results are reported in Table 1b, and generated images can be seen in Figure 2 and in the Appendix in Section A.7. Coulomb GANs tend to outperform standard GAN approaches like BEGAN and DCGAN, but are outperformed by the Improved Wasserstein GAN. However it is important to note that the Improved Wasserstein GAN used a more advanced network architecture based on ResNet blocks (Gulrajani et al., 2017), which we could not replicate due to runtime constraints. Overall, the low FID of Coulomb GANs stem from the fact that the images show a wide variety of different samples. E.g. on CelebA, Coulomb GAN exhibit a very wide variety of faces, backgrounds, eye colors and orientations. To further investigate how much variation the samples generated by the Coulomb GAN contains, we followed the advice of Arora and Zhang (Arora & Zhang, 2017) to estimate the support size of the generator’s distribution by checking how large a sample from the generator must be before we start generating duplicates. We were able to generate duplicates with a probability of around $50 \%$ when using samples of size 1024, which indicates that the support size learned by the Coulomb GAN would be around 1M. This is a strong indication that the Coulomb GAN was able to spread out its samples over the whole target distribution. A depiction is included in Figure 3, which also shows the nearest neighbor in the training set of the generated images, confirming that the Coulomb GAN does not just memorize training images. + +![](images/dc7625461a0dced041941f6df56258aa8ed1a05bcf3a0ec34978702ca93440d3.jpg) +Figure 2: Images from a Coulomb GAN after training on CelebA (first row), LSUN bedrooms (second row) and CIFAR 10 (last row). Further examples are located in the appendix in Section A.7 + +![](images/6a9c0c1703f8938cefd242d27336d75059e77e543cafd91e08231266b5b6539c.jpg) +Figure 3: The most similar pairs found in batches of 1024 generated faces sampled from the Coulomb GAN, and the nearest neighbor from the training data shown as third image. Distances were calculated as Euclidean distances on pixel level. + +# 3.2 LANGUAGE MODELING + +We repeated the experiments from Gulrajani et al. (2017), where Improved Wasserstein GANs (WGAN-GP) were trained to produce text samples after being trained on the Google Billion Word data set (Chelba et al., 2013), using the same network architecture as in the original publication. We use the Jensen-Shannon-divergence on 4-grams and 6-grams as an evaluation criterion. The results are summarized in Table 1a. + +(a) Normalized Jensen-ShanonDivergence for the Google Billion Word data. Values for WGANGP are without/with TTUR, taken from Heusel et al. (2017). + +
data setWGAN-GPours
4 grams0.38 / 0.350.35
6 grams0.77 / 0.740.74
+ +
data setBEGANDCGANWGAN-GPMMDours
CelebA29.2 /28.521.4 / 12.54.8 / 4.263.29.3
LSUN113 /11270.4 / 57.520.5 / 9.594.931.2
CIFAR10--29.3 /24.838.227.3
+ +(b) Performance comparison in FID (lower is better) on different data sets. Values for all methods except Coulomb GAN and MMD-GAN are from without/with TTUR, taken from Heusel et al. (2017). + +# 4 CONCLUSION + +The Coulomb GAN is a generative adversarial network with strong theoretical guarantees. Our theoretical results show that the Coulomb GAN will be able to approximate the real distribution perfectly if the networks have sufficient capacity and training does not get stuck in local minima. Our results show that the potential field used by the Coulomb GAN far outperforms MMD based approaches due to its low-dimensional Plummer kernel, which is better suited for modeling probability density functions, and is very effective at eliminating the mode collapse problem in GANs. This is because our loss function forces the generated samples to occupy different regions of the learned distribution. In practice, we have found that Coulomb GANs are able to produce a wide range of different samples. However, in our experience, this sometimes leads to a small number of generated samples that are non-sensical interpolations of existing data modes. While these are sometimes also present in other GAN models (Radford et al., 2016), we found that our model produces such images at a slightly higher rate. This issue might be solved by finding better ways of learning the discriminator, as learning the correct potential field is crucial for the Coulomb GAN’s performance. We also observed that increasing the capacity of the discriminator seems to always increase the generative performance. We thus hypothesize that the largest issue in learning Coulomb GANs is that the discriminator needs to approximate the potential field $\Phi$ very well in a high-dimensional space. In summary, instead of directly optimizing a criterion based on local differences of densities which can exhibit many local minima, Coulomb GANs are based on a potential field that has no local minima. The potential field is created by point charges in an analogy to electric field in physics. We have proved that if learning converges then it converges to the optimal solution if the samples can be moved freely. We showed that Coulomb GANs avoid mode collapsing, model the target distribution more truthfully than standard GANs, and do not overlook high probability regions of the target distribution. + +# ACKNOWLEDGMENTS + +This work was supported by Zalando SE with Research Agreement 01/2016, Audi.JKU Deep Learning Center, Audi Electronic Venture GmbH, IWT research grant IWT150865 (Exaptation), H2020 project grant 671555 (ExCAPE) and NVIDIA Corporation. The authors would like to thank Philipp Renz for fruitful discussions. + +# REFERENCES + +M. Arjovsky and L. Bottou. 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International Conference on Learning Representations (ICLR), 2017. + +# A APPENDIX + +A.1 EXAMPLE OF CONVERGENCE TO MODE COLLAPSE IN CONVENTIONAL GANS + +As an example of how a GAN can converge to a Nash Equilibrium that exhibits mode collapse, consider a target distribution that consists of two distinct/non-overlapping regions of support $C _ { 1 }$ and $C _ { 2 }$ that are distant from each other, i.e., the target probability is zero outside of $C _ { 1 }$ and $C _ { 2 }$ . Further assume that $50 \%$ of the probability mass is in $C _ { 1 }$ and $50 \%$ in $C _ { 2 }$ . Assume that the the generator has mode-collapsed onto $C _ { 1 }$ , which contains $100 \%$ of the generator’s probability mass. In this situation, the optimal discriminator classifies all points from $C _ { 2 }$ as ”real“ (pertaining to the target distribution) by supplying an output of 1 for them (1 is the target for real samples and 0 the target for generated samples). Within $C _ { 1 }$ , the other region, the discriminator sees twice as many generated data points as real ones, as $100 \%$ of the probability mass of the generator’s distribution is in $C _ { 1 }$ , but only $50 \%$ of the probability mass of the real data distribution. So one third of the points seen by the discriminator in $C _ { 1 }$ are real, the other 2 thirds are generated. Thus, to minimize its prediction error for a proper objective (squared or cross entropy), the discriminator has to output $1 / 3$ for every point from $C _ { 1 }$ . The optimal output is even independent of the exact form of the real distribution in $C _ { 1 }$ . The generator will match the shape of the target distribution locally. If the shape is not matched, local gradients of the discriminator with respect to its input would be present and the generator would improve locally. If local improvements of the generator are no longer possible, the shape of the target distribution is matched and the discriminator output is locally constant. In this situation, the expected gradient of the discriminator is the zero vector, because it has reached an optimum. Since the discriminator output is constant in $C _ { 1 }$ (and $C _ { 2 }$ ), the generator’s expected gradient is the zero vector, too. The situation is also stable even though we still have random fluctuations from the ongoing stochastic gradient (SGD) learning: whenever the generator produces data outside of (but close to) $C _ { 1 }$ , the discriminator can easily detect this and push the generator’s samples back. Inside $C _ { 1 }$ , small deviations of the generator from the shape of the real distribution are detected by the discriminator as well, by deviating slightly from $1 / 3$ . Subsequently, the generator is pushed back to the original shape. If the discriminator deviates from its optimum, it will also be forced back to its optimum. So overall, the GAN learning reached a local Nash equilibrium and has converged in the sense that the parameters fluctuate around the attractor point (fluctuations depend on learning rate, sample size, etc.). To achieve true mathematical convergence, Heusel et al. (2017) assume decaying learning rates to anneal the random fluctuations, similar to Robbins & Monro (1951) original convergence proof for SGD. + +# A.2 PROOF OF THEOREM 1 + +We first recall Theorem 1: + +Theorem (Convergence with low-dimensional Plummer kernel). For $\pmb { a } , \pmb { b } \in \mathbb { R } ^ { m }$ , $d \leqslant m - 2$ , and $\epsilon > 0$ the densities $p _ { x } ( . )$ and $p _ { y } ( . )$ equalize over time when minimizing energy $F$ with the lowdimensional Plummer kernel by gradient descent. The convergence is faster for larger $d$ . + +In a first step, we prove that for local maxima or local minima $\textbf { \em a }$ of $\rho$ , the expression $\sin ( \boldsymbol { \mathbf { V } } \cdot \mathrm { \partial _ { \mathrm { s i g n } } } ( \boldsymbol { \mathbf { V } } \cdot \mathrm { \partial _ { \mathrm { s i g n } } } )$ ${ \bf E } ( { \bf a } ) ) = \mathrm { s i g n } ( \rho ( { \bf a } ) )$ holds for $\epsilon$ small enough. For proving this equation, we apply the Laplace operator for spherical coordinates to the low-dimensional Plummer kernel. Using the result, we see that the integral $\begin{array} { r } { \nabla \cdot \mathbf { E } ( { a } ) = - \int \rho ( b ) \nabla _ { a } ^ { 2 } k \left( { a } , { b } \right) \mathrm { d } b } \end{array}$ is dominated by large negative values of $\nabla _ { a } ^ { 2 } k$ around $^ { a }$ . These negative values can even be decreased by decreasing $\epsilon$ . Therefore we can ensure by a small enough $\epsilon$ that at each local minimum and local maximum $\textbf { \em a }$ of $\rho \operatorname { s i g n } ( { \dot { \rho } } ( { \mathbf { \boldsymbol { a } } } ) ) = - \operatorname { s i g n } ( \rho ( { \mathbf { \boldsymbol { a } } } ) )$ . Thus, the maximal and minimal points of $\rho$ move toward zero. + +In a second step, we show that new maxima or minima cannot appear and that the movement of $\Phi$ toward zero stops at zero and not earlier. Since $\rho$ is continuously differentiable, all points in environments of maxima and minima move toward zero. Therefore the largest $| \rho ( a ) |$ moves toward zero. We have to ensure that moving toward zero does not converge to a point apart from zero. We derive that the movement toward zero is lower bounded by ${ \dot { \rho } } ( { \pmb a } ) = - \mathrm { s i g n } ( \rho ( { \pmb a } ) ) \lambda \rho ^ { 2 } ( { \pmb a } )$ . Thus, the movement slows down at $\rho ( { \pmb a } ) = 0$ . Solving the differential equation and applying it to the maximum of the absolute value of $\rho$ gives $| \rho | _ { \operatorname* { m a x } } ( \mathbf { \bar { \boldsymbol { t } } } ) = 1 / ( \lambda t + ( | \rho | _ { \operatorname* { m a x } } ( 0 ) ) ^ { - 1 } )$ . Thus, $\rho$ converges to zero over time. + +Proof. For $d = m - 2$ , we have $\pmb { \nabla } ^ { 2 } k ( \pmb { a } , \pmb { b } ) = \delta ( \pmb { a } - \pmb { b } )$ , where the theorem has already been proved for $\epsilon$ small enough (Hochreiter & Obermayer, 2005). + +At each local minimum and local maximum $\textbf { \em a }$ of $\rho$ we have $\pmb { \nabla } _ { a } \rho ( \pmb { a } ) = 0$ . Using the product rule for the divergence operator, Eq. (3) reduces to + +$$ +\dot { \rho } ( { \pmb a } ) = - \operatorname { s i g n } ( \rho ( { \pmb a } ) ) \rho ( { \pmb a } ) { \pmb \nabla } \cdot { \bf E } ( { \pmb a } ) . +$$ + +The term $\nabla \cdot \mathbf { E } ( a )$ can be expressed as + +$$ +\nabla \cdot { \bf E } ( a ) = - \nabla _ { a } ^ { 2 } \Phi ( a ) = - \int \rho ( b ) \nabla _ { a } ^ { 2 } k ( a , b ) \ \mathrm { d } b . +$$ + +We next consider $\nabla _ { a } ^ { 2 } k \left( a , b \right)$ for the low-dimensional Plummer kernel. We define the spherical Laplace operator in $( m - 1 )$ dimensions as ∇2Sm−1 , then the Laplace operator in spherical coordinates is (Proposition 2.5 in Frye & Efthimiou (Efthimiou & Frye, 2014)): + +$$ +{ \nabla } ^ { 2 } = { \frac { \partial ^ { 2 } } { \partial r ^ { 2 } } } + { \frac { m - 1 } { r } } { \frac { \partial } { \partial r } } + { \frac { m - 1 } { r ^ { 2 } } } { \nabla } _ { S ^ { m - 1 } } ^ { 2 } . +$$ + +Note that $\nabla _ { S ^ { m - 1 } } ^ { 2 }$ only has second order derivatives with respect to the angles of the spherical coordinates. + +With $r = \| a - b \|$ we obtain for the Laplace operator applied to the low-dimensional Plummer kernel: + +$$ +\nabla ^ { 2 } k ( a , b ) ~ = ~ d \left( - ~ \epsilon ^ { 2 } ~ m ~ + ~ ( 2 ~ + ~ d ~ - ~ m ) ~ r ^ { 2 } \right) ( \epsilon ^ { 2 } ~ + ~ r ^ { 2 } ) ^ { - 2 - d / 2 } ~ . +$$ + +and in particular + +$$ +{ \pmb { \nabla } } ^ { 2 } k ( { \pmb { a } } , { \pmb { a } } ) = - m d \epsilon ^ { - ( d + 2 ) } . +$$ + +For $d \leqslant m - 2$ we have $( 2 + d - m ) \leqslant 0$ , and obtain + +$$ +\nabla ^ { 2 } k ( { \pmb a } , { \pmb b } ) \ < \ 0 \ , +$$ + +and + +$$ +\frac { \partial } { \partial r } \nabla ^ { 2 } k ( a , b ) = d \left( 2 + d \right) r \left( \epsilon ^ { 2 } \left( 2 + m \right) + \left( - 2 - d + m \right) r ^ { 2 } \right) ( \epsilon ^ { 2 } + r ^ { 2 } ) ^ { - 3 - d / 2 } > 0 +$$ + +and + +$$ +{ \frac { \partial } { \partial \epsilon } } \nabla ^ { 2 } k ( { \pmb a } , { \pmb b } ) = { \cal d } \left( 2 + { \cal d } \right) \epsilon \left( \epsilon ^ { 2 } m + ( - 4 - { \cal d } + m ) r ^ { 2 } \right) ( \epsilon ^ { 2 } + r ^ { 2 } ) ^ { - 3 - { \cal d } / 2 } > 0 . +$$ + +Therefore, $\nabla ^ { 2 } k ( a , b )$ is negative with minimum $- m d \epsilon ^ { - ( d + 2 ) }$ at $r = 0$ and increasing with $r$ and increasing with $\epsilon$ for $d \leqslant m - 4$ . For $d = m - 3$ we have to restrict in the following the sphere $S _ { \tau } ( a )$ to $\tau < \sqrt { m } \epsilon$ and ensure increase of $\nabla ^ { 2 } k ( a , b )$ with $\epsilon$ . + +If $\rho ( \pmb { b } ) \neq 0$ , then we define a sphere $S _ { \tau } ( a )$ with radius $\tau$ around $\textbf { \em a }$ for which holds $\mathrm { s i g n } ( \rho ( { \boldsymbol { b } } ) ) =$ $\mathrm { s i g n } ( \rho ( a ) )$ for each $\pmb { b } \in S _ { \tau } ( \pmb { a } )$ . Note that $\dot { \nabla } ^ { 2 } k ( a , b )$ is continuous differentiable. We have + +$$ +\begin{array} { l } { \displaystyle \nabla \cdot { \bf E } ( a ) = - \int \rho ( b ) \nabla _ { a } ^ { 2 } k ( a , b ) \mathrm { d } b = } \\ { \displaystyle - \int _ { S _ { \tau } ( a ) } \rho ( b ) \nabla _ { a } ^ { 2 } k ( a , b ) \mathrm { d } b - \int _ { T \setminus S _ { \tau } ( a ) } \rho ( b ) \nabla _ { a } ^ { 2 } k ( a , b ) \mathrm { d } b . } \end{array} +$$ + +We bound $\nabla ^ { 2 } k ( a , b )$ by + +$$ +~ 1 ~ > ~ \nabla ^ { 2 } k ( a , b ) ~ = ~ d \left( - \epsilon ^ { 2 } ~ m ~ + ~ ( 2 ~ + ~ d ~ - ~ m ) ~ r ^ { 2 } \right) ( \epsilon ^ { 2 } ~ + ~ r ^ { 2 } ) ^ { - 2 - d / 2 } ~ > ~ d \left( 2 ~ + ~ d ~ - ~ m \right) r ^ { - 2 - d / 2 } ~ . +$$ + +Using $\tau$ , we now bound $\begin{array} { r l } { \bigg | \int _ { T \setminus S _ { \tau } ( \mathbf { a } ) } \rho ( \pmb { b } ) \ \pmb { \nabla } _ { a } ^ { 2 } k \left( \pmb { a } , \pmb { b } \right) \ \mathrm { d } \pmb { b } \bigg | } & { { } } \end{array}$ independently from $\epsilon$ , since $\rho$ is a difference of distributions. For small enough $\epsilon$ we can ensure + +$$ +\left| \int _ { S _ { \tau } ( \pmb { a } ) } \rho ( \pmb { b } ) \ \nabla _ { \ b { a } } ^ { 2 } k \left( \pmb { a } , \pmb { b } \right) \ \mathrm { d } \pmb { b } \right| > \ \left| \int _ { T \setminus S _ { \tau } ( \pmb { a } ) } \rho ( \pmb { b } ) \ \nabla _ { \ b { a } } ^ { 2 } k \left( \pmb { a } , \pmb { b } \right) \ \mathrm { d } \pmb { b } \right| \ . +$$ + +Therefore we have + +$$ +\mathrm { s i g n } ( \pmb { \nabla } \cdot \mathbf { E } ( \pmb { a } ) ) = \mathrm { s i g n } ( \rho ( \pmb { a } ) ) . +$$ + +Therefore we have at each local minimum and local maximum $^ { a }$ of $\rho$ + +$$ +\mathrm { s i g n } ( \dot { \rho } ( { \bf a } ) ) = - \mathrm { s i g n } ( \rho ( { \bf a } ) ) . +$$ + +Therefore the maximal and minimal points of $\rho$ move toward zero. Since $\rho$ is continuously differentiable as is the field, also the points in an environment of the maximal and minimal points move toward zero. Points that are not in an environment of the maximal or minimal points cannot become maximal points in an infinitesimal time step. + +Since the contribution of $\textbf { \em a }$ environment $S _ { \tau } ( a )$ dominates the integral Eq. (19), for $\epsilon$ small enough there exists a positive $0 < \lambda$ globally for all minima and maxima as well as for all time steps for which holds: + +$$ +\left| \nabla \cdot \mathbf { E } ( { \pmb a } ) \right| > \lambda \left| \rho ( { \pmb a } ) \right| . +$$ + +The factor $\lambda$ depends on $k$ and on the initial $\rho . \ \lambda$ is proportional to $d$ . Larger $d$ lead to larger $| \nabla \cdot \mathbf { E } ( a ) |$ since the maximum or minimum $\rho ( { \pmb a } )$ is upweighted. There might exist initial conditions $\rho$ for which $\lambda 0$ , e.g. for infinite many maxima and minima, but they are impossible in our applications. + +Therefore maximal or minimal points approach zero faster or equal than given by + +$$ +\dot { \rho } ( { \pmb a } ) = - \ \mathrm { s i g n } ( \rho ( { \pmb a } ) ) \lambda \rho ^ { 2 } ( { \pmb a } ) . +$$ + +In particular this differential equation dominates the global maximum $| \rho | _ { \mathrm { m a x } }$ of $| \rho ( . ) |$ . Solving the differential equation gives that at least + +$$ +| \rho | _ { \operatorname* { m a x } } ( t ) = \frac { 1 } { \lambda t + \left( | \rho | _ { \operatorname* { m a x } } ( 0 ) \right) ^ { - 1 } } . +$$ + +Thus $d$ influences the worst case rate of convergence, where larger $d$ with $d \leqslant m - 2$ leads to faster worst case convergence. + +Consequently, $\rho$ converges to the zero function over time, that is, $p _ { x } ( . )$ becomes equal to $p _ { y } ( . )$ . + +A.3 PROOF OF THEOREM 2 + +We first recall Theorem 2: + +Theorem (Optimal Solution). If the pair $( D ^ { * } , G ^ { * } )$ is a local Nash equilibrium for the Coulomb GAN objectives, then it is the global Nash equilibrium, and no other local Nash equilibria exist, and $G ^ { * }$ has output distribution $p _ { x } = p _ { y }$ . + +Proof. $( D ^ { * } , G ^ { * } )$ being in a local Nash equilibrium means that $( D ^ { * } , G ^ { * } )$ fulfills the two conditions + +$$ +D ^ { * } = \underset { D \in U ( D ^ { * } ) } { \arg \operatorname* { m i n } } \mathcal { L } _ { D } ( D ; G ^ { * } ) \quad \mathrm { a n d } \quad G ^ { * } = \underset { G \in U ( G ^ { * } ) } { \arg \operatorname* { m i n } } \mathcal { L } _ { G } ( G ; D ^ { * } ) +$$ + +for some neighborhoods $U ( D ^ { * } )$ and $U ( G ^ { * } )$ . For Coulomb GANs that means, $D ^ { * }$ has learned the potential $\Phi$ induced by $G ^ { * }$ perfectly, because $\mathcal { L } _ { D }$ is convex in $D$ , thus if $D ^ { * }$ is optimal within an neighborhood $U ( D ^ { * } )$ , it must be the global optimum. This means that $G ^ { * }$ is directly minimizing $\begin{array} { r } { \mathcal { L } _ { G } ( G ; D ) = - \frac { 1 } { 2 } \mathrm { E } _ { p _ { z } } } \end{array}$ $\tilde { \Phi } ( G ( z ) ) )$ . The Coulomb potential energy is according to Eq. (7) + +$$ +F ( \rho ) = \frac { 1 } { 2 } \int \rho ( a ) \Phi ( a ) \mathrm { d } a = \frac { 1 } { 2 } \int p _ { y } ( a ) \Phi ( a ) \mathrm { d } a - \frac { 1 } { 2 } \int p _ { x } ( a ) \Phi ( a ) \mathrm { d } a . +$$ + +Only the samples from $p _ { x }$ stem from the generator, where $\begin{array} { r } { p _ { x } ( \pmb { a } ) = \int \delta ( \pmb { a } - \pmb { G } ( \pmb { z } ) ) p _ { z } ( z ) \mathrm { d } z } \end{array}$ . Here $\delta$ is the $\delta$ -distribution centered at zero. The part of the energy which depends on the generator is + +$$ +\begin{array} { l } { \displaystyle - \mathrm { \small \displaystyle \frac { 1 } { 2 } } \int p _ { x } ( \pmb { a } ) \Phi ( \pmb { a } ) \mathrm { d } \pmb { a } = - \frac { 1 } { 2 } \int \int \delta ( \pmb { a } - G ( \pmb { z } ) ) p _ { z } ( \pmb { z } ) \mathrm { d } z \Phi ( \pmb { a } ) \mathrm { d } \pmb { a } } \\ { = - \frac { 1 } { 2 } \int \big ( \int \delta ( \pmb { a } - G ( \pmb { z } ) ) \Phi ( \pmb { a } ) \mathrm { d } \pmb { a } \big ) p _ { z } ( \pmb { z } ) \mathrm { d } z } \\ { = - \displaystyle \frac { 1 } { 2 } \int p _ { z } ( \pmb { z } ) \Phi ( G ( \pmb { z } ) ) \mathrm { d } z = - \frac { 1 } { 2 } \mathrm { E } _ { p _ { z } } \left( \Phi ( G ( \pmb { z } ) ) \right) . } \end{array} +$$ + +Theorem 1 guarantees that there are no other local minima except the global one when minimizing $F$ . $F$ has one minimum, $F = 0$ , which implies $\Phi ( { \pmb a } ) = 0$ and $\rho ( { \pmb a } ) = 0$ for all $\textbf { \em a }$ , therefore also $p _ { y } = p _ { x }$ according to Theorem 1. Each $\Phi ( { \pmb a } ) \neq \mathrm { ~ 0 ~ }$ would mean there exist potential differences which in turn would cause forces on generator samples that allow to further minimize the energy. Since we assumed that the generator can reach the minimum $p _ { y } = p _ { x }$ for any $p _ { y }$ , it will be reached by local (stepwise) optimization of $- { \textstyle \frac { 1 } { 2 } } \mathrm { E } _ { p _ { z } } \left( \Phi ( G ( z ) ) \right)$ with respect to $G$ . Since the pair $( D ^ { * } , G ^ { * } )$ is optimal within their neighborhood, the generator has reached this minimum as there is not other local minimum than the global one. Therefore $G ^ { * }$ has model density $p _ { x }$ with $p _ { y } = p _ { x }$ . The convergence point is a global Nash equilibrium, because there is no approximation error and zero energy $F = 0$ is a global minimum for discriminator and generator, respectively. Theorem 1 ensures that other local Nash equilibria are not possible. □ + +# A.4 COULOMB EQUATIONS IN THE CASE OF FINITE SAMPLES + +GANs are sample-based, that is, samples are drawn from the model for learning (Hochreiter & Obermayer, 2005; Gutmann & Hyvärinen, 2012). Typically this is done in mini-batches, where each mini-batch consists of two sets of samples, the target samples $\mathcal { Y } = \{ \pmb { y } _ { i } | i = 1 \ldots N _ { y } \}$ , and the model samples $\mathcal { X } = \{ \pmb { x } _ { i } | i = 1 \ldots N _ { x } \}$ . + +For such finite samples, i.e. point charges, we have to use delta distributions to obtain unbiased estimates of the the model distribution $p _ { x } ( . )$ and the target distribution $p _ { y } ( . )$ : + +$$ +\begin{array} { r l r l r } { \displaystyle { \mathrm { ~ \mu ~ } _ { y } ( { \pmb a } ; { \mathcal { X } } ) = \frac { 1 } { N _ { y } } \sum _ { i = 1 } ^ { N _ { y } } \delta \left( { \pmb a } - { \pmb y } _ { i } \right) \ , } } & { { } \hat { p } _ { x } ( { \pmb a } ; { \mathcal { X } } ) = \frac { 1 } { N _ { x } } \sum _ { i = 1 } ^ { N _ { x } } \delta \left( { \pmb a } - { \pmb x } _ { i } \right) \ , } & { \hat { \rho } ( { \pmb a } ; { \mathcal { X } } , { \mathcal { Y } } ) } & { { } = p _ { y } ( { \pmb a } ; { \mathcal { Y } } ) - p _ { x } ( { \pmb a } ; { \mathcal { X } } , { \mathcal { Y } } ) } \end{array} +$$ + +where $\delta$ is the Dirac $\delta$ -distribution centered at zero. These are unbiased estimates of the underlying distribution, as can be seen by: + +$$ +\operatorname { E } _ { \mathcal { X } } \left( \frac { 1 } { N _ { x } } \sum _ { i = 1 } ^ { N _ { x } } \delta ( { \pmb a } - { \pmb x } _ { i } ) \right) = \frac { 1 } { N _ { x } } \sum _ { i = 1 } ^ { N _ { x } } \operatorname { E } _ { x _ { i } } \left( \delta ( { \pmb a } - { \pmb x } _ { i } ) \right) = \frac { 1 } { N _ { x } } \sum _ { i = 1 } ^ { N _ { x } } p _ { x } ( { \pmb a } ) = p _ { x } ( { \pmb a } ) . +$$ + +In the rest of the paper, we will drop the explicit parameterization with $\mathcal { X }$ and $\mathcal { V }$ for all estimates to unclutter notation, and instead just use the hat sign to denote estimates. In the same fashion as for the distributions, when we use fixed samples $\mathcal { X }$ and $\mathcal { V }$ , we obtain the following unbiased estimates for the potential, energy and field given by Eq. (5), Eq. (6), and Eq. (7): + +$$ +\begin{array} { l } { \displaystyle \bar { \Phi } ( a ) = \frac { 1 } { N _ { y } } \displaystyle \sum _ { i = 1 } ^ { N _ { y } } k \left( a , y _ { i } \right) - \frac { 1 } { N _ { z } } \displaystyle \sum _ { i = 1 } ^ { N _ { z } } k \left( a , { \boldsymbol x } _ { i } \right) \ : , } \\ { \displaystyle \hat { F } \left( \boldsymbol \rho \right) = \frac { 1 } { 2 } \left( \frac { N _ { y } } { N _ { y } ^ { 2 } } \displaystyle \sum _ { i = 1 } ^ { N _ { y } } \sum _ { j = 1 } ^ { N _ { y } } k \left( y _ { i } , y _ { i } \right) - \frac { 2 } { N _ { y } N _ { x } } \displaystyle \sum _ { i = 1 } ^ { N _ { y } } \sum _ { j = 1 } ^ { N _ { x } } k \left( y _ { i } , { \boldsymbol x } _ { j } \right) + \frac { 1 } { N _ { z } ^ { 2 } } \displaystyle \sum _ { i = 1 } ^ { N _ { z } } \sum _ { j = 1 } ^ { N _ { x } } k \left( { \boldsymbol x } _ { i } , { \boldsymbol x } _ { j } \right) \right) } \\ { \displaystyle \qquad = \frac { 1 } { 2 } \left( \frac { 1 } { N _ { y } } \displaystyle \sum _ { i = 1 } ^ { N _ { y } } \hat { \Phi } \left( y _ { i } \right) - \frac { 1 } { N _ { x } } \displaystyle \sum _ { i = 1 } ^ { N _ { z } } \hat { \Phi } \left( { \boldsymbol x } _ { i } \right) \right) \ : , } \\ { \displaystyle \hat { \mathbf { E } } ( a ) = - \nabla _ { a } \hat { \Phi } ( a ) = - \frac { 1 } { N _ { y } } \displaystyle \sum _ { i = 1 } ^ { N _ { y } } \nabla _ { a } k \left( a , y _ { i } \right) + \frac { 1 } { N _ { z } } \displaystyle \sum _ { i = 1 } ^ { N _ { z } } \nabla _ { a } k \left( a , { \boldsymbol x } _ { i } \right) } \\ { \displaystyle \hat { \mathbf { E } } ( y _ { i } ) = - N _ { y } \nabla _ { y } \hat { \boldsymbol { F } } ( \boldsymbol \rho ) \ : , \quad \hat { \mathbf { E } } ( { \boldsymbol x } _ { i } ) = N _ { x } \nabla _ { x } \hat { \boldsymbol { F } } ( \boldsymbol \rho ) \ : . } \end{array} +$$ + +These are again unbiased, e.g.: + +$$ +\begin{array} { l } { \displaystyle \mathbb { E } _ { \boldsymbol { x } , \boldsymbol { y } } ( \hat { \boldsymbol { \Phi } } ( a ) ) = \frac { 1 } { N _ { y } } \sum _ { i = 1 } ^ { N _ { y } } E _ { y _ { i } } \left( { k ( a , y _ { i } ) } \right) - \frac { 1 } { N _ { x } } \sum _ { i = 1 } ^ { N _ { x } } E _ { \alpha _ { \boldsymbol { x } } } \left( { k ( a , x _ { i } ) } \right) } \\ { \displaystyle = \frac { 1 } { N _ { y } } \sum _ { i = 1 } ^ { N _ { y } } \int p _ { y } ( y _ { i } ) k ( a , y _ { i } ) \mathrm { d } y _ { i } - \frac { 1 } { N _ { x } } \sum _ { i = 1 } ^ { N _ { x } } \int p _ { x } ( x _ { i } ) k ( a , x _ { i } ) \mathrm { d } x _ { i } } \\ { \displaystyle = \int p _ { y } ( y ) k ( a , y ) \mathrm { d } y - \int p _ { x } ( x ) k ( a , x ) \mathrm { d } x } \\ { \displaystyle = \int \left( p _ { y } ( { \boldsymbol x } ) - p _ { x } ( { \boldsymbol x } ) \right) k ( a , x ) \mathrm { d } x = \int \rho ( { \boldsymbol x } ) k ( a , x ) \mathrm { d } x = \Phi ( a ) } \end{array} +$$ + +If we draw samples of infinite size, all these expressions for a fixed sample size lead to the equivalent statements for densities. The sample-based formulation, that is, point charges in physical terms, can only have local energy minima or maxima at locations of samples (Dembo & Zeitouni, 1988). Furthermore the field lines originate and end at samples, therefore the field guides model samples $_ { \textbf { \em x } }$ toward real world samples $\textbf { { y } }$ , as depicted in Figure 1. The factors $N _ { y }$ and $N _ { x }$ in the last equations arise from the fact that $- \nabla _ { a } F$ gives the force which is applied to a sample with charge. A sample $\mathbf { \nabla } _ { \mathbf { \boldsymbol { y } } _ { i } }$ is positively charged with $1 / N _ { y }$ and follows $- \nabla _ { y _ { i } } F$ while a sample $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ is negatively charged with $- 1 / N _ { x }$ and therefore follows $- \nabla _ { x _ { i } } F$ , too. Thus, following the force induced on a sample by the field is equivalent to gradient descent of the energy $F$ with respect to samples $\mathbf { \nabla } _ { \mathbf { \boldsymbol { y } } _ { i } }$ and $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ . + +# A.5 MIXTURE OF GAUSSIANS + +We use the synthetic data set introduced by Lim & Ye (2017) to show that Coulomb GANs avoid mode collapse and that all modes of the target distribution are captured by the generative model. This data set comprises 100K data points drawn from a Gaussian mixture model of 25 components which are spread out evenly in the range $[ - 2 1 , 2 1 ] \times [ - 2 1 , 2 1 ]$ , with each component having a variance of 1. To make results comparable with Lim & Ye (2017), the Coulomb GAN used a discriminator network with 2 hidden layers of 128 units, however we avoided batch normalization by using the ELU activation function (Clevert et al., 2016). We used the Plummer kernel in 3 dimensions $\left( d = 3 \right)$ ) with an epsilon of 3 $( \epsilon = 3$ ) and a learning rate of 0.01, both of which were exponentially decayed during the 1M update steps of the Adam optimizer. + +As can be seen in Figure 4, samples from the learned Coulomb GAN very well approximate the target distribution. All components of the original distribution are present at the model distribution at approximately the correct ratio, as shown in Figure 5. Moreover, the generated samples are distributed approximately according to the same spread for each component of the real world distribution. Coulomb GANs outperform other compared methods, which either fail to learn the distribution completely, ignore some of the modes, or do not capture the within-mode spread of a Gaussian. The Coulomb GAN is the only GAN approach that manages to avoid a within-cluster collapse leading to insufficient variance within a cluster. + +![](images/fa5e6f290d134c857c7dc9309463595b79e6cd093e140da31f28e96cef3db92a.jpg) +Figure 4: Scatter plots of generated samples from different GAN variants for the mixture of 25 Gaussians and the true data distribution. “WD” indicates weight decay and “proj.” means projection. Results and images for all methods except the Coulomb GAN are taken from Lim & Ye (2017). + +![](images/dbaa54afbfcd435cfee5c4d78e4151bd0163c3bdc9fac64f71caeb8e585bb71d.jpg) +Figure 5: 2D histogram of the density of generated and the training st data for the mixture of 25 Gaussians. For constructing the histogram, 10k samples were drawn from the target and the model distribution. The Coulomb GAN captures the underlying distribution well, does not miss any modes, and places almost all probability mass on the modes. Only the Coulomb GAN captured the withinmode spread of the Gaussians. + +# A.6 PSEUDOCODE FOR COULOMB GANS + +The following gives the pseudo code for training GANs. Note that when calculating the derivative of $\hat { \Phi } ( a _ { i } ; \mathcal { X } , \mathcal { Y } )$ , it is important to only derive with respect to $\textbf { \em a }$ , and not wrt. $\mathcal { X } , \mathcal { y }$ , even if it can happen that e.g. $\mathbf { \pmb { a } } \in \mathcal { X }$ . In frameworks that offer automatic differentiation such as Tensorflow or Theano, this means stopping the possible gradient back-propagation through those parameters. + +Algorithm 1 Minibatch stochastic gradient descent training of Coulomb GANs for updating the the discriminator weights $\textbf { \em w }$ and the generator weights $\pmb { \theta }$ . + +while Stopping criterion not met do + +• Sample minibatch of $N _ { x }$ training samples $\{ \pmb { x } _ { 1 } , \ldots , \pmb { x } _ { N _ { x } } \}$ from training set • Sample minibatch of $N _ { y }$ generator samples $\{ \pmb { y } _ { 1 } , \hdots , \pmb { y } _ { N _ { y } } \}$ from the generator • Calculate the gradient for the discriminator weights: + +$$ +d \pmb { w } \gets \nabla _ { \pmb { w } } \left[ \frac { 1 } { 2 } \sum _ { i = 1 } ^ { N _ { x } } \left( D ( \pmb { x } _ { i } ) - \hat { \Phi } ( \pmb { x } _ { i } ) \right) ^ { 2 } + \frac { 1 } { 2 } \sum _ { i = 1 } ^ { N _ { y } } \left( D ( \pmb { y } _ { i } ) - \hat { \Phi } ( \pmb { y } _ { i } ) \right) ^ { 2 } \right] +$$ + +• Calculate the gradient for the generator weights: + +$$ +d \pmb \theta \gets \nabla _ { \pmb \theta } \left[ - \frac { 1 } { 2 } \frac { 1 } { N _ { x } } \sum _ { i = 1 } ^ { N _ { x } } D \left( \pmb x _ { i } \right) \right] +$$ + +• Update weights according to optimizer rule (e.g. Adam): + +$$ +\begin{array} { c } { { { \pmb w } _ { n + 1 } = { \pmb w } _ { n } + \tt { \mathbb { A D A M } } ( d { \pmb w } , n ) } } \\ { { { \pmb \theta } _ { n + 1 } = { \pmb \theta } _ { n } + \tt { \mathbb { A D A M } } ( d { \pmb \theta } , n ) } } \end{array} +$$ + +# end while + +# A.7 MORE SAMPLES FROM COULOMB GANS + +# CELEBA + +Images from a Coulomb GAN after training on CelebA data set. The low FID stems from the fact that the images show a wide variety of different faces, backgrounds, eye colors and orientations. + +![](images/ec91dd18e28e2c15242beed056e837da884f72fbd0a6a284d92120e29ee12473.jpg) + +# LSUN BEDROOMS + +Images from a Coulomb GAN after training on the LSUN bedroom data set. + +![](images/c59127bf47e0c4a5a6b28ed944bc10e060f21202d5ffd8c1805aa234d8090a11.jpg) + +# CIFAR 10 + +Images from a Coulomb GAN after training on the CIFAR 10 data set. + +![](images/179eacfb5a2647ef4180f7ee98394ad8bede0e448baa8389ca7b2fdb000e7dd3.jpg) \ No newline at end of file diff --git a/parse/train/SkVqXOxCb/SkVqXOxCb_content_list.json b/parse/train/SkVqXOxCb/SkVqXOxCb_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..44fe0a6b3421d4ad4bcd60119f3d547c2a11b553 --- /dev/null +++ b/parse/train/SkVqXOxCb/SkVqXOxCb_content_list.json @@ -0,0 +1,2336 @@ +[ + { + "type": "text", + "text": "COULOMB GANS: PROVABLY OPTIMAL NASH EQUILIBRIA VIA POTENTIAL FIELDS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Thomas Unterthiner1 Bernhard Nessler1 Calvin Seward1,2 Günter Klambauer1 ", + "bbox": [ + 184, + 167, + 812, + 184 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Martin Heusel1 ", + "bbox": [ + 184, + 204, + 290, + 219 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Hubert Ramsauer1 ", + "bbox": [ + 424, + 204, + 558, + 219 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sepp Hochreiter1 ", + "bbox": [ + 689, + 205, + 810, + 220 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1LIT AI Lab & Institute of Bioinformatics, Johannes Kepler University Linz, Austria 2Zalando Research, Mühlenstraße 25, 10243 Berlin, Germany {unterthiner,nessler,seward,klambauer,mhe,ramsauer,hochreit}@bioinf.jku.at ", + "bbox": [ + 186, + 255, + 908, + 299 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 335, + 544, + 351 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Generative adversarial networks (GANs) evolved into one of the most successful unsupervised techniques for generating realistic images. Even though it has recently been shown that GAN training converges, GAN models often end up in local Nash equilibria that are associated with mode collapse or otherwise fail to model the target distribution. We introduce Coulomb GANs, which pose the GAN learning problem as a potential field, where generated samples are attracted to training set samples but repel each other. The discriminator learns a potential field while the generator decreases the energy by moving its samples along the vector (force) field determined by the gradient of the potential field. Through decreasing the energy, the GAN model learns to generate samples according to the whole target distribution and does not only cover some of its modes. We prove that Coulomb GANs possess only one Nash equilibrium which is optimal in the sense that the model distribution equals the target distribution. We show the efficacy of Coulomb GANs on LSUN bedrooms, CelebA faces, CIFAR-10 and the Google Billion Word text generation. ", + "bbox": [ + 233, + 367, + 764, + 575 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 606, + 336, + 622 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Generative adversarial networks (GANs) (Goodfellow et al., 2014) excel at constructing realistic images (Radford et al., 2016; Ledig et al., 2016; Isola et al., 2017; Arjovsky et al., 2017; Berthelot et al., 2017) and text (Gulrajani et al., 2017). In GAN learning, a discriminator network guides the learning of another, generative network. This procedure can be considered as a game between the generator which constructs synthetic data and the discriminator which separates synthetic data from training set data (Goodfellow, 2017). The generator’s goal is to construct data which the discriminator cannot tell apart from training set data. GAN convergence points are local Nash equilibria. At these local Nash equilibria neither the discriminator nor the generator can locally improve its objective. ", + "bbox": [ + 174, + 638, + 825, + 763 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Despite their recent successes, GANs have several problems. First (I), until recently it was not clear if in general gradient-based GAN learning could converge to one of the local Nash equilibria (Salimans et al., 2016; Goodfellow, 2014; Goodfellow et al., 2014). It is even possible to construct counterexamples (Goodfellow, 2017). Second (II), GANs suffer from “mode collapsing”, where the model generates samples only in certain regions which are called modes. While these modes contain realistic samples, the variety is low and only a few prototypes are generated. Mode collapsing is less likely if the generator is trained with batch normalization, since the network is bound to create a certain variance among its generated samples within one batch (Radford et al., 2016; Chintala et al., 2016). However batch normalization introduces fluctuations of normalizing constants which can be harmful (Klambauer et al., 2017; Goodfellow, 2017). To avoid mode collapsing without batch normalization, several methods have been proposed (Che et al., 2017; Metz et al., 2016; Salimans et al., ", + "bbox": [ + 174, + 771, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "2016). Third (III), GANs cannot assure that the density of training samples is correctly modeled by the generator. The discriminator only tells the generator whether a region is more likely to contain samples from the training set or synthetic samples. Therefore the discriminator can only distinguish the support of the model distribution from the support of the target distribution. Beyond matching the support of distributions, GANs with proper objectives may learn to locally align model and target densities via averaging over many training examples. On a global scale, however, GANs fail to equalize model and target densities. The discriminator does not inform the generator globally where probability mass is missing. Consequently, standard GANs are not assured to capture the global sample density and are prone to neglect large parts of the target distribution. The next paragraph gives an example of this. Fourth (IV), the discriminator of GANs may forget previous modeling errors of the generator which then may reappear, a property that leads to oscillatory behavior instead of convergence (Goodfellow, 2017). ", + "bbox": [ + 174, + 103, + 825, + 270 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Recently, problem (I) was solved by proving that GAN learning does indeed converge when discriminator and generator are learned using a two time-scale learning rule (Heusel et al., 2017). Convergence means that the expected SGD-gradient of both the discriminator objective and the generator objective are zero. Thus, neither the generator nor the discriminator can locally improve, i.e., learning has reached a local Nash equilibrium. However, convergence alone does not guarantee good generative performance. It is possible to converge to sub-optimal solutions which are local Nash equilibria. Mode collapse is a special case of a local Nash equilibrium associated with suboptimal generative performance. For example, assume a two mode real world distribution where one mode contains too few and the other mode too many generator samples. If no real world samples are between these two distinct modes, then the discriminator penalizes to move generated samples outside the modes. Therefore the generated samples cannot be correctly distributed over the modes. Thus, standard GANs cannot capture the global sample density such that the resulting generators are prone to neglect large parts of the real world distribution. A more detailed example is listed in the Appendix in Section A.1. ", + "bbox": [ + 174, + 277, + 825, + 472 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we introduce a novel GAN model, the Coulomb GAN, which has only one Nash equilibrium. We are later going to show that this Nash equilibrium is optimal, i.e., the model distribution matches the target distribution. We propose Coulomb GANs to avoid the GAN shortcoming (II) to (IV) by using a potential field created by point charges analogously to the electric field in physics. The next section will introduce the idea of learning in a potential field and prove that its only solution is optimal. We will then show how learning the discriminator and generator works in a Coulomb GAN and discuss the assumptions needed for our optimality proof. In Section 3 we will then see that the Coulomb GAN does indeed work well in practice and that the samples it produces have very large variability and appear to capture the original distribution very well. ", + "bbox": [ + 174, + 478, + 825, + 603 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Related Work. Several GAN approaches have been suggested for bringing the target and model distributions in alignment using not just local discriminator information: Geometric GANs combine samples via a linear support vector machine which uses the discriminator outputs as samples, therefore they are much more robust to mode collapsing (Lim & Ye, 2017). Energy-Based GANs (Zhao et al., 2017) and their later improvement BEGANs (Berthelot et al., 2017) optimize an energy landscape based on auto-encoders. McGANs match mean and covariance of synthetic and target data, therefore are more suited than standard GANs to approximate the target distribution (Mroueh et al., 2017). In a similar fashion, Generative Moment Matching Networks (Li et al., 2015) and MMD nets (Dziugaite et al., 2015) directly optimize a generator network to match a training distribution by using a loss function based on the maximum mean discrepancy (MMD) criterion (Gretton et al., 2012). These approaches were later expanded to include an MMD criterion with learnable kernels and discriminators (Li et al., 2017). The MMD criterion that these later approaches optimize has a form similar to the energy function that Coulomb GANs optimize (cf. Eq. (33)). However, all MMD approaches end up using either Gaussian or Laplace kernels, which are not guaranteed to find the optimal solution where the model distribution matches the target distribution. In contrast, the Plummer kernel which is employed in this work has been shown to lead to the optimal solution (Hochreiter & Obermayer, 2005). We show that even a simplified version of the Plummer kernel, the low-dimensional Plummer kernel, ensures that gradient descent convergences to the optimal solution as stated by Theorem 1. Furthermore, most MMD GAN approaches use the MMD directly as loss function though the number of possible samples in a mini-batch is limited. Therefore MMD approaches face a sampling problem in high-dimensional spaces. The Coulomb GAN instead learns a discriminator network that gradually improves its approximation of the potential field via learning on many mini-batches. The discriminator network also tracks the slowly changing generator distribution during learning. Most importantly however, our approach is, to the best of our knowledge, the first one for which optimality, i.e., ability to perfectly learn a target distribution, can be proved. ", + "bbox": [ + 173, + 618, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/1dd35099253bbe238ca744ba58c3e5789a9a64144bbf15b96950a65d14400fb8.jpg", + "image_caption": [ + "Figure 1: The vector field of a Coulomb GAN. The basic idea behind the Coulomb GAN: true samples (blue) and generated samples (red) create a potential field (scalar field). Blue samples act as sinks that attract the red samples, which repel each other. The superimposed vector field shows the forces acting on the generator samples to equalize potential differences, and the background color shows the potential at each position. Best viewed in color. " + ], + "image_footnote": [], + "bbox": [ + 176, + 101, + 789, + 291 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 402, + 821, + 445 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The use of the Coulomb potential for learning is not new. Coulomb Potential Learning was proposed to store arbitrary many patterns in a potential field with perfect recall and without spurious patterns (Perrone & Cooper, 1995). Another related work is the Potential Support Vector Machine (PSVM), which minimizes Coulomb potential differences (Hochreiter & Mozer, 2001; Hochreiter et al., 2003). Hochreiter & Obermayer (2005) also used a potential function based on Plummer kernels for optimal unsupervised learning, on which we base our work on Coulomb GANs. ", + "bbox": [ + 174, + 452, + 825, + 536 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 COULOMB GANS ", + "text_level": 1, + "bbox": [ + 176, + 561, + 352, + 578 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 GENERAL CONSIDERATIONS ON GANS ", + "text_level": 1, + "bbox": [ + 176, + 597, + 486, + 611 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We assume data samples $\\pmb { a } \\in \\mathbb { R } ^ { m }$ for a model density $p _ { x } ( . )$ and a target density $p _ { y } ( . )$ . The goal of GAN learning is to modify the model in a way to obtain $\\dot { p } _ { x } ( . ) = p _ { y } \\bar { ( . ) }$ . We define the difference of densities $\\rho ( { \\pmb a } ) = p _ { y } ( { \\pmb a } ) - p _ { x } ( { \\pmb a } )$ which should be pushed toward zero for all $\\pmb { a } \\in \\mathbb { R } ^ { m }$ during learning. In the GAN setting, the discriminator $D ( { \\pmb a } )$ is a function $D : \\mathbb { R } ^ { m } \\mathbb { R }$ that learns to discriminate between generated and target samples and predicts how likely it is that $\\textbf { \\em a }$ is sampled from the target distribution. In conventional GANs, $D ( { \\pmb a } )$ is usually optimized to approximate the probability of seeing a target sample, or $\\rho ( { \\pmb a } )$ or some similar function. The generator $G ( z )$ is a continuous function $G : \\mathbb { R } ^ { n } \\mathbb { R } ^ { m }$ which maps some $n$ -dimensional random variable $_ z$ into the space of target samples. $_ z$ is typically sampled from a multivariate Gaussian or Uniform distribution. ", + "bbox": [ + 173, + 625, + 825, + 750 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In order to improve the generator, a GAN uses the gradient of the discriminator $\\nabla _ { a } D ( { \\pmb a } )$ with respect to the discriminator input $\\begin{array} { r } { \\pmb { a } = G ( \\pmb { z } ) } \\end{array}$ for learning. The objective of the generator is a scalar function $D ( G ( z ) )$ , therefore the gradient of the objective function is just a scaled version of the gradient $\\nabla _ { a } D ( { \\pmb a } )$ which would then propagate further to the parameters of $G$ . This gradient $\\nabla _ { a } D ( { \\pmb a } )$ tells the generator in which direction $\\rho ( { \\pmb a } )$ becomes larger, i.e., in which direction the ratio of target examples increases. The generator changes slightly so that $_ { z }$ is now mapped to a new $\\pmb { a } ^ { \\prime } = G ^ { \\prime } ( \\pmb { z } )$ , moving the sample generated by $_ { z }$ a little bit towards the direction where $\\rho ( { \\pmb a } )$ was larger, i.e., where target examples were more likely. However, $\\rho ( { \\pmb a } )$ and its derivative only take into account the local neighborhood of $\\textbf { \\em a }$ , since regions of the sample space that are distant from $\\textbf { \\em a }$ do not have much influence on $\\rho ( { \\pmb a } )$ . Regions of data space that have strong support in $p _ { y }$ but not in $p _ { x }$ will not be noticed by the generator via discriminator gradients. The restriction to local environments hampers GAN learning significantly (Arjovsky & Bottou, 2017; Arjovsky et al., 2017). ", + "bbox": [ + 173, + 757, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The theoretical analysis of GAN learning can be done at three different levels: (1) in the space of distributions $p _ { x }$ and $p _ { y }$ regardless of the fact that $p _ { x }$ is realized by $G$ and $p _ { z }$ , (2) in the space of functions $G$ and $D$ regardless of the fact that $G$ and $D$ are typically realized by a parametric form, i.e., as neural networks, or (3) in the space of the parameters of $G$ and $D$ . Goodfellow et al. (2014) use (1) to prove convergence of GAN learning in their Proposition 2 in a hypothetical scenario where the learning algorithm operates by making small, local moves in $p _ { x }$ space. In order to see that level (1) and (2) should both be understood as hypothetical scenarios, remember that in all practical implementations, $p _ { x }$ can only be altered implicitly by making small changes to the generator function G, which in turn can only be changed implicitly by small steps in its parameters. Even if we assume that the mapping from a distribution $p _ { x }$ to the generator $G$ that induced it exists and is unique, this mapping from $p _ { x }$ to the space of $G$ is not continuous. To see this, consider changing a distribution $p 1 _ { x }$ to a new distribution $p 2 _ { x }$ by moving a small amount $\\epsilon$ of its density to an isolated region in space where $p 1 _ { x }$ has no support. Let’s further assume this region has distance $d$ to any other regions of support of $p 1 _ { x }$ . By letting $\\epsilon 0$ , the distance between $p 1 _ { x }$ and $p 2 _ { x }$ becomes smaller, yet the distance between the inducing generator functions $G _ { 1 }$ and $G _ { 2 }$ (e.g. using the supremum norm on bounded functions) will not tend to zero because for at least one function input $_ { z }$ we have: $| G _ { 1 } ( z ) - G _ { 2 } ( z ) | \\geqslant d$ . Because of this, we need to go further than the distribution space when analyzing GAN learning. In practice, when learning GANs, we are restricted to small steps in parameter space, which in turn lead to small steps in function space and finally to small steps in distribution space. But not all small steps in distribution space can be realized this way as shown in the example above. This causes local Nash equilibria in the function space, because even though in distribution space it would be easy to escape by making small steps, such a step would require very large changes in function space and is thus not realizable. In this paper we show that Coulomb GANs do not exhibit any local Nash equilibria in the space of the functions $G$ and $D$ . To the best of our knowledge, this is the first formulation of GAN learning that can guarantee this property. Of course, Coulomb GANs are learned as parametrized neural networks, and as we will discuss in Subsection 2.4.2, Coulomb GANs are not immune to the usual issues that arise from parameter learning, such as over- and underfitting, which can cause local Nash Equilibria due to a bad choice of parameters. ", + "bbox": [ + 173, + 103, + 825, + 506 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 FROM CONVENTIONAL GANS TO POTENTIALS ", + "text_level": 1, + "bbox": [ + 174, + 537, + 539, + 551 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "If the density $p _ { x } ( . )$ or $p _ { y } ( . )$ approaches a Dirac delta-distribution, gradients vanish since the density approaches zero except for the exact location of data points. Similarly, electric point charges are often represented by Dirac delta-distributions, however the electric potential created by a point charge has influence everywhere in the space, not just locally. The electric potential (Coulomb potential) created by the point charge $Q$ is $\\begin{array} { r } { \\Phi _ { C } = \\frac { 1 } { 4 \\pi \\varepsilon _ { 0 } } \\frac { Q } { r } } \\end{array}$ , where $r$ is the distance to the location of $Q$ and $\\varepsilon _ { \\mathrm { 0 } }$ is the dielectric constant. Motivated by this electric potential, we introduce a similar concept for GAN learning: Instead of the difference of densities $\\rho ( { \\pmb a } )$ , we rather consider a potential function $\\Phi ( a )$ defined as ", + "bbox": [ + 173, + 569, + 825, + 685 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/3ae2241c970ffda5a67dbce6323a9674f212f803f67b31ee38cfa7655581d2f7.jpg", + "text": "$$\n\\Phi ( { \\pmb a } ) = \\int \\rho ( { \\pmb b } ) \\ k ( { \\pmb a } , { \\pmb b } ) \\ \\mathrm { d } { \\pmb b } ,\n$$", + "text_format": "latex", + "bbox": [ + 398, + 702, + 598, + 734 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "with some kernel $k \\left( a , b \\right)$ which defines the influence of a point at $^ { b }$ onto a point at $^ { a }$ . The crucial advantage of potentials $\\Phi ( a )$ is that each point can influence each other point in space if $k$ is chosen properly. If we minimize this potential $\\Phi ( a )$ we are at the same time minimizing the difference of densities $\\rho ( { \\pmb a } )$ : For all kernels $k$ it holds that if $\\rho ( \\pmb { b } ) = 0$ for all $^ { b }$ then $\\Phi ( { \\pmb a } ) = 0$ for all $^ { a }$ . We must still show that (i) $\\Phi ( { \\pmb a } ) = 0$ for all $\\textbf { \\em a }$ then $\\rho ( \\pmb { b } ) = 0$ for all $^ { b }$ , and even more importantly, (ii) whether a gradient optimization of $\\Phi ( a )$ leads to $\\Phi ( { \\pmb a } ) = 0$ for all $\\textbf { \\em a }$ . This is not the case for every kernel. Indeed only for particular kernels $k$ gradient optimization of $\\Phi ( a )$ leads to $\\rho ( \\pmb { b } ) = 0$ for all $^ { b }$ , that is, $p _ { x } ( b ) \\stackrel { \\cdot } { = } p _ { y } ( \\bar { b } )$ for all $^ { b }$ (Hochreiter & Obermayer, 2005) (see also Theorem 1 below). An example for such a kernel $k$ is the one leading to the Coulomb potential $\\Phi _ { C }$ from above, where $\\begin{array} { r } { k \\left( \\mathbf { { a } } , \\mathbf { { b } } \\right) \\ { \\stackrel { * } { = } } \\ { \\frac { 1 } { \\left\\| \\mathbf { { a } } - \\mathbf { { b } } \\right\\| } } } \\end{array}$ for $m = 3$ . As we will see in the following, the ability to have samples that influence each other over long distances, like charges in a Coulomb potential, will lead to GANs with a single, optimal Nash equilibrium. ", + "bbox": [ + 173, + 753, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 GANS AS ELECTRICAL FIELDS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 431, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For Coulomb GANs, the generator objective is derived from electrical field dynamics: real and generated samples generate a potential field, where samples of the same class (real vs. generated) repel each other, but attract samples of the opposite class. However, real data points are fixed in space, so the only samples that can move are the generated ones. In turn, the gradient of the potential with respect to the input samples creates a vector field in the space of samples. The generator can move its samples along the forces generated by this field. Such a field is depicted in Figure 1. The discriminator learns to predict the potential function, in order to approximate the current potential landscape of all samples, not just the ones in the current mini-batch. Meanwhile, the generator learns to distribute its samples across the whole field in such a way that the energy is minimized, thus naturally avoids mode collapse and covering the whole region of support of the data. The energy is minimal and equal to zero only if all potential differences are zero and the model distribution is equal to the target distribution. ", + "bbox": [ + 173, + 130, + 825, + 296 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Within an electrostatic field, the strength of the force on one particle depends on its distance to other particles and their charges. If left to move freely, the particles will organize themselves into a constellation where all forces equal out and no potential differences are present. For continuous charge distributions, the potential field is constant without potential differences if charges no longer move since forces are equaled out. If the potential field is constant, then the difference of densities $\\rho$ is constant, too. Otherwise the potential field would have local bumps. The same behavior is modeled within our Coulomb GAN, except that real and generated samples replace the positive and negative particles, respectively, and that the real data points remain fixed. Only the generated samples are allowed to move freely, in order to minimize $\\rho$ . The generated samples are attracted by real samples, so they move towards them. At the same time, generated samples should repel each other, so they do not clump together, which would lead to mode collapsing. ", + "bbox": [ + 173, + 303, + 825, + 457 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Analogously to electrostatics, the potential $\\Phi ( a )$ from Eq. (1) gives rise to a field $\\begin{array} { l l } { \\mathbf { E } ( \\pmb { a } ) } & { = } \\end{array}$ $- \\nabla _ { a } \\bar { \\Phi } ( a )$ . and to an energy function $\\begin{array} { r } { F \\left( \\rho \\right) = \\frac { 1 } { 2 } \\int \\rho ( \\mathbf { { a } } ) \\Phi ( \\mathbf { { a } } ) \\mathrm { { d } } \\mathbf { { a } } } \\end{array}$ . The field ${ \\bf E } ( a )$ applies a force on charges at $^ { a }$ which pushes the charges toward lower energy constellations. Ultimately, the Coulomb GAN aims to make the potential $\\Phi$ zero everywhere via the field ${ \\bf E } ( a )$ , which is the negative gradient of $\\Phi$ . For proper kernels $k$ , it can be shown that (i) $\\Phi$ can be pushed to zero via its negative gradient given by the field and (ii) that $\\Phi ( { \\pmb a } ) = 0$ for all $\\textbf { \\em a }$ implies $\\bar { { \\rho } } ( { \\pmb a } ) = 0$ for all $^ { a }$ , therefore, $\\bar { p } _ { x } ( { \\pmb a } ) = \\bar { p } _ { y } ( { \\pmb a } )$ for all $^ { a }$ (Hochreiter & Obermayer, 2005) (see also Theorem 1 below). ", + "bbox": [ + 174, + 463, + 825, + 560 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.3.1 LEARNING PROCESS ", + "text_level": 1, + "bbox": [ + 174, + 577, + 370, + 590 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "During learning we do not change $\\Phi$ or $\\rho$ directly. Instead, the location $\\begin{array} { r } { \\pmb { a } = G ( \\pmb { z } ) } \\end{array}$ to which the random variable $_ z$ is mapped changes to a new location $\\pmb { a } ^ { \\prime } = G ^ { \\prime } ( \\pmb { z } )$ . For the GAN optimization dynamics, we assume that generator samples $\\pmb { a } = G ( \\pmb { z } )$ can move freely, which is ensured by a sufficiently complex generator. Importantly, generator samples originating from random variables $_ { z }$ do neither disappear nor are they newly created but are conserved. This conservation is expressed by the continuity equation (Schwartz, 1972) that describes how the difference between distributions $\\rho ( { \\pmb a } )$ changes as the particles are moving along the field, i.e., how moving samples during the learning process changes our densities: ", + "bbox": [ + 173, + 601, + 825, + 712 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/712b6d9fbb6f08ef9ae9c964070bbad5f0161d092772ddf3ffe9cdf34b8d98a1.jpg", + "text": "$$\n{ \\dot { \\rho } } ( { \\mathbf { a } } ) \\ = \\ - \\nabla \\cdot ( \\rho ( { \\mathbf { a } } ) \\ v ( { \\mathbf { a } } ) )\n$$", + "text_format": "latex", + "bbox": [ + 401, + 719, + 594, + 737 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "for sample density difference $\\rho$ and unit charges that move with “velocity” ${ \\pmb v } ( { \\pmb a } ) = \\mathrm { s i g n } ( \\rho ( { \\pmb a } ) ) { \\bf E } ( { \\pmb a } )$ . The continuity equation is crucial as it establishes the connection between moving samples and changing the generator density and thereby $\\rho$ . The sign function of the velocity indicates whether positive or negative charges are present at $\\textbf { \\em a }$ . The divergence operator $\\mathbf { \\omega } ^ { 6 6 } \\nabla \\cdot \\mathbf { \\vec { \\mu } } ^ { 5 }$ determines whether samples move toward or outward of $\\textbf { \\em a }$ for a given field. Basically, the continuity equation says that if the generator density increases, then generator samples must flow into the region and if the generator density decreases, they flow outwards. We assume that differently charged particles cancel each other. If generator samples are moved away from a location $\\textbf { \\em a }$ then $\\rho ( { \\pmb a } )$ is increasing while $\\rho ( { \\pmb a } )$ is decreasing when generator samples are moved toward $\\textbf { \\em a }$ . The continuity equation is also obtained as a first order ODE to move particles in a potential field (Dembo & Zeitouni, 1988), therefore describes the dynamics how the densities are changing. We obtain ", + "bbox": [ + 173, + 742, + 825, + 897 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6ba26ccc4ecb901e2bcd5fc975516ee4416a1c26e3d1d53abdc1824620d2dbcd.jpg", + "text": "$$\n\\dot { \\rho } ( a ) = - \\mathrm { s i g n } ( \\rho ( a ) ) \\nabla \\cdot \\left( \\rho ( a ) { \\bf E } ( a ) \\right) = - \\nabla \\cdot \\left( | \\rho ( a ) | { \\bf E } ( a ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 277, + 904, + 720, + 921 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The density difference $\\rho ( { \\pmb a } )$ indicates how many samples are locally available for being moved. At each local minimum and local maximum $\\textbf { \\em a }$ of $\\rho$ we obtain $\\pmb { \\nabla } _ { a } \\rho ( \\pmb { a } ) = 0$ . Using the product rule for the divergence operator, at points $^ { a }$ that are minima or maxima, Eq. (3) reduces to ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/88ed120b3851cb0c678506878af2392cc7881ee8ca90e04abb48442463aa11be.jpg", + "text": "$$\n\\dot { \\rho } ( { \\pmb a } ) = - \\operatorname { s i g n } ( \\rho ( { \\pmb a } ) ) \\rho ( { \\pmb a } ) { \\pmb \\nabla } \\cdot { \\bf E } ( { \\pmb a } ) .\n$$", + "text_format": "latex", + "bbox": [ + 366, + 151, + 630, + 169 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In order to ensure that $\\rho$ converges to zero, it is necessary and sufficient that $\\mathrm { s i g n } ( \\pmb { \\nabla } \\cdot \\mathbf { E } ( \\pmb { a } ) ) =$ $\\mathrm { s i g n } ( \\rho ( a ) )$ , where $\\div a \\rho ( { \\pmb a } ) = 0$ , as this condition ensures the uniform decrease of the maximal absolute density differences $\\left| \\rho ( \\pmb { a } _ { \\mathrm { m a x } } ) \\right|$ . ", + "bbox": [ + 173, + 174, + 825, + 218 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2.3.2 CHOICE OF KERNEL ", + "text_level": 1, + "bbox": [ + 174, + 231, + 369, + 246 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "As discussed before, the choice of kernel is crucial for Coulomb GANs. The $m$ -dimensional Coulomb kernel and the $m$ -dimensional Plummer kernel lead to (i) $\\Phi$ that is pushed to zero via the field it creates and (ii) that $\\Phi ( { \\pmb a } ) = 0$ for all $^ { a }$ implies $\\rho ( { \\pmb a } ) = 0$ for all $\\textbf { \\em a }$ , therefore, $p _ { x } ( { \\pmb a } ) = p _ { y } ( { \\pmb a } )$ for all $\\textbf { \\em a }$ (Hochreiter & Obermayer, 2005). Thus, gradient learning with these kernels has been proved to converge to an optimal solution. However, both the $m$ -dimensional Coulomb and the $m$ - dimensional Plummer kernel lead to numerical instabilities if $m$ is large. Therefore the Coulomb potential $\\Phi ( a )$ for the Coulomb GAN was constructed by a low-dimensional Plummer kernel $k$ with parameters $d \\leqslant m - 2$ and $\\epsilon$ : ", + "bbox": [ + 173, + 255, + 825, + 367 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/56db4c17c45804a613052896709a2f17549f1be20516bfba6e49480e2963ac21.jpg", + "text": "$$\n\\Phi ( { \\pmb a } ) = \\int \\rho ( { \\pmb b } ) ~ k ~ ( { \\pmb a } , { \\pmb b } ) ~ \\mathrm { d } { \\pmb b } , \\quad k ( { \\pmb a } , { \\pmb b } ) = \\frac { 1 } { ( \\sqrt { \\| { \\pmb a } - { \\pmb b } \\| ^ { 2 } + \\epsilon ^ { 2 } } ) ^ { d } } .\n$$", + "text_format": "latex", + "bbox": [ + 271, + 372, + 725, + 409 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The original Plummer kernel is obtained with $d = m - 2$ . The resulting field and potential energy is ", + "bbox": [ + 176, + 412, + 821, + 429 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/f6d1d16079797a929b6637715e8a5eb46b11c0b1ea7460f33a61f49edde26536.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbf { E } ( \\pmb { a } ) = - \\int \\rho ( \\pmb { b } ) \\nabla _ { a } k ( \\pmb { a } , \\pmb { b } ) \\mathrm { d } \\pmb { b } = - \\nabla _ { a } \\Phi ( \\pmb { a } ) \\ , } \\\\ { \\displaystyle F ( \\rho ) = \\frac { 1 } { 2 } \\int \\rho ( \\pmb { a } ) \\Phi ( \\pmb { a } ) \\mathrm { d } \\pmb { a } = \\frac { 1 } { 2 } \\int \\displaystyle \\int \\rho ( \\pmb { a } ) \\rho ( \\pmb { b } ) k ( \\pmb { a } , \\pmb { b } ) \\mathrm { d } \\pmb { b } \\mathrm { d } \\pmb { a } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 434, + 733, + 501 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The next theorem states that for freely moving generated samples, $\\rho$ converges to zero, that is, $p _ { x } ( . ) = p _ { y } ( . )$ , when using this potential function $\\Phi ( a )$ . ", + "bbox": [ + 173, + 503, + 823, + 532 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 1 (Convergence with low-dimensional Plummer kernel). For $\\pmb { a } , \\pmb { b } \\in \\mathbb { R } ^ { m }$ , $d \\leqslant m - 2$ , and $\\epsilon > 0$ the densities $p _ { x } ( . )$ and $p _ { y } ( . )$ equalize over time when minimizing energy $F$ with the low-dimensional Plummer kernel by gradient descent. The convergence is faster for larger $d$ . ", + "bbox": [ + 174, + 536, + 825, + 579 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proof. See Section A.2. ", + "bbox": [ + 174, + 593, + 333, + 608 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "2.4 DEFINITION OF THE COULOMB GAN ", + "text_level": 1, + "bbox": [ + 176, + 625, + 472, + 640 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The Coulomb GAN minimizes the electric potential energy from Eq. (6) using a stochastic gradient descent based approach using mini-batches. Appendix Section A.4 contains the equations for the Coulomb potential, field, and energy in this case. Generator samples are obtained by drawing $N _ { x }$ random numbers $z _ { i }$ and transforming them into outputs $\\pmb { x } _ { i } = G ( z _ { i } )$ . Each mini-batch also includes $N _ { y }$ real world samples $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { i } }$ . This gives rise to a mini-batch specific potential, where in Eq. (5) we use $\\rho ( { \\mathbf { \\bar { a } } } ) = p _ { y } ( { \\mathbf { \\bar { a } } } ) - \\bar { p _ { x } } ( { \\mathbf { \\bar { a } } } )$ and replace the expectations by empirical means using the drawn samples: ", + "bbox": [ + 173, + 650, + 825, + 736 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d6e3502e25c387929a6ea75f2ce5f5658eef5f5264cedaffd4510ae497057d6a.jpg", + "text": "$$\n\\hat { \\Phi } ( { \\pmb a } ) = \\frac { 1 } { N _ { y } } \\sum _ { i = 1 } ^ { N _ { y } } k \\left( { \\pmb a } , { \\pmb y } _ { i } \\right) - \\frac { 1 } { N _ { x } } \\sum _ { i = 1 } ^ { N _ { x } } k \\left( { \\pmb a } , { \\pmb x } _ { i } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 742, + 663, + 786 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "It is tempting to have a generator network that directly minimizes this potential $\\hat { \\Phi }$ between generated and training set points. In fact, we show that $\\hat { \\Phi }$ is an unbiased estimate for $\\Phi$ in Appendix Section A.4. However, the estimate has very high variance: for example, if a mini-batch fails to sample training data from an existing mode, the field would drive all generated samples that have been generated at this mode to move elsewhere. The high variance has to be counteracted by extremely low learning rates, which makes learning infeasible in practice, as confirmed by initial experiments. Our solution to this problem is to have a network that generalizes over the mini-batch specific potentials: each mini-batch contains different generator samples $\\mathcal { X } = \\pmb { x } _ { i }$ for $i = 1 , \\ldots , N _ { x }$ and real world samples $\\mathbf { \\boldsymbol { y } } = \\mathbf { \\boldsymbol { y } } _ { i }$ for $i = 1 , \\ldots , N _ { y }$ , they create a batch-specific potential $\\hat { \\Phi }$ . The goal of the discriminator is to learn $\\operatorname { E } _ { \\mathcal { X } , \\mathcal { Y } } ( \\hat { \\Phi } ( \\pmb { a } ) ) = \\Phi ( \\pmb { a } )$ , i.e., the potential averaged over many mini-batches. Thus the discriminator function $D$ fulfills a similar role as other typical GAN discriminator functions, i.e., it discriminates between real and generated data such that for any point in space $^ { a }$ , $D ( { \\pmb a } )$ should be greater than zero if the $p _ { y } ( { \\pmb a } ) > p _ { x } ( { \\pmb a } )$ and smaller than zero otherwise. In particular $D ( { \\pmb a } )$ also indicates, via its gradient and its potential properties, directions toward regions where training set samples are predominant and where generator samples are predominant. ", + "bbox": [ + 173, + 792, + 825, + 925 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 102, + 825, + 188 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The generator in turn tries to move all of its samples according to the vector field into areas where generator samples are missing and training set samples are predominant. The generator minimizes the approximated energy $F$ as predicted by the discriminator. The loss $\\mathcal { L } _ { D }$ for the discriminator and $\\mathcal { L } _ { G }$ for the generator are given by: ", + "bbox": [ + 176, + 194, + 823, + 251 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/97f2ba32147917e970f1648ca0f9a1733d666f3cb8197644b1dd6e4bf68b157d.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\mathcal L } _ { D } ( D ; G ) = \\frac { 1 } { 2 } \\mathrm { E } _ { p _ { a } } \\left( ( D ( { \\pmb a } ) - \\frac { \\hat { \\Phi } ( { \\pmb a } ) ) ^ { 2 } } { 2 } \\right) } } \\\\ { { \\displaystyle { \\mathcal L } _ { G } ( G ; D ) = - \\frac { 1 } { 2 } \\mathrm { E } _ { p _ { z } } \\left( D ( G ( { \\pmb z } ) ) \\right) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 256, + 642, + 319 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Where $\\begin{array} { r } { p ( { \\pmb a } ) = 1 / 2 \\int { \\mathcal N } ( { \\pmb a } ; G ( z ) , \\epsilon I ) p _ { z } ( z ) \\mathrm { d } z + 1 / 2 \\int { \\mathcal N } ( { \\pmb a } ; { \\pmb y } , \\epsilon I ) p _ { y } ( { \\pmb y } ) \\mathrm { d } { \\pmb y } , } \\end{array}$ , i.e., a distribution where each point of support both of the generator and the real world distribution is surrounded with a Gaussian ball of width $\\epsilon I$ similar to Bishop et al. (1998), in order to overcome the problem that the generator distribution is only a sub-manifold of $\\mathbb { R } ^ { m }$ . These loss functions cause the approximated potential values $D ( { \\pmb a } )$ that are negative are pushed toward zero. Finally, the Coulomb GAN, like all other GANs, consists of two parts: a generator to generate model samples, and a discriminator that provides its learning signal. Without a discriminator, our would be very similar to GMMNs (Li et al., 2015), as can be seen in Eq. (33), but with an optimal Kernel specifically tailored to the problem of estimating differences between probability distributions. ", + "bbox": [ + 173, + 321, + 825, + 450 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We use each mini-batch only for one update of the discriminator and the generator. It is important to note that the discriminator uses each sample in the mini batch twice: once as a point to generate the mini-batch specific potential $\\hat { \\Phi }$ , and once as a point in space for the evaluation of the potential $\\hat { \\Phi }$ and its approximation $D$ . Using each sample twice is done for performance reasons, but not strictly necessary: the discriminator could learn the potential field by sampling points that lie between generator and real samples as in Gulrajani et al. (2017), but we are mainly interested in correct predictions in the vicinity of generator samples. Pseudocode for the learning algorithm is detailed in Algorithm 1 in the appendix. ", + "bbox": [ + 173, + 455, + 825, + 570 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "2.4.1 OPTIMALITY OF THE SOLUTION ", + "text_level": 1, + "bbox": [ + 176, + 585, + 449, + 599 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Convergence of the GAN learning process was proved for a two time-scales update rule by Heusel et al. (2017). A local Nash equilibrium is a pair of generator and discriminator $( D ^ { * } , G ^ { * } )$ that fulfills the two conditions ", + "bbox": [ + 174, + 609, + 825, + 651 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/510be43e1dd7e0eb883459d3754a4698dc11046ee12a843bdcf343ff363c173b.jpg", + "text": "$$\nD ^ { * } = \\arg \\operatorname* { m i n } _ { D \\in U ( D ^ { * } ) } \\mathcal { L } _ { D } ( D ; G ^ { * } ) \\quad \\mathrm { a n d } \\quad G ^ { * } = \\arg \\operatorname* { m i n } _ { G \\in U ( G ^ { * } ) } \\mathcal { L } _ { G } ( G ; D ^ { * } ) .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 657, + 712, + 688 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "for some neighborhoods $U ( D ^ { * } )$ and $U ( G ^ { * } )$ . We show in the following Theorem 2 that for Coulomb GANs every local Nash equilibrium necessarily is identical to the unique global Nash equilibrium. In other words, any equilibrium point of the Coulomb GAN that is found to be local optimal has to be the one global Nash equilibrium as the minimization of the energy $F ( \\rho )$ in Eq. (33) leads to a single, global optimum at $p _ { y } = p _ { x }$ . ", + "bbox": [ + 174, + 694, + 825, + 765 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 2 (Optimal Solution). If the pair $( D ^ { * } , G ^ { * } )$ is a local Nash equilibrium for the Coulomb GAN objectives, then it is the global Nash equilibrium, and no other local Nash equilibria exist, and G∗ has output distribution px = py. ", + "bbox": [ + 174, + 768, + 825, + 813 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Proof. See Appendix Section A.3. ", + "bbox": [ + 176, + 827, + 400, + 842 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "2.4.2 COULOMB GANS IN PRACTICE ", + "text_level": 1, + "bbox": [ + 176, + 856, + 446, + 871 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To implement GANs in practice, we need learnable models for $G$ and $D$ . We assume that our models for $G$ and $D$ are continuously differentiable with respect to their parameters and inputs. Toward this end, GANs are typically implemented as neural networks optimized by (some variant of) gradient descent. Thus we may not find the optimal $G ^ { * }$ or $D ^ { * }$ , since neural networks may suffer from capacity or optimization issues. Recent research indicates that the effect of local minima in deep learning vanishes with increasing depth (Dauphin et al., 2014; Choromanska et al., 2015; Kawaguchi, 2016), such that this limitation becomes less restrictive as our ability to train deep networks grows thanks to hardware and optimization improvements. ", + "bbox": [ + 176, + 881, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 172 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The main problem with learning Coulomb GANs is to approximate the potential function $\\Phi$ , which is a complex function in a high-dimensional space, since the potential can be very non-linear and non-smooth. When learning the discriminator, we must ensure that enough data is sampled and averaged over. We already lessened the non-linear function problem by using a low-dimensional Plummer kernel. But still, this kernel can introduce large non-linearities if samples are close to each other. It is crucial that the discriminator learns slow enough to accurately estimate the potential function which is induced by the current generator. The generator, in turn, must be even slower since it must be tracked by the discriminator. These approximation problems are supposed to be tackled by the research community in near future, which would enable optimal GAN learning. ", + "bbox": [ + 174, + 180, + 825, + 305 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The formulation of GAN learning as a potential field naturally solves the mode collapsing issue: the example described in Section A.1, where a normal GAN cannot get out of a local Nash equilibria is not a converged solution for the Coulomb GAN: If all probability mass of the generator lies in one of the modes, then both attracting forces from real-world samples located at the other mode as well as repelling forces from the over-represented generator mode will act upon the generator until it generates samples at the other mode as well. ", + "bbox": [ + 174, + 313, + 825, + 396 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 417, + 326, + 434 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In all of our experiments, we used a low-dimensional Plummer Kernel of dimensionality $d = 3$ . This kernel both gave best computational performance and has low risk of running into numerical issues. We used a batch size of 128. To evaluate the quality of a GAN, the FID metric as proposed by Heusel et al. (2017) was calculated by using $5 0 \\mathrm { k }$ samples drawn from the generator, while the training set statistics were calculated using the whole training set. We compare to BEGAN (Berthelot et al., 2017), DCGAN (Radford et al., 2016) and WGAN-GP (Gulrajani et al., 2017) both in their original version as well as when using the two-timescale update-rule (TTUR), using the settings from Heusel et al. (2017). We additionally compare to MMD-GAN (Li et al., 2017), which is conceptually very similar to the Coulomb GAN, but uses a Gaussian Kernel instead of the Plummer Kernel. We use the dataset-specific settings recommended in (Li et al., 2017) and report the highest FID score over the course of training. All images shown in this paper were produced with a random seed and not cherry picked. The implementation used for these experiments is available online1. The appendix Section A.5 contains an additional toy example demonstrating that Coulomb GANs do not suffer from mode collapse when fitting a simple Gaussian Mixture of 25 components. ", + "bbox": [ + 174, + 449, + 825, + 643 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.1 IMAGE DATASETS ", + "text_level": 1, + "bbox": [ + 176, + 662, + 338, + 676 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To demonstrate the ability of the Coulomb GAN to learn distributions in high dimensional spaces, we trained a Coulomb GAN on several popular image data sets: The cropped and centered images of celebrities from the Large-scale CelebFaces Attributes (“CelebA”) data set (Liu et al., 2015), the LSUN bedrooms data set consists of over 3 million 64x64 pixel images of the bedrooms category of the large scale image database LSUN (Yu et al., 2015) as well as the CIFAR-10 data set. For these experiments, we used the DCGAN architecture (Radford et al., 2016) with a few modifications: our convolutional kernels all have a kernel size of 5x5, our random seed that serves as input to the generator has fewer dimensions: 32 for CelebA and LSUN bedrooms, and 16 for CIFAR-10. Furthermore, the discriminator uses twice as many feature channels in each layer as in the DCGAN architecture. For the Plummer kernel, $\\epsilon$ was set to 1. We used the Adam optimizer with a learning rate of $1 0 ^ { - 4 }$ for the generator and $5 \\cdot 1 0 ^ { - 5 }$ for the discriminator. To improve convergence performance, we used the tanh output activation function (LeCun et al., 1998). For regularization we used an L2 weight decay term with a weighting factor of $1 0 ^ { - 7 }$ . Learning was stopped by monitoring the FID metric (Heusel et al., 2017). Once learning plateaus, we scaled the learning rate down by a factor of 10 and let it continue once more until the FID plateaus. The results are reported in Table 1b, and generated images can be seen in Figure 2 and in the Appendix in Section A.7. Coulomb GANs tend to outperform standard GAN approaches like BEGAN and DCGAN, but are outperformed by the Improved Wasserstein GAN. However it is important to note that the Improved Wasserstein GAN used a more advanced network architecture based on ResNet blocks (Gulrajani et al., 2017), which we could not replicate due to runtime constraints. Overall, the low FID of Coulomb GANs stem from the fact that the images show a wide variety of different samples. E.g. on CelebA, Coulomb GAN exhibit a very wide variety of faces, backgrounds, eye colors and orientations. To further investigate how much variation the samples generated by the Coulomb GAN contains, we followed the advice of Arora and Zhang (Arora & Zhang, 2017) to estimate the support size of the generator’s distribution by checking how large a sample from the generator must be before we start generating duplicates. We were able to generate duplicates with a probability of around $50 \\%$ when using samples of size 1024, which indicates that the support size learned by the Coulomb GAN would be around 1M. This is a strong indication that the Coulomb GAN was able to spread out its samples over the whole target distribution. A depiction is included in Figure 3, which also shows the nearest neighbor in the training set of the generated images, confirming that the Coulomb GAN does not just memorize training images. ", + "bbox": [ + 174, + 689, + 825, + 897 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 202 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/dc7625461a0dced041941f6df56258aa8ed1a05bcf3a0ec34978702ca93440d3.jpg", + "image_caption": [ + "Figure 2: Images from a Coulomb GAN after training on CelebA (first row), LSUN bedrooms (second row) and CIFAR 10 (last row). Further examples are located in the appendix in Section A.7 " + ], + "image_footnote": [], + "bbox": [ + 238, + 218, + 759, + 342 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/6a9c0c1703f8938cefd242d27336d75059e77e543cafd91e08231266b5b6539c.jpg", + "image_caption": [ + "Figure 3: The most similar pairs found in batches of 1024 generated faces sampled from the Coulomb GAN, and the nearest neighbor from the training data shown as third image. Distances were calculated as Euclidean distances on pixel level. " + ], + "image_footnote": [], + "bbox": [ + 240, + 420, + 759, + 604 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 679, + 825, + 805 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.2 LANGUAGE MODELING ", + "text_level": 1, + "bbox": [ + 176, + 825, + 377, + 840 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We repeated the experiments from Gulrajani et al. (2017), where Improved Wasserstein GANs (WGAN-GP) were trained to produce text samples after being trained on the Google Billion Word data set (Chelba et al., 2013), using the same network architecture as in the original publication. We use the Jensen-Shannon-divergence on 4-grams and 6-grams as an evaluation criterion. The results are summarized in Table 1a. ", + "bbox": [ + 174, + 854, + 823, + 922 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/545a507dc4ce24d4ec35e0176eb35b655c8d7520253617dc38d3ef7baad02bb7.jpg", + "table_caption": [ + "(a) Normalized Jensen-ShanonDivergence for the Google Billion Word data. Values for WGANGP are without/with TTUR, taken from Heusel et al. (2017). " + ], + "table_footnote": [], + "table_body": "
data setWGAN-GPours
4 grams0.38 / 0.350.35
6 grams0.77 / 0.740.74
", + "bbox": [ + 173, + 101, + 377, + 166 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/dc88946d72897e179b6227c166320bf0f58e1c444365ae513aeb472fcd3a0d46.jpg", + "table_caption": [], + "table_footnote": [ + "(b) Performance comparison in FID (lower is better) on different data sets. Values for all methods except Coulomb GAN and MMD-GAN are from without/with TTUR, taken from Heusel et al. (2017). " + ], + "table_body": "
data setBEGANDCGANWGAN-GPMMDours
CelebA29.2 /28.521.4 / 12.54.8 / 4.263.29.3
LSUN113 /11270.4 / 57.520.5 / 9.594.931.2
CIFAR10--29.3 /24.838.227.3
", + "bbox": [ + 406, + 102, + 841, + 167 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "4 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 263, + 318, + 279 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The Coulomb GAN is a generative adversarial network with strong theoretical guarantees. Our theoretical results show that the Coulomb GAN will be able to approximate the real distribution perfectly if the networks have sufficient capacity and training does not get stuck in local minima. Our results show that the potential field used by the Coulomb GAN far outperforms MMD based approaches due to its low-dimensional Plummer kernel, which is better suited for modeling probability density functions, and is very effective at eliminating the mode collapse problem in GANs. This is because our loss function forces the generated samples to occupy different regions of the learned distribution. In practice, we have found that Coulomb GANs are able to produce a wide range of different samples. However, in our experience, this sometimes leads to a small number of generated samples that are non-sensical interpolations of existing data modes. While these are sometimes also present in other GAN models (Radford et al., 2016), we found that our model produces such images at a slightly higher rate. This issue might be solved by finding better ways of learning the discriminator, as learning the correct potential field is crucial for the Coulomb GAN’s performance. We also observed that increasing the capacity of the discriminator seems to always increase the generative performance. We thus hypothesize that the largest issue in learning Coulomb GANs is that the discriminator needs to approximate the potential field $\\Phi$ very well in a high-dimensional space. In summary, instead of directly optimizing a criterion based on local differences of densities which can exhibit many local minima, Coulomb GANs are based on a potential field that has no local minima. The potential field is created by point charges in an analogy to electric field in physics. We have proved that if learning converges then it converges to the optimal solution if the samples can be moved freely. We showed that Coulomb GANs avoid mode collapsing, model the target distribution more truthfully than standard GANs, and do not overlook high probability regions of the target distribution. ", + "bbox": [ + 173, + 295, + 825, + 614 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 638, + 356, + 652 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was supported by Zalando SE with Research Agreement 01/2016, Audi.JKU Deep Learning Center, Audi Electronic Venture GmbH, IWT research grant IWT150865 (Exaptation), H2020 project grant 671555 (ExCAPE) and NVIDIA Corporation. The authors would like to thank Philipp Renz for fruitful discussions. ", + "bbox": [ + 174, + 670, + 825, + 726 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 748, + 285, + 763 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "M. Arjovsky and L. Bottou. Towards principled methods for training generative adversarial networks. International Conference on Learning Representations (ICLR), 2017. \nM. Arjovsky, S. Chintala, and L. Bottou. 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GANs trained by a two time-scale update rule converge to a Nash equilibrium. ArXiv e-prints, 2017. ", + "bbox": [ + 168, + 751, + 825, + 781 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "S. Hochreiter and M. C. Mozer. Coulomb classifiers: Reinterpreting SVMs as electrostatic systems. Technical Report CU-CS-921-01, Department of Computer Science, University of Colorado, Boulder, 2001. ", + "bbox": [ + 171, + 790, + 823, + 833 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "S. Hochreiter and K. Obermayer. Optimal kernels for unsupervised learning. In Proceedings of the IEEE International Joint Conference on Neural Networks, volume 3, pp. 1895–1899, 2005. ", + "bbox": [ + 169, + 842, + 825, + 872 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "S. Hochreiter, M. C. Mozer, and K. Obermayer. Coulomb classifiers: Generalizing support vector machines via an analogy to electrostatic systems. 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", + "bbox": [ + 169, + 103, + 830, + 809 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 102, + 297, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 EXAMPLE OF CONVERGENCE TO MODE COLLAPSE IN CONVENTIONAL GANS ", + "bbox": [ + 173, + 137, + 758, + 151 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "As an example of how a GAN can converge to a Nash Equilibrium that exhibits mode collapse, consider a target distribution that consists of two distinct/non-overlapping regions of support $C _ { 1 }$ and $C _ { 2 }$ that are distant from each other, i.e., the target probability is zero outside of $C _ { 1 }$ and $C _ { 2 }$ . Further assume that $50 \\%$ of the probability mass is in $C _ { 1 }$ and $50 \\%$ in $C _ { 2 }$ . Assume that the the generator has mode-collapsed onto $C _ { 1 }$ , which contains $100 \\%$ of the generator’s probability mass. In this situation, the optimal discriminator classifies all points from $C _ { 2 }$ as ”real“ (pertaining to the target distribution) by supplying an output of 1 for them (1 is the target for real samples and 0 the target for generated samples). Within $C _ { 1 }$ , the other region, the discriminator sees twice as many generated data points as real ones, as $100 \\%$ of the probability mass of the generator’s distribution is in $C _ { 1 }$ , but only $50 \\%$ of the probability mass of the real data distribution. So one third of the points seen by the discriminator in $C _ { 1 }$ are real, the other 2 thirds are generated. Thus, to minimize its prediction error for a proper objective (squared or cross entropy), the discriminator has to output $1 / 3$ for every point from $C _ { 1 }$ . The optimal output is even independent of the exact form of the real distribution in $C _ { 1 }$ . The generator will match the shape of the target distribution locally. If the shape is not matched, local gradients of the discriminator with respect to its input would be present and the generator would improve locally. If local improvements of the generator are no longer possible, the shape of the target distribution is matched and the discriminator output is locally constant. In this situation, the expected gradient of the discriminator is the zero vector, because it has reached an optimum. Since the discriminator output is constant in $C _ { 1 }$ (and $C _ { 2 }$ ), the generator’s expected gradient is the zero vector, too. The situation is also stable even though we still have random fluctuations from the ongoing stochastic gradient (SGD) learning: whenever the generator produces data outside of (but close to) $C _ { 1 }$ , the discriminator can easily detect this and push the generator’s samples back. Inside $C _ { 1 }$ , small deviations of the generator from the shape of the real distribution are detected by the discriminator as well, by deviating slightly from $1 / 3$ . Subsequently, the generator is pushed back to the original shape. If the discriminator deviates from its optimum, it will also be forced back to its optimum. So overall, the GAN learning reached a local Nash equilibrium and has converged in the sense that the parameters fluctuate around the attractor point (fluctuations depend on learning rate, sample size, etc.). To achieve true mathematical convergence, Heusel et al. (2017) assume decaying learning rates to anneal the random fluctuations, similar to Robbins & Monro (1951) original convergence proof for SGD. ", + "bbox": [ + 174, + 165, + 825, + 580 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.2 PROOF OF THEOREM 1 ", + "text_level": 1, + "bbox": [ + 176, + 602, + 374, + 616 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "We first recall Theorem 1: ", + "bbox": [ + 176, + 631, + 346, + 645 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Theorem (Convergence with low-dimensional Plummer kernel). For $\\pmb { a } , \\pmb { b } \\in \\mathbb { R } ^ { m }$ , $d \\leqslant m - 2$ , and $\\epsilon > 0$ the densities $p _ { x } ( . )$ and $p _ { y } ( . )$ equalize over time when minimizing energy $F$ with the lowdimensional Plummer kernel by gradient descent. The convergence is faster for larger $d$ . ", + "bbox": [ + 176, + 651, + 823, + 694 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In a first step, we prove that for local maxima or local minima $\\textbf { \\em a }$ of $\\rho$ , the expression $\\sin ( \\boldsymbol { \\mathbf { V } } \\cdot \\mathrm { \\partial _ { \\mathrm { s i g n } } } ( \\boldsymbol { \\mathbf { V } } \\cdot \\mathrm { \\partial _ { \\mathrm { s i g n } } } )$ ${ \\bf E } ( { \\bf a } ) ) = \\mathrm { s i g n } ( \\rho ( { \\bf a } ) )$ holds for $\\epsilon$ small enough. For proving this equation, we apply the Laplace operator for spherical coordinates to the low-dimensional Plummer kernel. Using the result, we see that the integral $\\begin{array} { r } { \\nabla \\cdot \\mathbf { E } ( { a } ) = - \\int \\rho ( b ) \\nabla _ { a } ^ { 2 } k \\left( { a } , { b } \\right) \\mathrm { d } b } \\end{array}$ is dominated by large negative values of $\\nabla _ { a } ^ { 2 } k$ around $^ { a }$ . These negative values can even be decreased by decreasing $\\epsilon$ . Therefore we can ensure by a small enough $\\epsilon$ that at each local minimum and local maximum $\\textbf { \\em a }$ of $\\rho \\operatorname { s i g n } ( { \\dot { \\rho } } ( { \\mathbf { \\boldsymbol { a } } } ) ) = - \\operatorname { s i g n } ( \\rho ( { \\mathbf { \\boldsymbol { a } } } ) )$ . Thus, the maximal and minimal points of $\\rho$ move toward zero. ", + "bbox": [ + 174, + 708, + 825, + 805 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In a second step, we show that new maxima or minima cannot appear and that the movement of $\\Phi$ toward zero stops at zero and not earlier. Since $\\rho$ is continuously differentiable, all points in environments of maxima and minima move toward zero. Therefore the largest $| \\rho ( a ) |$ moves toward zero. We have to ensure that moving toward zero does not converge to a point apart from zero. We derive that the movement toward zero is lower bounded by ${ \\dot { \\rho } } ( { \\pmb a } ) = - \\mathrm { s i g n } ( \\rho ( { \\pmb a } ) ) \\lambda \\rho ^ { 2 } ( { \\pmb a } )$ . Thus, the movement slows down at $\\rho ( { \\pmb a } ) = 0$ . Solving the differential equation and applying it to the maximum of the absolute value of $\\rho$ gives $| \\rho | _ { \\operatorname* { m a x } } ( \\mathbf { \\bar { \\boldsymbol { t } } } ) = 1 / ( \\lambda t + ( | \\rho | _ { \\operatorname* { m a x } } ( 0 ) ) ^ { - 1 } )$ . Thus, $\\rho$ converges to zero over time. ", + "bbox": [ + 174, + 813, + 825, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. For $d = m - 2$ , we have $\\pmb { \\nabla } ^ { 2 } k ( \\pmb { a } , \\pmb { b } ) = \\delta ( \\pmb { a } - \\pmb { b } )$ , where the theorem has already been proved for $\\epsilon$ small enough (Hochreiter & Obermayer, 2005). ", + "bbox": [ + 173, + 102, + 825, + 133 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "At each local minimum and local maximum $\\textbf { \\em a }$ of $\\rho$ we have $\\pmb { \\nabla } _ { a } \\rho ( \\pmb { a } ) = 0$ . Using the product rule for the divergence operator, Eq. (3) reduces to ", + "bbox": [ + 174, + 138, + 823, + 167 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/d7d3b90477e14640cb4d303121fdf2835c98620ad9ea3776e2d672e568db87dc.jpg", + "text": "$$\n\\dot { \\rho } ( { \\pmb a } ) = - \\operatorname { s i g n } ( \\rho ( { \\pmb a } ) ) \\rho ( { \\pmb a } ) { \\pmb \\nabla } \\cdot { \\bf E } ( { \\pmb a } ) .\n$$", + "text_format": "latex", + "bbox": [ + 366, + 171, + 630, + 189 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The term $\\nabla \\cdot \\mathbf { E } ( a )$ can be expressed as ", + "bbox": [ + 173, + 193, + 436, + 209 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/f01bfd2d7c5d068d6450fe73b26e39deae71068e35a9082e8f77a40f0a9f012e.jpg", + "text": "$$\n\\nabla \\cdot { \\bf E } ( a ) = - \\nabla _ { a } ^ { 2 } \\Phi ( a ) = - \\int \\rho ( b ) \\nabla _ { a } ^ { 2 } k ( a , b ) \\ \\mathrm { d } b .\n$$", + "text_format": "latex", + "bbox": [ + 307, + 213, + 691, + 246 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We next consider $\\nabla _ { a } ^ { 2 } k \\left( a , b \\right)$ for the low-dimensional Plummer kernel. We define the spherical Laplace operator in $( m - 1 )$ dimensions as ∇2Sm−1 , then the Laplace operator in spherical coordinates is (Proposition 2.5 in Frye & Efthimiou (Efthimiou & Frye, 2014)): ", + "bbox": [ + 174, + 258, + 825, + 301 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/d993d86d85ea8b147d0765c9c878d8f866e3d2d8c197dd091fd5c268be690c15.jpg", + "text": "$$\n{ \\nabla } ^ { 2 } = { \\frac { \\partial ^ { 2 } } { \\partial r ^ { 2 } } } + { \\frac { m - 1 } { r } } { \\frac { \\partial } { \\partial r } } + { \\frac { m - 1 } { r ^ { 2 } } } { \\nabla } _ { S ^ { m - 1 } } ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 336, + 305, + 660, + 339 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Note that $\\nabla _ { S ^ { m - 1 } } ^ { 2 }$ only has second order derivatives with respect to the angles of the spherical coordinates. ", + "bbox": [ + 174, + 343, + 823, + 371 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "With $r = \\| a - b \\|$ we obtain for the Laplace operator applied to the low-dimensional Plummer kernel: ", + "bbox": [ + 171, + 377, + 825, + 407 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/8199af9ccff22481631c60dd398bf42b0216466f11d7cdfb82573e98532c34e8.jpg", + "text": "$$\n\\nabla ^ { 2 } k ( a , b ) ~ = ~ d \\left( - ~ \\epsilon ^ { 2 } ~ m ~ + ~ ( 2 ~ + ~ d ~ - ~ m ) ~ r ^ { 2 } \\right) ( \\epsilon ^ { 2 } ~ + ~ r ^ { 2 } ) ^ { - 2 - d / 2 } ~ .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 409, + 727, + 429 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "and in particular ", + "bbox": [ + 173, + 433, + 284, + 448 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/8a00b63be63d61cc4d242f1b498d78cacebd601893360aa83bca9f0dead9de45.jpg", + "text": "$$\n{ \\pmb { \\nabla } } ^ { 2 } k ( { \\pmb { a } } , { \\pmb { a } } ) = - m d \\epsilon ^ { - ( d + 2 ) } .\n$$", + "text_format": "latex", + "bbox": [ + 393, + 452, + 604, + 472 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For $d \\leqslant m - 2$ we have $( 2 + d - m ) \\leqslant 0$ , and obtain ", + "bbox": [ + 173, + 476, + 529, + 491 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/be3a92c31112d526a040a4af54ea47bdd7acddf088d458f08c58b954f1a98136.jpg", + "text": "$$\n\\nabla ^ { 2 } k ( { \\pmb a } , { \\pmb b } ) \\ < \\ 0 \\ ,\n$$", + "text_format": "latex", + "bbox": [ + 437, + 496, + 558, + 515 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 520, + 200, + 532 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/7c1f49a0a7d6067ae14320df15d75906fe4148ecf91c0c8f813979ec8d0ea058.jpg", + "text": "$$\n\\frac { \\partial } { \\partial r } \\nabla ^ { 2 } k ( a , b ) = d \\left( 2 + d \\right) r \\left( \\epsilon ^ { 2 } \\left( 2 + m \\right) + \\left( - 2 - d + m \\right) r ^ { 2 } \\right) ( \\epsilon ^ { 2 } + r ^ { 2 } ) ^ { - 3 - d / 2 } > 0\n$$", + "text_format": "latex", + "bbox": [ + 194, + 536, + 776, + 566 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 570, + 200, + 583 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e841220cc8c2887268076094534e24d8b62228ea4398d7c5be3b94a8b96cbc43.jpg", + "text": "$$\n{ \\frac { \\partial } { \\partial \\epsilon } } \\nabla ^ { 2 } k ( { \\pmb a } , { \\pmb b } ) = { \\cal d } \\left( 2 + { \\cal d } \\right) \\epsilon \\left( \\epsilon ^ { 2 } m + ( - 4 - { \\cal d } + m ) r ^ { 2 } \\right) ( \\epsilon ^ { 2 } + r ^ { 2 } ) ^ { - 3 - { \\cal d } / 2 } > 0 .\n$$", + "text_format": "latex", + "bbox": [ + 228, + 587, + 769, + 617 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Therefore, $\\nabla ^ { 2 } k ( a , b )$ is negative with minimum $- m d \\epsilon ^ { - ( d + 2 ) }$ at $r = 0$ and increasing with $r$ and increasing with $\\epsilon$ for $d \\leqslant m - 4$ . For $d = m - 3$ we have to restrict in the following the sphere $S _ { \\tau } ( a )$ to $\\tau < \\sqrt { m } \\epsilon$ and ensure increase of $\\nabla ^ { 2 } k ( a , b )$ with $\\epsilon$ . ", + "bbox": [ + 173, + 622, + 825, + 666 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "If $\\rho ( \\pmb { b } ) \\neq 0$ , then we define a sphere $S _ { \\tau } ( a )$ with radius $\\tau$ around $\\textbf { \\em a }$ for which holds $\\mathrm { s i g n } ( \\rho ( { \\boldsymbol { b } } ) ) =$ $\\mathrm { s i g n } ( \\rho ( a ) )$ for each $\\pmb { b } \\in S _ { \\tau } ( \\pmb { a } )$ . Note that $\\dot { \\nabla } ^ { 2 } k ( a , b )$ is continuous differentiable. We have ", + "bbox": [ + 173, + 671, + 825, + 702 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/c8a10002dace45c189d6d842a1c4edecf1cca628ef4226cc12429a81372c6416.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\nabla \\cdot { \\bf E } ( a ) = - \\int \\rho ( b ) \\nabla _ { a } ^ { 2 } k ( a , b ) \\mathrm { d } b = } \\\\ { \\displaystyle - \\int _ { S _ { \\tau } ( a ) } \\rho ( b ) \\nabla _ { a } ^ { 2 } k ( a , b ) \\mathrm { d } b - \\int _ { T \\setminus S _ { \\tau } ( a ) } \\rho ( b ) \\nabla _ { a } ^ { 2 } k ( a , b ) \\mathrm { d } b . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 705, + 751, + 775 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We bound $\\nabla ^ { 2 } k ( a , b )$ by ", + "bbox": [ + 174, + 779, + 338, + 794 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/e1e529778b89bfe25108dd770c941a8e90190630f0d2d400b35b542992a50376.jpg", + "text": "$$\n~ 1 ~ > ~ \\nabla ^ { 2 } k ( a , b ) ~ = ~ d \\left( - \\epsilon ^ { 2 } ~ m ~ + ~ ( 2 ~ + ~ d ~ - ~ m ) ~ r ^ { 2 } \\right) ( \\epsilon ^ { 2 } ~ + ~ r ^ { 2 } ) ^ { - 2 - d / 2 } ~ > ~ d \\left( 2 ~ + ~ d ~ - ~ m \\right) r ^ { - 2 - d / 2 } ~ .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 797, + 839, + 819 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Using $\\tau$ , we now bound $\\begin{array} { r l } { \\bigg | \\int _ { T \\setminus S _ { \\tau } ( \\mathbf { a } ) } \\rho ( \\pmb { b } ) \\ \\pmb { \\nabla } _ { a } ^ { 2 } k \\left( \\pmb { a } , \\pmb { b } \\right) \\ \\mathrm { d } \\pmb { b } \\bigg | } & { { } } \\end{array}$ independently from $\\epsilon$ , since $\\rho$ is a difference of distributions. For small enough $\\epsilon$ we can ensure ", + "bbox": [ + 174, + 838, + 826, + 876 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/47b1119669d94b75c9bac292b0165017e57563954513e2d491897fe219a0d63e.jpg", + "text": "$$\n\\left| \\int _ { S _ { \\tau } ( \\pmb { a } ) } \\rho ( \\pmb { b } ) \\ \\nabla _ { \\ b { a } } ^ { 2 } k \\left( \\pmb { a } , \\pmb { b } \\right) \\ \\mathrm { d } \\pmb { b } \\right| > \\ \\left| \\int _ { T \\setminus S _ { \\tau } ( \\pmb { a } ) } \\rho ( \\pmb { b } ) \\ \\nabla _ { \\ b { a } } ^ { 2 } k \\left( \\pmb { a } , \\pmb { b } \\right) \\ \\mathrm { d } \\pmb { b } \\right| \\ .\n$$", + "text_format": "latex", + "bbox": [ + 277, + 878, + 718, + 922 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Therefore we have ", + "bbox": [ + 174, + 103, + 299, + 117 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/ad498838741166521ce0a079ba7f6bcaed69245f7a0f39a5d2787bc9e5fb0b60.jpg", + "text": "$$\n\\mathrm { s i g n } ( \\pmb { \\nabla } \\cdot \\mathbf { E } ( \\pmb { a } ) ) = \\mathrm { s i g n } ( \\rho ( \\pmb { a } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 390, + 121, + 606, + 138 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Therefore we have at each local minimum and local maximum $^ { a }$ of $\\rho$ ", + "bbox": [ + 174, + 148, + 630, + 164 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/5a6dab9c7baf9482b5a825cef7260edbce7a6050e88d563764e4b4433e083f9f.jpg", + "text": "$$\n\\mathrm { s i g n } ( \\dot { \\rho } ( { \\bf a } ) ) = - \\mathrm { s i g n } ( \\rho ( { \\bf a } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 166, + 599, + 184 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Therefore the maximal and minimal points of $\\rho$ move toward zero. Since $\\rho$ is continuously differentiable as is the field, also the points in an environment of the maximal and minimal points move toward zero. Points that are not in an environment of the maximal or minimal points cannot become maximal points in an infinitesimal time step. ", + "bbox": [ + 174, + 185, + 825, + 242 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Since the contribution of $\\textbf { \\em a }$ environment $S _ { \\tau } ( a )$ dominates the integral Eq. (19), for $\\epsilon$ small enough there exists a positive $0 < \\lambda$ globally for all minima and maxima as well as for all time steps for which holds: ", + "bbox": [ + 173, + 247, + 825, + 290 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/a73efe20e32eb78d1fcd87919ef97a03429bdb76e8b4c4a81976a4ddd7dbef42.jpg", + "text": "$$\n\\left| \\nabla \\cdot \\mathbf { E } ( { \\pmb a } ) \\right| > \\lambda \\left| \\rho ( { \\pmb a } ) \\right| .\n$$", + "text_format": "latex", + "bbox": [ + 415, + 294, + 583, + 311 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The factor $\\lambda$ depends on $k$ and on the initial $\\rho . \\ \\lambda$ is proportional to $d$ . Larger $d$ lead to larger $| \\nabla \\cdot \\mathbf { E } ( a ) |$ since the maximum or minimum $\\rho ( { \\pmb a } )$ is upweighted. There might exist initial conditions $\\rho$ for which $\\lambda 0$ , e.g. for infinite many maxima and minima, but they are impossible in our applications. ", + "bbox": [ + 174, + 313, + 825, + 369 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Therefore maximal or minimal points approach zero faster or equal than given by ", + "bbox": [ + 173, + 376, + 709, + 391 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/995c619fb1c6b127c9a983bacf0ccbd97e400a8041b567cf0ba339f4ca985b95.jpg", + "text": "$$\n\\dot { \\rho } ( { \\pmb a } ) = - \\ \\mathrm { s i g n } ( \\rho ( { \\pmb a } ) ) \\lambda \\rho ^ { 2 } ( { \\pmb a } ) .\n$$", + "text_format": "latex", + "bbox": [ + 388, + 392, + 607, + 411 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "In particular this differential equation dominates the global maximum $| \\rho | _ { \\mathrm { m a x } }$ of $| \\rho ( . ) |$ . Solving the differential equation gives that at least ", + "bbox": [ + 176, + 414, + 823, + 441 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/3b631fc54412c55b4f70c05952fb72fcf5b2714d96229acfa092f9701877ce4c.jpg", + "text": "$$\n| \\rho | _ { \\operatorname* { m a x } } ( t ) = \\frac { 1 } { \\lambda t + \\left( | \\rho | _ { \\operatorname* { m a x } } ( 0 ) \\right) ^ { - 1 } } .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 444, + 624, + 479 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Thus $d$ influences the worst case rate of convergence, where larger $d$ with $d \\leqslant m - 2$ leads to faster worst case convergence. ", + "bbox": [ + 176, + 482, + 823, + 511 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Consequently, $\\rho$ converges to the zero function over time, that is, $p _ { x } ( . )$ becomes equal to $p _ { y } ( . )$ . ", + "bbox": [ + 176, + 516, + 799, + 531 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.3 PROOF OF THEOREM 2 ", + "bbox": [ + 176, + 546, + 377, + 561 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We first recall Theorem 2: ", + "bbox": [ + 174, + 573, + 348, + 587 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Theorem (Optimal Solution). If the pair $( D ^ { * } , G ^ { * } )$ is a local Nash equilibrium for the Coulomb GAN objectives, then it is the global Nash equilibrium, and no other local Nash equilibria exist, and $G ^ { * }$ has output distribution $p _ { x } = p _ { y }$ . ", + "bbox": [ + 176, + 589, + 825, + 635 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. $( D ^ { * } , G ^ { * } )$ being in a local Nash equilibrium means that $( D ^ { * } , G ^ { * } )$ fulfills the two conditions ", + "bbox": [ + 174, + 647, + 820, + 662 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/dae6cb4677afa4a624cb49999a1d5a209cb3e192ff978981245ba5fad8f53f9c.jpg", + "text": "$$\nD ^ { * } = \\underset { D \\in U ( D ^ { * } ) } { \\arg \\operatorname* { m i n } } \\mathcal { L } _ { D } ( D ; G ^ { * } ) \\quad \\mathrm { a n d } \\quad G ^ { * } = \\underset { G \\in U ( G ^ { * } ) } { \\arg \\operatorname* { m i n } } \\mathcal { L } _ { G } ( G ; D ^ { * } )\n$$", + "text_format": "latex", + "bbox": [ + 289, + 665, + 709, + 694 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "for some neighborhoods $U ( D ^ { * } )$ and $U ( G ^ { * } )$ . For Coulomb GANs that means, $D ^ { * }$ has learned the potential $\\Phi$ induced by $G ^ { * }$ perfectly, because $\\mathcal { L } _ { D }$ is convex in $D$ , thus if $D ^ { * }$ is optimal within an neighborhood $U ( D ^ { * } )$ , it must be the global optimum. This means that $G ^ { * }$ is directly minimizing $\\begin{array} { r } { \\mathcal { L } _ { G } ( G ; D ) = - \\frac { 1 } { 2 } \\mathrm { E } _ { p _ { z } } } \\end{array}$ $\\tilde { \\Phi } ( G ( z ) ) )$ . The Coulomb potential energy is according to Eq. (7) ", + "bbox": [ + 173, + 695, + 825, + 753 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/301e7d60fc6cd4bae651eba81c59bc6561b1a19e844853e9c71caef397faa342.jpg", + "text": "$$\nF ( \\rho ) = \\frac { 1 } { 2 } \\int \\rho ( a ) \\Phi ( a ) \\mathrm { d } a = \\frac { 1 } { 2 } \\int p _ { y } ( a ) \\Phi ( a ) \\mathrm { d } a - \\frac { 1 } { 2 } \\int p _ { x } ( a ) \\Phi ( a ) \\mathrm { d } a .\n$$", + "text_format": "latex", + "bbox": [ + 243, + 756, + 754, + 790 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Only the samples from $p _ { x }$ stem from the generator, where $\\begin{array} { r } { p _ { x } ( \\pmb { a } ) = \\int \\delta ( \\pmb { a } - \\pmb { G } ( \\pmb { z } ) ) p _ { z } ( z ) \\mathrm { d } z } \\end{array}$ . Here $\\delta$ is the $\\delta$ -distribution centered at zero. The part of the energy which depends on the generator is ", + "bbox": [ + 174, + 792, + 825, + 821 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/32ea803adb6723b2a2e44a855b4a5207f96051bd9396a8eecdf47876da634dd9.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle - \\mathrm { \\small \\displaystyle \\frac { 1 } { 2 } } \\int p _ { x } ( \\pmb { a } ) \\Phi ( \\pmb { a } ) \\mathrm { d } \\pmb { a } = - \\frac { 1 } { 2 } \\int \\int \\delta ( \\pmb { a } - G ( \\pmb { z } ) ) p _ { z } ( \\pmb { z } ) \\mathrm { d } z \\Phi ( \\pmb { a } ) \\mathrm { d } \\pmb { a } } \\\\ { = - \\frac { 1 } { 2 } \\int \\big ( \\int \\delta ( \\pmb { a } - G ( \\pmb { z } ) ) \\Phi ( \\pmb { a } ) \\mathrm { d } \\pmb { a } \\big ) p _ { z } ( \\pmb { z } ) \\mathrm { d } z } \\\\ { = - \\displaystyle \\frac { 1 } { 2 } \\int p _ { z } ( \\pmb { z } ) \\Phi ( G ( \\pmb { z } ) ) \\mathrm { d } z = - \\frac { 1 } { 2 } \\mathrm { E } _ { p _ { z } } \\left( \\Phi ( G ( \\pmb { z } ) ) \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 823, + 743, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Theorem 1 guarantees that there are no other local minima except the global one when minimizing $F$ . $F$ has one minimum, $F = 0$ , which implies $\\Phi ( { \\pmb a } ) = 0$ and $\\rho ( { \\pmb a } ) = 0$ for all $\\textbf { \\em a }$ , therefore also $p _ { y } = p _ { x }$ according to Theorem 1. Each $\\Phi ( { \\pmb a } ) \\neq \\mathrm { ~ 0 ~ }$ would mean there exist potential differences which in turn would cause forces on generator samples that allow to further minimize the energy. Since we assumed that the generator can reach the minimum $p _ { y } = p _ { x }$ for any $p _ { y }$ , it will be reached by local (stepwise) optimization of $- { \\textstyle \\frac { 1 } { 2 } } \\mathrm { E } _ { p _ { z } } \\left( \\Phi ( G ( z ) ) \\right)$ with respect to $G$ . Since the pair $( D ^ { * } , G ^ { * } )$ is optimal within their neighborhood, the generator has reached this minimum as there is not other local minimum than the global one. Therefore $G ^ { * }$ has model density $p _ { x }$ with $p _ { y } = p _ { x }$ . The convergence point is a global Nash equilibrium, because there is no approximation error and zero energy $F = 0$ is a global minimum for discriminator and generator, respectively. Theorem 1 ensures that other local Nash equilibria are not possible. □ ", + "bbox": [ + 173, + 103, + 825, + 258 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.4 COULOMB EQUATIONS IN THE CASE OF FINITE SAMPLES ", + "text_level": 1, + "bbox": [ + 174, + 297, + 612, + 313 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "GANs are sample-based, that is, samples are drawn from the model for learning (Hochreiter & Obermayer, 2005; Gutmann & Hyvärinen, 2012). Typically this is done in mini-batches, where each mini-batch consists of two sets of samples, the target samples $\\mathcal { Y } = \\{ \\pmb { y } _ { i } | i = 1 \\ldots N _ { y } \\}$ , and the model samples $\\mathcal { X } = \\{ \\pmb { x } _ { i } | i = 1 \\ldots N _ { x } \\}$ . ", + "bbox": [ + 174, + 332, + 825, + 390 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "For such finite samples, i.e. point charges, we have to use delta distributions to obtain unbiased estimates of the the model distribution $p _ { x } ( . )$ and the target distribution $p _ { y } ( . )$ : ", + "bbox": [ + 173, + 393, + 823, + 422 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/6942d7d9759c058f55200b25f4677eb5aa3ba839d5e1234750def59250ea6c83.jpg", + "text": "$$\n\\begin{array} { r l r l r } { \\displaystyle { \\mathrm { ~ \\mu ~ } _ { y } ( { \\pmb a } ; { \\mathcal { X } } ) = \\frac { 1 } { N _ { y } } \\sum _ { i = 1 } ^ { N _ { y } } \\delta \\left( { \\pmb a } - { \\pmb y } _ { i } \\right) \\ , } } & { { } \\hat { p } _ { x } ( { \\pmb a } ; { \\mathcal { X } } ) = \\frac { 1 } { N _ { x } } \\sum _ { i = 1 } ^ { N _ { x } } \\delta \\left( { \\pmb a } - { \\pmb x } _ { i } \\right) \\ , } & { \\hat { \\rho } ( { \\pmb a } ; { \\mathcal { X } } , { \\mathcal { Y } } ) } & { { } = p _ { y } ( { \\pmb a } ; { \\mathcal { Y } } ) - p _ { x } ( { \\pmb a } ; { \\mathcal { X } } , { \\mathcal { Y } } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 450, + 839, + 489 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $\\delta$ is the Dirac $\\delta$ -distribution centered at zero. These are unbiased estimates of the underlying distribution, as can be seen by: ", + "bbox": [ + 173, + 530, + 825, + 558 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/cda2ad5c8c7b095b05b7f826b1ede8f77daa5cd71d89e6516d17896a5d3c2ae7.jpg", + "text": "$$\n\\operatorname { E } _ { \\mathcal { X } } \\left( \\frac { 1 } { N _ { x } } \\sum _ { i = 1 } ^ { N _ { x } } \\delta ( { \\pmb a } - { \\pmb x } _ { i } ) \\right) = \\frac { 1 } { N _ { x } } \\sum _ { i = 1 } ^ { N _ { x } } \\operatorname { E } _ { x _ { i } } \\left( \\delta ( { \\pmb a } - { \\pmb x } _ { i } ) \\right) = \\frac { 1 } { N _ { x } } \\sum _ { i = 1 } ^ { N _ { x } } p _ { x } ( { \\pmb a } ) = p _ { x } ( { \\pmb a } ) .\n$$", + "text_format": "latex", + "bbox": [ + 235, + 583, + 763, + 623 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In the rest of the paper, we will drop the explicit parameterization with $\\mathcal { X }$ and $\\mathcal { V }$ for all estimates to unclutter notation, and instead just use the hat sign to denote estimates. In the same fashion as for the distributions, when we use fixed samples $\\mathcal { X }$ and $\\mathcal { V }$ , we obtain the following unbiased estimates for the potential, energy and field given by Eq. (5), Eq. (6), and Eq. (7): ", + "bbox": [ + 174, + 648, + 825, + 703 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/fb0df3f0fa7aaf282c64d5e8f6ebce74e4b9d4e0c50086f98981c5a140c0ebb0.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\bar { \\Phi } ( a ) = \\frac { 1 } { N _ { y } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { y } } k \\left( a , y _ { i } \\right) - \\frac { 1 } { N _ { z } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { z } } k \\left( a , { \\boldsymbol x } _ { i } \\right) \\ : , } \\\\ { \\displaystyle \\hat { F } \\left( \\boldsymbol \\rho \\right) = \\frac { 1 } { 2 } \\left( \\frac { N _ { y } } { N _ { y } ^ { 2 } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { y } } \\sum _ { j = 1 } ^ { N _ { y } } k \\left( y _ { i } , y _ { i } \\right) - \\frac { 2 } { N _ { y } N _ { x } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { y } } \\sum _ { j = 1 } ^ { N _ { x } } k \\left( y _ { i } , { \\boldsymbol x } _ { j } \\right) + \\frac { 1 } { N _ { z } ^ { 2 } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { z } } \\sum _ { j = 1 } ^ { N _ { x } } k \\left( { \\boldsymbol x } _ { i } , { \\boldsymbol x } _ { j } \\right) \\right) } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { 2 } \\left( \\frac { 1 } { N _ { y } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { y } } \\hat { \\Phi } \\left( y _ { i } \\right) - \\frac { 1 } { N _ { x } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { z } } \\hat { \\Phi } \\left( { \\boldsymbol x } _ { i } \\right) \\right) \\ : , } \\\\ { \\displaystyle \\hat { \\mathbf { E } } ( a ) = - \\nabla _ { a } \\hat { \\Phi } ( a ) = - \\frac { 1 } { N _ { y } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { y } } \\nabla _ { a } k \\left( a , y _ { i } \\right) + \\frac { 1 } { N _ { z } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { z } } \\nabla _ { a } k \\left( a , { \\boldsymbol x } _ { i } \\right) } \\\\ { \\displaystyle \\hat { \\mathbf { E } } ( y _ { i } ) = - N _ { y } \\nabla _ { y } \\hat { \\boldsymbol { F } } ( \\boldsymbol \\rho ) \\ : , \\quad \\hat { \\mathbf { E } } ( { \\boldsymbol x } _ { i } ) = N _ { x } \\nabla _ { x } \\hat { \\boldsymbol { F } } ( \\boldsymbol \\rho ) \\ : . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 192, + 729, + 779, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "These are again unbiased, e.g.: ", + "bbox": [ + 179, + 103, + 380, + 118 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/8ba689dfdecd6736b2d4be3d85f143b19238f5663dfb1f2c6654ae53df65f4a3.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { E } _ { \\boldsymbol { x } , \\boldsymbol { y } } ( \\hat { \\boldsymbol { \\Phi } } ( a ) ) = \\frac { 1 } { N _ { y } } \\sum _ { i = 1 } ^ { N _ { y } } E _ { y _ { i } } \\left( { k ( a , y _ { i } ) } \\right) - \\frac { 1 } { N _ { x } } \\sum _ { i = 1 } ^ { N _ { x } } E _ { \\alpha _ { \\boldsymbol { x } } } \\left( { k ( a , x _ { i } ) } \\right) } \\\\ { \\displaystyle = \\frac { 1 } { N _ { y } } \\sum _ { i = 1 } ^ { N _ { y } } \\int p _ { y } ( y _ { i } ) k ( a , y _ { i } ) \\mathrm { d } y _ { i } - \\frac { 1 } { N _ { x } } \\sum _ { i = 1 } ^ { N _ { x } } \\int p _ { x } ( x _ { i } ) k ( a , x _ { i } ) \\mathrm { d } x _ { i } } \\\\ { \\displaystyle = \\int p _ { y } ( y ) k ( a , y ) \\mathrm { d } y - \\int p _ { x } ( x ) k ( a , x ) \\mathrm { d } x } \\\\ { \\displaystyle = \\int \\left( p _ { y } ( { \\boldsymbol x } ) - p _ { x } ( { \\boldsymbol x } ) \\right) k ( a , x ) \\mathrm { d } x = \\int \\rho ( { \\boldsymbol x } ) k ( a , x ) \\mathrm { d } x = \\Phi ( a ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 125, + 781, + 282 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "If we draw samples of infinite size, all these expressions for a fixed sample size lead to the equivalent statements for densities. The sample-based formulation, that is, point charges in physical terms, can only have local energy minima or maxima at locations of samples (Dembo & Zeitouni, 1988). Furthermore the field lines originate and end at samples, therefore the field guides model samples $_ { \\textbf { \\em x } }$ toward real world samples $\\textbf { { y } }$ , as depicted in Figure 1. The factors $N _ { y }$ and $N _ { x }$ in the last equations arise from the fact that $- \\nabla _ { a } F$ gives the force which is applied to a sample with charge. A sample $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { i } }$ is positively charged with $1 / N _ { y }$ and follows $- \\nabla _ { y _ { i } } F$ while a sample $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ is negatively charged with $- 1 / N _ { x }$ and therefore follows $- \\nabla _ { x _ { i } } F$ , too. Thus, following the force induced on a sample by the field is equivalent to gradient descent of the energy $F$ with respect to samples $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { i } }$ and $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ . ", + "bbox": [ + 173, + 286, + 825, + 412 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "A.5 MIXTURE OF GAUSSIANS ", + "text_level": 1, + "bbox": [ + 176, + 429, + 395, + 443 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We use the synthetic data set introduced by Lim & Ye (2017) to show that Coulomb GANs avoid mode collapse and that all modes of the target distribution are captured by the generative model. This data set comprises 100K data points drawn from a Gaussian mixture model of 25 components which are spread out evenly in the range $[ - 2 1 , 2 1 ] \\times [ - 2 1 , 2 1 ]$ , with each component having a variance of 1. To make results comparable with Lim & Ye (2017), the Coulomb GAN used a discriminator network with 2 hidden layers of 128 units, however we avoided batch normalization by using the ELU activation function (Clevert et al., 2016). We used the Plummer kernel in 3 dimensions $\\left( d = 3 \\right)$ ) with an epsilon of 3 $( \\epsilon = 3$ ) and a learning rate of 0.01, both of which were exponentially decayed during the 1M update steps of the Adam optimizer. ", + "bbox": [ + 173, + 454, + 825, + 580 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "As can be seen in Figure 4, samples from the learned Coulomb GAN very well approximate the target distribution. All components of the original distribution are present at the model distribution at approximately the correct ratio, as shown in Figure 5. Moreover, the generated samples are distributed approximately according to the same spread for each component of the real world distribution. Coulomb GANs outperform other compared methods, which either fail to learn the distribution completely, ignore some of the modes, or do not capture the within-mode spread of a Gaussian. The Coulomb GAN is the only GAN approach that manages to avoid a within-cluster collapse leading to insufficient variance within a cluster. ", + "bbox": [ + 173, + 587, + 825, + 699 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/fa5e6f290d134c857c7dc9309463595b79e6cd093e140da31f28e96cef3db92a.jpg", + "image_caption": [ + "Figure 4: Scatter plots of generated samples from different GAN variants for the mixture of 25 Gaussians and the true data distribution. “WD” indicates weight decay and “proj.” means projection. Results and images for all methods except the Coulomb GAN are taken from Lim & Ye (2017). " + ], + "image_footnote": [], + "bbox": [ + 174, + 185, + 825, + 368 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/dbaa54afbfcd435cfee5c4d78e4151bd0163c3bdc9fac64f71caeb8e585bb71d.jpg", + "image_caption": [ + "Figure 5: 2D histogram of the density of generated and the training st data for the mixture of 25 Gaussians. For constructing the histogram, 10k samples were drawn from the target and the model distribution. The Coulomb GAN captures the underlying distribution well, does not miss any modes, and places almost all probability mass on the modes. Only the Coulomb GAN captured the withinmode spread of the Gaussians. " + ], + "image_footnote": [], + "bbox": [ + 250, + 594, + 764, + 753 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "A.6 PSEUDOCODE FOR COULOMB GANS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 473, + 118 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The following gives the pseudo code for training GANs. Note that when calculating the derivative of $\\hat { \\Phi } ( a _ { i } ; \\mathcal { X } , \\mathcal { Y } )$ , it is important to only derive with respect to $\\textbf { \\em a }$ , and not wrt. $\\mathcal { X } , \\mathcal { y }$ , even if it can happen that e.g. $\\mathbf { \\pmb { a } } \\in \\mathcal { X }$ . In frameworks that offer automatic differentiation such as Tensorflow or Theano, this means stopping the possible gradient back-propagation through those parameters. ", + "bbox": [ + 173, + 128, + 825, + 188 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Algorithm 1 Minibatch stochastic gradient descent training of Coulomb GANs for updating the the discriminator weights $\\textbf { \\em w }$ and the generator weights $\\pmb { \\theta }$ . ", + "bbox": [ + 168, + 203, + 823, + 231 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "while Stopping criterion not met do ", + "bbox": [ + 191, + 236, + 428, + 250 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• Sample minibatch of $N _ { x }$ training samples $\\{ \\pmb { x } _ { 1 } , \\ldots , \\pmb { x } _ { N _ { x } } \\}$ from training set • Sample minibatch of $N _ { y }$ generator samples $\\{ \\pmb { y } _ { 1 } , \\hdots , \\pmb { y } _ { N _ { y } } \\}$ from the generator • Calculate the gradient for the discriminator weights: ", + "bbox": [ + 204, + 250, + 730, + 291 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/aa32338f8030d50fcfc3421ddae0a64ec73d18ef6e7d417c6589d1b4071835c4.jpg", + "text": "$$\nd \\pmb { w } \\gets \\nabla _ { \\pmb { w } } \\left[ \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { N _ { x } } \\left( D ( \\pmb { x } _ { i } ) - \\hat { \\Phi } ( \\pmb { x } _ { i } ) \\right) ^ { 2 } + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { N _ { y } } \\left( D ( \\pmb { y } _ { i } ) - \\hat { \\Phi } ( \\pmb { y } _ { i } ) \\right) ^ { 2 } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 300, + 297, + 728, + 343 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• Calculate the gradient for the generator weights: ", + "bbox": [ + 207, + 354, + 535, + 369 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/3ca9aa66df6c5f95d6a9b53cf36cb4bdc9720d81d2b1ce82250736e34f2fd61f.jpg", + "text": "$$\nd \\pmb \\theta \\gets \\nabla _ { \\pmb \\theta } \\left[ - \\frac { 1 } { 2 } \\frac { 1 } { N _ { x } } \\sum _ { i = 1 } ^ { N _ { x } } D \\left( \\pmb x _ { i } \\right) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 406, + 375, + 622, + 420 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• Update weights according to optimizer rule (e.g. Adam): ", + "bbox": [ + 207, + 428, + 591, + 444 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/4ac3092b6aba7dff37c2da782743d6eb472bce112db11e2e8f195dbf52a487b0.jpg", + "text": "$$\n\\begin{array} { c } { { { \\pmb w } _ { n + 1 } = { \\pmb w } _ { n } + \\tt { \\mathbb { A D A M } } ( d { \\pmb w } , n ) } } \\\\ { { { \\pmb \\theta } _ { n + 1 } = { \\pmb \\theta } _ { n } + \\tt { \\mathbb { A D A M } } ( d { \\pmb \\theta } , n ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 423, + 449, + 607, + 483 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "end while ", + "text_level": 1, + "bbox": [ + 189, + 503, + 259, + 517 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "A.7 MORE SAMPLES FROM COULOMB GANS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 501, + 118 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "CELEBA ", + "text_level": 1, + "bbox": [ + 174, + 128, + 235, + 143 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Images from a Coulomb GAN after training on CelebA data set. The low FID stems from the fact that the images show a wide variety of different faces, backgrounds, eye colors and orientations. ", + "bbox": [ + 176, + 154, + 825, + 183 + ], + "page_idx": 19 + }, + { + "type": "image", + "img_path": "images/ec91dd18e28e2c15242beed056e837da884f72fbd0a6a284d92120e29ee12473.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 254, + 196, + 743, + 570 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "LSUN BEDROOMS ", + "text_level": 1, + "bbox": [ + 173, + 103, + 303, + 117 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Images from a Coulomb GAN after training on the LSUN bedroom data set. ", + "bbox": [ + 174, + 127, + 673, + 142 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/c59127bf47e0c4a5a6b28ed944bc10e060f21202d5ffd8c1805aa234d8090a11.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 254, + 156, + 743, + 531 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "CIFAR 10 ", + "text_level": 1, + "bbox": [ + 174, + 549, + 250, + 563 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Images from a Coulomb GAN after training on the CIFAR 10 data set. 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Even though it has", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 313, + 470, + 327 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 470, + 327 + ], + "score": 1.0, + "content": "recently been shown that GAN training converges, GAN models often end up", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 325, + 469, + 337 + ], + "spans": [ + { + "bbox": [ + 141, + 325, + 469, + 337 + ], + "score": 1.0, + "content": "in local Nash equilibria that are associated with mode collapse or otherwise fail", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 335, + 470, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 470, + 348 + ], + "score": 1.0, + "content": "to model the target distribution. We introduce Coulomb GANs, which pose the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 346, + 470, + 359 + ], + "spans": [ + { + "bbox": [ + 142, + 346, + 470, + 359 + ], + "score": 1.0, + "content": "GAN learning problem as a potential field, where generated samples are attracted", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 357, + 470, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 470, + 370 + ], + "score": 1.0, + "content": "to training set samples but repel each other. The discriminator learns a potential", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 369, + 470, + 381 + ], + "spans": [ + { + "bbox": [ + 141, + 369, + 470, + 381 + ], + "score": 1.0, + "content": "field while the generator decreases the energy by moving its samples along the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 379, + 470, + 392 + ], + "spans": [ + { + "bbox": [ + 141, + 379, + 470, + 392 + ], + "score": 1.0, + "content": "vector (force) field determined by the gradient of the potential field. Through de-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 390, + 470, + 403 + ], + "spans": [ + { + "bbox": [ + 141, + 390, + 470, + 403 + ], + "score": 1.0, + "content": "creasing the energy, the GAN model learns to generate samples according to the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 401, + 470, + 414 + ], + "spans": [ + { + "bbox": [ + 141, + 401, + 470, + 414 + ], + "score": 1.0, + "content": "whole target distribution and does not only cover some of its modes. We prove", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 412, + 470, + 425 + ], + "spans": [ + { + "bbox": [ + 141, + 412, + 470, + 425 + ], + "score": 1.0, + "content": "that Coulomb GANs possess only one Nash equilibrium which is optimal in the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 423, + 470, + 436 + ], + "spans": [ + { + "bbox": [ + 141, + 423, + 470, + 436 + ], + "score": 1.0, + "content": "sense that the model distribution equals the target distribution. We show the ef-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 434, + 469, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 434, + 469, + 446 + ], + "score": 1.0, + "content": "ficacy of Coulomb GANs on LSUN bedrooms, CelebA faces, CIFAR-10 and the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 444, + 293, + 458 + ], + "spans": [ + { + "bbox": [ + 141, + 444, + 293, + 458 + ], + "score": 1.0, + "content": "Google Billion Word text generation.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 480, + 206, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 479, + 208, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 208, + 496 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "Generative adversarial networks (GANs) (Goodfellow et al., 2014) excel at constructing realistic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "images (Radford et al., 2016; Ledig et al., 2016; Isola et al., 2017; Arjovsky et al., 2017; Berthelot", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "et al., 2017) and text (Gulrajani et al., 2017). In GAN learning, a discriminator network guides the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "learning of another, generative network. This procedure can be considered as a game between the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "generator which constructs synthetic data and the discriminator which separates synthetic data from", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "score": 1.0, + "content": "training set data (Goodfellow, 2017). The generator’s goal is to construct data which the discrim-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "inator cannot tell apart from training set data. GAN convergence points are local Nash equilibria.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "At these local Nash equilibria neither the discriminator nor the generator can locally improve its", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 594, + 148, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 148, + 607 + ], + "score": 1.0, + "content": "objective.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Despite their recent successes, GANs have several problems. First (I), until recently it was not clear", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "if in general gradient-based GAN learning could converge to one of the local Nash equilibria (Sal-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "imans et al., 2016; Goodfellow, 2014; Goodfellow et al., 2014). It is even possible to construct", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "counterexamples (Goodfellow, 2017). Second (II), GANs suffer from “mode collapsing”, where the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "model generates samples only in certain regions which are called modes. While these modes contain", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "realistic samples, the variety is low and only a few prototypes are generated. Mode collapsing is less", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "likely if the generator is trained with batch normalization, since the network is bound to create a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "certain variance among its generated samples within one batch (Radford et al., 2016; Chintala et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "2016). However batch normalization introduces fluctuations of normalizing constants which can be", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "harmful (Klambauer et al., 2017; Goodfellow, 2017). To avoid mode collapsing without batch nor-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "malization, several methods have been proposed (Che et al., 2017; Metz et al., 2016; Salimans et al.,", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40 + } + ], + "page_idx": 0, + "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": [ + 303, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 308, + 761 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 77, + 505, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 505, + 99 + ], + "score": 1.0, + "content": "COULOMB GANS: PROVABLY OPTIMAL NASH EQUI-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 339, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 339, + 117 + ], + "score": 1.0, + "content": "LIBRIA VIA POTENTIAL FIELDS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 133, + 497, + 146 + ], + "lines": [ + { + "bbox": [ + 113, + 132, + 499, + 148 + ], + "spans": [ + { + "bbox": [ + 113, + 132, + 499, + 148 + ], + "score": 1.0, + "content": "Thomas Unterthiner1 Bernhard Nessler1 Calvin Seward1,2 Günter Klambauer1", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 113, + 132, + 499, + 148 + ] + }, + { + "type": "text", + "bbox": [ + 113, + 162, + 178, + 174 + ], + "lines": [ + { + "bbox": [ + 111, + 162, + 180, + 175 + ], + "spans": [ + { + "bbox": [ + 111, + 162, + 180, + 175 + ], + "score": 1.0, + "content": "Martin Heusel1", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 111, + 162, + 180, + 175 + ] + }, + { + "type": "text", + "bbox": [ + 260, + 162, + 342, + 174 + ], + "lines": [ + { + "bbox": [ + 258, + 162, + 343, + 176 + ], + "spans": [ + { + "bbox": [ + 258, + 162, + 343, + 176 + ], + "score": 1.0, + "content": "Hubert Ramsauer1", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 258, + 162, + 343, + 176 + ] + }, + { + "type": "text", + "bbox": [ + 422, + 163, + 496, + 175 + ], + "lines": [ + { + "bbox": [ + 421, + 162, + 498, + 176 + ], + "spans": [ + { + "bbox": [ + 421, + 162, + 498, + 176 + ], + "score": 1.0, + "content": "Sepp Hochreiter1", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 421, + 162, + 498, + 176 + ] + }, + { + "type": "text", + "bbox": [ + 114, + 202, + 556, + 237 + ], + "lines": [ + { + "bbox": [ + 111, + 200, + 455, + 216 + ], + "spans": [ + { + "bbox": [ + 111, + 200, + 455, + 216 + ], + "score": 1.0, + "content": "1LIT AI Lab & Institute of Bioinformatics, Johannes Kepler University Linz, Austria", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 111, + 212, + 362, + 228 + ], + "spans": [ + { + "bbox": [ + 111, + 212, + 362, + 228 + ], + "score": 1.0, + "content": "2Zalando Research, Mühlenstraße 25, 10243 Berlin, Germany", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 113, + 226, + 556, + 238 + ], + "spans": [ + { + "bbox": [ + 113, + 226, + 556, + 238 + ], + "score": 1.0, + "content": "{unterthiner,nessler,seward,klambauer,mhe,ramsauer,hochreit}@bioinf.jku.at", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 111, + 200, + 556, + 238 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 266, + 333, + 278 + ], + "lines": [ + { + "bbox": [ + 276, + 264, + 335, + 279 + ], + "spans": [ + { + "bbox": [ + 276, + 264, + 335, + 279 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 143, + 291, + 468, + 456 + ], + "lines": [ + { + "bbox": [ + 142, + 291, + 469, + 304 + ], + "spans": [ + { + "bbox": [ + 142, + 291, + 469, + 304 + ], + "score": 1.0, + "content": "Generative adversarial networks (GANs) evolved into one of the most success-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 303, + 469, + 315 + ], + "spans": [ + { + "bbox": [ + 141, + 303, + 469, + 315 + ], + "score": 1.0, + "content": "ful unsupervised techniques for generating realistic images. Even though it has", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 313, + 470, + 327 + ], + "spans": [ + { + "bbox": [ + 141, + 313, + 470, + 327 + ], + "score": 1.0, + "content": "recently been shown that GAN training converges, GAN models often end up", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 325, + 469, + 337 + ], + "spans": [ + { + "bbox": [ + 141, + 325, + 469, + 337 + ], + "score": 1.0, + "content": "in local Nash equilibria that are associated with mode collapse or otherwise fail", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 335, + 470, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 470, + 348 + ], + "score": 1.0, + "content": "to model the target distribution. We introduce Coulomb GANs, which pose the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 346, + 470, + 359 + ], + "spans": [ + { + "bbox": [ + 142, + 346, + 470, + 359 + ], + "score": 1.0, + "content": "GAN learning problem as a potential field, where generated samples are attracted", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 357, + 470, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 470, + 370 + ], + "score": 1.0, + "content": "to training set samples but repel each other. The discriminator learns a potential", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 369, + 470, + 381 + ], + "spans": [ + { + "bbox": [ + 141, + 369, + 470, + 381 + ], + "score": 1.0, + "content": "field while the generator decreases the energy by moving its samples along the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 379, + 470, + 392 + ], + "spans": [ + { + "bbox": [ + 141, + 379, + 470, + 392 + ], + "score": 1.0, + "content": "vector (force) field determined by the gradient of the potential field. Through de-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 390, + 470, + 403 + ], + "spans": [ + { + "bbox": [ + 141, + 390, + 470, + 403 + ], + "score": 1.0, + "content": "creasing the energy, the GAN model learns to generate samples according to the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 401, + 470, + 414 + ], + "spans": [ + { + "bbox": [ + 141, + 401, + 470, + 414 + ], + "score": 1.0, + "content": "whole target distribution and does not only cover some of its modes. We prove", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 412, + 470, + 425 + ], + "spans": [ + { + "bbox": [ + 141, + 412, + 470, + 425 + ], + "score": 1.0, + "content": "that Coulomb GANs possess only one Nash equilibrium which is optimal in the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 423, + 470, + 436 + ], + "spans": [ + { + "bbox": [ + 141, + 423, + 470, + 436 + ], + "score": 1.0, + "content": "sense that the model distribution equals the target distribution. We show the ef-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 434, + 469, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 434, + 469, + 446 + ], + "score": 1.0, + "content": "ficacy of Coulomb GANs on LSUN bedrooms, CelebA faces, CIFAR-10 and the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 444, + 293, + 458 + ], + "spans": [ + { + "bbox": [ + 141, + 444, + 293, + 458 + ], + "score": 1.0, + "content": "Google Billion Word text generation.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 17, + "bbox_fs": [ + 141, + 291, + 470, + 458 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 480, + 206, + 493 + ], + "lines": [ + { + "bbox": [ + 105, + 479, + 208, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 208, + 496 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "Generative adversarial networks (GANs) (Goodfellow et al., 2014) excel at constructing realistic", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "images (Radford et al., 2016; Ledig et al., 2016; Isola et al., 2017; Arjovsky et al., 2017; Berthelot", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "et al., 2017) and text (Gulrajani et al., 2017). In GAN learning, a discriminator network guides the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "learning of another, generative network. This procedure can be considered as a game between the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "generator which constructs synthetic data and the discriminator which separates synthetic data from", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "score": 1.0, + "content": "training set data (Goodfellow, 2017). The generator’s goal is to construct data which the discrim-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 504, + 584 + ], + "score": 1.0, + "content": "inator cannot tell apart from training set data. GAN convergence points are local Nash equilibria.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "At these local Nash equilibria neither the discriminator nor the generator can locally improve its", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 594, + 148, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 148, + 607 + ], + "score": 1.0, + "content": "objective.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 506, + 505, + 607 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 611, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "Despite their recent successes, GANs have several problems. First (I), until recently it was not clear", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "if in general gradient-based GAN learning could converge to one of the local Nash equilibria (Sal-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "imans et al., 2016; Goodfellow, 2014; Goodfellow et al., 2014). It is even possible to construct", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "counterexamples (Goodfellow, 2017). Second (II), GANs suffer from “mode collapsing”, where the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "model generates samples only in certain regions which are called modes. While these modes contain", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "realistic samples, the variety is low and only a few prototypes are generated. Mode collapsing is less", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 689 + ], + "score": 1.0, + "content": "likely if the generator is trained with batch normalization, since the network is bound to create a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "certain variance among its generated samples within one batch (Radford et al., 2016; Chintala et al.,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "2016). However batch normalization introduces fluctuations of normalizing constants which can be", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "harmful (Klambauer et al., 2017; Goodfellow, 2017). To avoid mode collapsing without batch nor-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 504, + 732 + ], + "score": 1.0, + "content": "malization, several methods have been proposed (Che et al., 2017; Metz et al., 2016; Salimans et al.,", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 611, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "2016). Third (III), GANs cannot assure that the density of training samples is correctly modeled by", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "the generator. The discriminator only tells the generator whether a region is more likely to contain", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "samples from the training set or synthetic samples. Therefore the discriminator can only distinguish", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "the support of the model distribution from the support of the target distribution. Beyond matching", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "the support of distributions, GANs with proper objectives may learn to locally align model and tar-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "get densities via averaging over many training examples. On a global scale, however, GANs fail to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "equalize model and target densities. The discriminator does not inform the generator globally where", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "probability mass is missing. Consequently, standard GANs are not assured to capture the global", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "sample density and are prone to neglect large parts of the target distribution. The next paragraph", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "gives an example of this. Fourth (IV), the discriminator of GANs may forget previous modeling", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "errors of the generator which then may reappear, a property that leads to oscillatory behavior instead", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 252, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 252, + 216 + ], + "score": 1.0, + "content": "of convergence (Goodfellow, 2017).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 220, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "Recently, problem (I) was solved by proving that GAN learning does indeed converge when dis-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "criminator and generator are learned using a two time-scale learning rule (Heusel et al., 2017).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "Convergence means that the expected SGD-gradient of both the discriminator objective and the gen-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "erator objective are zero. Thus, neither the generator nor the discriminator can locally improve,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "i.e., learning has reached a local Nash equilibrium. However, convergence alone does not guarantee", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "good generative performance. It is possible to converge to sub-optimal solutions which are local", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "Nash equilibria. Mode collapse is a special case of a local Nash equilibrium associated with sub-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "optimal generative performance. For example, assume a two mode real world distribution where one", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "mode contains too few and the other mode too many generator samples. If no real world samples", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "are between these two distinct modes, then the discriminator penalizes to move generated samples", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "outside the modes. Therefore the generated samples cannot be correctly distributed over the modes.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Thus, standard GANs cannot capture the global sample density such that the resulting generators", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "are prone to neglect large parts of the real world distribution. A more detailed example is listed in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 363, + 225, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 225, + 375 + ], + "score": 1.0, + "content": "the Appendix in Section A.1.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "In this paper, we introduce a novel GAN model, the Coulomb GAN, which has only one Nash equi-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "librium. We are later going to show that this Nash equilibrium is optimal, i.e., the model distribution", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "matches the target distribution. We propose Coulomb GANs to avoid the GAN shortcoming (II) to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "(IV) by using a potential field created by point charges analogously to the electric field in physics.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "The next section will introduce the idea of learning in a potential field and prove that its only solution", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "is optimal. We will then show how learning the discriminator and generator works in a Coulomb", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "score": 1.0, + "content": "GAN and discuss the assumptions needed for our optimality proof. In Section 3 we will then see", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 454, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 470 + ], + "score": 1.0, + "content": "that the Coulomb GAN does indeed work well in practice and that the samples it produces have very", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 467, + 398, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 398, + 480 + ], + "score": 1.0, + "content": "large variability and appear to capture the original distribution very well.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "Related Work. Several GAN approaches have been suggested for bringing the target and model", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "distributions in alignment using not just local discriminator information: Geometric GANs combine", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "samples via a linear support vector machine which uses the discriminator outputs as samples, there-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "fore they are much more robust to mode collapsing (Lim & Ye, 2017). Energy-Based GANs (Zhao", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "et al., 2017) and their later improvement BEGANs (Berthelot et al., 2017) optimize an energy land-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "scape based on auto-encoders. McGANs match mean and covariance of synthetic and target data,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "therefore are more suited than standard GANs to approximate the target distribution (Mroueh et al.,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "2017). In a similar fashion, Generative Moment Matching Networks (Li et al., 2015) and MMD", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "nets (Dziugaite et al., 2015) directly optimize a generator network to match a training distribution", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "by using a loss function based on the maximum mean discrepancy (MMD) criterion (Gretton et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "2012). These approaches were later expanded to include an MMD criterion with learnable kernels", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "and discriminators (Li et al., 2017). The MMD criterion that these later approaches optimize has", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "a form similar to the energy function that Coulomb GANs optimize (cf. Eq. (33)). However, all", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "MMD approaches end up using either Gaussian or Laplace kernels, which are not guaranteed to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "find the optimal solution where the model distribution matches the target distribution. In contrast,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "the Plummer kernel which is employed in this work has been shown to lead to the optimal solution", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "(Hochreiter & Obermayer, 2005). We show that even a simplified version of the Plummer kernel,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "the low-dimensional Plummer kernel, ensures that gradient descent convergences to the optimal so-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "lution as stated by Theorem 1. Furthermore, most MMD GAN approaches use the MMD directly", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "as loss function though the number of possible samples in a mini-batch is limited. Therefore MMD", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "approaches face a sampling problem in high-dimensional spaces. The Coulomb GAN instead learns", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 104, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "a discriminator network that gradually improves its approximation of the potential field via learning", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 45.5 + } + ], + "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, + 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": [ + 107, + 82, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "2016). Third (III), GANs cannot assure that the density of training samples is correctly modeled by", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "the generator. The discriminator only tells the generator whether a region is more likely to contain", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "samples from the training set or synthetic samples. Therefore the discriminator can only distinguish", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "the support of the model distribution from the support of the target distribution. Beyond matching", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "the support of distributions, GANs with proper objectives may learn to locally align model and tar-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "get densities via averaging over many training examples. On a global scale, however, GANs fail to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "equalize model and target densities. The discriminator does not inform the generator globally where", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "probability mass is missing. Consequently, standard GANs are not assured to capture the global", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "sample density and are prone to neglect large parts of the target distribution. The next paragraph", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "gives an example of this. Fourth (IV), the discriminator of GANs may forget previous modeling", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "errors of the generator which then may reappear, a property that leads to oscillatory behavior instead", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 203, + 252, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 252, + 216 + ], + "score": 1.0, + "content": "of convergence (Goodfellow, 2017).", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 81, + 506, + 216 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 220, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "Recently, problem (I) was solved by proving that GAN learning does indeed converge when dis-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "criminator and generator are learned using a two time-scale learning rule (Heusel et al., 2017).", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "Convergence means that the expected SGD-gradient of both the discriminator objective and the gen-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "erator objective are zero. Thus, neither the generator nor the discriminator can locally improve,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "i.e., learning has reached a local Nash equilibrium. However, convergence alone does not guarantee", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 287 + ], + "score": 1.0, + "content": "good generative performance. It is possible to converge to sub-optimal solutions which are local", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "Nash equilibria. Mode collapse is a special case of a local Nash equilibrium associated with sub-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "optimal generative performance. For example, assume a two mode real world distribution where one", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "mode contains too few and the other mode too many generator samples. If no real world samples", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "are between these two distinct modes, then the discriminator penalizes to move generated samples", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "outside the modes. Therefore the generated samples cannot be correctly distributed over the modes.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "Thus, standard GANs cannot capture the global sample density such that the resulting generators", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 364 + ], + "score": 1.0, + "content": "are prone to neglect large parts of the real world distribution. A more detailed example is listed in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 363, + 225, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 225, + 375 + ], + "score": 1.0, + "content": "the Appendix in Section A.1.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 219, + 506, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "In this paper, we introduce a novel GAN model, the Coulomb GAN, which has only one Nash equi-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 403 + ], + "score": 1.0, + "content": "librium. We are later going to show that this Nash equilibrium is optimal, i.e., the model distribution", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "matches the target distribution. We propose Coulomb GANs to avoid the GAN shortcoming (II) to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "(IV) by using a potential field created by point charges analogously to the electric field in physics.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "score": 1.0, + "content": "The next section will introduce the idea of learning in a potential field and prove that its only solution", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "is optimal. We will then show how learning the discriminator and generator works in a Coulomb", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 459 + ], + "score": 1.0, + "content": "GAN and discuss the assumptions needed for our optimality proof. In Section 3 we will then see", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 454, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 470 + ], + "score": 1.0, + "content": "that the Coulomb GAN does indeed work well in practice and that the samples it produces have very", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 467, + 398, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 398, + 480 + ], + "score": 1.0, + "content": "large variability and appear to capture the original distribution very well.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 379, + 505, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 490, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 503 + ], + "score": 1.0, + "content": "Related Work. Several GAN approaches have been suggested for bringing the target and model", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "distributions in alignment using not just local discriminator information: Geometric GANs combine", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 505, + 525 + ], + "score": 1.0, + "content": "samples via a linear support vector machine which uses the discriminator outputs as samples, there-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "fore they are much more robust to mode collapsing (Lim & Ye, 2017). Energy-Based GANs (Zhao", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "et al., 2017) and their later improvement BEGANs (Berthelot et al., 2017) optimize an energy land-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "scape based on auto-encoders. McGANs match mean and covariance of synthetic and target data,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "therefore are more suited than standard GANs to approximate the target distribution (Mroueh et al.,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "2017). In a similar fashion, Generative Moment Matching Networks (Li et al., 2015) and MMD", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "nets (Dziugaite et al., 2015) directly optimize a generator network to match a training distribution", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "by using a loss function based on the maximum mean discrepancy (MMD) criterion (Gretton et al.,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 505, + 612 + ], + "score": 1.0, + "content": "2012). These approaches were later expanded to include an MMD criterion with learnable kernels", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "and discriminators (Li et al., 2017). The MMD criterion that these later approaches optimize has", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 633 + ], + "score": 1.0, + "content": "a form similar to the energy function that Coulomb GANs optimize (cf. Eq. (33)). However, all", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "MMD approaches end up using either Gaussian or Laplace kernels, which are not guaranteed to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "find the optimal solution where the model distribution matches the target distribution. In contrast,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "the Plummer kernel which is employed in this work has been shown to lead to the optimal solution", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "(Hochreiter & Obermayer, 2005). We show that even a simplified version of the Plummer kernel,", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "the low-dimensional Plummer kernel, ensures that gradient descent convergences to the optimal so-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "lution as stated by Theorem 1. Furthermore, most MMD GAN approaches use the MMD directly", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "as loss function though the number of possible samples in a mini-batch is limited. Therefore MMD", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "approaches face a sampling problem in high-dimensional spaces. The Coulomb GAN instead learns", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 104, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "a discriminator network that gradually improves its approximation of the potential field via learning", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "on many mini-batches. The discriminator network also tracks the slowly changing generator distri-", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "bution during learning. Most importantly however, our approach is, to the best of our knowledge,", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 341, + 501, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 501, + 354 + ], + "score": 1.0, + "content": "the first one for which optimality, i.e., ability to perfectly learn a target distribution, can be proved.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 45.5, + "bbox_fs": [ + 104, + 489, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 80, + 483, + 231 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 80, + 483, + 231 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 80, + 483, + 231 + ], + "spans": [ + { + "bbox": [ + 108, + 80, + 483, + 231 + ], + "score": 0.968, + "type": "image", + "image_path": "1dd35099253bbe238ca744ba58c3e5789a9a64144bbf15b96950a65d14400fb8.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 80, + 483, + 130.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 130.33333333333334, + 483, + 180.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 180.66666666666669, + 483, + 231.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 238, + 505, + 294 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "score": 1.0, + "content": "Figure 1: The vector field of a Coulomb GAN. The basic idea behind the Coulomb GAN: true", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "samples (blue) and generated samples (red) create a potential field (scalar field). Blue samples act as", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "sinks that attract the red samples, which repel each other. The superimposed vector field shows the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "score": 1.0, + "content": "forces acting on the generator samples to equalize potential differences, and the background color", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 283, + 340, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 340, + 295 + ], + "score": 1.0, + "content": "shows the potential at each position. Best viewed in color.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 319, + 503, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 505, + 332 + ], + "score": 1.0, + "content": "on many mini-batches. The discriminator network also tracks the slowly changing generator distri-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "bution during learning. Most importantly however, our approach is, to the best of our knowledge,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 341, + 501, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 501, + 354 + ], + "score": 1.0, + "content": "the first one for which optimality, i.e., ability to perfectly learn a target distribution, can be proved.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "score": 1.0, + "content": "The use of the Coulomb potential for learning is not new. Coulomb Potential Learning was pro-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "posed to store arbitrary many patterns in a potential field with perfect recall and without spurious", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "patterns (Perrone & Cooper, 1995). Another related work is the Potential Support Vector Machine", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "(PSVM), which minimizes Coulomb potential differences (Hochreiter & Mozer, 2001; Hochreiter", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "et al., 2003). Hochreiter & Obermayer (2005) also used a potential function based on Plummer", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 413, + 473, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 473, + 426 + ], + "score": 1.0, + "content": "kernels for optimal unsupervised learning, on which we base our work on Coulomb GANs.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 445, + 216, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 218, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 218, + 461 + ], + "score": 1.0, + "content": "2 COULOMB GANS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 473, + 298, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 299, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 299, + 486 + ], + "score": 1.0, + "content": "2.1 GENERAL CONSIDERATIONS ON GANS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 494, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 208, + 509 + ], + "score": 1.0, + "content": "We assume data samples", + "type": "text" + }, + { + "bbox": [ + 208, + 496, + 243, + 506 + ], + "score": 0.91, + "content": "\\pmb { a } \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 494, + 325, + 509 + ], + "score": 1.0, + "content": "for a model density", + "type": "text" + }, + { + "bbox": [ + 325, + 495, + 347, + 507 + ], + "score": 0.91, + "content": "p _ { x } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 494, + 429, + 509 + ], + "score": 1.0, + "content": "and a target density", + "type": "text" + }, + { + "bbox": [ + 429, + 495, + 451, + 507 + ], + "score": 0.91, + "content": "p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 494, + 506, + 509 + ], + "score": 1.0, + "content": ". The goal of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 338, + 519 + ], + "score": 1.0, + "content": "GAN learning is to modify the model in a way to obtain", + "type": "text" + }, + { + "bbox": [ + 339, + 506, + 396, + 519 + ], + "score": 0.93, + "content": "\\dot { p } _ { x } ( . ) = p _ { y } \\bar { ( . ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 505, + 506, + 519 + ], + "score": 1.0, + "content": ". We define the difference", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 515, + 507, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 158, + 531 + ], + "score": 1.0, + "content": "of densities", + "type": "text" + }, + { + "bbox": [ + 158, + 517, + 255, + 529 + ], + "score": 0.93, + "content": "\\rho ( { \\pmb a } ) = p _ { y } ( { \\pmb a } ) - p _ { x } ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 515, + 437, + 531 + ], + "score": 1.0, + "content": "which should be pushed toward zero for all", + "type": "text" + }, + { + "bbox": [ + 438, + 518, + 474, + 528 + ], + "score": 0.91, + "content": "\\pmb { a } \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 515, + 507, + 531 + ], + "score": 1.0, + "content": "during", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 526, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 304, + 542 + ], + "score": 1.0, + "content": "learning. In the GAN setting, the discriminator", + "type": "text" + }, + { + "bbox": [ + 305, + 528, + 328, + 540 + ], + "score": 0.93, + "content": "D ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 526, + 385, + 542 + ], + "score": 1.0, + "content": "is a function", + "type": "text" + }, + { + "bbox": [ + 385, + 528, + 447, + 538 + ], + "score": 0.91, + "content": "D : \\mathbb { R } ^ { m } \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 526, + 506, + 542 + ], + "score": 1.0, + "content": "that learns to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 450, + 552 + ], + "score": 1.0, + "content": "discriminate between generated and target samples and predicts how likely it is that", + "type": "text" + }, + { + "bbox": [ + 450, + 541, + 458, + 550 + ], + "score": 0.73, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "is sampled", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 317, + 563 + ], + "score": 1.0, + "content": "from the target distribution. In conventional GANs,", + "type": "text" + }, + { + "bbox": [ + 318, + 550, + 341, + 562 + ], + "score": 0.92, + "content": "D ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "is usually optimized to approximate the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 273, + 574 + ], + "score": 1.0, + "content": "probability of seeing a target sample, or", + "type": "text" + }, + { + "bbox": [ + 273, + 561, + 293, + 573 + ], + "score": 0.92, + "content": "\\rho ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 561, + 463, + 574 + ], + "score": 1.0, + "content": "or some similar function. The generator", + "type": "text" + }, + { + "bbox": [ + 464, + 561, + 486, + 573 + ], + "score": 0.92, + "content": "G ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 190, + 585 + ], + "score": 1.0, + "content": "continuous function", + "type": "text" + }, + { + "bbox": [ + 190, + 572, + 255, + 582 + ], + "score": 0.91, + "content": "G : \\mathbb { R } ^ { n } \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 571, + 333, + 585 + ], + "score": 1.0, + "content": "which maps some", + "type": "text" + }, + { + "bbox": [ + 333, + 574, + 340, + 582 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 571, + 462, + 585 + ], + "score": 1.0, + "content": "-dimensional random variable", + "type": "text" + }, + { + "bbox": [ + 462, + 574, + 470, + 582 + ], + "score": 0.76, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "into the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 203, + 595 + ], + "score": 1.0, + "content": "space of target samples.", + "type": "text" + }, + { + "bbox": [ + 203, + 585, + 210, + 593 + ], + "score": 0.76, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "is typically sampled from a multivariate Gaussian or Uniform distribution.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 504, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 431, + 612 + ], + "score": 1.0, + "content": "In order to improve the generator, a GAN uses the gradient of the discriminator", + "type": "text" + }, + { + "bbox": [ + 431, + 600, + 469, + 612 + ], + "score": 0.93, + "content": "\\nabla _ { a } D ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 600, + 504, + 612 + ], + "score": 1.0, + "content": "with re-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 236, + 624 + ], + "score": 1.0, + "content": "spect to the discriminator input", + "type": "text" + }, + { + "bbox": [ + 236, + 611, + 280, + 623 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\pmb { a } = G ( \\pmb { z } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "for learning. 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However,", + "type": "text" + }, + { + "bbox": [ + 285, + 677, + 305, + 689 + ], + "score": 0.91, + "content": "\\rho ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "and its derivative only take into account the local", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 177, + 700 + ], + "score": 1.0, + "content": "neighborhood of", + "type": "text" + }, + { + "bbox": [ + 177, + 690, + 185, + 698 + ], + "score": 0.74, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 687, + 419, + 700 + ], + "score": 1.0, + "content": ", since regions of the sample space that are distant from", + "type": "text" + }, + { + "bbox": [ + 419, + 690, + 426, + 698 + ], + "score": 0.68, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "do not have much", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 159, + 712 + ], + "score": 1.0, + "content": "influence on", + "type": "text" + }, + { + "bbox": [ + 159, + 699, + 180, + 711 + ], + "score": 0.92, + "content": "\\rho ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 699, + 389, + 712 + ], + "score": 1.0, + "content": ". Regions of data space that have strong support in", + "type": "text" + }, + { + "bbox": [ + 390, + 700, + 401, + 711 + ], + "score": 0.86, + "content": "p _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 699, + 445, + 712 + ], + "score": 1.0, + "content": "but not in", + "type": "text" + }, + { + "bbox": [ + 445, + 700, + 456, + 711 + ], + "score": 0.84, + "content": "p _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "will not be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "noticed by the generator via discriminator gradients. 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The basic idea behind the Coulomb GAN: true", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "samples (blue) and generated samples (red) create a potential field (scalar field). Blue samples act as", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "sinks that attract the red samples, which repel each other. The superimposed vector field shows the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 285 + ], + "score": 1.0, + "content": "forces acting on the generator samples to equalize potential differences, and the background color", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 283, + 340, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 340, + 295 + ], + "score": 1.0, + "content": "shows the potential at each position. Best viewed in color.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 319, + 503, + 353 + ], + "lines": [], + "index": 9, + "bbox_fs": [ + 105, + 320, + 505, + 354 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 358, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "score": 1.0, + "content": "The use of the Coulomb potential for learning is not new. Coulomb Potential Learning was pro-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "posed to store arbitrary many patterns in a potential field with perfect recall and without spurious", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "patterns (Perrone & Cooper, 1995). Another related work is the Potential Support Vector Machine", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "(PSVM), which minimizes Coulomb potential differences (Hochreiter & Mozer, 2001; Hochreiter", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 414 + ], + "score": 1.0, + "content": "et al., 2003). Hochreiter & Obermayer (2005) also used a potential function based on Plummer", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 413, + 473, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 473, + 426 + ], + "score": 1.0, + "content": "kernels for optimal unsupervised learning, on which we base our work on Coulomb GANs.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 357, + 505, + 426 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 445, + 216, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 218, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 218, + 461 + ], + "score": 1.0, + "content": "2 COULOMB GANS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 473, + 298, + 484 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 299, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 299, + 486 + ], + "score": 1.0, + "content": "2.1 GENERAL CONSIDERATIONS ON GANS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 494, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 494, + 208, + 509 + ], + "score": 1.0, + "content": "We assume data samples", + "type": "text" + }, + { + "bbox": [ + 208, + 496, + 243, + 506 + ], + "score": 0.91, + "content": "\\pmb { a } \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 494, + 325, + 509 + ], + "score": 1.0, + "content": "for a model density", + "type": "text" + }, + { + "bbox": [ + 325, + 495, + 347, + 507 + ], + "score": 0.91, + "content": "p _ { x } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 494, + 429, + 509 + ], + "score": 1.0, + "content": "and a target density", + "type": "text" + }, + { + "bbox": [ + 429, + 495, + 451, + 507 + ], + "score": 0.91, + "content": "p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 494, + 506, + 509 + ], + "score": 1.0, + "content": ". The goal of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 338, + 519 + ], + "score": 1.0, + "content": "GAN learning is to modify the model in a way to obtain", + "type": "text" + }, + { + "bbox": [ + 339, + 506, + 396, + 519 + ], + "score": 0.93, + "content": "\\dot { p } _ { x } ( . ) = p _ { y } \\bar { ( . ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 505, + 506, + 519 + ], + "score": 1.0, + "content": ". We define the difference", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 515, + 507, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 158, + 531 + ], + "score": 1.0, + "content": "of densities", + "type": "text" + }, + { + "bbox": [ + 158, + 517, + 255, + 529 + ], + "score": 0.93, + "content": "\\rho ( { \\pmb a } ) = p _ { y } ( { \\pmb a } ) - p _ { x } ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 515, + 437, + 531 + ], + "score": 1.0, + "content": "which should be pushed toward zero for all", + "type": "text" + }, + { + "bbox": [ + 438, + 518, + 474, + 528 + ], + "score": 0.91, + "content": "\\pmb { a } \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 515, + 507, + 531 + ], + "score": 1.0, + "content": "during", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 526, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 526, + 304, + 542 + ], + "score": 1.0, + "content": "learning. 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The restriction to local environments hampers", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 419, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 419, + 733 + ], + "score": 1.0, + "content": "GAN learning significantly (Arjovsky & Bottou, 2017; Arjovsky et al., 2017).", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 600, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "The theoretical analysis of GAN learning can be done at three different levels: (1) in the space", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 171, + 108 + ], + "score": 1.0, + "content": "of distributions", + "type": "text" + }, + { + "bbox": [ + 171, + 95, + 182, + 105 + ], + "score": 0.87, + "content": "p _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 92, + 201, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 202, + 95, + 213, + 106 + ], + "score": 0.84, + "content": "p _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 92, + 322, + 108 + ], + "score": 1.0, + "content": "regardless of the fact that", + "type": "text" + }, + { + "bbox": [ + 322, + 95, + 334, + 105 + ], + "score": 0.87, + "content": "p _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 92, + 394, + 108 + ], + "score": 1.0, + "content": "is realized by", + "type": "text" + }, + { + "bbox": [ + 394, + 94, + 403, + 104 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 92, + 423, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 95, + 434, + 105 + ], + "score": 0.85, + "content": "p _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 92, + 506, + 108 + ], + "score": 1.0, + "content": ", (2) in the space", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 159, + 117 + ], + "score": 1.0, + "content": "of functions", + "type": "text" + }, + { + "bbox": [ + 159, + 105, + 168, + 115 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 104, + 188, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 188, + 105, + 198, + 114 + ], + "score": 0.76, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 104, + 308, + 117 + ], + "score": 1.0, + "content": "regardless of the fact that", + "type": "text" + }, + { + "bbox": [ + 308, + 105, + 317, + 115 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 104, + 337, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 338, + 105, + 347, + 114 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "are typically realized by a parametric", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 504, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 406, + 129 + ], + "score": 1.0, + "content": "form, i.e., as neural networks, or (3) in the space of the parameters of", + "type": "text" + }, + { + "bbox": [ + 406, + 116, + 416, + 126 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 114, + 436, + 129 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 437, + 116, + 446, + 125 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 114, + 504, + 129 + ], + "score": 1.0, + "content": ". 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By letting", + "type": "text" + }, + { + "bbox": [ + 327, + 226, + 357, + 235 + ], + "score": 0.89, + "content": "\\epsilon 0", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 224, + 451, + 239 + ], + "score": 1.0, + "content": ", the distance between", + "type": "text" + }, + { + "bbox": [ + 451, + 226, + 467, + 236 + ], + "score": 0.89, + "content": "p 1 _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 224, + 487, + 239 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 487, + 225, + 504, + 237 + ], + "score": 0.89, + "content": "p 2 _ { x }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 234, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 104, + 234, + 412, + 250 + ], + "score": 1.0, + "content": "becomes smaller, yet the distance between the inducing generator functions", + "type": "text" + }, + { + "bbox": [ + 412, + 236, + 426, + 247 + ], + "score": 0.89, + "content": "G _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 234, + 444, + 250 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 236, + 458, + 247 + ], + "score": 0.88, + "content": "G _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 234, + 506, + 250 + ], + "score": 1.0, + "content": "(e.g. using", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "the supremum norm on bounded functions) will not tend to zero because for at least one function", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 129, + 271 + ], + "score": 1.0, + "content": "input", + "type": "text" + }, + { + "bbox": [ + 129, + 259, + 137, + 268 + ], + "score": 0.71, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 258, + 176, + 271 + ], + "score": 1.0, + "content": "we have:", + "type": "text" + }, + { + "bbox": [ + 176, + 258, + 265, + 270 + ], + "score": 0.93, + "content": "| G _ { 1 } ( z ) - G _ { 2 } ( z ) | \\geqslant d", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 258, + 505, + 271 + ], + "score": 1.0, + "content": ". Because of this, we need to go further than the distribution", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "space when analyzing GAN learning. In practice, when learning GANs, we are restricted to small", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "steps in parameter space, which in turn lead to small steps in function space and finally to small", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 104, + 290, + 506, + 304 + ], + "score": 1.0, + "content": "steps in distribution space. But not all small steps in distribution space can be realized this way", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 104, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "as shown in the example above. This causes local Nash equilibria in the function space, because", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 311, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 506, + 327 + ], + "score": 1.0, + "content": "even though in distribution space it would be easy to escape by making small steps, such a step", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "would require very large changes in function space and is thus not realizable. In this paper we show", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 477, + 347 + ], + "score": 1.0, + "content": "that Coulomb GANs do not exhibit any local Nash equilibria in the space of the functions", + "type": "text" + }, + { + "bbox": [ + 477, + 335, + 487, + 345 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 116, + 356 + ], + "score": 0.74, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 345, + 505, + 358 + ], + "score": 1.0, + "content": ". To the best of our knowledge, this is the first formulation of GAN learning that can guarantee", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "this property. Of course, Coulomb GANs are learned as parametrized neural networks, and as we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "will discuss in Subsection 2.4.2, Coulomb GANs are not immune to the usual issues that arise from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "parameter learning, such as over- and underfitting, which can cause local Nash Equilibria due to a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 390, + 211, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 211, + 402 + ], + "score": 1.0, + "content": "bad choice of parameters.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 107, + 426, + 330, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 332, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 332, + 439 + ], + "score": 1.0, + "content": "2.2 FROM CONVENTIONAL GANS TO POTENTIALS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 160, + 465 + ], + "score": 1.0, + "content": "If the density", + "type": "text" + }, + { + "bbox": [ + 161, + 452, + 183, + 464 + ], + "score": 0.91, + "content": "p _ { x } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 451, + 194, + 465 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 195, + 452, + 216, + 464 + ], + "score": 0.91, + "content": "p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "approaches a Dirac delta-distribution, gradients vanish since the density", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "approaches zero except for the exact location of data points. 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As we will see in the following, the ability to have samples that", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "influence each other over long distances, like charges in a Coulomb potential, will lead to GANs", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 270, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 270, + 733 + ], + "score": 1.0, + "content": "with a single, optimal Nash equilibrium.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 44.5 + } + ], + "page_idx": 3, + "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, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "score": 1.0, + "content": "The theoretical analysis of GAN learning can be done at three different levels: (1) in the space", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 171, + 108 + ], + "score": 1.0, + "content": "of distributions", + "type": "text" + }, + { + "bbox": [ + 171, + 95, + 182, + 105 + ], + "score": 0.87, + "content": "p _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 92, + 201, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 202, + 95, + 213, + 106 + ], + "score": 0.84, + "content": "p _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 92, + 322, + 108 + ], + "score": 1.0, + "content": "regardless of the fact that", + "type": "text" + }, + { + "bbox": [ + 322, + 95, + 334, + 105 + ], + "score": 0.87, + "content": "p _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 92, + 394, + 108 + ], + "score": 1.0, + "content": "is realized by", + "type": "text" + }, + { + "bbox": [ + 394, + 94, + 403, + 104 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 92, + 423, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 423, + 95, + 434, + 105 + ], + "score": 0.85, + "content": "p _ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 92, + 506, + 108 + ], + "score": 1.0, + "content": ", (2) in the space", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 159, + 117 + ], + "score": 1.0, + "content": "of functions", + "type": "text" + }, + { + "bbox": [ + 159, + 105, + 168, + 115 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 104, + 188, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 188, + 105, + 198, + 114 + ], + "score": 0.76, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 104, + 308, + 117 + ], + "score": 1.0, + "content": "regardless of the fact that", + "type": "text" + }, + { + "bbox": [ + 308, + 105, + 317, + 115 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 104, + 337, + 117 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 338, + 105, + 347, + 114 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "are typically realized by a parametric", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 504, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 406, + 129 + ], + "score": 1.0, + "content": "form, i.e., as neural networks, or (3) in the space of the parameters of", + "type": "text" + }, + { + "bbox": [ + 406, + 116, + 416, + 126 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 114, + 436, + 129 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 437, + 116, + 446, + 125 + ], + "score": 0.81, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 114, + 504, + 129 + ], + "score": 1.0, + "content": ". 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By letting", + "type": "text" + }, + { + "bbox": [ + 327, + 226, + 357, + 235 + ], + "score": 0.89, + "content": "\\epsilon 0", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 224, + 451, + 239 + ], + "score": 1.0, + "content": ", the distance between", + "type": "text" + }, + { + "bbox": [ + 451, + 226, + 467, + 236 + ], + "score": 0.89, + "content": "p 1 _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 224, + 487, + 239 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 487, + 225, + 504, + 237 + ], + "score": 0.89, + "content": "p 2 _ { x }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 234, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 104, + 234, + 412, + 250 + ], + "score": 1.0, + "content": "becomes smaller, yet the distance between the inducing generator functions", + "type": "text" + }, + { + "bbox": [ + 412, + 236, + 426, + 247 + ], + "score": 0.89, + "content": "G _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 234, + 444, + 250 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 236, + 458, + 247 + ], + "score": 0.88, + "content": "G _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 234, + 506, + 250 + ], + "score": 1.0, + "content": "(e.g. using", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "the supremum norm on bounded functions) will not tend to zero because for at least one function", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 129, + 271 + ], + "score": 1.0, + "content": "input", + "type": "text" + }, + { + "bbox": [ + 129, + 259, + 137, + 268 + ], + "score": 0.71, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 258, + 176, + 271 + ], + "score": 1.0, + "content": "we have:", + "type": "text" + }, + { + "bbox": [ + 176, + 258, + 265, + 270 + ], + "score": 0.93, + "content": "| G _ { 1 } ( z ) - G _ { 2 } ( z ) | \\geqslant d", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 258, + 505, + 271 + ], + "score": 1.0, + "content": ". Because of this, we need to go further than the distribution", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 281 + ], + "score": 1.0, + "content": "space when analyzing GAN learning. In practice, when learning GANs, we are restricted to small", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 292 + ], + "score": 1.0, + "content": "steps in parameter space, which in turn lead to small steps in function space and finally to small", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 104, + 290, + 506, + 304 + ], + "score": 1.0, + "content": "steps in distribution space. But not all small steps in distribution space can be realized this way", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 301, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 104, + 301, + 506, + 315 + ], + "score": 1.0, + "content": "as shown in the example above. This causes local Nash equilibria in the function space, because", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 311, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 506, + 327 + ], + "score": 1.0, + "content": "even though in distribution space it would be easy to escape by making small steps, such a step", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "would require very large changes in function space and is thus not realizable. In this paper we show", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 477, + 347 + ], + "score": 1.0, + "content": "that Coulomb GANs do not exhibit any local Nash equilibria in the space of the functions", + "type": "text" + }, + { + "bbox": [ + 477, + 335, + 487, + 345 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 116, + 356 + ], + "score": 0.74, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 345, + 505, + 358 + ], + "score": 1.0, + "content": ". To the best of our knowledge, this is the first formulation of GAN learning that can guarantee", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 369 + ], + "score": 1.0, + "content": "this property. Of course, Coulomb GANs are learned as parametrized neural networks, and as we", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "will discuss in Subsection 2.4.2, Coulomb GANs are not immune to the usual issues that arise from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "parameter learning, such as over- and underfitting, which can cause local Nash Equilibria due to a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 390, + 211, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 211, + 402 + ], + "score": 1.0, + "content": "bad choice of parameters.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 14, + "bbox_fs": [ + 104, + 82, + 506, + 402 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 426, + 330, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 332, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 332, + 439 + ], + "score": 1.0, + "content": "2.2 FROM CONVENTIONAL GANS TO POTENTIALS", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 451, + 505, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 451, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 160, + 465 + ], + "score": 1.0, + "content": "If the density", + "type": "text" + }, + { + "bbox": [ + 161, + 452, + 183, + 464 + ], + "score": 0.91, + "content": "p _ { x } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 451, + 194, + 465 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 195, + 452, + 216, + 464 + ], + "score": 0.91, + "content": "p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 451, + 505, + 465 + ], + "score": 1.0, + "content": "approaches a Dirac delta-distribution, gradients vanish since the density", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 475 + ], + "score": 1.0, + "content": "approaches zero except for the exact location of data points. Similarly, electric point charges are of-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 473, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 104, + 473, + 506, + 487 + ], + "score": 1.0, + "content": "ten represented by Dirac delta-distributions, however the electric potential created by a point charge", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "has influence everywhere in the space, not just locally. The electric potential (Coulomb potential)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 493, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 216, + 513 + ], + "score": 1.0, + "content": "created by the point charge", + "type": "text" + }, + { + "bbox": [ + 216, + 497, + 225, + 509 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 493, + 236, + 513 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 236, + 496, + 291, + 511 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\Phi _ { C } = \\frac { 1 } { 4 \\pi \\varepsilon _ { 0 } } \\frac { Q } { r } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 493, + 321, + 513 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 322, + 499, + 328, + 507 + ], + "score": 0.76, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 493, + 457, + 513 + ], + "score": 1.0, + "content": "is the distance to the location of", + "type": "text" + }, + { + "bbox": [ + 457, + 497, + 466, + 509 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 493, + 484, + 513 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 484, + 498, + 495, + 508 + ], + "score": 0.84, + "content": "\\varepsilon _ { \\mathrm { 0 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 493, + 507, + 513 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 521 + ], + "score": 1.0, + "content": "the dielectric constant. Motivated by this electric potential, we introduce a similar concept for GAN", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 519, + 504, + 534 + ], + "spans": [ + { + "bbox": [ + 104, + 519, + 294, + 534 + ], + "score": 1.0, + "content": "learning: Instead of the difference of densities", + "type": "text" + }, + { + "bbox": [ + 294, + 520, + 315, + 532 + ], + "score": 0.92, + "content": "\\rho ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 519, + 481, + 534 + ], + "score": 1.0, + "content": ", we rather consider a potential function", + "type": "text" + }, + { + "bbox": [ + 482, + 520, + 504, + 532 + ], + "score": 0.9, + "content": "\\Phi ( a )", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 529, + 150, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 150, + 543 + ], + "score": 1.0, + "content": "defined as", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 451, + 507, + 543 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 556, + 366, + 582 + ], + "lines": [ + { + "bbox": [ + 244, + 556, + 366, + 582 + ], + "spans": [ + { + "bbox": [ + 244, + 556, + 366, + 582 + ], + "score": 0.93, + "content": "\\Phi ( { \\pmb a } ) = \\int \\rho ( { \\pmb b } ) \\ k ( { \\pmb a } , { \\pmb b } ) \\ \\mathrm { d } { \\pmb b } ,", + "type": "interline_equation", + "image_path": "3ae2241c970ffda5a67dbce6323a9674f212f803f67b31ee38cfa7655581d2f7.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 244, + 556, + 366, + 582 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 597, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 178, + 610 + ], + "score": 1.0, + "content": "with some kernel", + "type": "text" + }, + { + "bbox": [ + 178, + 598, + 210, + 610 + ], + "score": 0.92, + "content": "k \\left( a , b \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 598, + 375, + 610 + ], + "score": 1.0, + "content": "which defines the influence of a point at", + "type": "text" + }, + { + "bbox": [ + 376, + 599, + 382, + 608 + ], + "score": 0.71, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 598, + 444, + 610 + ], + "score": 1.0, + "content": "onto a point at", + "type": "text" + }, + { + "bbox": [ + 445, + 600, + 452, + 608 + ], + "score": 0.75, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 598, + 505, + 610 + ], + "score": 1.0, + "content": ". The crucial", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 201, + 622 + ], + "score": 1.0, + "content": "advantage of potentials", + "type": "text" + }, + { + "bbox": [ + 201, + 609, + 223, + 621 + ], + "score": 0.92, + "content": "\\Phi ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 609, + 458, + 622 + ], + "score": 1.0, + "content": "is that each point can influence each other point in space if", + "type": "text" + }, + { + "bbox": [ + 458, + 610, + 465, + 619 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "is chosen", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 620, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 265, + 632 + ], + "score": 1.0, + "content": "properly. If we minimize this potential", + "type": "text" + }, + { + "bbox": [ + 266, + 620, + 288, + 632 + ], + "score": 0.92, + "content": "\\Phi ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 620, + 506, + 632 + ], + "score": 1.0, + "content": "we are at the same time minimizing the difference of", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 144, + 642 + ], + "score": 1.0, + "content": "densities", + "type": "text" + }, + { + "bbox": [ + 145, + 631, + 165, + 643 + ], + "score": 0.91, + "content": "\\rho ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 631, + 233, + 642 + ], + "score": 1.0, + "content": ": For all kernels", + "type": "text" + }, + { + "bbox": [ + 233, + 631, + 240, + 641 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 631, + 303, + 642 + ], + "score": 1.0, + "content": "it holds that if", + "type": "text" + }, + { + "bbox": [ + 303, + 631, + 343, + 642 + ], + "score": 0.9, + "content": "\\rho ( \\pmb { b } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 631, + 373, + 642 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 373, + 631, + 380, + 641 + ], + "score": 0.69, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 631, + 402, + 642 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 402, + 631, + 446, + 643 + ], + "score": 0.93, + "content": "\\Phi ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 631, + 475, + 642 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 476, + 633, + 483, + 641 + ], + "score": 0.73, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 631, + 505, + 642 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 200, + 654 + ], + "score": 1.0, + "content": "must still show that (i)", + "type": "text" + }, + { + "bbox": [ + 200, + 642, + 242, + 654 + ], + "score": 0.91, + "content": "\\Phi ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 642, + 270, + 654 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 271, + 644, + 278, + 652 + ], + "score": 0.77, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 642, + 299, + 654 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 300, + 642, + 338, + 653 + ], + "score": 0.89, + "content": "\\rho ( \\pmb { b } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 642, + 367, + 654 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 367, + 642, + 373, + 652 + ], + "score": 0.72, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 642, + 505, + 654 + ], + "score": 1.0, + "content": ", and even more importantly, (ii)", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 248, + 666 + ], + "score": 1.0, + "content": "whether a gradient optimization of", + "type": "text" + }, + { + "bbox": [ + 248, + 653, + 270, + 664 + ], + "score": 0.92, + "content": "\\Phi ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 652, + 305, + 666 + ], + "score": 1.0, + "content": "leads to", + "type": "text" + }, + { + "bbox": [ + 305, + 653, + 346, + 665 + ], + "score": 0.91, + "content": "\\Phi ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 652, + 375, + 666 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 375, + 655, + 382, + 663 + ], + "score": 0.71, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 652, + 505, + 666 + ], + "score": 1.0, + "content": ". This is not the case for every", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 663, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 273, + 677 + ], + "score": 1.0, + "content": "kernel. Indeed only for particular kernels", + "type": "text" + }, + { + "bbox": [ + 274, + 664, + 281, + 674 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 663, + 381, + 677 + ], + "score": 1.0, + "content": "gradient optimization of", + "type": "text" + }, + { + "bbox": [ + 382, + 664, + 403, + 676 + ], + "score": 0.92, + "content": "\\Phi ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 663, + 438, + 677 + ], + "score": 1.0, + "content": "leads to", + "type": "text" + }, + { + "bbox": [ + 439, + 664, + 477, + 676 + ], + "score": 0.92, + "content": "\\rho ( \\pmb { b } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 663, + 506, + 677 + ], + "score": 1.0, + "content": "for all", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 673, + 506, + 688 + ], + "spans": [ + { + "bbox": [ + 106, + 675, + 113, + 685 + ], + "score": 0.62, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 673, + 148, + 688 + ], + "score": 1.0, + "content": ", that is,", + "type": "text" + }, + { + "bbox": [ + 148, + 675, + 211, + 687 + ], + "score": 0.94, + "content": "p _ { x } ( b ) \\stackrel { \\cdot } { = } p _ { y } ( \\bar { b } )", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 673, + 241, + 688 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 241, + 675, + 248, + 685 + ], + "score": 0.69, + "content": "^ { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 673, + 506, + 688 + ], + "score": 1.0, + "content": "(Hochreiter & Obermayer, 2005) (see also Theorem 1 below).", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 686, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 227, + 698 + ], + "score": 1.0, + "content": "An example for such a kernel", + "type": "text" + }, + { + "bbox": [ + 227, + 686, + 234, + 696 + ], + "score": 0.83, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 686, + 411, + 698 + ], + "score": 1.0, + "content": "is the one leading to the Coulomb potential", + "type": "text" + }, + { + "bbox": [ + 411, + 686, + 426, + 697 + ], + "score": 0.89, + "content": "\\Phi _ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 686, + 505, + 698 + ], + "score": 1.0, + "content": "from above, where", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 107, + 696, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 107, + 696, + 181, + 712 + ], + "score": 0.93, + "content": "\\begin{array} { r } { k \\left( \\mathbf { { a } } , \\mathbf { { b } } \\right) \\ { \\stackrel { * } { = } } \\ { \\frac { 1 } { \\left\\| \\mathbf { { a } } - \\mathbf { { b } } \\right\\| } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 697, + 198, + 709 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 199, + 697, + 231, + 707 + ], + "score": 0.89, + "content": "m = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 697, + 506, + 709 + ], + "score": 1.0, + "content": ". As we will see in the following, the ability to have samples that", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "influence each other over long distances, like charges in a Coulomb potential, will lead to GANs", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 270, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 270, + 733 + ], + "score": 1.0, + "content": "with a single, optimal Nash equilibrium.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 598, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 264, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 266, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 266, + 95 + ], + "score": 1.0, + "content": "2.3 GANS AS ELECTRICAL FIELDS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 116 + ], + "score": 1.0, + "content": "For Coulomb GANs, the generator objective is derived from electrical field dynamics: real and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "score": 1.0, + "content": "generated samples generate a potential field, where samples of the same class (real vs. generated)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 137 + ], + "score": 1.0, + "content": "repel each other, but attract samples of the opposite class. However, real data points are fixed in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "space, so the only samples that can move are the generated ones. In turn, the gradient of the potential", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 160 + ], + "score": 1.0, + "content": "with respect to the input samples creates a vector field in the space of samples. The generator can", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "move its samples along the forces generated by this field. Such a field is depicted in Figure 1. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "discriminator learns to predict the potential function, in order to approximate the current potential", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "landscape of all samples, not just the ones in the current mini-batch. Meanwhile, the generator", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "score": 1.0, + "content": "learns to distribute its samples across the whole field in such a way that the energy is minimized,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 200, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 216 + ], + "score": 1.0, + "content": "thus naturally avoids mode collapse and covering the whole region of support of the data. The energy", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "is minimal and equal to zero only if all potential differences are zero and the model distribution is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 224, + 231, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 231, + 236 + ], + "score": 1.0, + "content": "equal to the target distribution.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "Within an electrostatic field, the strength of the force on one particle depends on its distance to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "score": 1.0, + "content": "other particles and their charges. If left to move freely, the particles will organize themselves into", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "a constellation where all forces equal out and no potential differences are present. For continuous", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "charge distributions, the potential field is constant without potential differences if charges no longer", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "move since forces are equaled out. If the potential field is constant, then the difference of densities", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 113, + 307 + ], + "score": 0.77, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "is constant, too. Otherwise the potential field would have local bumps. The same behavior is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "modeled within our Coulomb GAN, except that real and generated samples replace the positive", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "and negative particles, respectively, and that the real data points remain fixed. Only the generated", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 336, + 341 + ], + "score": 1.0, + "content": "samples are allowed to move freely, in order to minimize", + "type": "text" + }, + { + "bbox": [ + 336, + 330, + 343, + 340 + ], + "score": 0.78, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 329, + 505, + 341 + ], + "score": 1.0, + "content": ". The generated samples are attracted by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "real samples, so they move towards them. At the same time, generated samples should repel each", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 350, + 408, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 408, + 364 + ], + "score": 1.0, + "content": "other, so they do not clump together, which would lead to mode collapsing.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 504, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 291, + 380 + ], + "score": 1.0, + "content": "Analogously to electrostatics, the potential", + "type": "text" + }, + { + "bbox": [ + 292, + 367, + 314, + 378 + ], + "score": 0.88, + "content": "\\Phi ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 366, + 466, + 380 + ], + "score": 1.0, + "content": "from Eq. (1) gives rise to a field", + "type": "text" + }, + { + "bbox": [ + 467, + 367, + 504, + 379 + ], + "score": 0.89, + "content": "\\begin{array} { l l } { \\mathbf { E } ( \\pmb { a } ) } & { = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 376, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 107, + 378, + 151, + 390 + ], + "score": 0.92, + "content": "- \\nabla _ { a } \\bar { \\Phi } ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 376, + 262, + 392 + ], + "score": 1.0, + "content": ". and to an energy function", + "type": "text" + }, + { + "bbox": [ + 262, + 378, + 367, + 391 + ], + "score": 0.92, + "content": "\\begin{array} { r } { F \\left( \\rho \\right) = \\frac { 1 } { 2 } \\int \\rho ( \\mathbf { { a } } ) \\Phi ( \\mathbf { { a } } ) \\mathrm { { d } } \\mathbf { { a } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 376, + 409, + 392 + ], + "score": 1.0, + "content": ". The field", + "type": "text" + }, + { + "bbox": [ + 410, + 378, + 432, + 390 + ], + "score": 0.9, + "content": "{ \\bf E } ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 376, + 506, + 392 + ], + "score": 1.0, + "content": "applies a force on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 149, + 402 + ], + "score": 1.0, + "content": "charges at", + "type": "text" + }, + { + "bbox": [ + 149, + 392, + 156, + 399 + ], + "score": 0.75, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "which pushes the charges toward lower energy constellations. Ultimately, the Coulomb", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 241, + 414 + ], + "score": 1.0, + "content": "GAN aims to make the potential", + "type": "text" + }, + { + "bbox": [ + 241, + 401, + 250, + 410 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 399, + 371, + 414 + ], + "score": 1.0, + "content": "zero everywhere via the field", + "type": "text" + }, + { + "bbox": [ + 372, + 400, + 393, + 412 + ], + "score": 0.9, + "content": "{ \\bf E } ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 399, + 505, + 414 + ], + "score": 1.0, + "content": ", which is the negative gra-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 141, + 424 + ], + "score": 1.0, + "content": "dient of", + "type": "text" + }, + { + "bbox": [ + 141, + 412, + 149, + 421 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 410, + 232, + 424 + ], + "score": 1.0, + "content": ". For proper kernels", + "type": "text" + }, + { + "bbox": [ + 232, + 412, + 239, + 421 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 410, + 340, + 424 + ], + "score": 1.0, + "content": ", it can be shown that (i)", + "type": "text" + }, + { + "bbox": [ + 340, + 412, + 349, + 421 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "can be pushed to zero via its negative", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 267, + 435 + ], + "score": 1.0, + "content": "gradient given by the field and (ii) that", + "type": "text" + }, + { + "bbox": [ + 267, + 422, + 310, + 434 + ], + "score": 0.92, + "content": "\\Phi ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 421, + 339, + 435 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 340, + 424, + 347, + 432 + ], + "score": 0.75, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 421, + 381, + 435 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 381, + 422, + 423, + 434 + ], + "score": 0.92, + "content": "\\bar { { \\rho } } ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 421, + 452, + 435 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 452, + 424, + 460, + 432 + ], + "score": 0.7, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 421, + 506, + 435 + ], + "score": 1.0, + "content": ", therefore,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 433, + 456, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 168, + 446 + ], + "score": 0.93, + "content": "\\bar { p } _ { x } ( { \\pmb a } ) = \\bar { p } _ { y } ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 433, + 197, + 446 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 197, + 434, + 204, + 443 + ], + "score": 0.73, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 433, + 456, + 446 + ], + "score": 1.0, + "content": "(Hochreiter & Obermayer, 2005) (see also Theorem 1 below).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 457, + 227, + 468 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 228, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 228, + 469 + ], + "score": 1.0, + "content": "2.3.1 LEARNING PROCESS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 249, + 488 + ], + "score": 1.0, + "content": "During learning we do not change", + "type": "text" + }, + { + "bbox": [ + 250, + 477, + 258, + 487 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 477, + 272, + 488 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 272, + 478, + 279, + 488 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 477, + 404, + 488 + ], + "score": 1.0, + "content": "directly. Instead, the location", + "type": "text" + }, + { + "bbox": [ + 405, + 476, + 449, + 488 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\pmb { a } = G ( \\pmb { z } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 477, + 505, + 488 + ], + "score": 1.0, + "content": "to which the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 174, + 500 + ], + "score": 1.0, + "content": "random variable", + "type": "text" + }, + { + "bbox": [ + 175, + 489, + 183, + 497 + ], + "score": 0.75, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 486, + 338, + 500 + ], + "score": 1.0, + "content": "is mapped changes to a new location", + "type": "text" + }, + { + "bbox": [ + 338, + 487, + 388, + 499 + ], + "score": 0.93, + "content": "\\pmb { a } ^ { \\prime } = G ^ { \\prime } ( \\pmb { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 486, + 506, + 500 + ], + "score": 1.0, + "content": ". For the GAN optimization", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 287, + 511 + ], + "score": 1.0, + "content": "dynamics, we assume that generator samples", + "type": "text" + }, + { + "bbox": [ + 288, + 498, + 329, + 510 + ], + "score": 0.93, + "content": "\\pmb { a } = G ( \\pmb { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "can move freely, which is ensured by a suf-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 484, + 522 + ], + "score": 1.0, + "content": "ficiently complex generator. Importantly, generator samples originating from random variables", + "type": "text" + }, + { + "bbox": [ + 484, + 511, + 492, + 519 + ], + "score": 0.71, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "do", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "neither disappear nor are they newly created but are conserved. This conservation is expressed by the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 531, + 504, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 483, + 544 + ], + "score": 1.0, + "content": "continuity equation (Schwartz, 1972) that describes how the difference between distributions", + "type": "text" + }, + { + "bbox": [ + 483, + 531, + 504, + 543 + ], + "score": 0.91, + "content": "\\rho ( { \\pmb a } )", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 541, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 556 + ], + "score": 1.0, + "content": "changes as the particles are moving along the field, i.e., how moving samples during the learning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 554, + 230, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 230, + 565 + ], + "score": 1.0, + "content": "process changes our densities:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 570, + 364, + 584 + ], + "lines": [ + { + "bbox": [ + 246, + 570, + 364, + 584 + ], + "spans": [ + { + "bbox": [ + 246, + 570, + 364, + 584 + ], + "score": 0.91, + "content": "{ \\dot { \\rho } } ( { \\mathbf { a } } ) \\ = \\ - \\nabla \\cdot ( \\rho ( { \\mathbf { a } } ) \\ v ( { \\mathbf { a } } ) )", + "type": "interline_equation", + "image_path": "712b6d9fbb6f08ef9ae9c964070bbad5f0161d092772ddf3ffe9cdf34b8d98a1.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 246, + 570, + 364, + 584 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 588, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 224, + 603 + ], + "score": 1.0, + "content": "for sample density difference", + "type": "text" + }, + { + "bbox": [ + 224, + 591, + 231, + 601 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 588, + 402, + 603 + ], + "score": 1.0, + "content": "and unit charges that move with “velocity”", + "type": "text" + }, + { + "bbox": [ + 403, + 589, + 501, + 601 + ], + "score": 0.92, + "content": "{ \\pmb v } ( { \\pmb a } ) = \\mathrm { s i g n } ( \\rho ( { \\pmb a } ) ) { \\bf E } ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 588, + 505, + 603 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "The continuity equation is crucial as it establishes the connection between moving samples and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 612, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 284, + 624 + ], + "score": 1.0, + "content": "changing the generator density and thereby", + "type": "text" + }, + { + "bbox": [ + 284, + 613, + 290, + 623 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 612, + 506, + 624 + ], + "score": 1.0, + "content": ". The sign function of the velocity indicates whether", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 623, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 623, + 281, + 635 + ], + "score": 1.0, + "content": "positive or negative charges are present at", + "type": "text" + }, + { + "bbox": [ + 282, + 625, + 289, + 632 + ], + "score": 0.77, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 623, + 398, + 635 + ], + "score": 1.0, + "content": ". The divergence operator", + "type": "text" + }, + { + "bbox": [ + 398, + 623, + 422, + 633 + ], + "score": 0.31, + "content": "\\mathbf { \\omega } ^ { 6 6 } \\nabla \\cdot \\mathbf { \\vec { \\mu } } ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 623, + 506, + 635 + ], + "score": 1.0, + "content": "determines whether", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 632, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 259, + 647 + ], + "score": 1.0, + "content": "samples move toward or outward of", + "type": "text" + }, + { + "bbox": [ + 259, + 635, + 267, + 643 + ], + "score": 0.76, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 632, + 506, + 647 + ], + "score": 1.0, + "content": "for a given field. Basically, the continuity equation says", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "that if the generator density increases, then generator samples must flow into the region and if the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "generator density decreases, they flow outwards. We assume that differently charged particles cancel", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 375, + 679 + ], + "score": 1.0, + "content": "each other. If generator samples are moved away from a location", + "type": "text" + }, + { + "bbox": [ + 375, + 668, + 383, + 676 + ], + "score": 0.73, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 666, + 404, + 679 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 405, + 666, + 425, + 678 + ], + "score": 0.92, + "content": "\\rho ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 666, + 506, + 679 + ], + "score": 1.0, + "content": "is increasing while", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 127, + 689 + ], + "score": 0.91, + "content": "\\rho ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 677, + 362, + 690 + ], + "score": 1.0, + "content": "is decreasing when generator samples are moved toward", + "type": "text" + }, + { + "bbox": [ + 362, + 679, + 370, + 687 + ], + "score": 0.72, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 677, + 506, + 690 + ], + "score": 1.0, + "content": ". The continuity equation is also", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "obtained as a first order ODE to move particles in a potential field (Dembo & Zeitouni, 1988),", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 699, + 411, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 411, + 712 + ], + "score": 1.0, + "content": "therefore describes the dynamics how the densities are changing. We obtain", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 46 + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 716, + 441, + 730 + ], + "lines": [ + { + "bbox": [ + 170, + 716, + 441, + 730 + ], + "spans": [ + { + "bbox": [ + 170, + 716, + 441, + 730 + ], + "score": 0.88, + "content": "\\dot { \\rho } ( a ) = - \\mathrm { s i g n } ( \\rho ( a ) ) \\nabla \\cdot \\left( \\rho ( a ) { \\bf E } ( a ) \\right) = - \\nabla \\cdot \\left( | \\rho ( a ) | { \\bf E } ( a ) \\right) .", + "type": "interline_equation", + "image_path": "6ba26ccc4ecb901e2bcd5fc975516ee4416a1c26e3d1d53abdc1824620d2dbcd.jpg" + } + ] + } + ], + "index": 52, + "virtual_lines": [ + { + "bbox": [ + 170, + 716, + 441, + 730 + ], + "spans": [], + "index": 52 + } + ] + } + ], + "page_idx": 4, + "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, + 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, + 264, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 266, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 266, + 95 + ], + "score": 1.0, + "content": "2.3 GANS AS ELECTRICAL FIELDS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 103, + 505, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 116 + ], + "score": 1.0, + "content": "For Coulomb GANs, the generator objective is derived from electrical field dynamics: real and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 126 + ], + "score": 1.0, + "content": "generated samples generate a potential field, where samples of the same class (real vs. generated)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 137 + ], + "score": 1.0, + "content": "repel each other, but attract samples of the opposite class. However, real data points are fixed in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "space, so the only samples that can move are the generated ones. In turn, the gradient of the potential", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 160 + ], + "score": 1.0, + "content": "with respect to the input samples creates a vector field in the space of samples. The generator can", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "move its samples along the forces generated by this field. Such a field is depicted in Figure 1. The", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "discriminator learns to predict the potential function, in order to approximate the current potential", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 192 + ], + "score": 1.0, + "content": "landscape of all samples, not just the ones in the current mini-batch. Meanwhile, the generator", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "score": 1.0, + "content": "learns to distribute its samples across the whole field in such a way that the energy is minimized,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 200, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 505, + 216 + ], + "score": 1.0, + "content": "thus naturally avoids mode collapse and covering the whole region of support of the data. The energy", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 225 + ], + "score": 1.0, + "content": "is minimal and equal to zero only if all potential differences are zero and the model distribution is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 224, + 231, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 231, + 236 + ], + "score": 1.0, + "content": "equal to the target distribution.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 102, + 506, + 236 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 240, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "Within an electrostatic field, the strength of the force on one particle depends on its distance to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 263 + ], + "score": 1.0, + "content": "other particles and their charges. If left to move freely, the particles will organize themselves into", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "a constellation where all forces equal out and no potential differences are present. For continuous", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 287 + ], + "score": 1.0, + "content": "charge distributions, the potential field is constant without potential differences if charges no longer", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "move since forces are equaled out. If the potential field is constant, then the difference of densities", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 296, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 113, + 307 + ], + "score": 0.77, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 296, + 506, + 308 + ], + "score": 1.0, + "content": "is constant, too. Otherwise the potential field would have local bumps. The same behavior is", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 319 + ], + "score": 1.0, + "content": "modeled within our Coulomb GAN, except that real and generated samples replace the positive", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "and negative particles, respectively, and that the real data points remain fixed. Only the generated", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 329, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 336, + 341 + ], + "score": 1.0, + "content": "samples are allowed to move freely, in order to minimize", + "type": "text" + }, + { + "bbox": [ + 336, + 330, + 343, + 340 + ], + "score": 0.78, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 329, + 505, + 341 + ], + "score": 1.0, + "content": ". The generated samples are attracted by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 353 + ], + "score": 1.0, + "content": "real samples, so they move towards them. At the same time, generated samples should repel each", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 350, + 408, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 408, + 364 + ], + "score": 1.0, + "content": "other, so they do not clump together, which would lead to mode collapsing.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 240, + 506, + 364 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 367, + 505, + 444 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 504, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 291, + 380 + ], + "score": 1.0, + "content": "Analogously to electrostatics, the potential", + "type": "text" + }, + { + "bbox": [ + 292, + 367, + 314, + 378 + ], + "score": 0.88, + "content": "\\Phi ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 366, + 466, + 380 + ], + "score": 1.0, + "content": "from Eq. (1) gives rise to a field", + "type": "text" + }, + { + "bbox": [ + 467, + 367, + 504, + 379 + ], + "score": 0.89, + "content": "\\begin{array} { l l } { \\mathbf { E } ( \\pmb { a } ) } & { = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 376, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 107, + 378, + 151, + 390 + ], + "score": 0.92, + "content": "- \\nabla _ { a } \\bar { \\Phi } ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 376, + 262, + 392 + ], + "score": 1.0, + "content": ". and to an energy function", + "type": "text" + }, + { + "bbox": [ + 262, + 378, + 367, + 391 + ], + "score": 0.92, + "content": "\\begin{array} { r } { F \\left( \\rho \\right) = \\frac { 1 } { 2 } \\int \\rho ( \\mathbf { { a } } ) \\Phi ( \\mathbf { { a } } ) \\mathrm { { d } } \\mathbf { { a } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 376, + 409, + 392 + ], + "score": 1.0, + "content": ". The field", + "type": "text" + }, + { + "bbox": [ + 410, + 378, + 432, + 390 + ], + "score": 0.9, + "content": "{ \\bf E } ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 376, + 506, + 392 + ], + "score": 1.0, + "content": "applies a force on", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 149, + 402 + ], + "score": 1.0, + "content": "charges at", + "type": "text" + }, + { + "bbox": [ + 149, + 392, + 156, + 399 + ], + "score": 0.75, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "which pushes the charges toward lower energy constellations. Ultimately, the Coulomb", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 399, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 241, + 414 + ], + "score": 1.0, + "content": "GAN aims to make the potential", + "type": "text" + }, + { + "bbox": [ + 241, + 401, + 250, + 410 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 399, + 371, + 414 + ], + "score": 1.0, + "content": "zero everywhere via the field", + "type": "text" + }, + { + "bbox": [ + 372, + 400, + 393, + 412 + ], + "score": 0.9, + "content": "{ \\bf E } ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 399, + 505, + 414 + ], + "score": 1.0, + "content": ", which is the negative gra-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 410, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 141, + 424 + ], + "score": 1.0, + "content": "dient of", + "type": "text" + }, + { + "bbox": [ + 141, + 412, + 149, + 421 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 410, + 232, + 424 + ], + "score": 1.0, + "content": ". For proper kernels", + "type": "text" + }, + { + "bbox": [ + 232, + 412, + 239, + 421 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 410, + 340, + 424 + ], + "score": 1.0, + "content": ", it can be shown that (i)", + "type": "text" + }, + { + "bbox": [ + 340, + 412, + 349, + 421 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 410, + 505, + 424 + ], + "score": 1.0, + "content": "can be pushed to zero via its negative", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 267, + 435 + ], + "score": 1.0, + "content": "gradient given by the field and (ii) that", + "type": "text" + }, + { + "bbox": [ + 267, + 422, + 310, + 434 + ], + "score": 0.92, + "content": "\\Phi ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 421, + 339, + 435 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 340, + 424, + 347, + 432 + ], + "score": 0.75, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 421, + 381, + 435 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 381, + 422, + 423, + 434 + ], + "score": 0.92, + "content": "\\bar { { \\rho } } ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 421, + 452, + 435 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 452, + 424, + 460, + 432 + ], + "score": 0.7, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 421, + 506, + 435 + ], + "score": 1.0, + "content": ", therefore,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 433, + 456, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 168, + 446 + ], + "score": 0.93, + "content": "\\bar { p } _ { x } ( { \\pmb a } ) = \\bar { p } _ { y } ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 433, + 197, + 446 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 197, + 434, + 204, + 443 + ], + "score": 0.73, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 433, + 456, + 446 + ], + "score": 1.0, + "content": "(Hochreiter & Obermayer, 2005) (see also Theorem 1 below).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 366, + 506, + 446 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 457, + 227, + 468 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 228, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 228, + 469 + ], + "score": 1.0, + "content": "2.3.1 LEARNING PROCESS", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 505, + 564 + ], + "lines": [ + { + "bbox": [ + 106, + 476, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 249, + 488 + ], + "score": 1.0, + "content": "During learning we do not change", + "type": "text" + }, + { + "bbox": [ + 250, + 477, + 258, + 487 + ], + "score": 0.82, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 477, + 272, + 488 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 272, + 478, + 279, + 488 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 477, + 404, + 488 + ], + "score": 1.0, + "content": "directly. Instead, the location", + "type": "text" + }, + { + "bbox": [ + 405, + 476, + 449, + 488 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\pmb { a } = G ( \\pmb { z } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 477, + 505, + 488 + ], + "score": 1.0, + "content": "to which the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 486, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 174, + 500 + ], + "score": 1.0, + "content": "random variable", + "type": "text" + }, + { + "bbox": [ + 175, + 489, + 183, + 497 + ], + "score": 0.75, + "content": "_ z", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 486, + 338, + 500 + ], + "score": 1.0, + "content": "is mapped changes to a new location", + "type": "text" + }, + { + "bbox": [ + 338, + 487, + 388, + 499 + ], + "score": 0.93, + "content": "\\pmb { a } ^ { \\prime } = G ^ { \\prime } ( \\pmb { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 486, + 506, + 500 + ], + "score": 1.0, + "content": ". For the GAN optimization", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 287, + 511 + ], + "score": 1.0, + "content": "dynamics, we assume that generator samples", + "type": "text" + }, + { + "bbox": [ + 288, + 498, + 329, + 510 + ], + "score": 0.93, + "content": "\\pmb { a } = G ( \\pmb { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "can move freely, which is ensured by a suf-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 484, + 522 + ], + "score": 1.0, + "content": "ficiently complex generator. Importantly, generator samples originating from random variables", + "type": "text" + }, + { + "bbox": [ + 484, + 511, + 492, + 519 + ], + "score": 0.71, + "content": "_ { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "do", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "neither disappear nor are they newly created but are conserved. This conservation is expressed by the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 531, + 504, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 483, + 544 + ], + "score": 1.0, + "content": "continuity equation (Schwartz, 1972) that describes how the difference between distributions", + "type": "text" + }, + { + "bbox": [ + 483, + 531, + 504, + 543 + ], + "score": 0.91, + "content": "\\rho ( { \\pmb a } )", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 541, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 556 + ], + "score": 1.0, + "content": "changes as the particles are moving along the field, i.e., how moving samples during the learning", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 554, + 230, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 230, + 565 + ], + "score": 1.0, + "content": "process changes our densities:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 476, + 506, + 565 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 570, + 364, + 584 + ], + "lines": [ + { + "bbox": [ + 246, + 570, + 364, + 584 + ], + "spans": [ + { + "bbox": [ + 246, + 570, + 364, + 584 + ], + "score": 0.91, + "content": "{ \\dot { \\rho } } ( { \\mathbf { a } } ) \\ = \\ - \\nabla \\cdot ( \\rho ( { \\mathbf { a } } ) \\ v ( { \\mathbf { a } } ) )", + "type": "interline_equation", + "image_path": "712b6d9fbb6f08ef9ae9c964070bbad5f0161d092772ddf3ffe9cdf34b8d98a1.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 246, + 570, + 364, + 584 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 588, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 224, + 603 + ], + "score": 1.0, + "content": "for sample density difference", + "type": "text" + }, + { + "bbox": [ + 224, + 591, + 231, + 601 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 588, + 402, + 603 + ], + "score": 1.0, + "content": "and unit charges that move with “velocity”", + "type": "text" + }, + { + "bbox": [ + 403, + 589, + 501, + 601 + ], + "score": 0.92, + "content": "{ \\pmb v } ( { \\pmb a } ) = \\mathrm { s i g n } ( \\rho ( { \\pmb a } ) ) { \\bf E } ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 588, + 505, + 603 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 506, + 613 + ], + "score": 1.0, + "content": "The continuity equation is crucial as it establishes the connection between moving samples and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 612, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 284, + 624 + ], + "score": 1.0, + "content": "changing the generator density and thereby", + "type": "text" + }, + { + "bbox": [ + 284, + 613, + 290, + 623 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 612, + 506, + 624 + ], + "score": 1.0, + "content": ". The sign function of the velocity indicates whether", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 623, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 623, + 281, + 635 + ], + "score": 1.0, + "content": "positive or negative charges are present at", + "type": "text" + }, + { + "bbox": [ + 282, + 625, + 289, + 632 + ], + "score": 0.77, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 623, + 398, + 635 + ], + "score": 1.0, + "content": ". The divergence operator", + "type": "text" + }, + { + "bbox": [ + 398, + 623, + 422, + 633 + ], + "score": 0.31, + "content": "\\mathbf { \\omega } ^ { 6 6 } \\nabla \\cdot \\mathbf { \\vec { \\mu } } ^ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 623, + 506, + 635 + ], + "score": 1.0, + "content": "determines whether", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 632, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 259, + 647 + ], + "score": 1.0, + "content": "samples move toward or outward of", + "type": "text" + }, + { + "bbox": [ + 259, + 635, + 267, + 643 + ], + "score": 0.76, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 632, + 506, + 647 + ], + "score": 1.0, + "content": "for a given field. Basically, the continuity equation says", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "that if the generator density increases, then generator samples must flow into the region and if the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 104, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "generator density decreases, they flow outwards. We assume that differently charged particles cancel", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 375, + 679 + ], + "score": 1.0, + "content": "each other. 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The continuity equation is also", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "obtained as a first order ODE to move particles in a potential field (Dembo & Zeitouni, 1988),", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 699, + 411, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 411, + 712 + ], + "score": 1.0, + "content": "therefore describes the dynamics how the densities are changing. We obtain", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 46, + "bbox_fs": [ + 104, + 588, + 506, + 712 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 170, + 716, + 441, + 730 + ], + "lines": [ + { + "bbox": [ + 170, + 716, + 441, + 730 + ], + "spans": [ + { + "bbox": [ + 170, + 716, + 441, + 730 + ], + "score": 0.88, + "content": "\\dot { \\rho } ( a ) = - \\mathrm { s i g n } ( \\rho ( a ) ) \\nabla \\cdot \\left( \\rho ( a ) { \\bf E } ( a ) \\right) = - \\nabla \\cdot \\left( | \\rho ( a ) | { \\bf E } ( a ) \\right) .", + "type": "interline_equation", + "image_path": "6ba26ccc4ecb901e2bcd5fc975516ee4416a1c26e3d1d53abdc1824620d2dbcd.jpg" + } + ] + } + ], + "index": 52, + "virtual_lines": [ + { + "bbox": [ + 170, + 716, + 441, + 730 + ], + "spans": [], + "index": 52 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 199, + 95 + ], + "score": 1.0, + "content": "The density difference", + "type": "text" + }, + { + "bbox": [ + 199, + 83, + 219, + 95 + ], + "score": 0.92, + "content": "\\rho ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "indicates how many samples are locally available for being moved. At", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 272, + 107 + ], + "score": 1.0, + "content": "each local minimum and local maximum", + "type": "text" + }, + { + "bbox": [ + 272, + 96, + 280, + 104 + ], + "score": 0.74, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 93, + 291, + 107 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 292, + 95, + 299, + 105 + ], + "score": 0.82, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 93, + 342, + 107 + ], + "score": 1.0, + "content": "we obtain", + "type": "text" + }, + { + "bbox": [ + 342, + 93, + 394, + 106 + ], + "score": 0.92, + "content": "\\pmb { \\nabla } _ { a } \\rho ( \\pmb { a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 93, + 506, + 107 + ], + "score": 1.0, + "content": ". Using the product rule for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 437, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 241, + 117 + ], + "score": 1.0, + "content": "the divergence operator, at points", + "type": "text" + }, + { + "bbox": [ + 241, + 106, + 249, + 114 + ], + "score": 0.79, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 105, + 437, + 117 + ], + "score": 1.0, + "content": "that are minima or maxima, Eq. (3) reduces to", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 224, + 120, + 386, + 134 + ], + "lines": [ + { + "bbox": [ + 224, + 120, + 386, + 134 + ], + "spans": [ + { + "bbox": [ + 224, + 120, + 386, + 134 + ], + "score": 0.91, + "content": "\\dot { \\rho } ( { \\pmb a } ) = - \\operatorname { s i g n } ( \\rho ( { \\pmb a } ) ) \\rho ( { \\pmb a } ) { \\pmb \\nabla } \\cdot { \\bf E } ( { \\pmb a } ) .", + "type": "interline_equation", + "image_path": "88ed120b3851cb0c678506878af2392cc7881ee8ca90e04abb48442463aa11be.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 224, + 120, + 386, + 134 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 505, + 173 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 504, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 201, + 151 + ], + "score": 1.0, + "content": "In order to ensure that", + "type": "text" + }, + { + "bbox": [ + 202, + 141, + 209, + 150 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 138, + 426, + 151 + ], + "score": 1.0, + "content": "converges to zero, it is necessary and sufficient that", + "type": "text" + }, + { + "bbox": [ + 427, + 138, + 504, + 151 + ], + "score": 0.87, + "content": "\\mathrm { s i g n } ( \\pmb { \\nabla } \\cdot \\mathbf { E } ( \\pmb { a } ) ) =", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 107, + 149, + 151, + 162 + ], + "score": 0.9, + "content": "\\mathrm { s i g n } ( \\rho ( a ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 149, + 184, + 163 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 185, + 150, + 241, + 162 + ], + "score": 0.92, + "content": "\\div a \\rho ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 149, + 506, + 163 + ], + "score": 1.0, + "content": ", as this condition ensures the uniform decrease of the maximal", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 159, + 265, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 220, + 174 + ], + "score": 1.0, + "content": "absolute density differences", + "type": "text" + }, + { + "bbox": [ + 220, + 161, + 261, + 173 + ], + "score": 0.9, + "content": "\\left| \\rho ( \\pmb { a } _ { \\mathrm { m a x } } ) \\right|", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 159, + 265, + 174 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 183, + 226, + 195 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 228, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 228, + 197 + ], + "score": 1.0, + "content": "2.3.2 CHOICE OF KERNEL", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 202, + 505, + 291 + ], + "lines": [ + { + "bbox": [ + 106, + 203, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 203, + 441, + 214 + ], + "score": 1.0, + "content": "As discussed before, the choice of kernel is crucial for Coulomb GANs. The", + "type": "text" + }, + { + "bbox": [ + 442, + 205, + 451, + 213 + ], + "score": 0.78, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 203, + 505, + 214 + ], + "score": 1.0, + "content": "-dimensional", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 205, + 226 + ], + "score": 1.0, + "content": "Coulomb kernel and the", + "type": "text" + }, + { + "bbox": [ + 205, + 216, + 216, + 225 + ], + "score": 0.78, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 214, + 377, + 226 + ], + "score": 1.0, + "content": "-dimensional Plummer kernel lead to (i)", + "type": "text" + }, + { + "bbox": [ + 378, + 215, + 386, + 224 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "that is pushed to zero via the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 225, + 504, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 214, + 237 + ], + "score": 1.0, + "content": "field it creates and (ii) that", + "type": "text" + }, + { + "bbox": [ + 214, + 225, + 254, + 237 + ], + "score": 0.93, + "content": "\\Phi ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 225, + 282, + 237 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 282, + 227, + 290, + 235 + ], + "score": 0.75, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 225, + 322, + 237 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 323, + 225, + 362, + 237 + ], + "score": 0.93, + "content": "\\rho ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 225, + 389, + 237 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 389, + 227, + 397, + 235 + ], + "score": 0.75, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 225, + 441, + 237 + ], + "score": 1.0, + "content": ", therefore,", + "type": "text" + }, + { + "bbox": [ + 442, + 225, + 504, + 237 + ], + "score": 0.93, + "content": "p _ { x } ( { \\pmb a } ) = p _ { y } ( { \\pmb a } )", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 135, + 248 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 136, + 238, + 143, + 246 + ], + "score": 0.72, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "(Hochreiter & Obermayer, 2005). 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The resulting field and potential energy is", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 344, + 449, + 397 + ], + "lines": [ + { + "bbox": [ + 162, + 344, + 449, + 397 + ], + "spans": [ + { + "bbox": [ + 162, + 344, + 449, + 397 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\mathbf { E } ( \\pmb { a } ) = - \\int \\rho ( \\pmb { b } ) \\nabla _ { a } k ( \\pmb { a } , \\pmb { b } ) \\mathrm { d } \\pmb { b } = - \\nabla _ { a } \\Phi ( \\pmb { a } ) \\ , } \\\\ { \\displaystyle F ( \\rho ) = \\frac { 1 } { 2 } \\int \\rho ( \\pmb { a } ) \\Phi ( \\pmb { a } ) \\mathrm { d } \\pmb { a } = \\frac { 1 } { 2 } \\int \\displaystyle \\int \\rho ( \\pmb { a } ) \\rho ( \\pmb { b } ) k ( \\pmb { a } , \\pmb { b } ) \\mathrm { d } \\pmb { b } \\mathrm { d } \\pmb { a } . } \\end{array}", + "type": "interline_equation", + "image_path": "f6d1d16079797a929b6637715e8a5eb46b11c0b1ea7460f33a61f49edde26536.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 162, + 344, + 449, + 361.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 162, + 361.6666666666667, + 449, + 379.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 162, + 379.33333333333337, + 449, + 397.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 504, + 422 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 386, + 413 + ], + "score": 1.0, + "content": "The next theorem states that for freely moving generated samples,", + "type": "text" + }, + { + "bbox": [ + 386, + 402, + 393, + 411 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 398, + 505, + 413 + ], + "score": 1.0, + "content": "converges to zero, that is,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 410, + 330, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 162, + 423 + ], + "score": 0.92, + "content": "p _ { x } ( . ) = p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 410, + 304, + 424 + ], + "score": 1.0, + "content": ", when using this potential function", + "type": "text" + }, + { + "bbox": [ + 304, + 411, + 326, + 423 + ], + "score": 0.92, + "content": "\\Phi ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 410, + 330, + 424 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 399, + 439 + ], + "score": 1.0, + "content": "Theorem 1 (Convergence with low-dimensional Plummer kernel). For", + "type": "text" + }, + { + "bbox": [ + 399, + 425, + 446, + 437 + ], + "score": 0.88, + "content": "\\pmb { a } , \\pmb { b } \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 423, + 451, + 439 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 451, + 425, + 501, + 436 + ], + "score": 0.87, + "content": "d \\leqslant m - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 423, + 505, + 439 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 124, + 449 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 437, + 153, + 447 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 435, + 209, + 449 + ], + "score": 1.0, + "content": "the densities", + "type": "text" + }, + { + "bbox": [ + 209, + 436, + 231, + 448 + ], + "score": 0.91, + "content": "p _ { x } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 435, + 252, + 449 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 436, + 274, + 448 + ], + "score": 0.91, + "content": "p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 435, + 458, + 449 + ], + "score": 1.0, + "content": "equalize over time when minimizing energy", + "type": "text" + }, + { + "bbox": [ + 458, + 437, + 467, + 446 + ], + "score": 0.81, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "with the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 447, + 479, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 470, + 460 + ], + "score": 1.0, + "content": "low-dimensional Plummer kernel by gradient descent. The convergence is faster for larger", + "type": "text" + }, + { + "bbox": [ + 470, + 448, + 477, + 457 + ], + "score": 0.59, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 447, + 479, + 460 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 204, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 205, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 205, + 484 + ], + "score": 1.0, + "content": "Proof. See Section A.2.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 108, + 495, + 289, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 289, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 289, + 508 + ], + "score": 1.0, + "content": "2.4 DEFINITION OF THE COULOMB GAN", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 515, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "The Coulomb GAN minimizes the electric potential energy from Eq. (6) using a stochastic gradient", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "score": 1.0, + "content": "descent based approach using mini-batches. Appendix Section A.4 contains the equations for the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 536, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 489, + 551 + ], + "score": 1.0, + "content": "Coulomb potential, field, and energy in this case. Generator samples are obtained by drawing", + "type": "text" + }, + { + "bbox": [ + 490, + 539, + 504, + 549 + ], + "score": 0.86, + "content": "N _ { x }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 176, + 562 + ], + "score": 1.0, + "content": "random numbers", + "type": "text" + }, + { + "bbox": [ + 176, + 551, + 186, + 560 + ], + "score": 0.86, + "content": "z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 549, + 330, + 562 + ], + "score": 1.0, + "content": "and transforming them into outputs", + "type": "text" + }, + { + "bbox": [ + 330, + 549, + 378, + 561 + ], + "score": 0.92, + "content": "\\pmb { x } _ { i } = G ( z _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 549, + 505, + 562 + ], + "score": 1.0, + "content": ". Each mini-batch also includes", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 107, + 560, + 120, + 572 + ], + "score": 0.87, + "content": "N _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 559, + 199, + 573 + ], + "score": 1.0, + "content": "real world samples", + "type": "text" + }, + { + "bbox": [ + 199, + 562, + 209, + 572 + ], + "score": 0.83, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 559, + 505, + 573 + ], + "score": 1.0, + "content": ". This gives rise to a mini-batch specific potential, where in Eq. (5) we use", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 571, + 501, + 583 + ], + "spans": [ + { + "bbox": [ + 107, + 571, + 200, + 583 + ], + "score": 0.93, + "content": "\\rho ( { \\mathbf { \\bar { a } } } ) = p _ { y } ( { \\mathbf { \\bar { a } } } ) - \\bar { p _ { x } } ( { \\mathbf { \\bar { a } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 571, + 501, + 583 + ], + "score": 1.0, + "content": "and replace the expectations by empirical means using the drawn samples:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 588, + 406, + 623 + ], + "lines": [ + { + "bbox": [ + 205, + 588, + 406, + 623 + ], + "spans": [ + { + "bbox": [ + 205, + 588, + 406, + 623 + ], + "score": 0.94, + "content": "\\hat { \\Phi } ( { \\pmb a } ) = \\frac { 1 } { N _ { y } } \\sum _ { i = 1 } ^ { N _ { y } } k \\left( { \\pmb a } , { \\pmb y } _ { i } \\right) - \\frac { 1 } { N _ { x } } \\sum _ { i = 1 } ^ { N _ { x } } k \\left( { \\pmb a } , { \\pmb x } _ { i } \\right) .", + "type": "interline_equation", + "image_path": "d6e3502e25c387929a6ea75f2ce5f5658eef5f5264cedaffd4510ae497057d6a.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 588, + 406, + 605.5 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 205, + 605.5, + 406, + 623.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 437, + 642 + ], + "score": 1.0, + "content": "It is tempting to have a generator network that directly minimizes this potential", + "type": "text" + }, + { + "bbox": [ + 438, + 628, + 446, + 640 + ], + "score": 0.86, + "content": "\\hat { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "between gen-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 320, + 654 + ], + "score": 1.0, + "content": "erated and training set points. 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The resulting field and potential energy is", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 326, + 505, + 342 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 162, + 344, + 449, + 397 + ], + "lines": [ + { + "bbox": [ + 162, + 344, + 449, + 397 + ], + "spans": [ + { + "bbox": [ + 162, + 344, + 449, + 397 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\mathbf { E } ( \\pmb { a } ) = - \\int \\rho ( \\pmb { b } ) \\nabla _ { a } k ( \\pmb { a } , \\pmb { b } ) \\mathrm { d } \\pmb { b } = - \\nabla _ { a } \\Phi ( \\pmb { a } ) \\ , } \\\\ { \\displaystyle F ( \\rho ) = \\frac { 1 } { 2 } \\int \\rho ( \\pmb { a } ) \\Phi ( \\pmb { a } ) \\mathrm { d } \\pmb { a } = \\frac { 1 } { 2 } \\int \\displaystyle \\int \\rho ( \\pmb { a } ) \\rho ( \\pmb { b } ) k ( \\pmb { a } , \\pmb { b } ) \\mathrm { d } \\pmb { b } \\mathrm { d } \\pmb { a } . } \\end{array}", + "type": "interline_equation", + "image_path": "f6d1d16079797a929b6637715e8a5eb46b11c0b1ea7460f33a61f49edde26536.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 162, + 344, + 449, + 361.6666666666667 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 162, + 361.6666666666667, + 449, + 379.33333333333337 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 162, + 379.33333333333337, + 449, + 397.00000000000006 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 504, + 422 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 386, + 413 + ], + "score": 1.0, + "content": "The next theorem states that for freely moving generated samples,", + "type": "text" + }, + { + "bbox": [ + 386, + 402, + 393, + 411 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 398, + 505, + 413 + ], + "score": 1.0, + "content": "converges to zero, that is,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 410, + 330, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 162, + 423 + ], + "score": 0.92, + "content": "p _ { x } ( . ) = p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 410, + 304, + 424 + ], + "score": 1.0, + "content": ", when using this potential function", + "type": "text" + }, + { + "bbox": [ + 304, + 411, + 326, + 423 + ], + "score": 0.92, + "content": "\\Phi ( a )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 410, + 330, + 424 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 398, + 505, + 424 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 399, + 439 + ], + "score": 1.0, + "content": "Theorem 1 (Convergence with low-dimensional Plummer kernel). For", + "type": "text" + }, + { + "bbox": [ + 399, + 425, + 446, + 437 + ], + "score": 0.88, + "content": "\\pmb { a } , \\pmb { b } \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 423, + 451, + 439 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 451, + 425, + 501, + 436 + ], + "score": 0.87, + "content": "d \\leqslant m - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 423, + 505, + 439 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 435, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 124, + 449 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 125, + 437, + 153, + 447 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 435, + 209, + 449 + ], + "score": 1.0, + "content": "the densities", + "type": "text" + }, + { + "bbox": [ + 209, + 436, + 231, + 448 + ], + "score": 0.91, + "content": "p _ { x } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 435, + 252, + 449 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 436, + 274, + 448 + ], + "score": 0.91, + "content": "p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 435, + 458, + 449 + ], + "score": 1.0, + "content": "equalize over time when minimizing energy", + "type": "text" + }, + { + "bbox": [ + 458, + 437, + 467, + 446 + ], + "score": 0.81, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 435, + 505, + 449 + ], + "score": 1.0, + "content": "with the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 447, + 479, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 470, + 460 + ], + "score": 1.0, + "content": "low-dimensional Plummer kernel by gradient descent. The convergence is faster for larger", + "type": "text" + }, + { + "bbox": [ + 470, + 448, + 477, + 457 + ], + "score": 0.59, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 447, + 479, + 460 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 423, + 505, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 204, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 205, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 205, + 484 + ], + "score": 1.0, + "content": "Proof. See Section A.2.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 469, + 205, + 484 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 495, + 289, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 289, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 289, + 508 + ], + "score": 1.0, + "content": "2.4 DEFINITION OF THE COULOMB GAN", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 515, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "The Coulomb GAN minimizes the electric potential energy from Eq. (6) using a stochastic gradient", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 540 + ], + "score": 1.0, + "content": "descent based approach using mini-batches. Appendix Section A.4 contains the equations for the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 536, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 489, + 551 + ], + "score": 1.0, + "content": "Coulomb potential, field, and energy in this case. Generator samples are obtained by drawing", + "type": "text" + }, + { + "bbox": [ + 490, + 539, + 504, + 549 + ], + "score": 0.86, + "content": "N _ { x }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 176, + 562 + ], + "score": 1.0, + "content": "random numbers", + "type": "text" + }, + { + "bbox": [ + 176, + 551, + 186, + 560 + ], + "score": 0.86, + "content": "z _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 549, + 330, + 562 + ], + "score": 1.0, + "content": "and transforming them into outputs", + "type": "text" + }, + { + "bbox": [ + 330, + 549, + 378, + 561 + ], + "score": 0.92, + "content": "\\pmb { x } _ { i } = G ( z _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 549, + 505, + 562 + ], + "score": 1.0, + "content": ". Each mini-batch also includes", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 107, + 560, + 120, + 572 + ], + "score": 0.87, + "content": "N _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 559, + 199, + 573 + ], + "score": 1.0, + "content": "real world samples", + "type": "text" + }, + { + "bbox": [ + 199, + 562, + 209, + 572 + ], + "score": 0.83, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 559, + 505, + 573 + ], + "score": 1.0, + "content": ". This gives rise to a mini-batch specific potential, where in Eq. (5) we use", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 571, + 501, + 583 + ], + "spans": [ + { + "bbox": [ + 107, + 571, + 200, + 583 + ], + "score": 0.93, + "content": "\\rho ( { \\mathbf { \\bar { a } } } ) = p _ { y } ( { \\mathbf { \\bar { a } } } ) - \\bar { p _ { x } } ( { \\mathbf { \\bar { a } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 571, + 501, + 583 + ], + "score": 1.0, + "content": "and replace the expectations by empirical means using the drawn samples:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 515, + 505, + 583 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 588, + 406, + 623 + ], + "lines": [ + { + "bbox": [ + 205, + 588, + 406, + 623 + ], + "spans": [ + { + "bbox": [ + 205, + 588, + 406, + 623 + ], + "score": 0.94, + "content": "\\hat { \\Phi } ( { \\pmb a } ) = \\frac { 1 } { N _ { y } } \\sum _ { i = 1 } ^ { N _ { y } } k \\left( { \\pmb a } , { \\pmb y } _ { i } \\right) - \\frac { 1 } { N _ { x } } \\sum _ { i = 1 } ^ { N _ { x } } k \\left( { \\pmb a } , { \\pmb x } _ { i } \\right) .", + "type": "interline_equation", + "image_path": "d6e3502e25c387929a6ea75f2ce5f5658eef5f5264cedaffd4510ae497057d6a.jpg" + } + ] + } + ], + "index": 36.5, + "virtual_lines": [ + { + "bbox": [ + 205, + 588, + 406, + 605.5 + ], + "spans": [], + "index": 36 + }, + { + "bbox": [ + 205, + 605.5, + 406, + 623.0 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 628, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 437, + 642 + ], + "score": 1.0, + "content": "It is tempting to have a generator network that directly minimizes this potential", + "type": "text" + }, + { + "bbox": [ + 438, + 628, + 446, + 640 + ], + "score": 0.86, + "content": "\\hat { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "between gen-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 320, + 654 + ], + "score": 1.0, + "content": "erated and training set points. In fact, we show that", + "type": "text" + }, + { + "bbox": [ + 320, + 641, + 329, + 652 + ], + "score": 0.86, + "content": "\\hat { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 642, + 442, + 654 + ], + "score": 1.0, + "content": "is an unbiased estimate for", + "type": "text" + }, + { + "bbox": [ + 442, + 642, + 451, + 652 + ], + "score": 0.83, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "in Appendix", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "Section A.4. However, the estimate has very high variance: for example, if a mini-batch fails to sam-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 505, + 676 + ], + "score": 1.0, + "content": "ple training data from an existing mode, the field would drive all generated samples that have been", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 675, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 505, + 688 + ], + "score": 1.0, + "content": "generated at this mode to move elsewhere. 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(33), but with an optimal Kernel specifically tailored to the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 344, + 380, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 380, + 358 + ], + "score": 1.0, + "content": "problem of estimating differences between probability distributions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 504, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 504, + 374 + ], + "score": 1.0, + "content": "We use each mini-batch only for one update of the discriminator and the generator. 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Pseudocode for the learning algorithm is detailed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 441, + 235, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 235, + 453 + ], + "score": 1.0, + "content": "in Algorithm 1 in the appendix.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 108, + 464, + 275, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 277, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 277, + 477 + ], + "score": 1.0, + "content": "2.4.1 OPTIMALITY OF THE SOLUTION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "Convergence of the GAN learning process was proved for a two time-scales update rule by Heusel", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 420, + 506 + ], + "score": 1.0, + "content": "et al. (2017). A local Nash equilibrium is a pair of generator and discriminator", + "type": "text" + }, + { + "bbox": [ + 420, + 495, + 458, + 506 + ], + "score": 0.93, + "content": "( D ^ { * } , G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "that fulfills", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 505, + 182, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 182, + 517 + ], + "score": 1.0, + "content": "the two conditions", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 521, + 436, + 545 + ], + "lines": [ + { + "bbox": [ + 174, + 521, + 436, + 545 + ], + "spans": [ + { + "bbox": [ + 174, + 521, + 436, + 545 + ], + "score": 0.92, + "content": "D ^ { * } = \\arg \\operatorname* { m i n } _ { D \\in U ( D ^ { * } ) } \\mathcal { L } _ { D } ( D ; G ^ { * } ) \\quad \\mathrm { a n d } \\quad G ^ { * } = \\arg \\operatorname* { m i n } _ { G \\in U ( G ^ { * } ) } \\mathcal { L } _ { G } ( G ; D ^ { * } ) .", + "type": "interline_equation", + "image_path": "510be43e1dd7e0eb883459d3754a4698dc11046ee12a843bdcf343ff363c173b.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 174, + 521, + 436, + 545 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 205, + 563 + ], + "score": 1.0, + "content": "for some neighborhoods", + "type": "text" + }, + { + "bbox": [ + 205, + 550, + 235, + 562 + ], + "score": 0.93, + "content": "U ( D ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 550, + 252, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 550, + 281, + 563 + ], + "score": 0.92, + "content": "U ( G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 550, + 505, + 563 + ], + "score": 1.0, + "content": ". We show in the following Theorem 2 that for Coulomb", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "score": 1.0, + "content": "GANs every local Nash equilibrium necessarily is identical to the unique global Nash equilibrium.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "In other words, any equilibrium point of the Coulomb GAN that is found to be local optimal has to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 581, + 507, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 392, + 597 + ], + "score": 1.0, + "content": "be the one global Nash equilibrium as the minimization of the energy", + "type": "text" + }, + { + "bbox": [ + 393, + 583, + 414, + 595 + ], + "score": 0.92, + "content": "F ( \\rho )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 581, + 507, + 597 + ], + "score": 1.0, + "content": "in Eq. (33) leads to a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 592, + 250, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 210, + 609 + ], + "score": 1.0, + "content": "single, global optimum at", + "type": "text" + }, + { + "bbox": [ + 211, + 596, + 245, + 606 + ], + "score": 0.9, + "content": "p _ { y } = p _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 592, + 250, + 609 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 283, + 621 + ], + "score": 1.0, + "content": "Theorem 2 (Optimal Solution). If the pair", + "type": "text" + }, + { + "bbox": [ + 284, + 609, + 322, + 621 + ], + "score": 0.92, + "content": "( D ^ { * } , G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "is a local Nash equilibrium for the Coulomb", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "GAN objectives, then it is the global Nash equilibrium, and no other local Nash equilibria exist, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 631, + 253, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 253, + 645 + ], + "score": 1.0, + "content": "G∗ has output distribution px = py.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 655, + 245, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 247, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 247, + 667 + ], + "score": 1.0, + "content": "Proof. See Appendix Section A.3.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "title", + "bbox": [ + 108, + 678, + 273, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 275, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 275, + 691 + ], + "score": 1.0, + "content": "2.4.2 COULOMB GANS IN PRACTICE", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 372, + 711 + ], + "score": 1.0, + "content": "To implement GANs in practice, we need learnable models for", + "type": "text" + }, + { + "bbox": [ + 373, + 699, + 382, + 709 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 699, + 402, + 711 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 402, + 699, + 412, + 709 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". 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(1998), in order to overcome the problem", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 289, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 329, + 303 + ], + "score": 1.0, + "content": "that the generator distribution is only a sub-manifold of", + "type": "text" + }, + { + "bbox": [ + 329, + 291, + 344, + 300 + ], + "score": 0.87, + "content": "\\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 289, + 505, + 303 + ], + "score": 1.0, + "content": ". These loss functions cause the approx-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 201, + 315 + ], + "score": 1.0, + "content": "imated potential values", + "type": "text" + }, + { + "bbox": [ + 201, + 301, + 225, + 313 + ], + "score": 0.92, + "content": "D ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 301, + 505, + 315 + ], + "score": 1.0, + "content": "that are negative are pushed toward zero. Finally, the Coulomb GAN,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "like all other GANs, consists of two parts: a generator to generate model samples, and a discrimina-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 335 + ], + "score": 1.0, + "content": "tor that provides its learning signal. Without a discriminator, our would be very similar to GMMNs", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "(Li et al., 2015), as can be seen in Eq. (33), but with an optimal Kernel specifically tailored to the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 344, + 380, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 380, + 358 + ], + "score": 1.0, + "content": "problem of estimating differences between probability distributions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 256, + 506, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 452 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 504, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 504, + 374 + ], + "score": 1.0, + "content": "We use each mini-batch only for one update of the discriminator and the generator. It is important", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "to note that the discriminator uses each sample in the mini batch twice: once as a point to generate", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 384, + 504, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 236, + 398 + ], + "score": 1.0, + "content": "the mini-batch specific potential", + "type": "text" + }, + { + "bbox": [ + 236, + 384, + 245, + 396 + ], + "score": 0.84, + "content": "\\hat { \\Phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 385, + 495, + 398 + ], + "score": 1.0, + "content": ", and once as a point in space for the evaluation of the potential", + "type": "text" + }, + { + "bbox": [ + 496, + 384, + 504, + 396 + ], + "score": 0.83, + "content": "\\hat { \\Phi }", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 397, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 196, + 409 + ], + "score": 1.0, + "content": "and its approximation", + "type": "text" + }, + { + "bbox": [ + 196, + 397, + 205, + 406 + ], + "score": 0.78, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 397, + 505, + 409 + ], + "score": 1.0, + "content": ". Using each sample twice is done for performance reasons, but not strictly", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 407, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 506, + 420 + ], + "score": 1.0, + "content": "necessary: the discriminator could learn the potential field by sampling points that lie between", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 418, + 507, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 418, + 507, + 431 + ], + "score": 1.0, + "content": "generator and real samples as in Gulrajani et al. (2017), but we are mainly interested in correct", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 442 + ], + "score": 1.0, + "content": "predictions in the vicinity of generator samples. Pseudocode for the learning algorithm is detailed", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 441, + 235, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 235, + 453 + ], + "score": 1.0, + "content": "in Algorithm 1 in the appendix.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 362, + 507, + 453 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 464, + 275, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 464, + 277, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 277, + 477 + ], + "score": 1.0, + "content": "2.4.1 OPTIMALITY OF THE SOLUTION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 516 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 496 + ], + "score": 1.0, + "content": "Convergence of the GAN learning process was proved for a two time-scales update rule by Heusel", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 494, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 420, + 506 + ], + "score": 1.0, + "content": "et al. (2017). A local Nash equilibrium is a pair of generator and discriminator", + "type": "text" + }, + { + "bbox": [ + 420, + 495, + 458, + 506 + ], + "score": 0.93, + "content": "( D ^ { * } , G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 494, + 505, + 506 + ], + "score": 1.0, + "content": "that fulfills", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 505, + 182, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 182, + 517 + ], + "score": 1.0, + "content": "the two conditions", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 482, + 506, + 517 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 174, + 521, + 436, + 545 + ], + "lines": [ + { + "bbox": [ + 174, + 521, + 436, + 545 + ], + "spans": [ + { + "bbox": [ + 174, + 521, + 436, + 545 + ], + "score": 0.92, + "content": "D ^ { * } = \\arg \\operatorname* { m i n } _ { D \\in U ( D ^ { * } ) } \\mathcal { L } _ { D } ( D ; G ^ { * } ) \\quad \\mathrm { a n d } \\quad G ^ { * } = \\arg \\operatorname* { m i n } _ { G \\in U ( G ^ { * } ) } \\mathcal { L } _ { G } ( G ; D ^ { * } ) .", + "type": "interline_equation", + "image_path": "510be43e1dd7e0eb883459d3754a4698dc11046ee12a843bdcf343ff363c173b.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 174, + 521, + 436, + 545 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 205, + 563 + ], + "score": 1.0, + "content": "for some neighborhoods", + "type": "text" + }, + { + "bbox": [ + 205, + 550, + 235, + 562 + ], + "score": 0.93, + "content": "U ( D ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 550, + 252, + 563 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 253, + 550, + 281, + 563 + ], + "score": 0.92, + "content": "U ( G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 550, + 505, + 563 + ], + "score": 1.0, + "content": ". We show in the following Theorem 2 that for Coulomb", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 504, + 573 + ], + "score": 1.0, + "content": "GANs every local Nash equilibrium necessarily is identical to the unique global Nash equilibrium.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "In other words, any equilibrium point of the Coulomb GAN that is found to be local optimal has to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 581, + 507, + 597 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 392, + 597 + ], + "score": 1.0, + "content": "be the one global Nash equilibrium as the minimization of the energy", + "type": "text" + }, + { + "bbox": [ + 393, + 583, + 414, + 595 + ], + "score": 0.92, + "content": "F ( \\rho )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 581, + 507, + 597 + ], + "score": 1.0, + "content": "in Eq. (33) leads to a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 592, + 250, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 210, + 609 + ], + "score": 1.0, + "content": "single, global optimum at", + "type": "text" + }, + { + "bbox": [ + 211, + 596, + 245, + 606 + ], + "score": 0.9, + "content": "p _ { y } = p _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 592, + 250, + 609 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 550, + 507, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 609, + 505, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 283, + 621 + ], + "score": 1.0, + "content": "Theorem 2 (Optimal Solution). If the pair", + "type": "text" + }, + { + "bbox": [ + 284, + 609, + 322, + 621 + ], + "score": 0.92, + "content": "( D ^ { * } , G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "is a local Nash equilibrium for the Coulomb", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 506, + 633 + ], + "score": 1.0, + "content": "GAN objectives, then it is the global Nash equilibrium, and no other local Nash equilibria exist, and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 631, + 253, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 253, + 645 + ], + "score": 1.0, + "content": "G∗ has output distribution px = py.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 609, + 506, + 645 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 655, + 245, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 247, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 247, + 667 + ], + "score": 1.0, + "content": "Proof. See Appendix Section A.3.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 654, + 247, + 667 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 678, + 273, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 275, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 275, + 691 + ], + "score": 1.0, + "content": "2.4.2 COULOMB GANS IN PRACTICE", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 372, + 711 + ], + "score": 1.0, + "content": "To implement GANs in practice, we need learnable models for", + "type": "text" + }, + { + "bbox": [ + 373, + 699, + 382, + 709 + ], + "score": 0.83, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 699, + 402, + 711 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 402, + 699, + 412, + 709 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 699, + 505, + 711 + ], + "score": 1.0, + "content": ". We assume that our", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 154, + 722 + ], + "score": 1.0, + "content": "models for", + "type": "text" + }, + { + "bbox": [ + 154, + 710, + 164, + 720 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 709, + 183, + 722 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 184, + 710, + 194, + 720 + ], + "score": 0.8, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "are continuously differentiable with respect to their parameters and inputs.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "Toward this end, GANs are typically implemented as neural networks optimized by (some variant", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 344, + 95 + ], + "score": 1.0, + "content": "of) gradient descent. Thus we may not find the optimal", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 344, + 83, + 358, + 93 + ], + "score": 0.87, + "content": "G ^ { * }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 358, + 81, + 372, + 95 + ], + "score": 1.0, + "content": "or", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 372, + 83, + 386, + 93 + ], + "score": 0.88, + "content": "D ^ { * }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 387, + 81, + 505, + 95 + ], + "score": 1.0, + "content": ", since neural networks may", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "suffer from capacity or optimization issues. Recent research indicates that the effect of local minima", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "in deep learning vanishes with increasing depth (Dauphin et al., 2014; Choromanska et al., 2015;", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "Kawaguchi, 2016), such that this limitation becomes less restrictive as our ability to train deep", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 381, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 381, + 138 + ], + "score": 1.0, + "content": "networks grows thanks to hardware and optimization improvements.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 699, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 344, + 95 + ], + "score": 1.0, + "content": "of) gradient descent. Thus we may not find the optimal", + "type": "text" + }, + { + "bbox": [ + 344, + 83, + 358, + 93 + ], + "score": 0.87, + "content": "G ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 81, + 372, + 95 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 372, + 83, + 386, + 93 + ], + "score": 0.88, + "content": "D ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 81, + 505, + 95 + ], + "score": 1.0, + "content": ", since neural networks may", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "suffer from capacity or optimization issues. Recent research indicates that the effect of local minima", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "in deep learning vanishes with increasing depth (Dauphin et al., 2014; Choromanska et al., 2015;", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "Kawaguchi, 2016), such that this limitation becomes less restrictive as our ability to train deep", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 381, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 381, + 138 + ], + "score": 1.0, + "content": "networks grows thanks to hardware and optimization improvements.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 466, + 156 + ], + "score": 1.0, + "content": "The main problem with learning Coulomb GANs is to approximate the potential function", + "type": "text" + }, + { + "bbox": [ + 466, + 144, + 474, + 154 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 143, + 505, + 156 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "is a complex function in a high-dimensional space, since the potential can be very non-linear and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "non-smooth. When learning the discriminator, we must ensure that enough data is sampled and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "averaged over. We already lessened the non-linear function problem by using a low-dimensional", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "Plummer kernel. But still, this kernel can introduce large non-linearities if samples are close to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "each other. It is crucial that the discriminator learns slow enough to accurately estimate the potential", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "function which is induced by the current generator. The generator, in turn, must be even slower since", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "it must be tracked by the discriminator. These approximation problems are supposed to be tackled", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 452, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 452, + 244 + ], + "score": 1.0, + "content": "by the research community in near future, which would enable optimal GAN learning.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "The formulation of GAN learning as a potential field naturally solves the mode collapsing issue: the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "example described in Section A.1, where a normal GAN cannot get out of a local Nash equilibria", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "is not a converged solution for the Coulomb GAN: If all probability mass of the generator lies in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "one of the modes, then both attracting forces from real-world samples located at the other mode as", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "well as repelling forces from the over-represented generator mode will act upon the generator until", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 294, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 294, + 315 + ], + "score": 1.0, + "content": "it generates samples at the other mode as well.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 108, + 331, + 200, + 344 + ], + "lines": [ + { + "bbox": [ + 104, + 330, + 202, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 330, + 202, + 346 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 456, + 369 + ], + "score": 1.0, + "content": "In all of our experiments, we used a low-dimensional Plummer Kernel of dimensionality", + "type": "text" + }, + { + "bbox": [ + 456, + 357, + 480, + 367 + ], + "score": 0.88, + "content": "d = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 357, + 505, + 369 + ], + "score": 1.0, + "content": ". This", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "kernel both gave best computational performance and has low risk of running into numerical issues.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "We used a batch size of 128. To evaluate the quality of a GAN, the FID metric as proposed by Heusel", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 257, + 403 + ], + "score": 1.0, + "content": "et al. (2017) was calculated by using", + "type": "text" + }, + { + "bbox": [ + 257, + 390, + 274, + 401 + ], + "score": 0.48, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "samples drawn from the generator, while the training set", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "statistics were calculated using the whole training set. We compare to BEGAN (Berthelot et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "2017), DCGAN (Radford et al., 2016) and WGAN-GP (Gulrajani et al., 2017) both in their original", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "version as well as when using the two-timescale update-rule (TTUR), using the settings from Heusel", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 432, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 505, + 447 + ], + "score": 1.0, + "content": "et al. (2017). We additionally compare to MMD-GAN (Li et al., 2017), which is conceptually very", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "similar to the Coulomb GAN, but uses a Gaussian Kernel instead of the Plummer Kernel. We use", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "the dataset-specific settings recommended in (Li et al., 2017) and report the highest FID score over", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "the course of training. All images shown in this paper were produced with a random seed and not", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 478, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 489 + ], + "score": 1.0, + "content": "cherry picked. The implementation used for these experiments is available online1. The appendix", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "Section A.5 contains an additional toy example demonstrating that Coulomb GANs do not suffer", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 500, + 424, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 424, + 511 + ], + "score": 1.0, + "content": "from mode collapse when fitting a simple Gaussian Mixture of 25 components.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 525, + 207, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 208, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 208, + 537 + ], + "score": 1.0, + "content": "3.1 IMAGE DATASETS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "To demonstrate the ability of the Coulomb GAN to learn distributions in high dimensional spaces,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "we trained a Coulomb GAN on several popular image data sets: The cropped and centered images", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "score": 1.0, + "content": "of celebrities from the Large-scale CelebFaces Attributes (“CelebA”) data set (Liu et al., 2015), the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 578, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 592 + ], + "score": 1.0, + "content": "LSUN bedrooms data set consists of over 3 million 64x64 pixel images of the bedrooms category of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "the large scale image database LSUN (Yu et al., 2015) as well as the CIFAR-10 data set. For these", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "experiments, we used the DCGAN architecture (Radford et al., 2016) with a few modifications: our", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "convolutional kernels all have a kernel size of 5x5, our random seed that serves as input to the gener-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 621, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 636 + ], + "score": 1.0, + "content": "ator has fewer dimensions: 32 for CelebA and LSUN bedrooms, and 16 for CIFAR-10. Furthermore,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 634, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 504, + 645 + ], + "score": 1.0, + "content": "the discriminator uses twice as many feature channels in each layer as in the DCGAN architecture.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 643, + 504, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 643, + 207, + 658 + ], + "score": 1.0, + "content": "For the Plummer kernel,", + "type": "text" + }, + { + "bbox": [ + 208, + 647, + 213, + 655 + ], + "score": 0.58, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 643, + 482, + 658 + ], + "score": 1.0, + "content": "was set to 1. We used the Adam optimizer with a learning rate of", + "type": "text" + }, + { + "bbox": [ + 482, + 644, + 504, + 655 + ], + "score": 0.91, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 192, + 669 + ], + "score": 1.0, + "content": "for the generator and", + "type": "text" + }, + { + "bbox": [ + 192, + 655, + 226, + 666 + ], + "score": 0.91, + "content": "5 \\cdot 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "for the discriminator. To improve convergence performance, we used", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "the tanh output activation function (LeCun et al., 1998). For regularization we used an L2 weight", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 263, + 692 + ], + "score": 1.0, + "content": "decay term with a weighting factor of", + "type": "text" + }, + { + "bbox": [ + 263, + 677, + 285, + 688 + ], + "score": 0.91, + "content": "1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 676, + 506, + 692 + ], + "score": 1.0, + "content": ". Learning was stopped by monitoring the FID metric", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "(Heusel et al., 2017). Once learning plateaus, we scaled the learning rate down by a factor of 10 and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "let it continue once more until the FID plateaus. The results are reported in Table 1b, and generated", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 43 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 120, + 722, + 325, + 731 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 327, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 327, + 733 + ], + "score": 1.0, + "content": "1 www.github.com/bioinf-jku/coulomb_gan", + "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, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 137 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 81, + 506, + 138 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 466, + 156 + ], + "score": 1.0, + "content": "The main problem with learning Coulomb GANs is to approximate the potential function", + "type": "text" + }, + { + "bbox": [ + 466, + 144, + 474, + 154 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 143, + 505, + 156 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "is a complex function in a high-dimensional space, since the potential can be very non-linear and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "non-smooth. When learning the discriminator, we must ensure that enough data is sampled and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "averaged over. We already lessened the non-linear function problem by using a low-dimensional", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "Plummer kernel. But still, this kernel can introduce large non-linearities if samples are close to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "each other. It is crucial that the discriminator learns slow enough to accurately estimate the potential", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "function which is induced by the current generator. The generator, in turn, must be even slower since", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "it must be tracked by the discriminator. These approximation problems are supposed to be tackled", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 452, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 452, + 244 + ], + "score": 1.0, + "content": "by the research community in near future, which would enable optimal GAN learning.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 143, + 506, + 244 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 505, + 260 + ], + "score": 1.0, + "content": "The formulation of GAN learning as a potential field naturally solves the mode collapsing issue: the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "example described in Section A.1, where a normal GAN cannot get out of a local Nash equilibria", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "is not a converged solution for the Coulomb GAN: If all probability mass of the generator lies in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 294 + ], + "score": 1.0, + "content": "one of the modes, then both attracting forces from real-world samples located at the other mode as", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "well as repelling forces from the over-represented generator mode will act upon the generator until", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 294, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 294, + 315 + ], + "score": 1.0, + "content": "it generates samples at the other mode as well.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 247, + 505, + 315 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 331, + 200, + 344 + ], + "lines": [ + { + "bbox": [ + 104, + 330, + 202, + 346 + ], + "spans": [ + { + "bbox": [ + 104, + 330, + 202, + 346 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 356, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 456, + 369 + ], + "score": 1.0, + "content": "In all of our experiments, we used a low-dimensional Plummer Kernel of dimensionality", + "type": "text" + }, + { + "bbox": [ + 456, + 357, + 480, + 367 + ], + "score": 0.88, + "content": "d = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 357, + 505, + 369 + ], + "score": 1.0, + "content": ". This", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "kernel both gave best computational performance and has low risk of running into numerical issues.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "We used a batch size of 128. To evaluate the quality of a GAN, the FID metric as proposed by Heusel", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 257, + 403 + ], + "score": 1.0, + "content": "et al. (2017) was calculated by using", + "type": "text" + }, + { + "bbox": [ + 257, + 390, + 274, + 401 + ], + "score": 0.48, + "content": "5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "samples drawn from the generator, while the training set", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "statistics were calculated using the whole training set. We compare to BEGAN (Berthelot et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "2017), DCGAN (Radford et al., 2016) and WGAN-GP (Gulrajani et al., 2017) both in their original", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "version as well as when using the two-timescale update-rule (TTUR), using the settings from Heusel", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 432, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 505, + 447 + ], + "score": 1.0, + "content": "et al. (2017). We additionally compare to MMD-GAN (Li et al., 2017), which is conceptually very", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "similar to the Coulomb GAN, but uses a Gaussian Kernel instead of the Plummer Kernel. We use", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "the dataset-specific settings recommended in (Li et al., 2017) and report the highest FID score over", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "the course of training. All images shown in this paper were produced with a random seed and not", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 478, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 489 + ], + "score": 1.0, + "content": "cherry picked. The implementation used for these experiments is available online1. The appendix", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 501 + ], + "score": 1.0, + "content": "Section A.5 contains an additional toy example demonstrating that Coulomb GANs do not suffer", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 500, + 424, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 424, + 511 + ], + "score": 1.0, + "content": "from mode collapse when fitting a simple Gaussian Mixture of 25 components.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 357, + 505, + 511 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 525, + 207, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 208, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 208, + 537 + ], + "score": 1.0, + "content": "3.1 IMAGE DATASETS", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 559 + ], + "score": 1.0, + "content": "To demonstrate the ability of the Coulomb GAN to learn distributions in high dimensional spaces,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "we trained a Coulomb GAN on several popular image data sets: The cropped and centered images", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 505, + 580 + ], + "score": 1.0, + "content": "of celebrities from the Large-scale CelebFaces Attributes (“CelebA”) data set (Liu et al., 2015), the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 578, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 592 + ], + "score": 1.0, + "content": "LSUN bedrooms data set consists of over 3 million 64x64 pixel images of the bedrooms category of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "the large scale image database LSUN (Yu et al., 2015) as well as the CIFAR-10 data set. For these", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 613 + ], + "score": 1.0, + "content": "experiments, we used the DCGAN architecture (Radford et al., 2016) with a few modifications: our", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "convolutional kernels all have a kernel size of 5x5, our random seed that serves as input to the gener-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 621, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 636 + ], + "score": 1.0, + "content": "ator has fewer dimensions: 32 for CelebA and LSUN bedrooms, and 16 for CIFAR-10. Furthermore,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 634, + 504, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 504, + 645 + ], + "score": 1.0, + "content": "the discriminator uses twice as many feature channels in each layer as in the DCGAN architecture.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 643, + 504, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 643, + 207, + 658 + ], + "score": 1.0, + "content": "For the Plummer kernel,", + "type": "text" + }, + { + "bbox": [ + 208, + 647, + 213, + 655 + ], + "score": 0.58, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 643, + 482, + 658 + ], + "score": 1.0, + "content": "was set to 1. We used the Adam optimizer with a learning rate of", + "type": "text" + }, + { + "bbox": [ + 482, + 644, + 504, + 655 + ], + "score": 0.91, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 192, + 669 + ], + "score": 1.0, + "content": "for the generator and", + "type": "text" + }, + { + "bbox": [ + 192, + 655, + 226, + 666 + ], + "score": 0.91, + "content": "5 \\cdot 1 0 ^ { - 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "for the discriminator. To improve convergence performance, we used", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "the tanh output activation function (LeCun et al., 1998). For regularization we used an L2 weight", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 263, + 692 + ], + "score": 1.0, + "content": "decay term with a weighting factor of", + "type": "text" + }, + { + "bbox": [ + 263, + 677, + 285, + 688 + ], + "score": 0.91, + "content": "1 0 ^ { - 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 676, + 506, + 692 + ], + "score": 1.0, + "content": ". Learning was stopped by monitoring the FID metric", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "(Heusel et al., 2017). Once learning plateaus, we scaled the learning rate down by a factor of 10 and", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "let it continue once more until the FID plateaus. The results are reported in Table 1b, and generated", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "images can be seen in Figure 2 and in the Appendix in Section A.7. Coulomb GANs tend to outper-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "form standard GAN approaches like BEGAN and DCGAN, but are outperformed by the Improved", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "Wasserstein GAN. However it is important to note that the Improved Wasserstein GAN used a more", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "advanced network architecture based on ResNet blocks (Gulrajani et al., 2017), which we could not", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "replicate due to runtime constraints. Overall, the low FID of Coulomb GANs stem from the fact", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "that the images show a wide variety of different samples. E.g. on CelebA, Coulomb GAN exhibit", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "a very wide variety of faces, backgrounds, eye colors and orientations. To further investigate how", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 538, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 550 + ], + "score": 1.0, + "content": "much variation the samples generated by the Coulomb GAN contains, we followed the advice of", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "Arora and Zhang (Arora & Zhang, 2017) to estimate the support size of the generator’s distribution", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "by checking how large a sample from the generator must be before we start generating duplicates.", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 369, + 584 + ], + "score": 1.0, + "content": "We were able to generate duplicates with a probability of around", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 370, + 572, + 391, + 582 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 391, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "when using samples of size", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "1024, which indicates that the support size learned by the Coulomb GAN would be around 1M.", + "type": "text", + "cross_page": true + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "This is a strong indication that the Coulomb GAN was able to spread out its samples over the whole", + "type": "text", + "cross_page": true + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "target distribution. A depiction is included in Figure 3, which also shows the nearest neighbor in", + "type": "text", + "cross_page": true + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "the training set of the generated images, confirming that the Coulomb GAN does not just memorize", + "type": "text", + "cross_page": true + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 626, + 173, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 173, + 641 + ], + "score": 1.0, + "content": "training images.", + "type": "text", + "cross_page": true + } + ], + "index": 26 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 545, + 506, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "images can be seen in Figure 2 and in the Appendix in Section A.7. Coulomb GANs tend to outper-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "form standard GAN approaches like BEGAN and DCGAN, but are outperformed by the Improved", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "Wasserstein GAN. However it is important to note that the Improved Wasserstein GAN used a more", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "advanced network architecture based on ResNet blocks (Gulrajani et al., 2017), which we could not", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 138 + ], + "score": 1.0, + "content": "replicate due to runtime constraints. Overall, the low FID of Coulomb GANs stem from the fact", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 149 + ], + "score": 1.0, + "content": "that the images show a wide variety of different samples. E.g. on CelebA, Coulomb GAN exhibit", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "a very wide variety of faces, backgrounds, eye colors and orientations. To further investigate how", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "image", + "bbox": [ + 146, + 173, + 465, + 271 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 146, + 173, + 465, + 271 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 173, + 465, + 271 + ], + "spans": [ + { + "bbox": [ + 146, + 173, + 465, + 271 + ], + "score": 0.976, + "type": "image", + "image_path": "dc7625461a0dced041941f6df56258aa8ed1a05bcf3a0ec34978702ca93440d3.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 146, + 173, + 465, + 205.66666666666666 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 146, + 205.66666666666666, + 465, + 238.33333333333331 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 146, + 238.33333333333331, + 465, + 271.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 279, + 505, + 303 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 279, + 505, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 279, + 505, + 292 + ], + "score": 1.0, + "content": "Figure 2: Images from a Coulomb GAN after training on CelebA (first row), LSUN bedrooms", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 303 + ], + "score": 1.0, + "content": "(second row) and CIFAR 10 (last row). Further examples are located in the appendix in Section A.7", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + } + ], + "index": 9.25 + }, + { + "type": "image", + "bbox": [ + 147, + 333, + 465, + 479 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 147, + 333, + 465, + 479 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 147, + 333, + 465, + 479 + ], + "spans": [ + { + "bbox": [ + 147, + 333, + 465, + 479 + ], + "score": 0.978, + "type": "image", + "image_path": "6a9c0c1703f8938cefd242d27336d75059e77e543cafd91e08231266b5b6539c.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 147, + 333, + 465, + 381.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 147, + 381.6666666666667, + 465, + 430.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 147, + 430.33333333333337, + 465, + 479.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 489, + 504, + 522 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "Figure 3: The most similar pairs found in batches of 1024 generated faces sampled from the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "Coulomb GAN, and the nearest neighbor from the training data shown as third image. Distances", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 511, + 322, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 322, + 523 + ], + "score": 1.0, + "content": "were calculated as Euclidean distances on pixel level.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 538, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 550 + ], + "score": 1.0, + "content": "much variation the samples generated by the Coulomb GAN contains, we followed the advice of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "Arora and Zhang (Arora & Zhang, 2017) to estimate the support size of the generator’s distribution", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "by checking how large a sample from the generator must be before we start generating duplicates.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 369, + 584 + ], + "score": 1.0, + "content": "We were able to generate duplicates with a probability of around", + "type": "text" + }, + { + "bbox": [ + 370, + 572, + 391, + 582 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 571, + 505, + 584 + ], + "score": 1.0, + "content": "when using samples of size", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "1024, which indicates that the support size learned by the Coulomb GAN would be around 1M.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 606 + ], + "score": 1.0, + "content": "This is a strong indication that the Coulomb GAN was able to spread out its samples over the whole", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "target distribution. A depiction is included in Figure 3, which also shows the nearest neighbor in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "the training set of the generated images, confirming that the Coulomb GAN does not just memorize", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 626, + 173, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 173, + 641 + ], + "score": 1.0, + "content": "training images.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + }, + { + "type": "title", + "bbox": [ + 108, + 654, + 231, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 654, + 232, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 232, + 667 + ], + "score": 1.0, + "content": "3.2 LANGUAGE MODELING", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 107, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "We repeated the experiments from Gulrajani et al. 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data setWGAN-GPours
4 grams0.38 / 0.350.35
6 grams0.77 / 0.740.74
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data setBEGANDCGANWGAN-GPMMDours
CelebA29.2 /28.521.4 / 12.54.8 / 4.263.29.3
LSUN113 /11270.4 / 57.520.5 / 9.594.931.2
CIFAR10--29.3 /24.838.227.3
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Our theo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "retical results show that the Coulomb GAN will be able to approximate the real distribution perfectly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "if the networks have sufficient capacity and training does not get stuck in local minima. 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However, in our experience, this sometimes leads to a small number of generated samples", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "that are non-sensical interpolations of existing data modes. While these are sometimes also present", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "in other GAN models (Radford et al., 2016), we found that our model produces such images at a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 355, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 504, + 368 + ], + "score": 1.0, + "content": "slightly higher rate. 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We thus hypothesize that the largest issue in learning Coulomb GANs is that the dis-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 317, + 412 + ], + "score": 1.0, + "content": "criminator needs to approximate the potential field", + "type": "text" + }, + { + "bbox": [ + 318, + 400, + 327, + 410 + ], + "score": 0.79, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "very well in a high-dimensional space. In", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "score": 1.0, + "content": "summary, instead of directly optimizing a criterion based on local differences of densities which can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "exhibit many local minima, Coulomb GANs are based on a potential field that has no local minima.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "The potential field is created by point charges in an analogy to electric field in physics. 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data setWGAN-GPours
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data setBEGANDCGANWGAN-GPMMDours
CelebA29.2 /28.521.4 / 12.54.8 / 4.263.29.3
LSUN113 /11270.4 / 57.520.5 / 9.594.931.2
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Our theo-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "retical results show that the Coulomb GAN will be able to approximate the real distribution perfectly", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "score": 1.0, + "content": "if the networks have sufficient capacity and training does not get stuck in local minima. Our results", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 281 + ], + "score": 1.0, + "content": "show that the potential field used by the Coulomb GAN far outperforms MMD based approaches", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 277, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 293 + ], + "score": 1.0, + "content": "due to its low-dimensional Plummer kernel, which is better suited for modeling probability density", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 506, + 303 + ], + "score": 1.0, + "content": "functions, and is very effective at eliminating the mode collapse problem in GANs. 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However, in our experience, this sometimes leads to a small number of generated samples", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 333, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 333, + 505, + 347 + ], + "score": 1.0, + "content": "that are non-sensical interpolations of existing data modes. While these are sometimes also present", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 358 + ], + "score": 1.0, + "content": "in other GAN models (Radford et al., 2016), we found that our model produces such images at a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 355, + 504, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 355, + 504, + 368 + ], + "score": 1.0, + "content": "slightly higher rate. 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We thus hypothesize that the largest issue in learning Coulomb GANs is that the dis-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 399, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 317, + 412 + ], + "score": 1.0, + "content": "criminator needs to approximate the potential field", + "type": "text" + }, + { + "bbox": [ + 318, + 400, + 327, + 410 + ], + "score": 0.79, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 399, + 505, + 412 + ], + "score": 1.0, + "content": "very well in a high-dimensional space. In", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 506, + 422 + ], + "score": 1.0, + "content": "summary, instead of directly optimizing a criterion based on local differences of densities which can", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "exhibit many local minima, Coulomb GANs are based on a potential field that has no local minima.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 432, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 506, + 446 + ], + "score": 1.0, + "content": "The potential field is created by point charges in an analogy to electric field in physics. We have", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 443, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 443, + 505, + 456 + ], + "score": 1.0, + "content": "proved that if learning converges then it converges to the optimal solution if the samples can be", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 505, + 467 + ], + "score": 1.0, + "content": "moved freely. 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Assume that the the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 250, + 188 + ], + "score": 1.0, + "content": "generator has mode-collapsed onto", + "type": "text" + }, + { + "bbox": [ + 250, + 175, + 263, + 186 + ], + "score": 0.89, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 174, + 331, + 188 + ], + "score": 1.0, + "content": ", which contains", + "type": "text" + }, + { + "bbox": [ + 332, + 175, + 358, + 185 + ], + "score": 0.88, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "of the generator’s probability mass.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 380, + 198 + ], + "score": 1.0, + "content": "In this situation, the optimal discriminator classifies all points from", + "type": "text" + }, + { + "bbox": [ + 381, + 186, + 393, + 196 + ], + "score": 0.88, + "content": "C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "as ”real“ (pertaining to the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 209 + ], + "score": 1.0, + "content": "target distribution) by supplying an output of 1 for them (1 is the target for real samples and 0 the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 263, + 221 + ], + "score": 1.0, + "content": "target for generated samples). Within", + "type": "text" + }, + { + "bbox": [ + 264, + 208, + 276, + 218 + ], + "score": 0.87, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 208, + 505, + 221 + ], + "score": 1.0, + "content": ", the other region, the discriminator sees twice as many", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 259, + 231 + ], + "score": 1.0, + "content": "generated data points as real ones, as", + "type": "text" + }, + { + "bbox": [ + 260, + 219, + 286, + 229 + ], + "score": 0.89, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "of the probability mass of the generator’s distribution", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 127, + 242 + ], + "score": 1.0, + "content": "is in", + "type": "text" + }, + { + "bbox": [ + 128, + 230, + 141, + 240 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 229, + 182, + 242 + ], + "score": 1.0, + "content": ", but only", + "type": "text" + }, + { + "bbox": [ + 183, + 230, + 204, + 240 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 229, + 506, + 242 + ], + "score": 1.0, + "content": "of the probability mass of the real data distribution. So one third of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 248, + 253 + ], + "score": 1.0, + "content": "points seen by the discriminator in", + "type": "text" + }, + { + "bbox": [ + 248, + 240, + 261, + 252 + ], + "score": 0.89, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "are real, the other 2 thirds are generated. Thus, to minimize", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "its prediction error for a proper objective (squared or cross entropy), the discriminator has to output", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 123, + 274 + ], + "score": 0.46, + "content": "1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 262, + 213, + 276 + ], + "score": 1.0, + "content": "for every point from", + "type": "text" + }, + { + "bbox": [ + 214, + 263, + 226, + 273 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 262, + 506, + 276 + ], + "score": 1.0, + "content": ". The optimal output is even independent of the exact form of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 186, + 285 + ], + "score": 1.0, + "content": "real distribution in", + "type": "text" + }, + { + "bbox": [ + 186, + 273, + 199, + 284 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 274, + 505, + 285 + ], + "score": 1.0, + "content": ". The generator will match the shape of the target distribution locally. If", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "the shape is not matched, local gradients of the discriminator with respect to its input would be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "present and the generator would improve locally. If local improvements of the generator are no", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "longer possible, the shape of the target distribution is matched and the discriminator output is locally", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 318, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 329 + ], + "score": 1.0, + "content": "constant. In this situation, the expected gradient of the discriminator is the zero vector, because it", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 389, + 340 + ], + "score": 1.0, + "content": "has reached an optimum. Since the discriminator output is constant in", + "type": "text" + }, + { + "bbox": [ + 389, + 328, + 402, + 339 + ], + "score": 0.87, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 328, + 423, + 340 + ], + "score": 1.0, + "content": "(and", + "type": "text" + }, + { + "bbox": [ + 423, + 328, + 437, + 339 + ], + "score": 0.85, + "content": "C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "), the generator’s", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "expected gradient is the zero vector, too. The situation is also stable even though we still have", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "score": 1.0, + "content": "random fluctuations from the ongoing stochastic gradient (SGD) learning: whenever the generator", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 269, + 373 + ], + "score": 1.0, + "content": "produces data outside of (but close to)", + "type": "text" + }, + { + "bbox": [ + 269, + 361, + 282, + 372 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 362, + 505, + 373 + ], + "score": 1.0, + "content": ", the discriminator can easily detect this and push the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 243, + 385 + ], + "score": 1.0, + "content": "generator’s samples back. Inside", + "type": "text" + }, + { + "bbox": [ + 243, + 372, + 256, + 383 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 371, + 506, + 385 + ], + "score": 1.0, + "content": ", small deviations of the generator from the shape of the real", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 382, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 426, + 396 + ], + "score": 1.0, + "content": "distribution are detected by the discriminator as well, by deviating slightly from", + "type": "text" + }, + { + "bbox": [ + 426, + 383, + 443, + 395 + ], + "score": 0.51, + "content": "1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 382, + 505, + 396 + ], + "score": 1.0, + "content": ". Subsequently,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "the generator is pushed back to the original shape. If the discriminator deviates from its optimum,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "score": 1.0, + "content": "it will also be forced back to its optimum. So overall, the GAN learning reached a local Nash", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "equilibrium and has converged in the sense that the parameters fluctuate around the attractor point", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "(fluctuations depend on learning rate, sample size, etc.). To achieve true mathematical convergence,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "Heusel et al. (2017) assume decaying learning rates to anneal the random fluctuations, similar to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 448, + 360, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 360, + 461 + ], + "score": 1.0, + "content": "Robbins & Monro (1951) original convergence proof for SGD.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 108, + 477, + 229, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 231, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 231, + 491 + ], + "score": 1.0, + "content": "A.2 PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 500, + 212, + 511 + ], + "lines": [ + { + "bbox": [ + 107, + 500, + 213, + 511 + ], + "spans": [ + { + "bbox": [ + 107, + 500, + 213, + 511 + ], + "score": 1.0, + "content": "We first recall Theorem 1:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 108, + 516, + 504, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 387, + 529 + ], + "score": 1.0, + "content": "Theorem (Convergence with low-dimensional Plummer kernel). For", + "type": "text" + }, + { + "bbox": [ + 388, + 517, + 432, + 528 + ], + "score": 0.87, + "content": "\\pmb { a } , \\pmb { b } \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 516, + 437, + 529 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 437, + 517, + 483, + 528 + ], + "score": 0.84, + "content": "d \\leqslant m - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 516, + 506, + 529 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 107, + 528, + 134, + 538 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 528, + 189, + 540 + ], + "score": 1.0, + "content": "the densities", + "type": "text" + }, + { + "bbox": [ + 190, + 527, + 212, + 539 + ], + "score": 0.9, + "content": "p _ { x } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 528, + 232, + 540 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 527, + 254, + 540 + ], + "score": 0.91, + "content": "p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 528, + 437, + 540 + ], + "score": 1.0, + "content": "equalize over time when minimizing energy", + "type": "text" + }, + { + "bbox": [ + 437, + 528, + 446, + 537 + ], + "score": 0.79, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "with the low-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 537, + 462, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 452, + 551 + ], + "score": 1.0, + "content": "dimensional Plummer kernel by gradient descent. The convergence is faster for larger", + "type": "text" + }, + { + "bbox": [ + 452, + 539, + 459, + 548 + ], + "score": 0.66, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 537, + 462, + 551 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 371, + 574 + ], + "score": 1.0, + "content": "In a first step, we prove that for local maxima or local minima", + "type": "text" + }, + { + "bbox": [ + 371, + 564, + 379, + 571 + ], + "score": 0.67, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 560, + 392, + 574 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 393, + 563, + 399, + 573 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 560, + 466, + 574 + ], + "score": 1.0, + "content": ", the expression", + "type": "text" + }, + { + "bbox": [ + 466, + 561, + 505, + 573 + ], + "score": 0.8, + "content": "\\sin ( \\boldsymbol { \\mathbf { V } } \\cdot \\mathrm { \\partial _ { \\mathrm { s i g n } } } ( \\boldsymbol { \\mathbf { V } } \\cdot \\mathrm { \\partial _ { \\mathrm { s i g n } } } )", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 192, + 584 + ], + "score": 0.91, + "content": "{ \\bf E } ( { \\bf a } ) ) = \\mathrm { s i g n } ( \\rho ( { \\bf a } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 571, + 234, + 585 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + }, + { + "bbox": [ + 235, + 574, + 241, + 582 + ], + "score": 0.6, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "small enough. For proving this equation, we apply the Laplace", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "operator for spherical coordinates to the low-dimensional Plummer kernel. Using the result, we see", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 592, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 172, + 607 + ], + "score": 1.0, + "content": "that the integral", + "type": "text" + }, + { + "bbox": [ + 172, + 594, + 319, + 606 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nabla \\cdot \\mathbf { E } ( { a } ) = - \\int \\rho ( b ) \\nabla _ { a } ^ { 2 } k \\left( { a } , { b } \\right) \\mathrm { d } b } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 592, + 483, + 607 + ], + "score": 1.0, + "content": "is dominated by large negative values of", + "type": "text" + }, + { + "bbox": [ + 483, + 593, + 504, + 606 + ], + "score": 0.92, + "content": "\\nabla _ { a } ^ { 2 } k", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 136, + 618 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 137, + 606, + 144, + 615 + ], + "score": 0.7, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 604, + 383, + 618 + ], + "score": 1.0, + "content": ". These negative values can even be decreased by decreasing", + "type": "text" + }, + { + "bbox": [ + 384, + 607, + 389, + 615 + ], + "score": 0.5, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 604, + 505, + 618 + ], + "score": 1.0, + "content": ". Therefore we can ensure by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 168, + 630 + ], + "score": 1.0, + "content": "a small enough", + "type": "text" + }, + { + "bbox": [ + 168, + 618, + 174, + 626 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 616, + 365, + 630 + ], + "score": 1.0, + "content": "that at each local minimum and local maximum", + "type": "text" + }, + { + "bbox": [ + 366, + 618, + 373, + 626 + ], + "score": 0.7, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 616, + 384, + 630 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 385, + 616, + 501, + 628 + ], + "score": 0.92, + "content": "\\rho \\operatorname { s i g n } ( { \\dot { \\rho } } ( { \\mathbf { \\boldsymbol { a } } } ) ) = - \\operatorname { s i g n } ( \\rho ( { \\mathbf { \\boldsymbol { a } } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 616, + 505, + 630 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 626, + 358, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 273, + 640 + ], + "score": 1.0, + "content": "Thus, the maximal and minimal points of", + "type": "text" + }, + { + "bbox": [ + 274, + 629, + 280, + 639 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 626, + 358, + 640 + ], + "score": 1.0, + "content": "move toward zero.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "In a second step, we show that new maxima or minima cannot appear and that the movement of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 115, + 665 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 654, + 318, + 667 + ], + "score": 1.0, + "content": "toward zero stops at zero and not earlier. Since", + "type": "text" + }, + { + "bbox": [ + 319, + 657, + 326, + 667 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "is continuously differentiable, all points in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 420, + 678 + ], + "score": 1.0, + "content": "environments of maxima and minima move toward zero. Therefore the largest", + "type": "text" + }, + { + "bbox": [ + 420, + 666, + 446, + 678 + ], + "score": 0.93, + "content": "| \\rho ( a ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "moves toward", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "zero. We have to ensure that moving toward zero does not converge to a point apart from zero.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 360, + 700 + ], + "score": 1.0, + "content": "We derive that the movement toward zero is lower bounded by", + "type": "text" + }, + { + "bbox": [ + 360, + 687, + 475, + 700 + ], + "score": 0.93, + "content": "{ \\dot { \\rho } } ( { \\pmb a } ) = - \\mathrm { s i g n } ( \\rho ( { \\pmb a } ) ) \\lambda \\rho ^ { 2 } ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 687, + 505, + 700 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 229, + 712 + ], + "score": 1.0, + "content": "the movement slows down at", + "type": "text" + }, + { + "bbox": [ + 229, + 699, + 271, + 711 + ], + "score": 0.91, + "content": "\\rho ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ". Solving the differential equation and applying it to the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 245, + 723 + ], + "score": 1.0, + "content": "maximum of the absolute value of", + "type": "text" + }, + { + "bbox": [ + 246, + 712, + 253, + 721 + ], + "score": 0.83, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 709, + 277, + 723 + ], + "score": 1.0, + "content": "gives", + "type": "text" + }, + { + "bbox": [ + 277, + 709, + 425, + 722 + ], + "score": 0.93, + "content": "| \\rho | _ { \\operatorname* { m a x } } ( \\mathbf { \\bar { \\boldsymbol { t } } } ) = 1 / ( \\lambda t + ( | \\rho | _ { \\operatorname* { m a x } } ( 0 ) ) ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 709, + 455, + 723 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + }, + { + "bbox": [ + 455, + 712, + 462, + 721 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "converges", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 179, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 179, + 732 + ], + "score": 1.0, + "content": "to zero over time.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5 + } + ], + "page_idx": 12, + "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": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 182, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 184, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 184, + 96 + ], + "score": 1.0, + "content": "A APPENDIX", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 109, + 464, + 120 + ], + "lines": [ + { + "bbox": [ + 106, + 108, + 466, + 121 + ], + "spans": [ + { + "bbox": [ + 106, + 108, + 466, + 121 + ], + "score": 1.0, + "content": "A.1 EXAMPLE OF CONVERGENCE TO MODE COLLAPSE IN CONVENTIONAL GANS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 108, + 466, + 121 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 131, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 144 + ], + "score": 1.0, + "content": "As an example of how a GAN can converge to a Nash Equilibrium that exhibits mode collapse,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 142, + 504, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 491, + 155 + ], + "score": 1.0, + "content": "consider a target distribution that consists of two distinct/non-overlapping regions of support", + "type": "text" + }, + { + "bbox": [ + 491, + 142, + 504, + 153 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 151, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 124, + 166 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 153, + 137, + 164 + ], + "score": 0.88, + "content": "C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 151, + 455, + 166 + ], + "score": 1.0, + "content": "that are distant from each other, i.e., the target probability is zero outside of", + "type": "text" + }, + { + "bbox": [ + 456, + 153, + 469, + 164 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 469, + 151, + 488, + 166 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 488, + 153, + 501, + 164 + ], + "score": 0.86, + "content": "C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 151, + 505, + 166 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 163, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 190, + 177 + ], + "score": 1.0, + "content": "Further assume that", + "type": "text" + }, + { + "bbox": [ + 190, + 164, + 212, + 174 + ], + "score": 0.87, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 163, + 333, + 177 + ], + "score": 1.0, + "content": "of the probability mass is in", + "type": "text" + }, + { + "bbox": [ + 333, + 164, + 346, + 174 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 163, + 366, + 177 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 366, + 164, + 387, + 174 + ], + "score": 0.88, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 163, + 400, + 177 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 401, + 164, + 413, + 175 + ], + "score": 0.89, + "content": "C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 163, + 505, + 177 + ], + "score": 1.0, + "content": ". Assume that the the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 250, + 188 + ], + "score": 1.0, + "content": "generator has mode-collapsed onto", + "type": "text" + }, + { + "bbox": [ + 250, + 175, + 263, + 186 + ], + "score": 0.89, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 174, + 331, + 188 + ], + "score": 1.0, + "content": ", which contains", + "type": "text" + }, + { + "bbox": [ + 332, + 175, + 358, + 185 + ], + "score": 0.88, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "of the generator’s probability mass.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 380, + 198 + ], + "score": 1.0, + "content": "In this situation, the optimal discriminator classifies all points from", + "type": "text" + }, + { + "bbox": [ + 381, + 186, + 393, + 196 + ], + "score": 0.88, + "content": "C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "as ”real“ (pertaining to the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 209 + ], + "score": 1.0, + "content": "target distribution) by supplying an output of 1 for them (1 is the target for real samples and 0 the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 208, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 263, + 221 + ], + "score": 1.0, + "content": "target for generated samples). Within", + "type": "text" + }, + { + "bbox": [ + 264, + 208, + 276, + 218 + ], + "score": 0.87, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 208, + 505, + 221 + ], + "score": 1.0, + "content": ", the other region, the discriminator sees twice as many", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 259, + 231 + ], + "score": 1.0, + "content": "generated data points as real ones, as", + "type": "text" + }, + { + "bbox": [ + 260, + 219, + 286, + 229 + ], + "score": 0.89, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "of the probability mass of the generator’s distribution", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 506, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 127, + 242 + ], + "score": 1.0, + "content": "is in", + "type": "text" + }, + { + "bbox": [ + 128, + 230, + 141, + 240 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 229, + 182, + 242 + ], + "score": 1.0, + "content": ", but only", + "type": "text" + }, + { + "bbox": [ + 183, + 230, + 204, + 240 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 229, + 506, + 242 + ], + "score": 1.0, + "content": "of the probability mass of the real data distribution. So one third of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 248, + 253 + ], + "score": 1.0, + "content": "points seen by the discriminator in", + "type": "text" + }, + { + "bbox": [ + 248, + 240, + 261, + 252 + ], + "score": 0.89, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "are real, the other 2 thirds are generated. Thus, to minimize", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 506, + 264 + ], + "score": 1.0, + "content": "its prediction error for a proper objective (squared or cross entropy), the discriminator has to output", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 123, + 274 + ], + "score": 0.46, + "content": "1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 262, + 213, + 276 + ], + "score": 1.0, + "content": "for every point from", + "type": "text" + }, + { + "bbox": [ + 214, + 263, + 226, + 273 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 262, + 506, + 276 + ], + "score": 1.0, + "content": ". The optimal output is even independent of the exact form of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 274, + 186, + 285 + ], + "score": 1.0, + "content": "real distribution in", + "type": "text" + }, + { + "bbox": [ + 186, + 273, + 199, + 284 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 274, + 505, + 285 + ], + "score": 1.0, + "content": ". The generator will match the shape of the target distribution locally. If", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "score": 1.0, + "content": "the shape is not matched, local gradients of the discriminator with respect to its input would be", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 506, + 308 + ], + "score": 1.0, + "content": "present and the generator would improve locally. If local improvements of the generator are no", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 505, + 319 + ], + "score": 1.0, + "content": "longer possible, the shape of the target distribution is matched and the discriminator output is locally", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 318, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 329 + ], + "score": 1.0, + "content": "constant. In this situation, the expected gradient of the discriminator is the zero vector, because it", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 389, + 340 + ], + "score": 1.0, + "content": "has reached an optimum. Since the discriminator output is constant in", + "type": "text" + }, + { + "bbox": [ + 389, + 328, + 402, + 339 + ], + "score": 0.87, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 328, + 423, + 340 + ], + "score": 1.0, + "content": "(and", + "type": "text" + }, + { + "bbox": [ + 423, + 328, + 437, + 339 + ], + "score": 0.85, + "content": "C _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "), the generator’s", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "expected gradient is the zero vector, too. The situation is also stable even though we still have", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 363 + ], + "score": 1.0, + "content": "random fluctuations from the ongoing stochastic gradient (SGD) learning: whenever the generator", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 361, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 269, + 373 + ], + "score": 1.0, + "content": "produces data outside of (but close to)", + "type": "text" + }, + { + "bbox": [ + 269, + 361, + 282, + 372 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 362, + 505, + 373 + ], + "score": 1.0, + "content": ", the discriminator can easily detect this and push the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 243, + 385 + ], + "score": 1.0, + "content": "generator’s samples back. Inside", + "type": "text" + }, + { + "bbox": [ + 243, + 372, + 256, + 383 + ], + "score": 0.88, + "content": "C _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 371, + 506, + 385 + ], + "score": 1.0, + "content": ", small deviations of the generator from the shape of the real", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 382, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 426, + 396 + ], + "score": 1.0, + "content": "distribution are detected by the discriminator as well, by deviating slightly from", + "type": "text" + }, + { + "bbox": [ + 426, + 383, + 443, + 395 + ], + "score": 0.51, + "content": "1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 382, + 505, + 396 + ], + "score": 1.0, + "content": ". Subsequently,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 407 + ], + "score": 1.0, + "content": "the generator is pushed back to the original shape. If the discriminator deviates from its optimum,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 506, + 417 + ], + "score": 1.0, + "content": "it will also be forced back to its optimum. So overall, the GAN learning reached a local Nash", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 506, + 429 + ], + "score": 1.0, + "content": "equilibrium and has converged in the sense that the parameters fluctuate around the attractor point", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "(fluctuations depend on learning rate, sample size, etc.). To achieve true mathematical convergence,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "Heusel et al. (2017) assume decaying learning rates to anneal the random fluctuations, similar to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 448, + 360, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 360, + 461 + ], + "score": 1.0, + "content": "Robbins & Monro (1951) original convergence proof for SGD.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 16.5, + "bbox_fs": [ + 104, + 131, + 506, + 461 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 477, + 229, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 477, + 231, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 231, + 491 + ], + "score": 1.0, + "content": "A.2 PROOF OF THEOREM 1", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 500, + 212, + 511 + ], + "lines": [ + { + "bbox": [ + 107, + 500, + 213, + 511 + ], + "spans": [ + { + "bbox": [ + 107, + 500, + 213, + 511 + ], + "score": 1.0, + "content": "We first recall Theorem 1:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 107, + 500, + 213, + 511 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 516, + 504, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 387, + 529 + ], + "score": 1.0, + "content": "Theorem (Convergence with low-dimensional Plummer kernel). For", + "type": "text" + }, + { + "bbox": [ + 388, + 517, + 432, + 528 + ], + "score": 0.87, + "content": "\\pmb { a } , \\pmb { b } \\in \\mathbb { R } ^ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 516, + 437, + 529 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 437, + 517, + 483, + 528 + ], + "score": 0.84, + "content": "d \\leqslant m - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 516, + 506, + 529 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 107, + 528, + 134, + 538 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 528, + 189, + 540 + ], + "score": 1.0, + "content": "the densities", + "type": "text" + }, + { + "bbox": [ + 190, + 527, + 212, + 539 + ], + "score": 0.9, + "content": "p _ { x } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 528, + 232, + 540 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 527, + 254, + 540 + ], + "score": 0.91, + "content": "p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 528, + 437, + 540 + ], + "score": 1.0, + "content": "equalize over time when minimizing energy", + "type": "text" + }, + { + "bbox": [ + 437, + 528, + 446, + 537 + ], + "score": 0.79, + "content": "F", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 528, + 505, + 540 + ], + "score": 1.0, + "content": "with the low-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 537, + 462, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 452, + 551 + ], + "score": 1.0, + "content": "dimensional Plummer kernel by gradient descent. The convergence is faster for larger", + "type": "text" + }, + { + "bbox": [ + 452, + 539, + 459, + 548 + ], + "score": 0.66, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 537, + 462, + 551 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 516, + 506, + 551 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 371, + 574 + ], + "score": 1.0, + "content": "In a first step, we prove that for local maxima or local minima", + "type": "text" + }, + { + "bbox": [ + 371, + 564, + 379, + 571 + ], + "score": 0.67, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 560, + 392, + 574 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 393, + 563, + 399, + 573 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 560, + 466, + 574 + ], + "score": 1.0, + "content": ", the expression", + "type": "text" + }, + { + "bbox": [ + 466, + 561, + 505, + 573 + ], + "score": 0.8, + "content": "\\sin ( \\boldsymbol { \\mathbf { V } } \\cdot \\mathrm { \\partial _ { \\mathrm { s i g n } } } ( \\boldsymbol { \\mathbf { V } } \\cdot \\mathrm { \\partial _ { \\mathrm { s i g n } } } )", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 192, + 584 + ], + "score": 0.91, + "content": "{ \\bf E } ( { \\bf a } ) ) = \\mathrm { s i g n } ( \\rho ( { \\bf a } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 571, + 234, + 585 + ], + "score": 1.0, + "content": "holds for", + "type": "text" + }, + { + "bbox": [ + 235, + 574, + 241, + 582 + ], + "score": 0.6, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "small enough. For proving this equation, we apply the Laplace", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "operator for spherical coordinates to the low-dimensional Plummer kernel. Using the result, we see", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 592, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 172, + 607 + ], + "score": 1.0, + "content": "that the integral", + "type": "text" + }, + { + "bbox": [ + 172, + 594, + 319, + 606 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\nabla \\cdot \\mathbf { E } ( { a } ) = - \\int \\rho ( b ) \\nabla _ { a } ^ { 2 } k \\left( { a } , { b } \\right) \\mathrm { d } b } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 592, + 483, + 607 + ], + "score": 1.0, + "content": "is dominated by large negative values of", + "type": "text" + }, + { + "bbox": [ + 483, + 593, + 504, + 606 + ], + "score": 0.92, + "content": "\\nabla _ { a } ^ { 2 } k", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 136, + 618 + ], + "score": 1.0, + "content": "around", + "type": "text" + }, + { + "bbox": [ + 137, + 606, + 144, + 615 + ], + "score": 0.7, + "content": "^ { a }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 604, + 383, + 618 + ], + "score": 1.0, + "content": ". These negative values can even be decreased by decreasing", + "type": "text" + }, + { + "bbox": [ + 384, + 607, + 389, + 615 + ], + "score": 0.5, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 604, + 505, + 618 + ], + "score": 1.0, + "content": ". Therefore we can ensure by", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 616, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 168, + 630 + ], + "score": 1.0, + "content": "a small enough", + "type": "text" + }, + { + "bbox": [ + 168, + 618, + 174, + 626 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 616, + 365, + 630 + ], + "score": 1.0, + "content": "that at each local minimum and local maximum", + "type": "text" + }, + { + "bbox": [ + 366, + 618, + 373, + 626 + ], + "score": 0.7, + "content": "\\textbf { \\em a }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 616, + 384, + 630 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 385, + 616, + 501, + 628 + ], + "score": 0.92, + "content": "\\rho \\operatorname { s i g n } ( { \\dot { \\rho } } ( { \\mathbf { \\boldsymbol { a } } } ) ) = - \\operatorname { s i g n } ( \\rho ( { \\mathbf { \\boldsymbol { a } } } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 616, + 505, + 630 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 626, + 358, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 273, + 640 + ], + "score": 1.0, + "content": "Thus, the maximal and minimal points of", + "type": "text" + }, + { + "bbox": [ + 274, + 629, + 280, + 639 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 626, + 358, + 640 + ], + "score": 1.0, + "content": "move toward zero.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 560, + 506, + 640 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 644, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 656 + ], + "score": 1.0, + "content": "In a second step, we show that new maxima or minima cannot appear and that the movement of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 115, + 665 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 654, + 318, + 667 + ], + "score": 1.0, + "content": "toward zero stops at zero and not earlier. Since", + "type": "text" + }, + { + "bbox": [ + 319, + 657, + 326, + 667 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "is continuously differentiable, all points in", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 420, + 678 + ], + "score": 1.0, + "content": "environments of maxima and minima move toward zero. Therefore the largest", + "type": "text" + }, + { + "bbox": [ + 420, + 666, + 446, + 678 + ], + "score": 0.93, + "content": "| \\rho ( a ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "moves toward", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "zero. We have to ensure that moving toward zero does not converge to a point apart from zero.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 360, + 700 + ], + "score": 1.0, + "content": "We derive that the movement toward zero is lower bounded by", + "type": "text" + }, + { + "bbox": [ + 360, + 687, + 475, + 700 + ], + "score": 0.93, + "content": "{ \\dot { \\rho } } ( { \\pmb a } ) = - \\mathrm { s i g n } ( \\rho ( { \\pmb a } ) ) \\lambda \\rho ^ { 2 } ( { \\pmb a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 687, + 505, + 700 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 229, + 712 + ], + "score": 1.0, + "content": "the movement slows down at", + "type": "text" + }, + { + "bbox": [ + 229, + 699, + 271, + 711 + ], + "score": 0.91, + "content": "\\rho ( { \\pmb a } ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 698, + 505, + 712 + ], + "score": 1.0, + "content": ". Solving the differential equation and applying it to the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 104, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 104, + 709, + 245, + 723 + ], + "score": 1.0, + "content": "maximum of the absolute value of", + "type": "text" + }, + { + "bbox": [ + 246, + 712, + 253, + 721 + ], + "score": 0.83, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 709, + 277, + 723 + ], + "score": 1.0, + "content": "gives", + "type": "text" + }, + { + "bbox": [ + 277, + 709, + 425, + 722 + ], + "score": 0.93, + "content": "| \\rho | _ { \\operatorname* { m a x } } ( \\mathbf { \\bar { \\boldsymbol { t } } } ) = 1 / ( \\lambda t + ( | \\rho | _ { \\operatorname* { m a x } } ( 0 ) ) ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 709, + 455, + 723 + ], + "score": 1.0, + "content": ". Thus,", + "type": "text" + }, + { + "bbox": [ + 455, + 712, + 462, + 721 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "converges", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 179, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 179, + 732 + ], + "score": 1.0, + "content": "to zero over time.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5, + "bbox_fs": [ + 104, + 644, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 152, + 95 + ], + "score": 1.0, + "content": "Proof. 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Since", + "type": "text" + }, + { + "bbox": [ + 405, + 149, + 412, + 159 + ], + "score": 0.81, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 146, + 505, + 159 + ], + "score": 1.0, + "content": "is continuously differ-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 157, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 505, + 172 + ], + "score": 1.0, + "content": "entiable as is the field, also the points in an environment of the maximal and minimal points move", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 182 + ], + "score": 1.0, + "content": "toward zero. 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(19), for", + "type": "text" + }, + { + "bbox": [ + 442, + 200, + 448, + 207 + ], + "score": 0.7, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 197, + 504, + 209 + ], + "score": 1.0, + "content": "small enough", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 197, + 220 + ], + "score": 1.0, + "content": "there exists a positive", + "type": "text" + }, + { + "bbox": [ + 197, + 209, + 225, + 218 + ], + "score": 0.9, + "content": "0 < \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "globally for all minima and maxima as well as for all time steps for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 161, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 161, + 231 + ], + "score": 1.0, + "content": "which holds:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 233, + 357, + 247 + ], + "lines": [ + { + "bbox": [ + 254, + 233, + 357, + 247 + ], + "spans": [ + { + "bbox": [ + 254, + 233, + 357, + 247 + ], + "score": 0.91, + "content": "\\left| \\nabla \\cdot \\mathbf { E } ( { \\pmb a } ) \\right| > \\lambda \\left| \\rho ( { \\pmb a } ) \\right| .", + "type": "interline_equation", + "image_path": "a73efe20e32eb78d1fcd87919ef97a03429bdb76e8b4c4a81976a4ddd7dbef42.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 254, + 233, + 357, + 247 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 152, + 262 + ], + "score": 1.0, + "content": "The factor", + "type": "text" + }, + { + "bbox": [ + 153, + 249, + 160, + 259 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 248, + 212, + 262 + ], + "score": 1.0, + "content": "depends on", + "type": "text" + }, + { + "bbox": [ + 212, + 249, + 219, + 259 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 248, + 295, + 262 + ], + "score": 1.0, + "content": "and on the initial", + "type": "text" + }, + { + "bbox": [ + 296, + 249, + 317, + 261 + ], + "score": 0.27, + "content": "\\rho . \\ \\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 248, + 394, + 262 + ], + "score": 1.0, + "content": "is proportional to", + "type": "text" + }, + { + "bbox": [ + 394, + 249, + 401, + 259 + ], + "score": 0.75, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 248, + 439, + 262 + ], + "score": 1.0, + "content": ". 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There might exist initial conditions", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 270, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 113, + 282 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 270, + 158, + 284 + ], + "score": 1.0, + "content": "for which", + "type": "text" + }, + { + "bbox": [ + 159, + 271, + 191, + 281 + ], + "score": 0.9, + "content": "\\lambda 0", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 270, + 505, + 284 + ], + "score": 1.0, + "content": ", e.g. for infinite many maxima and minima, but they are impossible in our", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 160, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 160, + 294 + ], + "score": 1.0, + "content": "applications.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 106, + 298, + 434, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 433, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 433, + 313 + ], + "score": 1.0, + "content": "Therefore maximal or minimal points approach zero faster or equal than given by", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 311, + 372, + 326 + ], + "lines": [ + { + "bbox": [ + 238, + 311, + 372, + 326 + ], + "spans": [ + { + "bbox": [ + 238, + 311, + 372, + 326 + ], + "score": 0.92, + "content": "\\dot { \\rho } ( { \\pmb a } ) = - \\ \\mathrm { s i g n } ( \\rho ( { \\pmb a } ) ) \\lambda \\rho ^ { 2 } ( { \\pmb a } ) .", + "type": "interline_equation", + "image_path": "995c619fb1c6b127c9a983bacf0ccbd97e400a8041b567cf0ba339f4ca985b95.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 238, + 311, + 372, + 326 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 328, + 504, + 350 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 389, + 341 + ], + "score": 1.0, + "content": "In particular this differential equation dominates the global maximum", + "type": "text" + }, + { + "bbox": [ + 389, + 328, + 416, + 340 + ], + "score": 0.91, + "content": "| \\rho | _ { \\mathrm { m a x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 326, + 429, + 341 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 429, + 328, + 451, + 340 + ], + "score": 0.9, + "content": "| \\rho ( . ) |", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 326, + 505, + 341 + ], + "score": 1.0, + "content": ". 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If the pair", + "type": "text" + }, + { + "bbox": [ + 279, + 468, + 317, + 480 + ], + "score": 0.92, + "content": "( D ^ { * } , G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 467, + 505, + 481 + ], + "score": 1.0, + "content": "is a local Nash equilibrium for the Coulomb", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 478, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 493 + ], + "score": 1.0, + "content": "GAN objectives, then it is the global Nash equilibrium, and no other local Nash equilibria exist, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 488, + 252, + 505 + ], + "spans": [ + { + "bbox": [ + 107, + 490, + 120, + 500 + ], + "score": 0.83, + "content": "G ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 488, + 214, + 505 + ], + "score": 1.0, + "content": "has output distribution", + "type": "text" + }, + { + "bbox": [ + 214, + 492, + 248, + 502 + ], + "score": 0.9, + "content": "p _ { x } = p _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 488, + 252, + 505 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 502, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 502, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 137, + 527 + ], + "score": 1.0, + "content": "Proof.", + "type": "text" + }, + { + "bbox": [ + 137, + 514, + 175, + 526 + ], + "score": 0.92, + "content": "( D ^ { * } , G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 512, + 358, + 527 + ], + "score": 1.0, + "content": "being in a local Nash equilibrium means that", + "type": "text" + }, + { + "bbox": [ + 358, + 514, + 396, + 526 + ], + "score": 0.93, + "content": "( D ^ { * } , G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 512, + 502, + 527 + ], + "score": 1.0, + "content": "fulfills the two conditions", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 527, + 434, + 550 + ], + "lines": [ + { + "bbox": [ + 177, + 527, + 434, + 550 + ], + "spans": [ + { + "bbox": [ + 177, + 527, + 434, + 550 + ], + "score": 0.88, + "content": "D ^ { * } = \\underset { D \\in U ( D ^ { * } ) } { \\arg \\operatorname* { m i n } } \\mathcal { L } _ { D } ( D ; G ^ { * } ) \\quad \\mathrm { a n d } \\quad G ^ { * } = \\underset { G \\in U ( G ^ { * } ) } { \\arg \\operatorname* { m i n } } \\mathcal { L } _ { G } ( G ; D ^ { * } )", + "type": "interline_equation", + "image_path": "dae6cb4677afa4a624cb49999a1d5a209cb3e192ff978981245ba5fad8f53f9c.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 177, + 527, + 434, + 550 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 207, + 565 + ], + "score": 1.0, + "content": "for some neighborhoods", + "type": "text" + }, + { + "bbox": [ + 208, + 552, + 237, + 564 + ], + "score": 0.92, + "content": "U ( D ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 551, + 256, + 565 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 257, + 552, + 285, + 564 + ], + "score": 0.93, + "content": "U ( G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 551, + 425, + 565 + ], + "score": 1.0, + "content": ". 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This means that", + "type": "text" + }, + { + "bbox": [ + 398, + 574, + 411, + 585 + ], + "score": 0.88, + "content": "G ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "is directly minimizing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 584, + 459, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 192, + 598 + ], + "score": 0.53, + "content": "\\begin{array} { r } { \\mathcal { L } _ { G } ( G ; D ) = - \\frac { 1 } { 2 } \\mathrm { E } _ { p _ { z } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 585, + 238, + 597 + ], + "score": 0.75, + "content": "\\tilde { \\Phi } ( G ( z ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 584, + 459, + 599 + ], + "score": 1.0, + "content": ". The Coulomb potential energy is according to Eq. (7)", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 599, + 462, + 626 + ], + "lines": [ + { + "bbox": [ + 149, + 599, + 462, + 626 + ], + "spans": [ + { + "bbox": [ + 149, + 599, + 462, + 626 + ], + "score": 0.92, + "content": "F ( \\rho ) = \\frac { 1 } { 2 } \\int \\rho ( a ) \\Phi ( a ) \\mathrm { d } a = \\frac { 1 } { 2 } \\int p _ { y } ( a ) \\Phi ( a ) \\mathrm { d } a - \\frac { 1 } { 2 } \\int p _ { x } ( a ) \\Phi ( a ) \\mathrm { d } a .", + "type": "interline_equation", + "image_path": "301e7d60fc6cd4bae651eba81c59bc6561b1a19e844853e9c71caef397faa342.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 149, + 599, + 462, + 626 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 504, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 199, + 641 + ], + "score": 1.0, + "content": "Only the samples from", + "type": "text" + }, + { + "bbox": [ + 200, + 630, + 211, + 640 + ], + "score": 0.85, + "content": "p _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 627, + 338, + 641 + ], + "score": 1.0, + "content": "stem from the generator, where", + "type": "text" + }, + { + "bbox": [ + 339, + 628, + 472, + 641 + ], + "score": 0.92, + "content": "\\begin{array} { r } { p _ { x } ( \\pmb { a } ) = \\int \\delta ( \\pmb { a } - \\pmb { G } ( \\pmb { z } ) ) p _ { z } ( z ) \\mathrm { d } z } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 627, + 498, + 641 + ], + "score": 1.0, + "content": ". Here", + "type": "text" + }, + { + "bbox": [ + 498, + 629, + 504, + 638 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 639, + 485, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 130, + 651 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 131, + 640, + 136, + 649 + ], + "score": 0.84, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 639, + 485, + 651 + ], + "score": 1.0, + "content": "-distribution centered at zero. The part of the energy which depends on the generator is", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "interline_equation", + "bbox": [ + 156, + 652, + 455, + 732 + ], + "lines": [ + { + "bbox": [ + 156, + 652, + 455, + 732 + ], + "spans": [ + { + "bbox": [ + 156, + 652, + 455, + 732 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle - \\mathrm { \\small \\displaystyle \\frac { 1 } { 2 } } \\int p _ { x } ( \\pmb { a } ) \\Phi ( \\pmb { a } ) \\mathrm { d } \\pmb { a } = - \\frac { 1 } { 2 } \\int \\int \\delta ( \\pmb { a } - G ( \\pmb { z } ) ) p _ { z } ( \\pmb { z } ) \\mathrm { d } z \\Phi ( \\pmb { a } ) \\mathrm { d } \\pmb { a } } \\\\ { = - \\frac { 1 } { 2 } \\int \\big ( \\int \\delta ( \\pmb { a } - G ( \\pmb { z } ) ) \\Phi ( \\pmb { a } ) \\mathrm { d } \\pmb { a } \\big ) p _ { z } ( \\pmb { z } ) \\mathrm { d } z } \\\\ { = - \\displaystyle \\frac { 1 } { 2 } \\int p _ { z } ( \\pmb { z } ) \\Phi ( G ( \\pmb { z } ) ) \\mathrm { d } z = - 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Solving the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 339, + 262, + 351 + ], + "spans": [ + { + "bbox": [ + 107, + 339, + 262, + 351 + ], + "score": 1.0, + "content": "differential equation gives that at least", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 326, + 505, + 351 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 230, + 352, + 382, + 380 + ], + "lines": [ + { + "bbox": [ + 230, + 352, + 382, + 380 + ], + "spans": [ + { + "bbox": [ + 230, + 352, + 382, + 380 + ], + "score": 0.94, + "content": "| \\rho | _ { \\operatorname* { m a x } } ( t ) = \\frac { 1 } { \\lambda t + \\left( | \\rho | _ { \\operatorname* { m a x } } ( 0 ) \\right) ^ { - 1 } } .", + "type": "interline_equation", + "image_path": "3b631fc54412c55b4f70c05952fb72fcf5b2714d96229acfa092f9701877ce4c.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 230, + 352, + 382, + 380 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 382, + 504, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 380, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 129, + 394 + ], + "score": 1.0, + "content": "Thus", + "type": "text" + }, + { + "bbox": [ + 129, + 383, + 135, + 392 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 380, + 373, + 394 + ], + "score": 1.0, + "content": "influences the worst case rate of convergence, where larger", + "type": "text" + }, + { + "bbox": [ + 373, + 382, + 380, + 392 + ], + "score": 0.81, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 380, + 401, + 394 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 401, + 382, + 446, + 392 + ], + "score": 0.91, + "content": "d \\leqslant m - 2", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 380, + 506, + 394 + ], + "score": 1.0, + "content": "leads to faster", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 393, + 206, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 206, + 406 + ], + "score": 1.0, + "content": "worst case convergence.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 380, + 506, + 406 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 409, + 489, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 488, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 165, + 425 + ], + "score": 1.0, + "content": "Consequently,", + "type": "text" + }, + { + "bbox": [ + 166, + 412, + 172, + 421 + ], + "score": 0.76, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 406, + 367, + 425 + ], + "score": 1.0, + "content": "converges to the zero function over time, that is,", + "type": "text" + }, + { + "bbox": [ + 367, + 409, + 389, + 421 + ], + "score": 0.91, + "content": "p _ { x } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 406, + 462, + 425 + ], + "score": 1.0, + "content": "becomes equal to", + "type": "text" + }, + { + "bbox": [ + 462, + 409, + 483, + 422 + ], + "score": 0.91, + "content": "p _ { y } ( . )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 406, + 488, + 425 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 406, + 488, + 425 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 433, + 231, + 445 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 232, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 232, + 446 + ], + "score": 1.0, + "content": "A.3 PROOF OF THEOREM 2", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 432, + 232, + 446 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 454, + 213, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 214, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 214, + 467 + ], + "score": 1.0, + "content": "We first recall Theorem 2:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 452, + 214, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 467, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 279, + 481 + ], + "score": 1.0, + "content": "Theorem (Optimal Solution). If the pair", + "type": "text" + }, + { + "bbox": [ + 279, + 468, + 317, + 480 + ], + "score": 0.92, + "content": "( D ^ { * } , G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 467, + 505, + 481 + ], + "score": 1.0, + "content": "is a local Nash equilibrium for the Coulomb", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 478, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 493 + ], + "score": 1.0, + "content": "GAN objectives, then it is the global Nash equilibrium, and no other local Nash equilibria exist, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 488, + 252, + 505 + ], + "spans": [ + { + "bbox": [ + 107, + 490, + 120, + 500 + ], + "score": 0.83, + "content": "G ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 488, + 214, + 505 + ], + "score": 1.0, + "content": "has output distribution", + "type": "text" + }, + { + "bbox": [ + 214, + 492, + 248, + 502 + ], + "score": 0.9, + "content": "p _ { x } = p _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 488, + 252, + 505 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 467, + 506, + 505 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 502, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 502, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 137, + 527 + ], + "score": 1.0, + "content": "Proof.", + "type": "text" + }, + { + "bbox": [ + 137, + 514, + 175, + 526 + ], + "score": 0.92, + "content": "( D ^ { * } , G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 512, + 358, + 527 + ], + "score": 1.0, + "content": "being in a local Nash equilibrium means that", + "type": "text" + }, + { + "bbox": [ + 358, + 514, + 396, + 526 + ], + "score": 0.93, + "content": "( D ^ { * } , G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 512, + 502, + 527 + ], + "score": 1.0, + "content": "fulfills the two conditions", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 512, + 502, + 527 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 177, + 527, + 434, + 550 + ], + "lines": [ + { + "bbox": [ + 177, + 527, + 434, + 550 + ], + "spans": [ + { + "bbox": [ + 177, + 527, + 434, + 550 + ], + "score": 0.88, + "content": "D ^ { * } = \\underset { D \\in U ( D ^ { * } ) } { \\arg \\operatorname* { m i n } } \\mathcal { L } _ { D } ( D ; G ^ { * } ) \\quad \\mathrm { a n d } \\quad G ^ { * } = \\underset { G \\in U ( G ^ { * } ) } { \\arg \\operatorname* { m i n } } \\mathcal { L } _ { G } ( G ; D ^ { * } )", + "type": "interline_equation", + "image_path": "dae6cb4677afa4a624cb49999a1d5a209cb3e192ff978981245ba5fad8f53f9c.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 177, + 527, + 434, + 550 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 207, + 565 + ], + "score": 1.0, + "content": "for some neighborhoods", + "type": "text" + }, + { + "bbox": [ + 208, + 552, + 237, + 564 + ], + "score": 0.92, + "content": "U ( D ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 551, + 256, + 565 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 257, + 552, + 285, + 564 + ], + "score": 0.93, + "content": "U ( G ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 551, + 425, + 565 + ], + "score": 1.0, + "content": ". 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This means that", + "type": "text" + }, + { + "bbox": [ + 398, + 574, + 411, + 585 + ], + "score": 0.88, + "content": "G ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "is directly minimizing", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 584, + 459, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 192, + 598 + ], + "score": 0.53, + "content": "\\begin{array} { r } { \\mathcal { L } _ { G } ( G ; D ) = - \\frac { 1 } { 2 } \\mathrm { E } _ { p _ { z } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 585, + 238, + 597 + ], + "score": 0.75, + "content": "\\tilde { \\Phi } ( G ( z ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 584, + 459, + 599 + ], + "score": 1.0, + "content": ". The Coulomb potential energy is according to Eq. 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Each", + "type": "text" + }, + { + "bbox": [ + 276, + 105, + 320, + 117 + ], + "score": 0.92, + "content": "\\Phi ( { \\pmb a } ) \\neq \\mathrm { ~ 0 ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "would mean there exist potential differences", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "which in turn would cause forces on generator samples that allow to further minimize the energy.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 351, + 141 + ], + "score": 1.0, + "content": "Since we assumed that the generator can reach the minimum", + "type": "text" + }, + { + "bbox": [ + 351, + 128, + 386, + 139 + ], + "score": 0.9, + "content": "p _ { y } = p _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 125, + 418, + 141 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 418, + 128, + 429, + 139 + ], + "score": 0.86, + "content": "p _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 125, + 506, + 141 + ], + "score": 1.0, + "content": ", it will be reached", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 247, + 152 + ], + "score": 1.0, + "content": "by local (stepwise) optimization of", + "type": "text" + }, + { + "bbox": [ + 247, + 137, + 324, + 151 + ], + "score": 0.59, + "content": "- { \\textstyle \\frac { 1 } { 2 } } \\mathrm { E } _ { p _ { z } } \\left( \\Phi ( G ( z ) ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 138, + 385, + 152 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 386, + 139, + 395, + 149 + ], + "score": 0.77, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 138, + 456, + 152 + ], + "score": 1.0, + "content": ". 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The convergence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 172, + 504, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 476, + 183 + ], + "score": 1.0, + "content": "point is a global Nash equilibrium, because there is no approximation error and zero energy", + "type": "text" + }, + { + "bbox": [ + 476, + 172, + 504, + 182 + ], + "score": 0.89, + "content": "F = 0", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "is a global minimum for discriminator and generator, respectively. Theorem 1 ensures that other", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 260, + 206 + ], + "score": 1.0, + "content": "local Nash equilibria are not possible.", + "type": "text" + }, + { + "bbox": [ + 494, + 194, + 505, + 205 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 107, + 236, + 375, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 377, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 377, + 249 + ], + "score": 1.0, + "content": "A.4 COULOMB EQUATIONS IN THE CASE OF FINITE SAMPLES", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "GANs are sample-based, that is, samples are drawn from the model for learning (Hochreiter &", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "Obermayer, 2005; Gutmann & Hyvärinen, 2012). 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(7):", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 578, + 477, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 578, + 477, + 732 + ], + "spans": [ + { + "bbox": [ + 118, + 578, + 477, + 732 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\bar { \\Phi } ( a ) = \\frac { 1 } { N _ { y } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { y } } k \\left( a , y _ { i } \\right) - \\frac { 1 } { N _ { z } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { z } } k \\left( a , { \\boldsymbol x } _ { i } \\right) \\ : , } \\\\ { \\displaystyle \\hat { F } \\left( \\boldsymbol \\rho \\right) = \\frac { 1 } { 2 } \\left( \\frac { N _ { y } } { N _ { y } ^ { 2 } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { y } } \\sum _ { j = 1 } ^ { N _ { y } } k \\left( y _ { i } , y _ { i } \\right) - \\frac { 2 } { N _ { y } N _ { x } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { y } } \\sum _ { j = 1 } ^ { N _ { x } } k \\left( y _ { i } , { \\boldsymbol x } _ { j } \\right) + \\frac { 1 } { N _ { z } ^ { 2 } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { z } } \\sum _ { j = 1 } ^ { N _ { x } } k \\left( { \\boldsymbol x } _ { i } , { \\boldsymbol x } _ { j } \\right) \\right) } \\\\ { \\displaystyle \\qquad = \\frac { 1 } { 2 } \\left( \\frac { 1 } { N _ { y } } \\displaystyle \\sum _ { i = 1 } ^ { N _ { y } } \\hat { \\Phi } \\left( y _ { i } \\right) - 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Each", + "type": "text" + }, + { + "bbox": [ + 276, + 105, + 320, + 117 + ], + "score": 0.92, + "content": "\\Phi ( { \\pmb a } ) \\neq \\mathrm { ~ 0 ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "would mean there exist potential differences", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "which in turn would cause forces on generator samples that allow to further minimize the energy.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 351, + 141 + ], + "score": 1.0, + "content": "Since we assumed that the generator can reach the minimum", + "type": "text" + }, + { + "bbox": [ + 351, + 128, + 386, + 139 + ], + "score": 0.9, + "content": "p _ { y } = p _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 125, + 418, + 141 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 418, + 128, + 429, + 139 + ], + "score": 0.86, + "content": "p _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 125, + 506, + 141 + ], + "score": 1.0, + "content": ", it will be reached", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 247, + 152 + ], + "score": 1.0, + "content": "by local (stepwise) optimization of", + "type": "text" + }, + { + "bbox": [ + 247, + 137, + 324, + 151 + ], + "score": 0.59, + "content": "- { \\textstyle \\frac { 1 } { 2 } } \\mathrm { E } _ { p _ { z } } \\left( \\Phi ( G ( z ) ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 138, + 385, + 152 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 386, + 139, + 395, + 149 + ], + "score": 0.77, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 138, + 456, + 152 + ], + "score": 1.0, + "content": ". 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The convergence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 172, + 504, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 476, + 183 + ], + "score": 1.0, + "content": "point is a global Nash equilibrium, because there is no approximation error and zero energy", + "type": "text" + }, + { + "bbox": [ + 476, + 172, + 504, + 182 + ], + "score": 0.89, + "content": "F = 0", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "is a global minimum for discriminator and generator, respectively. Theorem 1 ensures that other", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 260, + 206 + ], + "score": 1.0, + "content": "local Nash equilibria are not possible.", + "type": "text" + }, + { + "bbox": [ + 494, + 194, + 505, + 205 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 82, + 506, + 206 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 236, + 375, + 248 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 377, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 377, + 249 + ], + "score": 1.0, + "content": "A.4 COULOMB EQUATIONS IN THE CASE OF FINITE SAMPLES", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 263, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "GANs are sample-based, that is, samples are drawn from the model for learning (Hochreiter &", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "Obermayer, 2005; Gutmann & Hyvärinen, 2012). 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In the same fashion as for", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 288, + 547 + ], + "score": 1.0, + "content": "the distributions, when we use fixed samples", + "type": "text" + }, + { + "bbox": [ + 288, + 536, + 298, + 545 + ], + "score": 0.83, + "content": "\\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 534, + 316, + 547 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 535, + 325, + 545 + ], + "score": 0.82, + "content": "\\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 534, + 505, + 547 + ], + "score": 1.0, + "content": ", we obtain the following unbiased estimates", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 545, + 393, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 393, + 557 + ], + "score": 1.0, + "content": "for the potential, energy and field given by Eq. 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The sample-based formulation, that is, point charges in physical terms,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 250, + 504, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 504, + 262 + ], + "score": 1.0, + "content": "can only have local energy minima or maxima at locations of samples (Dembo & Zeitouni, 1988).", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 259, + 504, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 496, + 274 + ], + "score": 1.0, + "content": "Furthermore the field lines originate and end at samples, therefore the field guides model samples", + "type": "text" + }, + { + "bbox": [ + 496, + 263, + 504, + 271 + ], + "score": 0.64, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 215, + 285 + ], + "score": 1.0, + "content": "toward real world samples", + "type": "text" + }, + { + "bbox": [ + 215, + 273, + 223, + 283 + ], + "score": 0.75, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 271, + 373, + 285 + ], + "score": 1.0, + "content": ", as depicted in Figure 1. The factors", + "type": "text" + }, + { + "bbox": [ + 374, + 272, + 388, + 284 + ], + "score": 0.9, + "content": "N _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 271, + 406, + 285 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 406, + 272, + 420, + 282 + ], + "score": 0.9, + "content": "N _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "in the last equations", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 200, + 295 + ], + "score": 1.0, + "content": "arise from the fact that", + "type": "text" + }, + { + "bbox": [ + 200, + 283, + 231, + 294 + ], + "score": 0.91, + "content": "- \\nabla _ { a } F", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "gives the force which is applied to a sample with charge. A sample", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 292, + 506, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 117, + 305 + ], + "score": 0.83, + "content": "\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { y } } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 292, + 227, + 308 + ], + "score": 1.0, + "content": "is positively charged with", + "type": "text" + }, + { + "bbox": [ + 227, + 294, + 250, + 306 + ], + "score": 0.91, + "content": "1 / N _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 292, + 303, + 308 + ], + "score": 1.0, + "content": "and follows", + "type": "text" + }, + { + "bbox": [ + 303, + 294, + 337, + 306 + ], + "score": 0.91, + "content": "- \\nabla _ { y _ { i } } F", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 292, + 403, + 308 + ], + "score": 1.0, + "content": "while a sample", + "type": "text" + }, + { + "bbox": [ + 403, + 295, + 415, + 304 + ], + "score": 0.87, + "content": "\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 292, + 506, + 308 + ], + "score": 1.0, + "content": "is negatively charged", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 127, + 318 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 304, + 158, + 316 + ], + "score": 0.92, + "content": "- 1 / N _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 303, + 247, + 318 + ], + "score": 1.0, + "content": "and therefore follows", + "type": "text" + }, + { + "bbox": [ + 248, + 305, + 282, + 316 + ], + "score": 0.92, + "content": "- \\nabla _ { x _ { i } } F", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 303, + 506, + 318 + ], + "score": 1.0, + "content": ", too. 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This", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 383, + 504, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 504, + 394 + ], + "score": 1.0, + "content": "data set comprises 100K data points drawn from a Gaussian mixture model of 25 components which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 248, + 406 + ], + "score": 1.0, + "content": "are spread out evenly in the range", + "type": "text" + }, + { + "bbox": [ + 248, + 394, + 336, + 406 + ], + "score": 0.9, + "content": "[ - 2 1 , 2 1 ] \\times [ - 2 1 , 2 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 393, + 506, + 406 + ], + "score": 1.0, + "content": ", with each component having a variance", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "of 1. To make results comparable with Lim & Ye (2017), the Coulomb GAN used a discriminator", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "network with 2 hidden layers of 128 units, however we avoided batch normalization by using the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 475, + 439 + ], + "score": 1.0, + "content": "ELU activation function (Clevert et al., 2016). We used the Plummer kernel in 3 dimensions", + "type": "text" + }, + { + "bbox": [ + 475, + 427, + 501, + 437 + ], + "score": 0.86, + "content": "\\left( d = 3 \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 427, + 505, + 439 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 437, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 191, + 451 + ], + "score": 1.0, + "content": "with an epsilon of 3", + "type": "text" + }, + { + "bbox": [ + 192, + 438, + 218, + 448 + ], + "score": 0.83, + "content": "( \\epsilon = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 437, + 505, + 451 + ], + "score": 1.0, + "content": ") and a learning rate of 0.01, both of which were exponentially decayed", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 449, + 312, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 312, + 461 + ], + "score": 1.0, + "content": "during the 1M update steps of the Adam optimizer.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 505, + 478 + ], + "score": 1.0, + "content": "As can be seen in Figure 4, samples from the learned Coulomb GAN very well approximate the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "target distribution. 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The sample-based formulation, that is, point charges in physical terms,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 250, + 504, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 504, + 262 + ], + "score": 1.0, + "content": "can only have local energy minima or maxima at locations of samples (Dembo & Zeitouni, 1988).", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 259, + 504, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 496, + 274 + ], + "score": 1.0, + "content": "Furthermore the field lines originate and end at samples, therefore the field guides model samples", + "type": "text" + }, + { + "bbox": [ + 496, + 263, + 504, + 271 + ], + "score": 0.64, + "content": "_ { \\textbf { \\em x } }", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 215, + 285 + ], + "score": 1.0, + "content": "toward real world samples", + "type": "text" + }, + { + "bbox": [ + 215, + 273, + 223, + 283 + ], + "score": 0.75, + "content": "\\textbf { { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 271, + 373, + 285 + ], + "score": 1.0, + "content": ", as depicted in Figure 1. 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This", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 383, + 504, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 504, + 394 + ], + "score": 1.0, + "content": "data set comprises 100K data points drawn from a Gaussian mixture model of 25 components which", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 248, + 406 + ], + "score": 1.0, + "content": "are spread out evenly in the range", + "type": "text" + }, + { + "bbox": [ + 248, + 394, + 336, + 406 + ], + "score": 0.9, + "content": "[ - 2 1 , 2 1 ] \\times [ - 2 1 , 2 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 393, + 506, + 406 + ], + "score": 1.0, + "content": ", with each component having a variance", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 416 + ], + "score": 1.0, + "content": "of 1. To make results comparable with Lim & Ye (2017), the Coulomb GAN used a discriminator", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "network with 2 hidden layers of 128 units, however we avoided batch normalization by using the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 427, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 427, + 475, + 439 + ], + "score": 1.0, + "content": "ELU activation function (Clevert et al., 2016). 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All components of the original distribution are present at the model distribu-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "tion at approximately the correct ratio, as shown in Figure 5. Moreover, the generated samples", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "are distributed approximately according to the same spread for each component of the real world", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "distribution. 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data setBEGANDCGANWGAN-GPMMDours
CelebA29.2 /28.521.4 / 12.54.8 / 4.263.29.3
LSUN113 /11270.4 / 57.520.5 / 9.594.931.2
CIFAR10--29.3 /24.838.227.3
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data setWGAN-GPours
4 grams0.38 / 0.350.35
6 grams0.77 / 0.740.74
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huan@huan-zhang.com xu.kaid@northeastern.edu xue.lin@northeastern.edu suman@cs.columbia.edu chohsieh@cs.ucla.edu zkolter@cs.cmu.edu + +\* Equal Contribution + +# Abstract + +Bound propagation based incomplete neural network verifiers such as CROWN are very efficient and can significantly accelerate branch-and-bound (BaB) based complete verification of neural networks. However, bound propagation cannot fully handle the neuron split constraints introduced by BaB commonly handled by expensive linear programming (LP) solvers, leading to loose bounds and hurting verification efficiency. In this work, we develop $\beta$ -CROWN, a new bound propagation based method that can fully encode neuron splits via optimizable parameters $\beta$ constructed from either primal or dual space. When jointly optimized in intermediate layers, $\beta$ -CROWN generally produces better bounds than typical LP verifiers with neuron split constraints, while being as efficient and parallelizable as CROWN on GPUs. Applied to complete robustness verification benchmarks, $\beta$ -CROWN with BaB is up to three orders of magnitude faster than LP-based BaB methods, and is notably faster than all existing approaches while producing lower timeout rates. By terminating BaB early, our method can also be used for efficient incomplete verification. We consistently achieve higher verified accuracy in many settings compared to powerful incomplete verifiers, including those based on convex barrier breaking techniques. Compared to the typically tightest but very costly semidefinite programming (SDP) based incomplete verifiers, we obtain higher verified accuracy with three orders of magnitudes less verification time. Our algorithm empowered the $\alpha , \beta$ -CROWN (alpha-beta-CROWN) verifier, the winning tool in VNN-COMP 2021. Our code is available at http://PaperCode.cc/BetaCROWN. + +# 1 Introduction + +As neural networks (NNs) are being deployed in safety-critical applications, it becomes increasingly important to formally verify their behaviors under potentially malicious inputs. Broadly speaking, the neural network verification problem involves proving certain desired relationships between inputs and outputs (often referred to as specifications), such as safety or robustness guarantees, for all inputs inside some domain. Canonically, the problem can be cast as finding the global minima of some functions on the network’s outputs (e.g., the difference between the predictions of the true label and another target label), within a bounded input set as constraints. This is a challenging problem due to the non-convexity and high dimensionality of neural networks. + +We first focus on complete verification: the verifier should give a definite “yes/no” answer given sufficient time. Many complete verifiers rely on the branch and bound (BaB) method $\textcircled { 8 } \textcircled { 1 8 }$ involving (1) branching by recursively splitting the original verification problem into subdomains (e.g., splitting a ReLU neuron into positive/negative linear regions by adding split constraints) and (2) bounding each subdomain with specialized incomplete verifiers. Traditional BaB-based verifiers use expensive linear programming (LP) solvers $\pm \boxed { 1 } \boxed { 2 3 } \boxed { 7 }$ as incomplete verifiers which can fully encode neuron split constraints. Meanwhile, a recent verifier, Fast-and-Complete $\mathbb { \lVert \boldsymbol { 4 5 } \rVert }$ , demonstrates that cheap incomplete verifiers can significantly accelerate complete verification on GPUs over LP-based ones thanks to their efficiency. Many cheap incomplete verifiers are based on bound propagation methods [46, 42, 41, 13, 17, 36, 44], i.e., maintaining and propagating tractable and sound bounds through networks, and CROWN [46] is a representative which propagates a linear or quadratic bound. + +However, unlike LP based verifiers, existing bound propagation methods lack the power to handle neuron split constraints introduced by BaB. For instance, given inputs $x , y \in [ - 1 , 1 ]$ , they can bound a ReLU’s input $x + y$ as $[ - 2 , 2 ]$ but they have no means to consider neuron split constraints such as $x - y \ge 0$ introduced by splitting another ReLU to the positive linear region. Such a problem causes looser bounds and unnecessary branching, hurting the verification efficiency. Even worse, without considering these split constraints, bound propagation methods cannot detect many infeasible subdomains in BaB $\lVert \boldsymbol { \mathsf { \Pi } } \rVert$ , leading to incompleteness unless costly checking is performed. + +In our work, we develop a new, fast bound propagation based incomplete verifier, $\beta$ -CROWN. It solves an optimization problem equivalent to the expensive LP based methods with neuron split constraints while still enjoying the efficiency of bound propagation methods. $\beta$ -CROWN contains optimizable parameters $\beta$ which come from propagation of Lagrangian multipliers, and any valid settings of these parameters yield sound bounds for verification. These parameters are optimized using a few steps of (super)gradient ascent to achieve bounds as tight as possible. Optimizing $\beta$ can also eliminate many infeasible subdomains and avoid further useless branching. Furthermore, we can jointly optimize intermediate layer bounds similar to $\pm \boxed { \boxed { 4 4 } }$ but also with the additional parameters $\beta$ , allowing $\beta$ - CROWN to tighten relaxations and outperform typical LP verifiers with fixed intermediate layer bounds. Unlike traditional LP-based BaB methods, $\beta$ -CROWN can be efficiently implemented with an automatic differentiation framework on GPUs to fully exploit the power of modern accelerators. The combination of $\beta$ -CROWN and BaB ( $\beta$ -CROWN BaB) produces a complete verifier with GPU acceleration, reducing the verification time of traditional LP based BaB verifiers $\textcircled { 8 }$ by up to three orders of magnitudes on a commonly used benchmark suite on CIFAR-10 [6, 10]. Compared to all state-of-the-art GPU-based complete verifiers [7, 45, 10, 23, 6, 11], our approach is noticeably faster with lower timeout rates. Our algorithm empowered the tool $\alpha , \beta$ -CROWN (alpha-beta-CROWN), which won the 2nd International Verification of Neural Networks Competition $\pmb { \mathbb { B } } \|$ (VNN-COMP 2021) with the highest total score and verified the most number of problem instances in 8 benchmarks. + +Finally, by terminating our complete verifier $\beta$ -CROWN BaB early, our approach can also function as a more accurate incomplete verifier by returning an incomplete but sound lower bound of all subdomains explored so far. We achieve better verified accuracy on a few benchmarking models over powerful incomplete verifiers including those based on tight linear relaxations $\overline { { \mathbb { B } 5 } } \overline { { \mathbb { B } \overline { { \lambda } } } } \overline { { \mathbb { B } 6 } }$ and semidefinite relaxations $\pmb { \Vert 9 \Vert }$ . Compared to the typically tightest but very costly incomplete verifier SDP-FO $\pmb { \Vert }$ based on the semidefinite programming (SDP) relaxations $[ \overbrace { 2 8 } , \overbrace { 1 4 } ]$ , our method obtains consistently higher verified accuracy while reducing verification time by three orders of magnitudes. + +# 2 Background + +# 2.1 The neural network verification problem and its LP relaxation + +We define the input of a neural network as $\begin{array} { r l r } { x } & { { } \in } & { \mathbb { R } ^ { d _ { 0 } } } \end{array}$ , and define the weights and biases of an $L$ -layer neural network as $\mathbf W ^ { ( i ) } \in \mathbb R ^ { d _ { i } \times d _ { i - 1 } }$ and $\mathbf { b } ^ { ( i ) } \in \mathbb { R } ^ { d _ { i } }$ $( i \in \{ 1 , \cdots , L \} )$ respectively. For simplicity we assume that $d _ { L } = 1$ so $\mathbf { W } ^ { ( L ) }$ is a vector and $\mathbf { b } ^ { ( L ) }$ is a scalar. The neural network function $f : \mathbb { R } ^ { d _ { 0 } } \mathbb { R }$ is defined as $f ( x ) = z ^ { ( L ) } ( x )$ , where $z ^ { ( i ) } ( x ) = \mathbf { W } ^ { ( i ) } \hat { z } ^ { ( i - 1 ) } ( x ) + \mathbf { b } ^ { ( i ) }$ , $\hat { z } ^ { ( i ) } ( x ) = \sigma ( z ^ { ( i ) } ( x ) )$ and $\hat { z } ^ { ( 0 ) } ( x ) = x$ . $\sigma$ is the activation function and we use ReLU throughout this paper. When the context is clear, we omit · $( x )$ and use $z _ { j } ^ { ( i ) }$ and $\hat { z } _ { j } ^ { ( i ) }$ to represent the pre-activation and post-activation values of the $j$ -th neuron in the $i$ -th layer. Neural network verification seeks the solution of the optimization problem in Eq. 1: + +$$ +\operatorname* { m i n } { f ( x ) : = z ^ { ( L ) } ( x ) } \quad \mathrm { s . t . ~ } z ^ { ( i ) } = \mathbf { W } ^ { ( i ) } \hat { z } ^ { ( i - 1 ) } + \mathbf { b } ^ { ( i ) } , \hat { z } ^ { ( i ) } = \sigma ( z ^ { ( i ) } ) , x \in \mathcal { C } , i \in \{ 1 , \cdots , L - 1 \} +$$ + +The set $\mathcal { C }$ defines the allowed input region and our aim is to find the minimum of $f ( x )$ for $x \in { \mathcal { C } }$ , and throughout this paper we consider $\mathcal { C }$ as an $\ell _ { \infty }$ ball around a data example $x _ { 0 }$ : $\mathcal { C } = \{ x \vert \| x - x _ { 0 } \| _ { \infty } \leq \epsilon \}$ but other $\ell _ { p }$ norms can also be supported. In practical settings, we typically have “specifications” to verify, which are (usually linear) functions of neural network outputs describing the desired behavior of neural networks. For example, to guarantee robustness we typically investigate the margin between logits. Because the specification can also be seen as an output layer of NN and merged into $f ( x )$ under verification, we do not discuss it in detail in this work. We consider the canonical specification $f ( x ) > 0$ : if we can prove that $f ( x ) > 0$ , $\forall x \in { \mathcal { C } }$ , we say $f ( x )$ is verified. + +When $\mathcal { C }$ is a convex set, Eq. 1 is still a non-convex problem because the constraints $\hat { z } ^ { ( i ) } = \sigma ( z ^ { ( i ) } )$ are non-convex. Given unlimited time, complete verifiers can solve Eq. 1 exactly: $f ^ { * } = \operatorname* { m i n } { f ( x ) }$ , $\forall x \in$ $\mathcal { C }$ , so we can always conclude if the specification holds or not for any problem instance. On the other hand, incomplete verifiers usually relax the non-convexity of neural networks to obtain a tractable lower bound of the solution $\underline { { f } } \le f ^ { * }$ . If $\underline { { f } } \geq 0$ , then $f ^ { * } > 0$ so $f ( x )$ can be verified; when $\underline { { f } } < 0$ , we are not able to infer the sign of $f ^ { * }$ so cannot conclude if the specification holds or not. + +A commonly used incomplete verification technique is to relax non-convex ReLU constraints with linear constraints and turn the verification problem into a linear programming (LP) problem, which can then be solved with linear solvers. We refer to it as the “LP verifier” in this paper. Specifically, given $\mathrm { R e L U } ( z _ { j } ^ { ( i ) } ) : = \operatorname* { m a x } ( 0 , z _ { j } ^ { ( i ) } )$ and its intermediate layer bounds $\mathbf { l } _ { j } ^ { ( i ) } \le z _ { j } ^ { ( i ) } \overset { \_ } { \le } \mathbf { u } _ { j } ^ { ( i ) }$ , each ReLU can be categorized into three cases: (1) if $\mathbf { l } _ { j } ^ { ( i ) } \geq 0$ (ReLU in linear region) then $\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) }$ ; (2) if $\mathbf { u } _ { j } ^ { ( i ) } \leq 0$ (ReLU in inactive region) then $\hat { z } _ { j } ^ { ( i ) } = 0$ ; (3) if $\mathbf { l } _ { j } ^ { ( i ) } \le 0 \le \mathbf { u } _ { j } ^ { ( i ) }$ (ReLU is unstable) then three linear bounds are used: $\hat { z } _ { j } ^ { ( i ) } \geq 0$ , $\hat { z } _ { j } ^ { ( i ) } \geq z _ { j } ^ { ( i ) }$ , and $\begin{array} { r } { \hat { z } _ { j } ^ { ( i ) } \leq \frac { \mathbf { u } _ { j } ^ { ( i ) } } { \mathbf { u } _ { j } ^ { ( i ) } - \mathbf { l } _ { j } ^ { ( i ) } } \left( z _ { j } ^ { ( i ) } - \mathbf { l } _ { j } ^ { ( i ) } \right) } \end{array}$ ; they are often referred to as the “triangle” relaxation $\mathbb { m } 5 \mathbb { k } 2 \mathbb { l }$ . The intermediate layer bounds $\mathbf { l } ^ { ( i ) }$ and $\mathbf { u } ^ { ( i ) }$ are usually obtained from a cheaper bound propagation method (see next subsection). LP verifiers can provide relatively tight bounds but linear solvers are still expensive especially when the network is large. Also, unlike our $\beta$ -CROWN, they have to use fixed intermediate bounds and cannot use the joint optimization of intermediate layer bounds (Section $3 . 3 )$ to tighten relaxation. + +# 2.2 CROWN: efficient incomplete verification by propagating linear bounds + +Another cheaper way to give a lower bound for the objective in Eq. $^ 1$ is through sound bound propagation. CROWN $\bar { \lVert 4 6 rVert }$ is a representative method that propagates a linear bound of $f ( x )$ w.r.t. every intermediate layer in a backward manner until reaching the input $x$ . CROWN uses two linear constraints to relax unstable ReLU neurons: a linear upper bound $\begin{array} { r } { \hat { z } _ { j } ^ { ( i ) } \leq \frac { \mathbf { u } _ { j } ^ { ( i ) } } { \mathbf { u } _ { j } ^ { ( i ) } - \mathbf { l } _ { j } ^ { ( i ) } } \left( z _ { j } ^ { ( i ) } - \mathbf { l } _ { j } ^ { ( i ) } \right) } \end{array}$ and a linear lower bound $\hat { z } _ { j } ^ { ( i ) } \geq \alpha _ { j } ^ { ( i ) } z _ { j } ^ { ( i ) }$ $0 \leq \alpha _ { j } ^ { ( i ) } \leq 1 $ ). We can then bound the output of a ReLU layer: Lemma 2.1 (ReLU relaxation in CROWN). Given $w , v \in \mathbb { R } ^ { d } , 1 \leq v \leq \mathbf { u }$ (element-wise), we have + +$$ +w ^ { \top } \mathrm { R e L U } ( v ) \geq w ^ { \top } \mathbf { D } v + b ^ { \prime } . +$$ + +where $\mathbf { D }$ is a diagonal matrix containing free variables $0 \leq \alpha _ { j } \leq 1$ only when $\mathbf { u } _ { j } > 0 > 1 _ { j }$ and $w _ { j } \geq 0$ , while its rest values as well as constant $b ^ { \prime }$ are determined by ${ \mathbf { l } } , { \mathbf { u } } , w$ . + +Detailed forms of each term are listed in Appendix $\mathbf { A } .$ Lemma $\boxed { 2 . 1 }$ can be repeatedly applied, resulting in an efficient back-substitution procedure to derive a linear lower bound of NN output w.r.t. $x$ : + +Lemma 2.2 (CROWN bound $[ \overline { { | 4 6 | } }$ ). Given an $L$ -layer ReLU NN $f ( \boldsymbol { x } ) : \mathbb { R } ^ { d _ { 0 } } \mathbb { R }$ with weights $\mathbf { W } ^ { ( i ) }$ , biases $\mathbf { b } ^ { ( i ) }$ , pre-ReLU bounds $\mathbf { l } ^ { ( i ) } \le z ^ { ( i ) } \le \mathbf { u } ^ { ( i ) }$ $1 \leq i \leq L )$ and input constraint $x \in { \mathcal { C } }$ . We have + +$$ +\operatorname* { m i n } _ { x \in \mathcal { C } } f ( x ) \geq \operatorname* { m i n } _ { x \in \mathcal { C } } { \pmb a } _ { \mathrm { c R o w n } } ^ { \top } x + c _ { \mathrm { C R O W N } } +$$ + +where $\mathbf { \pmb { a } } _ { \mathrm { C R O W N } }$ and $c _ { \mathrm { C R O W N } }$ can be computed using $\mathbf { W } ^ { ( i ) } , \mathbf { b } ^ { ( i ) } , \mathbf { l } ^ { ( i ) } , \mathbf { u } ^ { ( i ) }$ in polynomial time. + +When $\mathcal { C }$ is an $\ell _ { p }$ norm ball, minimization over the linear function can be easily solved using Hölder’s inequality. The main benefit of CROWN is its efficiency: CROWN can be efficiently implemented on machine learning accelerators such as GPUs $\pm \mathbb { H }$ and TPUs $\textcircled { \scriptsize { 1 4 7 } }$ , and it can be a few magnitudes faster than an LP verifier which is hard to parallelize on GPUs. CROWN was generalized to general architectures $\pm 4 \sqrt { 3 1 }$ while we only demonstrate it for feedforward ReLU networks for simplicity. Additionally, Xu et al. $\lVert \rVert \varTheta 5 \rVert$ showed that it is possible to optimize the slope of the lower bound, $_ { \pmb { \alpha } }$ , using gradient ascent, to further tighten the bound (sometimes referred to as $\alpha$ -CROWN). + +# 2.3 Branch and Bound and Neuron Split Constraints + +Branch and bound (BaB) method is widely adopted in complete verifiers $\textcircled { 8 }$ : we divide the domain of the verification problem $\mathcal { C }$ into two subdomains $\mathcal { C } _ { 1 } = \{ x \in \mathcal { C } , z _ { j } ^ { ( i ) } \geq 0 \}$ and $\mathcal { C } _ { 2 } = \{ x \in \mathcal { C } , z _ { j } ^ { ( i ) } < 0 \}$ where $z _ { j } ^ { ( i ) }$ is an unstable ReLU neuron in $\mathcal { C }$ but now becomes linear for each subdomain. Incomplete verifiers can then estimate the lower bound of each subdomain with relaxations. If the lower bound produced for subdomain $\mathcal { C } _ { i }$ (denoted by $\underline { { f } } _ { \mathcal { C } _ { i } } )$ is greater than 0, $\mathcal { C } _ { i }$ is verified; otherwise, we further branch over domain $\mathcal { C } _ { i }$ by splitting another unstable ReLU neuron. The process terminates when all subdomains are verified. The completeness is guaranteed when all unstable ReLU neurons are split. + +LP verifier with neuron split constraints. A popular incomplete verifier used in BaB is the LP verifier. Essentially, when we split the $j$ -th ReLU in layer $i$ , we can simply add $z _ { j } ^ { ( i ) } \geq 0$ or $z _ { j } ^ { ( i ) } < 0$ to Eq. $^ 1$ and get a linearly relaxed lower bound to each subdomain. We denote the $\mathcal { Z } ^ { + ( i ) }$ and $\mathcal { Z } ^ { - ( i ) }$ as the set of neuron indices with positive and negative split constraints in layer $i$ . We define the split constraints at layer $i$ as $\mathcal { Z } ^ { ( i ) } : = \{ z ^ { ( i ) } \mid z _ { j _ { 1 } } ^ { ( i ) } \geq 0 , z _ { j _ { 2 } } ^ { ( i ) } < 0 , \forall j _ { 1 } \in \mathcal { Z } ^ { + ( i ) } , \forall j _ { 2 } \in \mathcal { Z } ^ { - ( i ) } \}$ . We denote the vector of all pre-ReLU neurons as $z$ , and we define a set $\mathcal { Z }$ to represent the split constraints on $z$ : $\mathcal { Z } = \mathcal { Z } ^ { ( 1 ) } \cap \mathcal { Z } ^ { ( 2 ) } \cap \cdot \cdot \cdot \cap \mathcal { Z } ^ { ( L - 1 ) }$ . For convenience, we also use the shorthand $\tilde { \mathcal { Z } } ^ { ( i ) } : = \mathcal { Z } ^ { ( 1 ) } \cap \cdot \cdot \cdot \cap \mathcal { Z } ^ { ( i ) }$ and $\tilde { z } ^ { ( i ) } : = \{ z ^ { ( 1 ) } , z ^ { ( 2 ) } , \cdot \cdot \cdot , z ^ { ( i ) } \}$ . LP verifiers can easily handle these neuron split constraints but are more expensive than bound propagation methods like CROWN and cannot be accelerated on GPUs. + +Branching strategy. Branching strategies (selecting which ReLU neuron to split) are generally agnostic to the incomplete verifier used in BaB but do affect the overall BaB performance. BaBSR [7] is a widely used strategy in complete verifiers, which is based on an fast estimates on objective improvements after splitting each neuron. The neuron with highest estimated improvement is selected for branching. Recently, Filtered Smart Branching (FSB) $\bar { \mathbb { M } } \bar { \mathbb { I } }$ improves BaBSR by mimicking strong branching - it utilizes bound propagation methods to evaluate the best a few candidates proposed by BaBSR and chooses the one with largest improvement. Graph neural network (GNN) based branching was also proposed $\mathbb { \left[ \left. 2 3 \right] \right. }$ . Our $\beta$ -CROWN BaB is a general complete verification framework fit for any potential branching strategy, and we evaluate both BaBSR and FSB in experiments. + +# 3 $\beta$ -CROWN for Complete and Incomplete Verification + +In this section, we first give intuitions on how $\beta$ -CROWN handles neuron split constraints without costly LP solvers. Then we formally state the main theorem of $\beta$ -CROWN from both primal and dual spaces, and discuss how to tighten the bounds using free parameters $\alpha , \beta$ . Lastly, we propose $\beta$ -CROWN BaB, a complete verifier that is also a strong incomplete verifier when stopped early. + +# 3.1 $\beta$ -CROWN: Linear Bound Propagation with Neuron Split Constraints + +The NN verification problem under neuron split constraints can be seen as an optimization problem: + +$$ +\operatorname* { m i n } _ { x \in { \mathcal { C } } , z \in { \mathcal { Z } } } f ( x ) . +$$ + +Bound propagation methods like CROWN can give a relatively tight lower bound for $\scriptstyle \operatorname* { m i n } _ { x \in { \mathcal { C } } } f ( x )$ but they cannot handle the neuron split constraints $z \in { \mathcal { Z } }$ . Before we present our main theorem, we first show the intuition on how to apply split constraints to the bound propagation process. + +To encode the neuron splits, we first define diagonal matrix $\mathbf { S } ^ { ( i ) } ~ \in ~ \mathbb { R } ^ { d _ { i } \times d _ { i } }$ in Eq. $3$ where $i \in$ $[ 1 , \cdot \cdot \cdot L - 1 ] , j \in [ 1 , \cdot \cdot \cdot , d _ { i } ]$ are indices of layers and neurons, respectively: + +$$ +{ \bf S } _ { j , j } ^ { ( i ) } = - 1 ( \mathrm { i f ~ s p l i t ~ } z _ { j } ^ { ( i ) } \geq 0 ) ; \quad { \bf S } _ { j , j } ^ { ( i ) } = + 1 ( \mathrm { i f ~ s p l i t ~ } z _ { j } ^ { ( i ) } < 0 ) ; \quad { \bf S } _ { j , j } ^ { ( i ) } = 0 ( \mathrm { i f ~ n o ~ s p l i t ~ } z _ { j } ^ { ( i ) } ) +$$ + +We start from the last layer and derive linear bounds for each intermediate layer $z ^ { ( i ) }$ and $\hat { z } ^ { ( i ) }$ with both constraints $x \in { \mathcal { C } }$ and $z \in { \mathcal { Z } }$ . We also assume that pre-ReLU bounds $\mathbf { l } ^ { ( i ) } \dot { \leq } z ^ { ( i ) } \leq \mathbf { u } ^ { ( i ) }$ for each layer $i$ are available (see discussions in Sec. 3.3 on these intermediate layer bounds). We initially have: + +$$ +\operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) = \operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } \mathbf { W } ^ { ( L ) } \hat { z } ^ { ( L - 1 ) } + \mathbf { b } ^ { ( L ) } . +$$ + +Since $\hat { z } ^ { ( L - 1 ) } = \mathrm { R e L U } ( z ^ { ( L - 1 ) } )$ , we can apply Lemma $2 . 1$ to relax the ReLU neuron at layer $L - 1$ , and obtain a linear lower bound for $f ( x )$ w.r.t. $z ^ { ( L - 1 ) }$ (we omit all constant terms to avoid clutter): + +$$ +\operatorname* { m i n } _ { x \in \mathcal C , z \in \mathcal Z } f ( x ) \ge \operatorname* { m i n } _ { x \in \mathcal C , z \in \mathcal Z } \mathbf W ^ { ( L ) } \mathbf D ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \mathrm { c o n s t . } +$$ + +To enforce the split neurons at layer $L - 1$ , we use a Lagrange function with $\beta ^ { ( L - 1 ) \top } \mathbf { S } ^ { ( L - 1 ) }$ multiplied on z(L1): + +$$ +\begin{array} { l } { \displaystyle \operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \geq \operatorname* { m i n } _ { \bar { x } \in \mathcal { C } _ { \mathcal { C } } ^ { ( L - 2 ) } \in \mathcal { L } ^ { ( L - 1 ) } \geq 0 } \operatorname* { m a x } _ { 0 ^ { ( L - 1 ) } \geq 0 } \mathbf { W } ^ { ( L ) } \mathbf { D } ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \beta ^ { ( L - 1 ) ^ { \top } } \mathbf { S } ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \mathrm { c o n s t } } \\ { \displaystyle \qquad \geq \operatorname* { m a x } _ { \beta ^ { ( L - 1 ) } \geq 0 } \operatorname* { m i n } _ { \bar { x } \in \mathcal { Z } _ { \mathcal { C } } ^ { ( L - 2 ) } \in \mathcal { Z } ( L - 2 ) } \left( \mathbf { W } ^ { ( L ) } \mathbf { D } ^ { ( L - 1 ) } + \beta ^ { ( L - 1 ) ^ { \top } } \mathbf { S } ^ { ( L - 1 ) } \right) z ^ { ( L - 1 ) } + \mathrm { c o n s t } } \end{array} +$$ + +The first inequality is due to the definition of the Lagrange function: we remove the constraint 2 Zduality. Due to the design of S(L1), neuron split z (L1)j 0 has a negative multiplier (L1)j $z ^ { ( L - 1 ) } \in \mathcal { Z } ^ { ( \bar { L } - 1 ) }$ and use a multiplier to replace this constraint. The second inequality is due to weak and split $z _ { j } ^ { ( L - 1 ) } < 0$ has a positive multiplier $\beta _ { j } ^ { ( L - 1 ) }$ . Any $\beta ^ { ( L - 1 ) } \geq 0$ yields a lower bound for the constrained optimization problem. Then we substitute $z ^ { ( L - 1 ) }$ with $\mathbf { W } ^ { ( L - 1 ) } \hat { z } ^ { ( L - 2 ) } + \mathbf { b } ^ { ( L - 1 ) }$ for next layer: + +$$ +\operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \geq \operatorname* { m a x } _ { \beta ^ { ( L - 1 ) } \geq 0 } \operatorname* { m i n } _ { \tilde { x } ^ { ( L - 2 ) } \in \mathcal { C } ( L - 2 ) } \left( \mathbf { W } ^ { ( L ) } \mathbf { D } ^ { ( L - 1 ) } + \beta ^ { ( L - 1 ) ^ { \top } } \mathbf { S } ^ { ( L - 1 ) } \right) \mathbf { W } ^ { ( L - 1 ) } \hat { z } ^ { ( L - 2 ) } + \mathrm { c o n s t } +$$ + +We define a matrix $\mathbf { A } ^ { ( i ) }$ to represent the linear relationship between $f ( x )$ and $\hat { z } ^ { ( i ) }$ , where $\mathbf { A } ^ { ( L - 1 ) } =$ $\mathbf { W } ^ { ( L ) }$ according to Eq. $_ 4$ and $\mathbf { A } ^ { ( L - 2 ) } = \bigl ( \mathbf { A } ^ { ( L - 1 ) } \mathbf { D } ^ { ( L - 1 ) } + \beta ^ { ( L - 1 ) } { } ^ { \top } \mathbf { S } ^ { ( L - 1 ) } \bigr ) \mathbf { W } ^ { ( L - 1 ) }$ by Eq. 6. Considering 1-dimension output $f ( \boldsymbol { x } ) , \mathbf { A } ^ { ( i ) }$ has only 1 row. With $\mathbf { A } ^ { ( L - 2 ) }$ , Eq. 6 becomes: + +$$ +\operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \ge \operatorname* { m a x } _ { \beta ^ { ( L - 1 ) } \ge 0 } \operatorname* { m i n } _ { \tilde { x } \in \mathcal { C } } \quad \mathbf { A } ^ { ( L - 2 ) } \hat { z } ^ { ( L - 2 ) } + \mathrm { c o n s t } , +$$ + +which is in a form similar to Eq. $\boxed { 4 }$ except for the outer maximization over $\beta ^ { ( L - 1 ) }$ . This allows the back-substitution process (Eq. 4, Eq. 5, and Eq. $\textcircled { 6 }$ to continue. In each step, we swap max and min as in $\mathrm { E q . } | \overline { { 5 } } |$ so every maximization over $\beta ^ { ( i ) }$ is outside of $\operatorname* { m i n } _ { x \in { \mathcal { C } } }$ . Eventually, we have: + +$$ +\operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \ge \operatorname* { m a x } _ { \beta \ge 0 } \operatorname* { m i n } _ { x \in \mathcal { C } } { \mathbf { A } ^ { ( 0 ) } } x + \mathrm { c o n s t } , +$$ + +where := ⇥ (1)> (2)> (L1)>⇤> concatenates all $\beta ^ { ( i ) }$ vectors. Following the above idea, we present the main theorem in Theorem $3 . 1$ (proof is given in Appendix $\mathbf { A } )$ . + +Theorem 3.1 ( $\beta$ -CROWN bound). Given an $L$ -layer NN $f ( \boldsymbol { x } ) : \mathbb { R } ^ { d _ { 0 } } \mathbb { R }$ with weights $\mathbf { W } ^ { ( i ) }$ , biases $\mathbf { b } ^ { ( i ) }$ , pre-ReLU bounds $\mathbf { l } ^ { ( i ) } \le z ^ { ( i ) } \le \mathbf { u } ^ { ( i ) }$ ( $1 \leq i \leq L )$ ), input bounds $\mathcal { C }$ , split constraints $\mathcal { Z }$ . We have: + +$$ +\operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \ge \operatorname* { m a x } _ { \beta \ge 0 } \operatorname* { m i n } _ { x \in \mathcal { C } } ( \pmb { a } + \mathbf { P } \beta ) ^ { \top } x + \mathbf { q } ^ { \top } \beta + c , +$$ + +where $\pmb { a } \in \mathbb { R } ^ { d _ { 0 } } , \mathbf { P } \in \mathbb { R } ^ { d _ { 0 } \times ( \sum _ { i = 1 } ^ { L - 1 } d _ { i } ) } , \mathbf { q } \in \mathbb { R } ^ { \sum _ { i = 1 } ^ { L - 1 } d _ { i } }$ and $c \in \mathbb { R }$ are functions of $\mathbf { W } ^ { ( i ) } , \mathbf { b } ^ { ( i ) } , \mathbf { l } ^ { ( i ) } , \mathbf { u } ^ { ( i ) } .$ + +Detailed formulations for ${ \mathbf { } } _ { { \mathbf { } } _ { a , { \mathbf { P } } } }$ , $\mathbf { q }$ and $c$ are given in Appendix $\boxed { \mathrm { A } }$ Theorem $3 . 1$ shows that when neuron split constraints exist, $f ( x )$ can still be bounded by a linear equation containing optimizable multipliers $\beta$ . Observing Eq. 5, the main difference between CROWN and $\beta$ -CROWN lies in the relaxation of each ReLU layer, where we need an extra term $\beta ^ { ( i ) \top } \mathbf { S } ^ { ( i ) }$ in the linear relationship matrix (for example, $\mathbf { W } ^ { ( L ) } \dot { \mathbf { D } ^ { ( L - 1 ) } }$ in Eq. 5) between $f ( x )$ and $z ^ { ( i ) }$ to enforce neuron split constraints. This extra term in every ReLU layer yields $\mathbf { P }$ and $\mathbf { q }$ in Eq. $\perp$ after bound propagations. + +To solve the optimization problem in Eq. $^ { 7 , }$ we note that in the $\ell _ { p }$ norm robustness setting ${ \mathcal { C } } =$ $\{ x \mid \| x - x _ { 0 } \| _ { p } \leq \epsilon \} )$ ), the inner minimization has a closed solution: + +$$ +\operatorname* { m i n } _ { x \in { \mathcal { C } } , z \in { \mathcal { Z } } } f ( x ) \geq \operatorname* { m a x } _ { \beta \geq 0 } - \| a + \mathbf { P } \beta \| _ { q } \epsilon + ( \mathbf { P } ^ { \top } x _ { 0 } + \mathbf { q } ) ^ { \top } \beta + a ^ { \top } x _ { 0 } + c : = \operatorname* { m a x } _ { \beta \geq 0 } g ( \beta ) +$$ + +where $\textstyle { \frac { 1 } { p } } + { \frac { 1 } { q } } = 1$ . The maximization is concave in $\beta ( q \geq 1 )$ , so we can simply optimize it using projected (super)gradient ascent with gradients from an automatic differentiation library. Since any $\beta \geq 0$ yields a valid lower bound for $\scriptstyle \operatorname* { m i n } _ { x \in { \mathcal { C } } , z \in { \mathcal { Z } } } f ( x )$ , convergence is not necessary to guarantee soundness. $\beta$ -CROWN is efficient - it has the same asymptotic complexity as CROWN when $\beta$ is fixed. When $\beta = 0$ , $\beta$ -CROWN yields the same results as CROWN; however the additional optimizable $\beta$ allows us to maximize and tighten the lower bound due to neuron split constraints. + +We define $\pmb { \alpha } ^ { ( i ) } \in \mathbb { R } ^ { d _ { i } }$ for free variables associated with unstable ReLU neurons in Lemma $\boxed { 2 . 1 }$ for layer $i$ 2 and define all free variables ${ \pmb { \alpha } } = \{ { \pmb { \alpha } } ^ { ( 1 ) } \cdot \cdot \cdot { \pmb { \alpha } } ^ { ( L - 1 ) } \}$ . Since any $0 \mathop { \leq } \alpha _ { j } ^ { ( i ) } \mathop { \leq } 1$ yields a valid bound, we can optimize it to tighten the bound, similarly as done in $\lVert \rVert \dot { \boldsymbol { \mathrm { E } } } \rVert$ . Formally, we rewrite Eq. $8$ with $_ { \pmb { \alpha } }$ explicitly: + +$$ +\operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \geq \operatorname* { m a x } _ { 0 \leq \alpha \leq 1 , \beta \geq 0 } g ( \alpha , \beta ) . +$$ + +# 3.2 Connections to the Dual Problem + +In this subsection, we show that $\beta$ -CROWN can also be derived from a dual LP problem. Based on Eq. $^ { 1 } _ { . }$ and linear relaxations in Section $\bigstar ,$ we first construct an LP problem for $\ell _ { \infty }$ robustness verification in Eq. 10 where $i \in \{ 1 , \cdots , L - 1 \}$ . + +$\operatorname* { m i n } f ( x ) : = z ^ { ( L ) } ( x )$ s.t. +Network and Input Bounds: $z ^ { ( i ) } = \mathbf { W } ^ { ( i ) } \hat { z } ^ { ( i - 1 ) } + \mathbf { b } ^ { ( i ) } ; \hat { z } ^ { ( 0 ) } \geq x _ { 0 } - \epsilon ; \hat { z } ^ { ( 0 ) } \leq x _ { 0 } + \epsilon ;$ +Stable ReLUs: $\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) }$ (if $1 _ { j } ^ { ( i ) } \geq 0$ ); $\hat { z } _ { j } ^ { ( i ) } = 0$ (if $\begin{array} { r } { \mathbf { u } _ { j } ^ { ( i ) } \le 0 . } \end{array}$ ); +Unstable: $\begin{array} { r } { \hat { z } _ { j } ^ { ( i ) } \ge 0 , \hat { z } _ { j } ^ { ( i ) } \ge z _ { j } ^ { ( i ) } , \hat { z } _ { j } ^ { ( i ) } \le \frac { \mathbf { u } _ { j } ^ { ( i ) } } { \mathbf { u } _ { j } ^ { ( i ) } - \mathbf { 1 } _ { j } ^ { ( i ) } } \left( z _ { j } ^ { ( i ) } - \mathbf { 1 } _ { j } ^ { ( i ) } \right) ( \mathrm { i f } \mathbf { l } _ { j } ^ { ( i ) } < 0 < \mathbf { u } _ { j } ^ { ( i ) } , j \notin \mathcal { Z } ^ { + ( i ) } \cup \mathcal { Z } ^ { - ( i ) } ) } \end{array}$ +Neuron Split Constraints: $\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) } , z _ { j } ^ { ( i ) } \geq 0$ (if $j \in \mathcal { Z } ^ { + ( i ) }$ ); ˆz (i)j = 0, z(i)j < 0 (if j 2 Z (i) ) + +Compared to the formulation in $\mathbb { \lVert \rVert }$ , we have neuron split constraints. Many BaB based complete verifiers $\mathbb { B } \mathbb { B } \mathbb { 3 } \mathbb { I }$ use an LP solver for Eq. $\boxed { 1 0 }$ as the incomplete verifier. We first show that it is possible to derive Theorem $3 . 1$ from the dual of this LP, leading to Theorem 3.2: + +Theorem 3.2. The objective $d _ { L P }$ for the dual problem of Eq. 10 can be represented as + +$$ +d _ { L P } = - \| \pmb { a } + \mathbf { P } \pmb { \beta } \| _ { 1 } { \cdot } \epsilon + ( \mathbf { P } ^ { \top } x _ { 0 } + \mathbf { q } ) ^ { \top } \pmb { \beta } + \pmb { a } ^ { \top } x _ { 0 } + c , +$$ + +where ${ \mathbf { } } _ { { \mathbf { } } _ { a , { \mathrm { ~ P ~ } } } }$ , q and c are defined in the same way as in Theorem $3 . { { I } _ { : } }$ and $\beta \geq 0$ corresponds to the dual variables of neuron split constraints in Eq. $\boxed { I O }$ + +A similar connection between CROWN and dual LP based verifier $[ \mathbb { A } 2 ]$ was shown in $\pmb { \mathbb { B } } 0 \|$ , and their results can be seen as a special case of ours when $\beta = 0$ (none of the split constraints are active). An immediate consequence is that $\beta$ -CROWN can potentially solve Eq. 10 as well as using an LP solver: + +Corollary 3.2.1. When $_ { \pmb { \alpha } }$ and $\beta$ are optimally set and intermediate bounds l, u are fixed, $\beta$ -CROWN produces $p _ { L P } ^ { * }$ , the optimal objective of $L P$ with split constraints in Eq. 10: + +$$ +\operatorname* { m a x } _ { 0 \le \alpha \le 1 , \beta \ge 0 } g ( \alpha , \beta ) = p _ { L P } ^ { * } , +$$ + +In Appendix $\boxed { \mathrm { A } }$ we give detailed formulations for conversions between the variables $_ { \pmb { \alpha } }$ , $\beta$ in $\beta$ - CROWN and their corresponding dual variables in the LP problem. + +# 3.3 Joint Optimization of Free Variables in $\beta$ -CROWN + +In Eq. 9, $g$ is also a function of $\mathbf { l } _ { j } ^ { ( i ) }$ and $\mathbf { u } _ { j } ^ { ( i ) }$ , the intermediate layer bounds for each neuron z(i) . They are also computed using $\beta$ -CROWN. To obtain $\mathbf { l } _ { j } ^ { ( i ) }$ , we set $f ( x ) : = z _ { j } ^ { ( i ) } ( x )$ and apply Theorem $3 . 1$ : + +$$ +\operatorname* { m i n } _ { \substack { x \in \mathcal { C } , \tilde { z } ^ { ( i - 1 ) } \in \tilde { \mathcal { Z } } ^ { ( i - 1 ) } } } z _ { j } ^ { ( i ) } ( x ) \ge \operatorname* { m a x } _ { 0 \le \alpha ^ { \prime } \le 1 , \beta ^ { \prime } \ge 0 } g ^ { \prime } ( \alpha ^ { \prime } , \beta ^ { \prime } ) : = \mathbb { I } _ { j } ^ { ( i ) } +$$ + +For computing u(i) we simply set $f ( x ) : = - z _ { j } ^ { ( i ) } ( x )$ . Importantly, during solving these intermediate layer bounds, the $\mathbf { { \boldsymbol { \alpha } } ^ { \prime } }$ and $\beta ^ { \prime }$ are independent sets of variables, not the same ones for the objective $f ( x ) : = z ^ { ( L ) }$ . Since $g$ is a function of $\mathbf { l } _ { j } ^ { ( i ) }$ , it is also a function of $\alpha ^ { \prime }$ and $\beta ^ { \prime }$ . In fact, there are a total of $\textstyle \sum _ { i = 1 } ^ { L - 1 } d _ { i }$ intermediate layer neurons, and each neuron is associated with a set of independent $\mathbf { { \boldsymbol { \alpha } } ^ { \prime } }$ and $\beta ^ { \prime }$ variables. Optimizing these variables allowing us to tighten the relaxations on unstable ReLU neurons (which depend on $\mathbf { l } _ { j } ^ { ( i ) }$ and $\mathbf { u } _ { j } ^ { ( i ) }$ ) and produce tight final bounds, which is impossible in LP. In other words, we need to optimize $\hat { \alpha }$ and $\hat { \beta }$ , which are two vectors concatenating $\alpha , \beta$ as well as a large number of $\mathbf { { \boldsymbol { \alpha } } ^ { \prime } }$ and $\beta ^ { \prime }$ used to compute each intermediate layer bound: + +$$ +\operatorname* { m i n } _ { x \in \mathcal { C } , z \in \mathcal { Z } } f ( x ) \ge \operatorname* { m a x } _ { 0 \le \hat { \alpha } \le 1 , \hat { \beta } \ge 0 } g ( \hat { \alpha } , \hat { \beta } ) . +$$ + +This formulation is non-convex and has a large number of variables. Since any $0 \leq \hat { \alpha } \leq 1$ , ${ \hat { \beta } } \geq 0$ leads to a valid lower bound, the non-convexity does not affect soundness. When intermediate layer bounds are also allowed to be tightened during optimization, we can outperform the LP verifier for Eq. $\textcircled { 1 0 }$ using fixed intermediate layer bounds. Typically, in many previous works $\boxed { 8 } \boxed { 2 3 } \boxed { 6 }$ , when the LP formulation Eq. 10 is formed, intermediate layer bounds are pre-computed with bound propagation procedures $\boxed { 8 } \boxed { 2 3 }$ , which are far from optimal. To estimate the dimension of this problem, we denote the number of unstable neurons at layer $i$ as $s _ { i } : = \operatorname { T r } ( | \mathbf { S } ^ { ( i ) } | )$ . Each neuron in layer $i$ is associated with $2 \times \textstyle \sum _ { k = 1 } ^ { i - 1 } s _ { k }$ variables $\mathbf { { \boldsymbol { \alpha } } ^ { \prime } }$ . Suppose each hidden layer has $d$ neurons $( s _ { i } = { O } ( d ) )$ , then ↵ˆ has 2 ⇥ PL1i=1 d $\begin{array} { r } { 2 \times \sum _ { i = 1 } ^ { L - 1 } d _ { i } \sum _ { k = 1 } ^ { i - 1 } s _ { k } = O ( L ^ { 2 } d ^ { 2 } ) } \end{array}$ variables in total. This can be too large for efficientintermediate neurons of the same layer, leading to a $\mathbf { { \boldsymbol { \alpha } } ^ { \prime } }$ $\beta ^ { \prime }$ total number of $O ( L ^ { 2 } d )$ variables to optimize. Note that a weaker form of joint optimization was also discussed in $\lVert \boldsymbol { \mathsf { \overline { { 4 5 } } } } \rVert$ without $\beta$ , and a detailed analysis can be found in Appendix B.2. + +# 3.4 $\beta$ -CROWN with Branch and Bound ( $\beta$ -CROWN BaB) + +We perform complete verification following BaB framework $\pmb { \bigtriangledown }$ using $\beta$ -CROWN as the incomplete solver, and we use simple branching heuristics like BaBSR $\mathbb { U }$ or FSB $\mathbb { \ m }$ . To efficiently utilize GPU, we also use batch splits to evaluate multiple subdomains in the same batch as in [44, 10]. We list our full algorithm $\beta$ -CROWN BaB in Appendix B and we show it is sound and complete here: + +Theorem 3.3. $\beta$ -CROWN with Branch and Bound on splitting ReLUs is sound and complete. + +Soundness is trivial because $\beta$ -CROWN is a sound verifier. For completeness, it suffices to show that when all unstable ReLU neurons are split, $\beta$ -CROWN gives the global minimum for Eq. 10. In contrast, combining CROWN $\left[ \left[ 4 6 \right] \right]$ with BaB does not yield a complete verifier, as it cannot detect infeasible splits and a slow LP solver is still needed to guarantee completeness $\lVert \rVert \dot { \boldsymbol { \mathrm { E } } } \rVert$ . Instead, $\beta$ -CROWN can detect infeasible subdomains - according to duality theory, an infeasible primal problem leads to an unbounded dual objective, which can be detected (see Sec. B.3 for more details). + +Additionally, we show the potential of early stopping a complete verifier as an incomplete verifier. BaB approaches the exact solution of Eq. $^ 1$ by splitting the problem into multiple subdomains, and more subdomains give a tighter lower bound for Eq. $^ { 1 . }$ Unlike traditional complete verifiers, $\beta$ -CROWN is efficient to explore a large number of subdomains during a very short time, making $\beta$ -CROWN BaB an attractive solution for efficient incomplete verification. + +# 4 Experimental Results + +# 4.1 Comparison to Complete Verifiers + +We evaluate complete verification performance on dataset provided in $\mathbb { \left[ 2 3 , \overline { { 1 0 } } \right] }$ and used in VNNCOMP 2020 $\vec { \mathbb { B } 2 2 } \rvert \vec { | }$ . The benchmark contains three CIFAR-10 models (Base, Wide, and Deep) with 100 examples each. Each data example is associated with an $\ell _ { \infty }$ norm $\epsilon$ and a target label for verification (referred to as a property to verify). The details of neural network structures and experimental setups can be found in Appendix $\mathbf { C } .$ We compare against multiple baselines for complete verification: (1) BaBSR [7], a basic BaB and LP based verifier; (2) MIPplanet $\mathbb { \lVert \rVert \cdot \rVert }$ , a customized MIP solver for NN verification where unstable ReLU neurons are randomly selected for splitting; (3) ERAN [35, 33, 36, 34], an abstract interpretation based verifier which performs well on this benchmark in VNN-COMP 2020; (4) GNN-Online $\mathbb { \left[ \left. 2 3 \right| \right. }$ , a BaB and LP based verifiers using a learned Graph Neural Network (GNN) to guide the ReLU splits; (5) $\mathrm { B D D + }$ BaBSR $\pmb { \Vert 6 \Vert }$ , a verification framework based on Lagrangian decomposition on GPUs $\mathrm { ( B D D + ) }$ ) with BaBSR branching strategy; (6) OVAL (BDD+ GNN) [6, 23], a strong verifier in VNN-COMP 2020 using $\mathrm { B D D + }$ with GNN guiding the ReLU splits; (7) A.set BaBSR and (8) Big-M+A.set BaBSR $\mathbb { m }$ , very recent dual-space verifiers on GPUs with a tighter linear relaxation than triangle LP relaxations; (9) Fast-and-Complete [45], which uses CROWN (LiRPA) on GPUs as the incomplete verifier in BaB without neuron split constraints; (10) BaDNB $\mathrm { B D D + }$ FSB) $\mathbb { \ m }$ , a concurrent state-of-the-art complete verifier, using $\mathrm { B D D + }$ on GPUs with FSB branching strategy. $\beta$ -CROWN BaB can use either BaBSR or FSB branching heuristic, and we include both in evaluation. All methods use a 1 hour timeout threshold. + +Table 1: Average runtime and average number of branches on three CIFAR-10 models over 100 properties. + +
CIFAR-10 BaseCIFAR-10 WideCIFAR-10 Deep
Methodtime(s)branches%timeouttime(s)branches%timeouttime(s)branches%timeout
BaBSR2367.781020.5536.002871.14812.6549.002750.75401.2839.00
MIPplanet 152849.69-68.002417.53-46.002302.25140.00
ERAN*[35333634805.94=5.00632.20=9.00545.72=0.00
GNN-online [231794.85565.1333.001367.38372.7415.001055.33131.854.00
BDD+ BaBSR 6807.91195480.1420.00505.6574203.1110.00266.2812722.744.00
OVAL (BDD+ GNN)*[623]662.1767938.3816.00280.3817895.946.0094.691990.341.00
A.set BaBSR10381.7812004.607.00165.912233.103.00190.282491.552.00
BigM+A.setBaBSR[10390.4411938.757.00172.654050.593.00177.223275.252.00
Fast-and-Complete [45]695.01119522.6517.00495.8880519.859.00105.642455.111.00
BaDNB (BDD+ FSB)II309.2938239.047.00165.5311214.444.0010.50368.160.00
β-CROWNBaBSR226.06509608.506.00118.26217691.243.006.12204.660.00
β-CROWN FSB118.23208018.213.0078.32116912.572.005.6941.120.00
+ +![](images/e15f89e6f30fcd3a3be38f99210245d62207d65a1866c03408e05d561e72706d.jpg) +OVAL (BDD+ GNN) and ERAN results are from VNN-COMP 2020 report [22]. Other results were reported by their authors. +Figure 1: Percentage of solved properties with growing running time. $\beta$ -CROWN FSB (light green) and $\beta$ -CROWN BaBSR (dark green) clearly lead in all 3 settings and solve over $90 \%$ properties within 10 seconds. + +We report the average verification time and branch numbers in Table $^ 1$ and plot the percentage of solved properties over time in Figure 1. $\beta$ -CROWN FSB achieves the fastest average running time compared to all other baselines with minimal timeouts, and also clearly leads on the cactus plot. When using a weaker branching heuristic, $\beta$ -CROWN BaBSR still outperforms all literature baselines, including very recent ones such as A.set BaBSR $\mathbb { \ m }$ , Fast-and-Complete $\lVert \boldsymbol { \mathsf { A } } \boldsymbol { \mathsf { S } } \rVert$ and BaDNB [11]. Our benefits are more clearly shown in Figure $\bigstar \bigstar$ where we solve over $90 \%$ examples under 10s and most other verifiers can verify much less or none of the properties within 10s. We see a 2 to 3 orders of magitudes speedup in Figure 1 compared to CPU based verifiers such as MIPplanet and BaBSR. + +# 4.2 Comparison to Incomplete Verifiers + +Verified accuracy. In Table $\lfloor 2 \rfloor$ we compare against a few representative and strong incomplete verifiers on 5 convolutional networks and 4 MLP networks for MNIST and CIFAR-10 under the same set of 1000 images and perturbation $\epsilon$ as reported in [35, 37, 26]. Among the baselines, kPoly [35], OptC2V $ { \mathbb { I } } ^ { \smash { \sum } }$ and PRIMA $\left[ \left[ 2 6 \right] \right]$ utilize state-of-the-art multi-neuron linear relaxation for ReLUs and can bypass the single-neuron convex relaxation barrier $\pmb { \mathbb { B } } \pmb { \mathrm { O } } \|$ , and are among the strongest incomplete verifiers. $\beta$ -CROWN FSB achieves better verified accuracy on all models using a similar or less amount of time. Some models, such as MNIST ConvBig and CIFAR ConvBig, are quite challenging - the verified accuracy obtained by $\beta$ -CROWN FSB is close to the upper bound found via PGD attack. + +To make more comprehensive evaluations, in Table $\textcircled { 3 }$ we further compare against a state-of-the-art semidefinite programming (SDP) based verifier, SDP-FO $\pmb { \mathbb { Q } } \|$ , on one MNIST and six CIFAR-10 models reported in their paper. The models were trained using adversarial training, which posed a challenge for verification $[ [ 2 8 ] ]$ . The SDP formulation can be tighter than linear relaxation based ones, but it is computationally expensive - SDP-FO takes 2 to 3 hours to converge on one GPU for verifying a single property, resulting 5,400 GPU hours to verify 200 testing images with 10 labels each. Due to resource limitations, we directly quote SDP-FO results from $\blacktriangleleft$ on the same set of models. We evaluate verified accuracy on the same set of 200 test images for other baselines. We include a concurrent work PRIMA $\pmb { \mathbb { Z } } \pmb { 6 } \|$ , the strongest multi-neuron linear relaxation baseline in Table 2, which generally outperforms kPoly and OptC2V. Table $\textcircled { 3 }$ shows that overall we are 3 orders of magnitude faster than SDP-FO while still achieving consistently higher verified accuracy on average. + +Tightness of verification. In Figure $\bigstar ,$ we compare the tightness of verification bounds against SDP-FO on two adversarially trained networks from $\left[ \left[ 9 \right] \right]$ . Specifically, we use the verification objective $f ( x ) : = z _ { y } ^ { ( L ) } ( x ) - z _ { y ^ { \prime } } ^ { ( L ) } ( x ) ,$ , where $z ^ { ( L ) }$ is the logit layer output, $y$ and $y ^ { \prime }$ are the true label and the runner-up label. For each test image, a 200-step PGD attack $\mathbb { [ 2 4 ] }$ provides an adversarial upper bound $\overline { { f } }$ of the optimal objective: $f ^ { * } \leq { \overline { { f } } }$ . Verifiers, on the other hand, can provide a verified lower bound $f \leq f ^ { * }$ . Bounds from tighter verification methods lie closer to line $y = x$ in Figure $\bigtriangledown$ Figure $\left. 2 \right.$ shows that on both PGD adversarially trained networks, $\beta$ -CROWN FSB consistently outperforms SDP-FO for all 100 random test images. Importantly, for each point on the plots, $\beta$ -CROWN FSB needs 3 minutes while SDP-FO needs 178 minutes on average. LP verifier with triangle relaxations produces much looser bounds than $\beta$ -CROWN FSB and SDP-FO. Additional results are in Appendix C.2. + +Table 2: Verified accuracy $( \% )$ and avg. time (s) of 1000 images evaluated on the ERAN models in [35, 37, 26]. kPoly, OptC2V and PRIMA are strong incomplete verifiers that can break the convex relaxation barrier [30]. The average time reported by us excludes examples that are classified incorrectly. + +
DatasetModel (Same settings as 35.37261)CROWN/DeepPoly*1361 Verified%Time (s)kPoly 因 Ver.% Time(s)OptC2V Ver.%四 Time(s)PRIMA+ 园 Ver.% Time(s)β-CROWN FSB Ver.% Time(s)Upper bound
MNISTMLP5×100‡16.00.744.1307 42.913751.015969.9 10284.2
MLP8×10018.21.436.9 17138.475942.8301 62.010382.0
MLP5×20029.22.457.4 18760.140369.0 22477.48690.1
MLP8× 20025.95.650.6 46452.8345162.4 39573.59591.1
ConvSmall15.8334.7 47743.65559.872.77.073.2
ConvBig71.12173.6 4077.110277.542 11 79.33.180.4
CIFARConvSmall35.9439.986 39.810544.613 46.36.848.1
ConvBig42.14345.9 346No public code48.3176 51.615.355.0
ResNet24.1124.591cannot run24.81.724.8 1.624.8
+ +\* CROWN/DeepPoly evaluated on CPU. † PRIMA is a concurrent work and its results are from [26] (Oct 26, 2021 version), except that ResNet results are from personal communications with the authors due to a different input normalization used. ‡ Because these MLP models are fairly small, some of their intermediate layer bounds are computed by mixed integer programming (MIP) using $80 \%$ time budget before branch and bound starts and $\beta$ -CROWN FSB is used during the branch and bound process. We find that tighter intermediate bounds by MIP is beneficial for these small MLP models. + +Table 3: Verified accuracy $( \% )$ and avg. per-example verification time (s) on 7 models from SDP-FO [9] CROWN/DeepPoly are fast but loose bound propagation based methods, and they cannot be improved with more running time. SDP-FO uses stronger semidefinite relaxations, which can be very slow and sometimes has convergence issues. PRIMA, a concurrent work, is the state-of-the-art relaxation barrier breaking method; we did not include kPoly and $\mathrm { O p t C 2 V }$ because they are weaker than PRIMA (see Table 2). + +
DatasetModel ∈= 0.3 and ε= 2/255CROWN/DeepPoly Verified%SDP-FO*PRIMA [26]β-CROWN FSBUpper bound
MNISTTime (s)Ver.%Time(s)Ver.%Time(s)Ver.%Time(s)
CIFARCNN-A-Adv1.00.143.4>20h44.5135.970.521.176.5
CNN-B-Adv21.50.532.8>25h38.0343.646.532.265.0
CNN-B-Adv-443.50.946.0>25h53.543.854.011.663.5
CNN-A-Adv35.50.639.6>25h41.54.844.05.850.0
CNN-A-Adv-441.50.740.0>25h45.04.946.05.649.5
CNN-A-Mix CNN-A-Mix-423.5 38.00.4 0.539.6 47.8>25h >25h37.5 48.534.3 7.041.5 50.549.6 5.953.0 57.5
+ +SDP-FO results are directly from their paper due to its very long running time (>20h per example). † PRIMA experiments were done using commi ${ \widehat { 3 9 6 \mathrm { d c } 7 \mathrm { a } } } ,$ released on June 4, 2021. PRIMA and $\beta$ -CROWN FSB results are on the same set of 200 examples (first 200 examples of CIFAR-10 dataset) and we don’t run verifiers on examples that are classified incorrectly or can be attacked by a 200-step PGD. $\beta$ -CROWN uses 1 GPU and 1 CPU; PRIMA uses 1 GPU and 20 CPUs. + +VNN-COMP 2021 results. We encourage the readers to checkout the report of the Second International Verification of Neural Networks Competition (VNN-COMP 2021) $\pmb { \mathbb { B } } \|$ with 9 additional benchmarks and 12 competing methods evaluated in a standardized testing environment on AWS. Our entry $\alpha , \beta$ -CROWN is based on the $\beta$ -CROWN algorithm in this work and uses the same codebase. + +# 5 Related Work + +Many early complete verifiers for neural networks relied on existing solvers such as MILP or SMT solvers [20, 15, 19, 12, 38] and were limited to very small problem instances. Branch and bound (BaB) + +![](images/f268324c95f4c9216eb4f87162c23c7018811152e56851544933396dfb737acb.jpg) +Figure 2: Verified lower bound v.s. PGD adversarial upper bound. A lower bound closer to the upper bound (closer to the line $y = x$ ) is better. $\beta$ -CROWN FSB uses 3mins while SDP-FO needs 2 to 3 hours per point. + +based method was proposed to better exploit the network structure using LP-based incomplete verifier for bounding and ReLU splits for branching [8, 40, 23, 5]. Besides branching on ReLU neurons, input domain branching was also considered in [41, 29, 1] but limited by input dimensions $\pmb { \mathbb { B } } ] \mathbf l$ . + +Recently, a few approaches have been proposed to use efficient iterative solvers or bound propagation methods on GPUs without relying on LP solvers. Bunel et al. [6] decomposed the verification problem layer by layer, solved each layer in a closed form on GPUs, and used Lagrangian to enforce consistency between layers. However, their formulation only has the same power as LP and needs many iterations to converge. De Palma et al. $\mathbb { \ m }$ used a dual-space verifier with a linear relaxation [2, 37] tighter than triangle LP relaxation, but in most settings the extra computational costs and difficulties for an efficient implementation offset its benefits (more discussions in section $\mathbf { B } . 2 )$ . A concurrent work BaDNB [11] proposed a new branching strategy, filtered smart branching (FSB), combined with Lagrangian decomposition to get better verification performance. Xu et al. $\pm \boxed { \boxed { 4 4 } }$ used CROWN as a massively paralleled incomplete solver on GPUs for complete verification, but it cannot handle neuron split constraints, leading to suboptimal efficiency and high timeout rates. + +For incomplete verification, Salman et al. $\textcircled { \lvert 3 0 \rvert }$ shows the inherent limitation of using per-neuron convex relaxations for verification problems. Singh et al. $\pmb { \Vert 3 5 \Vert }$ and Müller et al. $[ [ 2 6 ] ]$ broke this barrier by considering constraints involving multiple ReLU neurons; Tjandraatmadja et al. $\pmb { \Vert 3 7 } \Vert$ proposed to relax a linear layer with a ReLU neuron together using a strong mixed-integer programming formulation $\mathbb { M }$ . SDP based relaxations $[ \sqrt { 2 8 } , \sqrt { 1 6 } , \sqrt { 1 4 } ]$ typically produce tight bounds but with significantly higher cost. The most recent GPU based SDP verifier $\bigstar \bigstar$ is still relatively slow and can take 2 hours to verify a single image. In this work, we impose neuron split constraints using $\beta$ -CROWN and combine it with branch and bound done in parallel on GPUs. Although for each subdomain in BaB, $\beta$ -CROWN is still subject to the convex relaxation barrier, the efficiency of $\beta$ -CROWN BaB allows it to quickly explore a very large number of subdomains and outperform existing convex barrier breaking incomplete verifiers under many scenarios in both runtime and tightness. + +Additionally, another line of works train networks to enhance verified accuracy, typically using cheap incomplete verifiers at training time [42, 39, 25, 43, 18, 25, 47, 4, 32]. Traditionally only these verification-customized networks can have reasonable verified accuracy, while $\beta$ -CROWN BaB can also give non-trivial verified accuracy on relatively large networks agnostic to verification. + +# 6 Conclusion + +We proposed $\beta$ -CROWN, a new bound propagation method that can fully encode the neuron split constraints introduced in BaB, which clearly leads in both complete and incomplete verification settings. The success of $\beta$ -CROWN comes from a few factors: (1) In Section $\boxed { 3 . 1 }$ we show that $\beta$ - CROWN is an GPU-friendly bound propagation algorithm significantly faster than $\bar { L } P$ solvers. (2) In Section $^ { 3 . 2 , }$ we show that $\beta$ -CROWN is solving an equivalent problem of the LP verifier with neuron split constraints. (3) In Section $^ { 3 . 3 , }$ we show that $\beta$ -CROWN can jointly optimize intermediate layer bounds and achieve tighter bounds than typical $L P$ verifiers using fixed intermediate layer bounds. + +Limitations. Our verifier has several limitations which are commonly shared by most existing BaB-based complete verifiers. First, we focused on ReLU which can be split into two linear cases. For other non-piecewise linear activation functions, although it is still possible to conduct branch and bound, it is difficult to guarantee completeness. Second, we discussed only the norm perturbations for input domains. In practice, the threat model may involve complicated and nonconvex perturbation specifications. Third, although our GPU accelerated verifier outperforms existing ones, all BaB based verifiers, including ours, are still limited to relatively small models far from the ImageNet scale. Finally, we have only demonstrated robustness verification of image classification tasks, and generalizing it to give verification guarantees for other tasks such as robust deep reinforcement learning $\frac { 1 } { 1 2 7 }$ 48, 49] is an interesting direction for future work. + +Societal Impact NNs have been used in an increasingly wide range of real-world applications and play an important role in artificial intelligence (AI). The trustworthiness and robustness of NNs have become crucial factors since AI plays an important role in modern society. $\beta$ -CROWN is a strong neural network verifier which can be used to check certain properties of neural networks, which can be helpful for guaranteeing the robustness, correctness, and fairness of NNs in applications that can directly or indirectly impact human life. We believe our work has overall positive societal impacts, although it may potentially be misused to identify the weakness of NNs and guide attacks. + +# Acknowledgement + +This work is supported by NSF grant CNS18-01426; an ARL Young Investigator (YIP) award; an NSF CAREER award; a Google Faculty Fellowship; a Capital One Research Grant; and a J.P. Morgan Faculty Award; Air Force Research Laboratory under FA8750-18-2-0058; NSF IIS-1901527, NSF IIS-2008173 and NSF CAREER-2048280; and NSF CNS-1932351. Huan Zhang is supported by funding from the Bosch Center for Artificial Intelligence. + +# References + +[1] G. Anderson, S. Pailoor, I. Dillig, and S. Chaudhuri. Optimization and abstraction: A synergistic approach for analyzing neural network robustness. In Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), 2019. [2] R. Anderson, J. Huchette, C. Tjandraatmadja, and J. P. Vielma. 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International Conference on Learning Representations (ICLR), 2021. \ No newline at end of file diff --git a/parse/train/ahYIlRBeCFw/ahYIlRBeCFw_content_list.json b/parse/train/ahYIlRBeCFw/ahYIlRBeCFw_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..b9d67bb42388e044d1061e654818663aefe51de9 --- /dev/null +++ b/parse/train/ahYIlRBeCFw/ahYIlRBeCFw_content_list.json @@ -0,0 +1,1383 @@ +[ + { + "type": "text", + "text": "Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Neural Network Robustness Verification ", + "text_level": 1, + "bbox": [ + 200, + 122, + 797, + 196 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Shiqi Wang\\*,1 Huan Zhang\\*,2 Kaidi Xu\\*,3 Xue Lin3 Suman Jana1 Cho-Jui Hsieh4 Zico Kolter2 ", + "bbox": [ + 290, + 246, + 702, + 280 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1Columbia University 2CMU 3Northeastern University 4UCLA sw3215@columbia.edu huan@huan-zhang.com xu.kaid@northeastern.edu xue.lin@northeastern.edu suman@cs.columbia.edu chohsieh@cs.ucla.edu zkolter@cs.cmu.edu ", + "bbox": [ + 202, + 284, + 795, + 342 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "\\* Equal Contribution ", + "bbox": [ + 428, + 349, + 570, + 363 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 400, + 535, + 416 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Bound propagation based incomplete neural network verifiers such as CROWN are very efficient and can significantly accelerate branch-and-bound (BaB) based complete verification of neural networks. However, bound propagation cannot fully handle the neuron split constraints introduced by BaB commonly handled by expensive linear programming (LP) solvers, leading to loose bounds and hurting verification efficiency. In this work, we develop $\\beta$ -CROWN, a new bound propagation based method that can fully encode neuron splits via optimizable parameters $\\beta$ constructed from either primal or dual space. When jointly optimized in intermediate layers, $\\beta$ -CROWN generally produces better bounds than typical LP verifiers with neuron split constraints, while being as efficient and parallelizable as CROWN on GPUs. Applied to complete robustness verification benchmarks, $\\beta$ -CROWN with BaB is up to three orders of magnitude faster than LP-based BaB methods, and is notably faster than all existing approaches while producing lower timeout rates. By terminating BaB early, our method can also be used for efficient incomplete verification. We consistently achieve higher verified accuracy in many settings compared to powerful incomplete verifiers, including those based on convex barrier breaking techniques. Compared to the typically tightest but very costly semidefinite programming (SDP) based incomplete verifiers, we obtain higher verified accuracy with three orders of magnitudes less verification time. Our algorithm empowered the $\\alpha , \\beta$ -CROWN (alpha-beta-CROWN) verifier, the winning tool in VNN-COMP 2021. Our code is available at http://PaperCode.cc/BetaCROWN. ", + "bbox": [ + 232, + 433, + 766, + 723 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 753, + 310, + 770 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As neural networks (NNs) are being deployed in safety-critical applications, it becomes increasingly important to formally verify their behaviors under potentially malicious inputs. Broadly speaking, the neural network verification problem involves proving certain desired relationships between inputs and outputs (often referred to as specifications), such as safety or robustness guarantees, for all inputs inside some domain. Canonically, the problem can be cast as finding the global minima of some functions on the network’s outputs (e.g., the difference between the predictions of the true label and another target label), within a bounded input set as constraints. This is a challenging problem due to the non-convexity and high dimensionality of neural networks. ", + "bbox": [ + 174, + 786, + 825, + 898 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We first focus on complete verification: the verifier should give a definite “yes/no” answer given sufficient time. Many complete verifiers rely on the branch and bound (BaB) method $\\textcircled { 8 } \\textcircled { 1 8 }$ involving (1) branching by recursively splitting the original verification problem into subdomains (e.g., splitting a ReLU neuron into positive/negative linear regions by adding split constraints) and (2) bounding each subdomain with specialized incomplete verifiers. Traditional BaB-based verifiers use expensive linear programming (LP) solvers $\\pm \\boxed { 1 } \\boxed { 2 3 } \\boxed { 7 }$ as incomplete verifiers which can fully encode neuron split constraints. Meanwhile, a recent verifier, Fast-and-Complete $\\mathbb { \\lVert \\boldsymbol { 4 5 } \\rVert }$ , demonstrates that cheap incomplete verifiers can significantly accelerate complete verification on GPUs over LP-based ones thanks to their efficiency. Many cheap incomplete verifiers are based on bound propagation methods [46, 42, 41, 13, 17, 36, 44], i.e., maintaining and propagating tractable and sound bounds through networks, and CROWN [46] is a representative which propagates a linear or quadratic bound. ", + "bbox": [ + 173, + 90, + 825, + 243 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "However, unlike LP based verifiers, existing bound propagation methods lack the power to handle neuron split constraints introduced by BaB. For instance, given inputs $x , y \\in [ - 1 , 1 ]$ , they can bound a ReLU’s input $x + y$ as $[ - 2 , 2 ]$ but they have no means to consider neuron split constraints such as $x - y \\ge 0$ introduced by splitting another ReLU to the positive linear region. Such a problem causes looser bounds and unnecessary branching, hurting the verification efficiency. Even worse, without considering these split constraints, bound propagation methods cannot detect many infeasible subdomains in BaB $\\lVert \\boldsymbol { \\mathsf { \\Pi } } \\rVert$ , leading to incompleteness unless costly checking is performed. ", + "bbox": [ + 173, + 250, + 825, + 347 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In our work, we develop a new, fast bound propagation based incomplete verifier, $\\beta$ -CROWN. It solves an optimization problem equivalent to the expensive LP based methods with neuron split constraints while still enjoying the efficiency of bound propagation methods. $\\beta$ -CROWN contains optimizable parameters $\\beta$ which come from propagation of Lagrangian multipliers, and any valid settings of these parameters yield sound bounds for verification. These parameters are optimized using a few steps of (super)gradient ascent to achieve bounds as tight as possible. Optimizing $\\beta$ can also eliminate many infeasible subdomains and avoid further useless branching. Furthermore, we can jointly optimize intermediate layer bounds similar to $\\pm \\boxed { \\boxed { 4 4 } }$ but also with the additional parameters $\\beta$ , allowing $\\beta$ - CROWN to tighten relaxations and outperform typical LP verifiers with fixed intermediate layer bounds. Unlike traditional LP-based BaB methods, $\\beta$ -CROWN can be efficiently implemented with an automatic differentiation framework on GPUs to fully exploit the power of modern accelerators. The combination of $\\beta$ -CROWN and BaB ( $\\beta$ -CROWN BaB) produces a complete verifier with GPU acceleration, reducing the verification time of traditional LP based BaB verifiers $\\textcircled { 8 }$ by up to three orders of magnitudes on a commonly used benchmark suite on CIFAR-10 [6, 10]. Compared to all state-of-the-art GPU-based complete verifiers [7, 45, 10, 23, 6, 11], our approach is noticeably faster with lower timeout rates. Our algorithm empowered the tool $\\alpha , \\beta$ -CROWN (alpha-beta-CROWN), which won the 2nd International Verification of Neural Networks Competition $\\pmb { \\mathbb { B } } \\|$ (VNN-COMP 2021) with the highest total score and verified the most number of problem instances in 8 benchmarks. ", + "bbox": [ + 173, + 354, + 825, + 602 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Finally, by terminating our complete verifier $\\beta$ -CROWN BaB early, our approach can also function as a more accurate incomplete verifier by returning an incomplete but sound lower bound of all subdomains explored so far. We achieve better verified accuracy on a few benchmarking models over powerful incomplete verifiers including those based on tight linear relaxations $\\overline { { \\mathbb { B } 5 } } \\overline { { \\mathbb { B } \\overline { { \\lambda } } } } \\overline { { \\mathbb { B } 6 } }$ and semidefinite relaxations $\\pmb { \\Vert 9 \\Vert }$ . Compared to the typically tightest but very costly incomplete verifier SDP-FO $\\pmb { \\Vert }$ based on the semidefinite programming (SDP) relaxations $[ \\overbrace { 2 8 } , \\overbrace { 1 4 } ]$ , our method obtains consistently higher verified accuracy while reducing verification time by three orders of magnitudes. ", + "bbox": [ + 174, + 607, + 825, + 705 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Background ", + "text_level": 1, + "bbox": [ + 174, + 719, + 308, + 737 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 The neural network verification problem and its LP relaxation ", + "text_level": 1, + "bbox": [ + 176, + 743, + 647, + 758 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We define the input of a neural network as $\\begin{array} { r l r } { x } & { { } \\in } & { \\mathbb { R } ^ { d _ { 0 } } } \\end{array}$ , and define the weights and biases of an $L$ -layer neural network as $\\mathbf W ^ { ( i ) } \\in \\mathbb R ^ { d _ { i } \\times d _ { i - 1 } }$ and $\\mathbf { b } ^ { ( i ) } \\in \\mathbb { R } ^ { d _ { i } }$ $( i \\in \\{ 1 , \\cdots , L \\} )$ respectively. For simplicity we assume that $d _ { L } = 1$ so $\\mathbf { W } ^ { ( L ) }$ is a vector and $\\mathbf { b } ^ { ( L ) }$ is a scalar. The neural network function $f : \\mathbb { R } ^ { d _ { 0 } } \\mathbb { R }$ is defined as $f ( x ) = z ^ { ( L ) } ( x )$ , where $z ^ { ( i ) } ( x ) = \\mathbf { W } ^ { ( i ) } \\hat { z } ^ { ( i - 1 ) } ( x ) + \\mathbf { b } ^ { ( i ) }$ , $\\hat { z } ^ { ( i ) } ( x ) = \\sigma ( z ^ { ( i ) } ( x ) )$ and $\\hat { z } ^ { ( 0 ) } ( x ) = x$ . $\\sigma$ is the activation function and we use ReLU throughout this paper. When the context is clear, we omit · $( x )$ and use $z _ { j } ^ { ( i ) }$ and $\\hat { z } _ { j } ^ { ( i ) }$ to represent the pre-activation and post-activation values of the $j$ -th neuron in the $i$ -th layer. Neural network verification seeks the solution of the optimization problem in Eq. 1: ", + "bbox": [ + 173, + 768, + 826, + 892 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/e1e2644eab43c8d8713c59e071d3c3166d8c857865addae35fc81633a6b12bcb.jpg", + "text": "$$\n\\operatorname* { m i n } { f ( x ) : = z ^ { ( L ) } ( x ) } \\quad \\mathrm { s . t . ~ } z ^ { ( i ) } = \\mathbf { W } ^ { ( i ) } \\hat { z } ^ { ( i - 1 ) } + \\mathbf { b } ^ { ( i ) } , \\hat { z } ^ { ( i ) } = \\sigma ( z ^ { ( i ) } ) , x \\in \\mathcal { C } , i \\in \\{ 1 , \\cdots , L - 1 \\}\n$$", + "text_format": "latex", + "bbox": [ + 179, + 893, + 799, + 912 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The set $\\mathcal { C }$ defines the allowed input region and our aim is to find the minimum of $f ( x )$ for $x \\in { \\mathcal { C } }$ , and throughout this paper we consider $\\mathcal { C }$ as an $\\ell _ { \\infty }$ ball around a data example $x _ { 0 }$ : $\\mathcal { C } = \\{ x \\vert \\| x - x _ { 0 } \\| _ { \\infty } \\leq \\epsilon \\}$ but other $\\ell _ { p }$ norms can also be supported. In practical settings, we typically have “specifications” to verify, which are (usually linear) functions of neural network outputs describing the desired behavior of neural networks. For example, to guarantee robustness we typically investigate the margin between logits. Because the specification can also be seen as an output layer of NN and merged into $f ( x )$ under verification, we do not discuss it in detail in this work. We consider the canonical specification $f ( x ) > 0$ : if we can prove that $f ( x ) > 0$ , $\\forall x \\in { \\mathcal { C } }$ , we say $f ( x )$ is verified. ", + "bbox": [ + 173, + 90, + 825, + 203 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "When $\\mathcal { C }$ is a convex set, Eq. 1 is still a non-convex problem because the constraints $\\hat { z } ^ { ( i ) } = \\sigma ( z ^ { ( i ) } )$ are non-convex. Given unlimited time, complete verifiers can solve Eq. 1 exactly: $f ^ { * } = \\operatorname* { m i n } { f ( x ) }$ , $\\forall x \\in$ $\\mathcal { C }$ , so we can always conclude if the specification holds or not for any problem instance. On the other hand, incomplete verifiers usually relax the non-convexity of neural networks to obtain a tractable lower bound of the solution $\\underline { { f } } \\le f ^ { * }$ . If $\\underline { { f } } \\geq 0$ , then $f ^ { * } > 0$ so $f ( x )$ can be verified; when $\\underline { { f } } < 0$ , we are not able to infer the sign of $f ^ { * }$ so cannot conclude if the specification holds or not. ", + "bbox": [ + 173, + 207, + 825, + 294 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A commonly used incomplete verification technique is to relax non-convex ReLU constraints with linear constraints and turn the verification problem into a linear programming (LP) problem, which can then be solved with linear solvers. We refer to it as the “LP verifier” in this paper. Specifically, given $\\mathrm { R e L U } ( z _ { j } ^ { ( i ) } ) : = \\operatorname* { m a x } ( 0 , z _ { j } ^ { ( i ) } )$ and its intermediate layer bounds $\\mathbf { l } _ { j } ^ { ( i ) } \\le z _ { j } ^ { ( i ) } \\overset { \\_ } { \\le } \\mathbf { u } _ { j } ^ { ( i ) }$ , each ReLU can be categorized into three cases: (1) if $\\mathbf { l } _ { j } ^ { ( i ) } \\geq 0$ (ReLU in linear region) then $\\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) }$ ; (2) if $\\mathbf { u } _ { j } ^ { ( i ) } \\leq 0$ (ReLU in inactive region) then $\\hat { z } _ { j } ^ { ( i ) } = 0$ ; (3) if $\\mathbf { l } _ { j } ^ { ( i ) } \\le 0 \\le \\mathbf { u } _ { j } ^ { ( i ) }$ (ReLU is unstable) then three linear bounds are used: $\\hat { z } _ { j } ^ { ( i ) } \\geq 0$ , $\\hat { z } _ { j } ^ { ( i ) } \\geq z _ { j } ^ { ( i ) }$ , and $\\begin{array} { r } { \\hat { z } _ { j } ^ { ( i ) } \\leq \\frac { \\mathbf { u } _ { j } ^ { ( i ) } } { \\mathbf { u } _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } } \\left( z _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } \\right) } \\end{array}$ ; they are often referred to as the “triangle” relaxation $\\mathbb { m } 5 \\mathbb { k } 2 \\mathbb { l }$ . The intermediate layer bounds $\\mathbf { l } ^ { ( i ) }$ and $\\mathbf { u } ^ { ( i ) }$ are usually obtained from a cheaper bound propagation method (see next subsection). LP verifiers can provide relatively tight bounds but linear solvers are still expensive especially when the network is large. Also, unlike our $\\beta$ -CROWN, they have to use fixed intermediate bounds and cannot use the joint optimization of intermediate layer bounds (Section $3 . 3 )$ to tighten relaxation. ", + "bbox": [ + 173, + 299, + 826, + 501 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 CROWN: efficient incomplete verification by propagating linear bounds ", + "text_level": 1, + "bbox": [ + 173, + 508, + 714, + 525 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Another cheaper way to give a lower bound for the objective in Eq. $^ 1$ is through sound bound propagation. CROWN $\\bar { \\lVert 4 6 rVert }$ is a representative method that propagates a linear bound of $f ( x )$ w.r.t. every intermediate layer in a backward manner until reaching the input $x$ . CROWN uses two linear constraints to relax unstable ReLU neurons: a linear upper bound $\\begin{array} { r } { \\hat { z } _ { j } ^ { ( i ) } \\leq \\frac { \\mathbf { u } _ { j } ^ { ( i ) } } { \\mathbf { u } _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } } \\left( z _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } \\right) } \\end{array}$ and a linear lower bound $\\hat { z } _ { j } ^ { ( i ) } \\geq \\alpha _ { j } ^ { ( i ) } z _ { j } ^ { ( i ) }$ $0 \\leq \\alpha _ { j } ^ { ( i ) } \\leq 1 $ ). We can then bound the output of a ReLU layer: Lemma 2.1 (ReLU relaxation in CROWN). Given $w , v \\in \\mathbb { R } ^ { d } , 1 \\leq v \\leq \\mathbf { u }$ (element-wise), we have ", + "bbox": [ + 173, + 527, + 826, + 637 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2068a804b82efa90a51fa0eb0b3bb6c8376efd5e229b79f91a053c2c7a168fda.jpg", + "text": "$$\nw ^ { \\top } \\mathrm { R e L U } ( v ) \\geq w ^ { \\top } \\mathbf { D } v + b ^ { \\prime } .\n$$", + "text_format": "latex", + "bbox": [ + 401, + 638, + 591, + 656 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { D }$ is a diagonal matrix containing free variables $0 \\leq \\alpha _ { j } \\leq 1$ only when $\\mathbf { u } _ { j } > 0 > 1 _ { j }$ and $w _ { j } \\geq 0$ , while its rest values as well as constant $b ^ { \\prime }$ are determined by ${ \\mathbf { l } } , { \\mathbf { u } } , w$ . ", + "bbox": [ + 174, + 659, + 823, + 688 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Detailed forms of each term are listed in Appendix $\\mathbf { A } .$ Lemma $\\boxed { 2 . 1 }$ can be repeatedly applied, resulting in an efficient back-substitution procedure to derive a linear lower bound of NN output w.r.t. $x$ : ", + "bbox": [ + 173, + 695, + 823, + 724 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Lemma 2.2 (CROWN bound $[ \\overline { { | 4 6 | } }$ ). Given an $L$ -layer ReLU NN $f ( \\boldsymbol { x } ) : \\mathbb { R } ^ { d _ { 0 } } \\mathbb { R }$ with weights $\\mathbf { W } ^ { ( i ) }$ , biases $\\mathbf { b } ^ { ( i ) }$ , pre-ReLU bounds $\\mathbf { l } ^ { ( i ) } \\le z ^ { ( i ) } \\le \\mathbf { u } ^ { ( i ) }$ $1 \\leq i \\leq L )$ and input constraint $x \\in { \\mathcal { C } }$ . We have ", + "bbox": [ + 171, + 728, + 825, + 760 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/9d89f0180e7f21f8db97b304f57d413dd9954ef38882d3bcf373cd146334d19a.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { C } } f ( x ) \\geq \\operatorname* { m i n } _ { x \\in \\mathcal { C } } { \\pmb a } _ { \\mathrm { c R o w n } } ^ { \\top } x + c _ { \\mathrm { C R O W N } }\n$$", + "text_format": "latex", + "bbox": [ + 372, + 762, + 622, + 786 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathbf { \\pmb { a } } _ { \\mathrm { C R O W N } }$ and $c _ { \\mathrm { C R O W N } }$ can be computed using $\\mathbf { W } ^ { ( i ) } , \\mathbf { b } ^ { ( i ) } , \\mathbf { l } ^ { ( i ) } , \\mathbf { u } ^ { ( i ) }$ in polynomial time. ", + "bbox": [ + 169, + 789, + 771, + 808 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "When $\\mathcal { C }$ is an $\\ell _ { p }$ norm ball, minimization over the linear function can be easily solved using Hölder’s inequality. The main benefit of CROWN is its efficiency: CROWN can be efficiently implemented on machine learning accelerators such as GPUs $\\pm \\mathbb { H }$ and TPUs $\\textcircled { \\scriptsize { 1 4 7 } }$ , and it can be a few magnitudes faster than an LP verifier which is hard to parallelize on GPUs. CROWN was generalized to general architectures $\\pm 4 \\sqrt { 3 1 }$ while we only demonstrate it for feedforward ReLU networks for simplicity. Additionally, Xu et al. $\\lVert \\rVert \\varTheta 5 \\rVert$ showed that it is possible to optimize the slope of the lower bound, $_ { \\pmb { \\alpha } }$ , using gradient ascent, to further tighten the bound (sometimes referred to as $\\alpha$ -CROWN). ", + "bbox": [ + 173, + 814, + 826, + 912 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.3 Branch and Bound and Neuron Split Constraints ", + "text_level": 1, + "bbox": [ + 173, + 90, + 553, + 106 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Branch and bound (BaB) method is widely adopted in complete verifiers $\\textcircled { 8 }$ : we divide the domain of the verification problem $\\mathcal { C }$ into two subdomains $\\mathcal { C } _ { 1 } = \\{ x \\in \\mathcal { C } , z _ { j } ^ { ( i ) } \\geq 0 \\}$ and $\\mathcal { C } _ { 2 } = \\{ x \\in \\mathcal { C } , z _ { j } ^ { ( i ) } < 0 \\}$ where $z _ { j } ^ { ( i ) }$ is an unstable ReLU neuron in $\\mathcal { C }$ but now becomes linear for each subdomain. Incomplete verifiers can then estimate the lower bound of each subdomain with relaxations. If the lower bound produced for subdomain $\\mathcal { C } _ { i }$ (denoted by $\\underline { { f } } _ { \\mathcal { C } _ { i } } )$ is greater than 0, $\\mathcal { C } _ { i }$ is verified; otherwise, we further branch over domain $\\mathcal { C } _ { i }$ by splitting another unstable ReLU neuron. The process terminates when all subdomains are verified. The completeness is guaranteed when all unstable ReLU neurons are split. ", + "bbox": [ + 173, + 112, + 825, + 223 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "LP verifier with neuron split constraints. A popular incomplete verifier used in BaB is the LP verifier. Essentially, when we split the $j$ -th ReLU in layer $i$ , we can simply add $z _ { j } ^ { ( i ) } \\geq 0$ or $z _ { j } ^ { ( i ) } < 0$ to Eq. $^ 1$ and get a linearly relaxed lower bound to each subdomain. We denote the $\\mathcal { Z } ^ { + ( i ) }$ and $\\mathcal { Z } ^ { - ( i ) }$ as the set of neuron indices with positive and negative split constraints in layer $i$ . We define the split constraints at layer $i$ as $\\mathcal { Z } ^ { ( i ) } : = \\{ z ^ { ( i ) } \\mid z _ { j _ { 1 } } ^ { ( i ) } \\geq 0 , z _ { j _ { 2 } } ^ { ( i ) } < 0 , \\forall j _ { 1 } \\in \\mathcal { Z } ^ { + ( i ) } , \\forall j _ { 2 } \\in \\mathcal { Z } ^ { - ( i ) } \\}$ . We denote the vector of all pre-ReLU neurons as $z$ , and we define a set $\\mathcal { Z }$ to represent the split constraints on $z$ : $\\mathcal { Z } = \\mathcal { Z } ^ { ( 1 ) } \\cap \\mathcal { Z } ^ { ( 2 ) } \\cap \\cdot \\cdot \\cdot \\cap \\mathcal { Z } ^ { ( L - 1 ) }$ . For convenience, we also use the shorthand $\\tilde { \\mathcal { Z } } ^ { ( i ) } : = \\mathcal { Z } ^ { ( 1 ) } \\cap \\cdot \\cdot \\cdot \\cap \\mathcal { Z } ^ { ( i ) }$ and $\\tilde { z } ^ { ( i ) } : = \\{ z ^ { ( 1 ) } , z ^ { ( 2 ) } , \\cdot \\cdot \\cdot , z ^ { ( i ) } \\}$ . LP verifiers can easily handle these neuron split constraints but are more expensive than bound propagation methods like CROWN and cannot be accelerated on GPUs. ", + "bbox": [ + 173, + 238, + 826, + 377 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Branching strategy. Branching strategies (selecting which ReLU neuron to split) are generally agnostic to the incomplete verifier used in BaB but do affect the overall BaB performance. BaBSR [7] is a widely used strategy in complete verifiers, which is based on an fast estimates on objective improvements after splitting each neuron. The neuron with highest estimated improvement is selected for branching. Recently, Filtered Smart Branching (FSB) $\\bar { \\mathbb { M } } \\bar { \\mathbb { I } }$ improves BaBSR by mimicking strong branching - it utilizes bound propagation methods to evaluate the best a few candidates proposed by BaBSR and chooses the one with largest improvement. Graph neural network (GNN) based branching was also proposed $\\mathbb { \\left[ \\left. 2 3 \\right] \\right. }$ . Our $\\beta$ -CROWN BaB is a general complete verification framework fit for any potential branching strategy, and we evaluate both BaBSR and FSB in experiments. ", + "bbox": [ + 173, + 392, + 825, + 518 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 $\\beta$ -CROWN for Complete and Incomplete Verification ", + "text_level": 1, + "bbox": [ + 173, + 537, + 650, + 556 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In this section, we first give intuitions on how $\\beta$ -CROWN handles neuron split constraints without costly LP solvers. Then we formally state the main theorem of $\\beta$ -CROWN from both primal and dual spaces, and discuss how to tighten the bounds using free parameters $\\alpha , \\beta$ . Lastly, we propose $\\beta$ -CROWN BaB, a complete verifier that is also a strong incomplete verifier when stopped early. ", + "bbox": [ + 174, + 564, + 825, + 621 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 $\\beta$ -CROWN: Linear Bound Propagation with Neuron Split Constraints ", + "text_level": 1, + "bbox": [ + 171, + 631, + 712, + 647 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The NN verification problem under neuron split constraints can be seen as an optimization problem: ", + "bbox": [ + 166, + 651, + 816, + 666 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/29b81d7a4774cfc4fc8d7a4d038ccb0ce94f5f906fe65723fe3528954e84b817.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in { \\mathcal { C } } , z \\in { \\mathcal { Z } } } f ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 450, + 674, + 547, + 698 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Bound propagation methods like CROWN can give a relatively tight lower bound for $\\scriptstyle \\operatorname* { m i n } _ { x \\in { \\mathcal { C } } } f ( x )$ but they cannot handle the neuron split constraints $z \\in { \\mathcal { Z } }$ . Before we present our main theorem, we first show the intuition on how to apply split constraints to the bound propagation process. ", + "bbox": [ + 174, + 707, + 825, + 750 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To encode the neuron splits, we first define diagonal matrix $\\mathbf { S } ^ { ( i ) } ~ \\in ~ \\mathbb { R } ^ { d _ { i } \\times d _ { i } }$ in Eq. $3$ where $i \\in$ $[ 1 , \\cdot \\cdot \\cdot L - 1 ] , j \\in [ 1 , \\cdot \\cdot \\cdot , d _ { i } ]$ are indices of layers and neurons, respectively: ", + "bbox": [ + 173, + 756, + 823, + 786 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/42c29c4edf3482f89d2d60c251dd0147602ce69cfed2a01e36de47e396d6a1ae.jpg", + "text": "$$\n{ \\bf S } _ { j , j } ^ { ( i ) } = - 1 ( \\mathrm { i f ~ s p l i t ~ } z _ { j } ^ { ( i ) } \\geq 0 ) ; \\quad { \\bf S } _ { j , j } ^ { ( i ) } = + 1 ( \\mathrm { i f ~ s p l i t ~ } z _ { j } ^ { ( i ) } < 0 ) ; \\quad { \\bf S } _ { j , j } ^ { ( i ) } = 0 ( \\mathrm { i f ~ n o ~ s p l i t ~ } z _ { j } ^ { ( i ) } )\n$$", + "text_format": "latex", + "bbox": [ + 212, + 792, + 785, + 816 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We start from the last layer and derive linear bounds for each intermediate layer $z ^ { ( i ) }$ and $\\hat { z } ^ { ( i ) }$ with both constraints $x \\in { \\mathcal { C } }$ and $z \\in { \\mathcal { Z } }$ . We also assume that pre-ReLU bounds $\\mathbf { l } ^ { ( i ) } \\dot { \\leq } z ^ { ( i ) } \\leq \\mathbf { u } ^ { ( i ) }$ for each layer $i$ are available (see discussions in Sec. 3.3 on these intermediate layer bounds). We initially have: ", + "bbox": [ + 173, + 825, + 825, + 883 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/783898ff9f46ede6c5eb8ec66bf7ec3be45dc99c7bc0cd36a850147e935d99cb.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) = \\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } \\mathbf { W } ^ { ( L ) } \\hat { z } ^ { ( L - 1 ) } + \\mathbf { b } ^ { ( L ) } .\n$$", + "text_format": "latex", + "bbox": [ + 343, + 888, + 655, + 915 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Since $\\hat { z } ^ { ( L - 1 ) } = \\mathrm { R e L U } ( z ^ { ( L - 1 ) } )$ , we can apply Lemma $2 . 1$ to relax the ReLU neuron at layer $L - 1$ , and obtain a linear lower bound for $f ( x )$ w.r.t. $z ^ { ( L - 1 ) }$ (we omit all constant terms to avoid clutter): ", + "bbox": [ + 173, + 88, + 825, + 122 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7f3e4767dc90b93cbbd550a079de48ee524afc6a2f7d63fe4c469b49059544ef.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal C , z \\in \\mathcal Z } f ( x ) \\ge \\operatorname* { m i n } _ { x \\in \\mathcal C , z \\in \\mathcal Z } \\mathbf W ^ { ( L ) } \\mathbf D ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \\mathrm { c o n s t . }\n$$", + "text_format": "latex", + "bbox": [ + 318, + 126, + 678, + 152 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To enforce the split neurons at layer $L - 1$ , we use a Lagrange function with $\\beta ^ { ( L - 1 ) \\top } \\mathbf { S } ^ { ( L - 1 ) }$ multiplied on z(L\u00001): ", + "bbox": [ + 173, + 160, + 826, + 190 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4881df06ffd2068cdaff4c089ecd7ab147f7dae2b8d0d373dae121415de3e9fa.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\geq \\operatorname* { m i n } _ { \\bar { x } \\in \\mathcal { C } _ { \\mathcal { C } } ^ { ( L - 2 ) } \\in \\mathcal { L } ^ { ( L - 1 ) } \\geq 0 } \\operatorname* { m a x } _ { 0 ^ { ( L - 1 ) } \\geq 0 } \\mathbf { W } ^ { ( L ) } \\mathbf { D } ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) ^ { \\top } } \\mathbf { S } ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \\mathrm { c o n s t } } \\\\ { \\displaystyle \\qquad \\geq \\operatorname* { m a x } _ { \\beta ^ { ( L - 1 ) } \\geq 0 } \\operatorname* { m i n } _ { \\bar { x } \\in \\mathcal { Z } _ { \\mathcal { C } } ^ { ( L - 2 ) } \\in \\mathcal { Z } ( L - 2 ) } \\left( \\mathbf { W } ^ { ( L ) } \\mathbf { D } ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) ^ { \\top } } \\mathbf { S } ^ { ( L - 1 ) } \\right) z ^ { ( L - 1 ) } + \\mathrm { c o n s t } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 193, + 782, + 266 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The first inequality is due to the definition of the Lagrange function: we remove the constraint 2 Zduality. Due to the design of S(L\u00001), neuron split z (L\u00001)j \u0000 0 has a negative multiplier \u0000\u0000(L\u00001)j $z ^ { ( L - 1 ) } \\in \\mathcal { Z } ^ { ( \\bar { L } - 1 ) }$ and use a multiplier to replace this constraint. The second inequality is due to weak and split $z _ { j } ^ { ( L - 1 ) } < 0$ has a positive multiplier $\\beta _ { j } ^ { ( L - 1 ) }$ . Any $\\beta ^ { ( L - 1 ) } \\geq 0$ yields a lower bound for the constrained optimization problem. Then we substitute $z ^ { ( L - 1 ) }$ with $\\mathbf { W } ^ { ( L - 1 ) } \\hat { z } ^ { ( L - 2 ) } + \\mathbf { b } ^ { ( L - 1 ) }$ for next layer: ", + "bbox": [ + 173, + 268, + 826, + 368 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/5458721cd6ce8eab141e678de6118d20e86f08ba767e7090d0b22e9ae7b2ef9a.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\geq \\operatorname* { m a x } _ { \\beta ^ { ( L - 1 ) } \\geq 0 } \\operatorname* { m i n } _ { \\tilde { x } ^ { ( L - 2 ) } \\in \\mathcal { C } ( L - 2 ) } \\left( \\mathbf { W } ^ { ( L ) } \\mathbf { D } ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) ^ { \\top } } \\mathbf { S } ^ { ( L - 1 ) } \\right) \\mathbf { W } ^ { ( L - 1 ) } \\hat { z } ^ { ( L - 2 ) } + \\mathrm { c o n s t }\n$$", + "text_format": "latex", + "bbox": [ + 183, + 371, + 799, + 407 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We define a matrix $\\mathbf { A } ^ { ( i ) }$ to represent the linear relationship between $f ( x )$ and $\\hat { z } ^ { ( i ) }$ , where $\\mathbf { A } ^ { ( L - 1 ) } =$ $\\mathbf { W } ^ { ( L ) }$ according to Eq. $_ 4$ and $\\mathbf { A } ^ { ( L - 2 ) } = \\bigl ( \\mathbf { A } ^ { ( L - 1 ) } \\mathbf { D } ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) } { } ^ { \\top } \\mathbf { S } ^ { ( L - 1 ) } \\bigr ) \\mathbf { W } ^ { ( L - 1 ) }$ by Eq. 6. Considering 1-dimension output $f ( \\boldsymbol { x } ) , \\mathbf { A } ^ { ( i ) }$ has only 1 row. With $\\mathbf { A } ^ { ( L - 2 ) }$ , Eq. 6 becomes: ", + "bbox": [ + 173, + 412, + 826, + 463 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6ce4bd64f77b01732437fb947acb02645ff938fcf0d5deb3a877e08d82097e11.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\ge \\operatorname* { m a x } _ { \\beta ^ { ( L - 1 ) } \\ge 0 } \\operatorname* { m i n } _ { \\tilde { x } \\in \\mathcal { C } } \\quad \\mathbf { A } ^ { ( L - 2 ) } \\hat { z } ^ { ( L - 2 ) } + \\mathrm { c o n s t } ,\n$$", + "text_format": "latex", + "bbox": [ + 292, + 467, + 702, + 501 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "which is in a form similar to Eq. $\\boxed { 4 }$ except for the outer maximization over $\\beta ^ { ( L - 1 ) }$ . This allows the back-substitution process (Eq. 4, Eq. 5, and Eq. $\\textcircled { 6 }$ to continue. In each step, we swap max and min as in $\\mathrm { E q . } | \\overline { { 5 } } |$ so every maximization over $\\beta ^ { ( i ) }$ is outside of $\\operatorname* { m i n } _ { x \\in { \\mathcal { C } } }$ . Eventually, we have: ", + "bbox": [ + 173, + 506, + 826, + 553 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/b408a6b971acefbfee90b3ed9956949fec23c7985ad7a029e3b06dee82491cd5.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\ge \\operatorname* { m a x } _ { \\beta \\ge 0 } \\operatorname* { m i n } _ { x \\in \\mathcal { C } } { \\mathbf { A } ^ { ( 0 ) } } x + \\mathrm { c o n s t } ,\n$$", + "text_format": "latex", + "bbox": [ + 362, + 555, + 632, + 582 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where \u0000 := ⇥ \u0000(1)> \u0000(2)> \u0000(L\u00001)>⇤> concatenates all $\\beta ^ { ( i ) }$ vectors. Following the above idea, we present the main theorem in Theorem $3 . 1$ (proof is given in Appendix $\\mathbf { A } )$ . ", + "bbox": [ + 173, + 587, + 825, + 621 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 3.1 ( $\\beta$ -CROWN bound). Given an $L$ -layer NN $f ( \\boldsymbol { x } ) : \\mathbb { R } ^ { d _ { 0 } } \\mathbb { R }$ with weights $\\mathbf { W } ^ { ( i ) }$ , biases $\\mathbf { b } ^ { ( i ) }$ , pre-ReLU bounds $\\mathbf { l } ^ { ( i ) } \\le z ^ { ( i ) } \\le \\mathbf { u } ^ { ( i ) }$ ( $1 \\leq i \\leq L )$ ), input bounds $\\mathcal { C }$ , split constraints $\\mathcal { Z }$ . We have: ", + "bbox": [ + 173, + 623, + 825, + 655 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/fd96a8b77bd9fa6a18934b30aab702f30eb5fc34a2e8d953e5f3d011c59f42e4.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\ge \\operatorname* { m a x } _ { \\beta \\ge 0 } \\operatorname* { m i n } _ { x \\in \\mathcal { C } } ( \\pmb { a } + \\mathbf { P } \\beta ) ^ { \\top } x + \\mathbf { q } ^ { \\top } \\beta + c ,\n$$", + "text_format": "latex", + "bbox": [ + 325, + 664, + 671, + 690 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\pmb { a } \\in \\mathbb { R } ^ { d _ { 0 } } , \\mathbf { P } \\in \\mathbb { R } ^ { d _ { 0 } \\times ( \\sum _ { i = 1 } ^ { L - 1 } d _ { i } ) } , \\mathbf { q } \\in \\mathbb { R } ^ { \\sum _ { i = 1 } ^ { L - 1 } d _ { i } }$ and $c \\in \\mathbb { R }$ are functions of $\\mathbf { W } ^ { ( i ) } , \\mathbf { b } ^ { ( i ) } , \\mathbf { l } ^ { ( i ) } , \\mathbf { u } ^ { ( i ) } .$ ", + "bbox": [ + 174, + 693, + 820, + 713 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Detailed formulations for ${ \\mathbf { } } _ { { \\mathbf { } } _ { a , { \\mathbf { P } } } }$ , $\\mathbf { q }$ and $c$ are given in Appendix $\\boxed { \\mathrm { A } }$ Theorem $3 . 1$ shows that when neuron split constraints exist, $f ( x )$ can still be bounded by a linear equation containing optimizable multipliers $\\beta$ . Observing Eq. 5, the main difference between CROWN and $\\beta$ -CROWN lies in the relaxation of each ReLU layer, where we need an extra term $\\beta ^ { ( i ) \\top } \\mathbf { S } ^ { ( i ) }$ in the linear relationship matrix (for example, $\\mathbf { W } ^ { ( L ) } \\dot { \\mathbf { D } ^ { ( L - 1 ) } }$ in Eq. 5) between $f ( x )$ and $z ^ { ( i ) }$ to enforce neuron split constraints. This extra term in every ReLU layer yields $\\mathbf { P }$ and $\\mathbf { q }$ in Eq. $\\perp$ after bound propagations. ", + "bbox": [ + 173, + 719, + 826, + 810 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To solve the optimization problem in Eq. $^ { 7 , }$ we note that in the $\\ell _ { p }$ norm robustness setting ${ \\mathcal { C } } =$ $\\{ x \\mid \\| x - x _ { 0 } \\| _ { p } \\leq \\epsilon \\} )$ ), the inner minimization has a closed solution: ", + "bbox": [ + 169, + 813, + 823, + 843 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/422ee7224a3dbb203b63d7712774736a3bc40020eb3447d3d0f760a44749773f.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in { \\mathcal { C } } , z \\in { \\mathcal { Z } } } f ( x ) \\geq \\operatorname* { m a x } _ { \\beta \\geq 0 } - \\| a + \\mathbf { P } \\beta \\| _ { q } \\epsilon + ( \\mathbf { P } ^ { \\top } x _ { 0 } + \\mathbf { q } ) ^ { \\top } \\beta + a ^ { \\top } x _ { 0 } + c : = \\operatorname* { m a x } _ { \\beta \\geq 0 } g ( \\beta )\n$$", + "text_format": "latex", + "bbox": [ + 225, + 847, + 771, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\textstyle { \\frac { 1 } { p } } + { \\frac { 1 } { q } } = 1$ . The maximization is concave in $\\beta ( q \\geq 1 )$ , so we can simply optimize it using projected (super)gradient ascent with gradients from an automatic differentiation library. Since any $\\beta \\geq 0$ yields a valid lower bound for $\\scriptstyle \\operatorname* { m i n } _ { x \\in { \\mathcal { C } } , z \\in { \\mathcal { Z } } } f ( x )$ , convergence is not necessary to guarantee soundness. $\\beta$ -CROWN is efficient - it has the same asymptotic complexity as CROWN when $\\beta$ is fixed. When $\\beta = 0$ , $\\beta$ -CROWN yields the same results as CROWN; however the additional optimizable $\\beta$ allows us to maximize and tighten the lower bound due to neuron split constraints. ", + "bbox": [ + 174, + 880, + 825, + 912 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 825, + 147 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We define $\\pmb { \\alpha } ^ { ( i ) } \\in \\mathbb { R } ^ { d _ { i } }$ for free variables associated with unstable ReLU neurons in Lemma $\\boxed { 2 . 1 }$ for layer $i$ 2 and define all free variables ${ \\pmb { \\alpha } } = \\{ { \\pmb { \\alpha } } ^ { ( 1 ) } \\cdot \\cdot \\cdot { \\pmb { \\alpha } } ^ { ( L - 1 ) } \\}$ . Since any $0 \\mathop { \\leq } \\alpha _ { j } ^ { ( i ) } \\mathop { \\leq } 1$ yields a valid bound, we can optimize it to tighten the bound, similarly as done in $\\lVert \\rVert \\dot { \\boldsymbol { \\mathrm { E } } } \\rVert$ . Formally, we rewrite Eq. $8$ with $_ { \\pmb { \\alpha } }$ explicitly: ", + "bbox": [ + 173, + 154, + 825, + 210 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/516eec9c5309da66e6f5308c46c24814291909930d575bdf06700cb77aa9a0d9.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\geq \\operatorname* { m a x } _ { 0 \\leq \\alpha \\leq 1 , \\beta \\geq 0 } g ( \\alpha , \\beta ) .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 208, + 624, + 233 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 Connections to the Dual Problem ", + "text_level": 1, + "bbox": [ + 174, + 247, + 444, + 263 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this subsection, we show that $\\beta$ -CROWN can also be derived from a dual LP problem. Based on Eq. $^ { 1 } _ { . }$ and linear relaxations in Section $\\bigstar ,$ we first construct an LP problem for $\\ell _ { \\infty }$ robustness verification in Eq. 10 where $i \\in \\{ 1 , \\cdots , L - 1 \\}$ . ", + "bbox": [ + 173, + 272, + 826, + 316 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$\\operatorname* { m i n } f ( x ) : = z ^ { ( L ) } ( x )$ s.t. \nNetwork and Input Bounds: $z ^ { ( i ) } = \\mathbf { W } ^ { ( i ) } \\hat { z } ^ { ( i - 1 ) } + \\mathbf { b } ^ { ( i ) } ; \\hat { z } ^ { ( 0 ) } \\geq x _ { 0 } - \\epsilon ; \\hat { z } ^ { ( 0 ) } \\leq x _ { 0 } + \\epsilon ;$ \nStable ReLUs: $\\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) }$ (if $1 _ { j } ^ { ( i ) } \\geq 0$ ); $\\hat { z } _ { j } ^ { ( i ) } = 0$ (if $\\begin{array} { r } { \\mathbf { u } _ { j } ^ { ( i ) } \\le 0 . } \\end{array}$ ); \nUnstable: $\\begin{array} { r } { \\hat { z } _ { j } ^ { ( i ) } \\ge 0 , \\hat { z } _ { j } ^ { ( i ) } \\ge z _ { j } ^ { ( i ) } , \\hat { z } _ { j } ^ { ( i ) } \\le \\frac { \\mathbf { u } _ { j } ^ { ( i ) } } { \\mathbf { u } _ { j } ^ { ( i ) } - \\mathbf { 1 } _ { j } ^ { ( i ) } } \\left( z _ { j } ^ { ( i ) } - \\mathbf { 1 } _ { j } ^ { ( i ) } \\right) ( \\mathrm { i f } \\mathbf { l } _ { j } ^ { ( i ) } < 0 < \\mathbf { u } _ { j } ^ { ( i ) } , j \\notin \\mathcal { Z } ^ { + ( i ) } \\cup \\mathcal { Z } ^ { - ( i ) } ) } \\end{array}$ \nNeuron Split Constraints: $\\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) } , z _ { j } ^ { ( i ) } \\geq 0$ (if $j \\in \\mathcal { Z } ^ { + ( i ) }$ ); ˆz (i)j = 0, z(i)j < 0 (if j 2 Z \u0000(i) ) ", + "bbox": [ + 173, + 323, + 828, + 445 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Compared to the formulation in $\\mathbb { \\lVert \\rVert }$ , we have neuron split constraints. Many BaB based complete verifiers $\\mathbb { B } \\mathbb { B } \\mathbb { 3 } \\mathbb { I }$ use an LP solver for Eq. $\\boxed { 1 0 }$ as the incomplete verifier. We first show that it is possible to derive Theorem $3 . 1$ from the dual of this LP, leading to Theorem 3.2: ", + "bbox": [ + 174, + 464, + 828, + 508 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3.2. The objective $d _ { L P }$ for the dual problem of Eq. 10 can be represented as ", + "bbox": [ + 178, + 511, + 738, + 527 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/2a36bdf8f2ad41bafb23bfdf80952dffaa574b59fd39c50715d90166d950287b.jpg", + "text": "$$\nd _ { L P } = - \\| \\pmb { a } + \\mathbf { P } \\pmb { \\beta } \\| _ { 1 } { \\cdot } \\epsilon + ( \\mathbf { P } ^ { \\top } x _ { 0 } + \\mathbf { q } ) ^ { \\top } \\pmb { \\beta } + \\pmb { a } ^ { \\top } x _ { 0 } + c ,\n$$", + "text_format": "latex", + "bbox": [ + 316, + 532, + 679, + 551 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where ${ \\mathbf { } } _ { { \\mathbf { } } _ { a , { \\mathrm { ~ P ~ } } } }$ , q and c are defined in the same way as in Theorem $3 . { { I } _ { : } }$ and $\\beta \\geq 0$ corresponds to the dual variables of neuron split constraints in Eq. $\\boxed { I O }$ ", + "bbox": [ + 173, + 558, + 825, + 588 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "A similar connection between CROWN and dual LP based verifier $[ \\mathbb { A } 2 ]$ was shown in $\\pmb { \\mathbb { B } } 0 \\|$ , and their results can be seen as a special case of ours when $\\beta = 0$ (none of the split constraints are active). An immediate consequence is that $\\beta$ -CROWN can potentially solve Eq. 10 as well as using an LP solver: ", + "bbox": [ + 173, + 597, + 826, + 641 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Corollary 3.2.1. When $_ { \\pmb { \\alpha } }$ and $\\beta$ are optimally set and intermediate bounds l, u are fixed, $\\beta$ -CROWN produces $p _ { L P } ^ { * }$ , the optimal objective of $L P$ with split constraints in Eq. 10: ", + "bbox": [ + 174, + 643, + 823, + 674 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/5f6ef334a6b39408bf0eb7a0b69591924252c8ddb842d5daf34df7ff146d49e0.jpg", + "text": "$$\n\\operatorname* { m a x } _ { 0 \\le \\alpha \\le 1 , \\beta \\ge 0 } g ( \\alpha , \\beta ) = p _ { L P } ^ { * } ,\n$$", + "text_format": "latex", + "bbox": [ + 408, + 679, + 588, + 704 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Appendix $\\boxed { \\mathrm { A } }$ we give detailed formulations for conversions between the variables $_ { \\pmb { \\alpha } }$ , $\\beta$ in $\\beta$ - CROWN and their corresponding dual variables in the LP problem. ", + "bbox": [ + 168, + 718, + 826, + 747 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 Joint Optimization of Free Variables in $\\beta$ -CROWN ", + "text_level": 1, + "bbox": [ + 173, + 756, + 568, + 771 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In Eq. 9, $g$ is also a function of $\\mathbf { l } _ { j } ^ { ( i ) }$ and $\\mathbf { u } _ { j } ^ { ( i ) }$ , the intermediate layer bounds for each neuron z(i) . They are also computed using $\\beta$ -CROWN. To obtain $\\mathbf { l } _ { j } ^ { ( i ) }$ , we set $f ( x ) : = z _ { j } ^ { ( i ) } ( x )$ and apply Theorem $3 . 1$ : ", + "bbox": [ + 173, + 775, + 825, + 815 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/dbc9bb849d61d1e67f11847bb7e5d647f6f8cd792617342f8085722f43f2268a.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\substack { x \\in \\mathcal { C } , \\tilde { z } ^ { ( i - 1 ) } \\in \\tilde { \\mathcal { Z } } ^ { ( i - 1 ) } } } z _ { j } ^ { ( i ) } ( x ) \\ge \\operatorname* { m a x } _ { 0 \\le \\alpha ^ { \\prime } \\le 1 , \\beta ^ { \\prime } \\ge 0 } g ^ { \\prime } ( \\alpha ^ { \\prime } , \\beta ^ { \\prime } ) : = \\mathbb { I } _ { j } ^ { ( i ) }\n$$", + "text_format": "latex", + "bbox": [ + 305, + 821, + 692, + 852 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For computing u(i) we simply set $f ( x ) : = - z _ { j } ^ { ( i ) } ( x )$ . Importantly, during solving these intermediate layer bounds, the $\\mathbf { { \\boldsymbol { \\alpha } } ^ { \\prime } }$ and $\\beta ^ { \\prime }$ are independent sets of variables, not the same ones for the objective $f ( x ) : = z ^ { ( L ) }$ . Since $g$ is a function of $\\mathbf { l } _ { j } ^ { ( i ) }$ , it is also a function of $\\alpha ^ { \\prime }$ and $\\beta ^ { \\prime }$ . In fact, there are a total of $\\textstyle \\sum _ { i = 1 } ^ { L - 1 } d _ { i }$ intermediate layer neurons, and each neuron is associated with a set of independent $\\mathbf { { \\boldsymbol { \\alpha } } ^ { \\prime } }$ and $\\beta ^ { \\prime }$ variables. Optimizing these variables allowing us to tighten the relaxations on unstable ReLU neurons (which depend on $\\mathbf { l } _ { j } ^ { ( i ) }$ and $\\mathbf { u } _ { j } ^ { ( i ) }$ ) and produce tight final bounds, which is impossible in LP. In other words, we need to optimize $\\hat { \\alpha }$ and $\\hat { \\beta }$ , which are two vectors concatenating $\\alpha , \\beta$ as well as a large number of $\\mathbf { { \\boldsymbol { \\alpha } } ^ { \\prime } }$ and $\\beta ^ { \\prime }$ used to compute each intermediate layer bound: ", + "bbox": [ + 174, + 861, + 825, + 914 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 88, + 825, + 169 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/91c26298ee356f3c64643fc6cb6013983c83e08211032e5710172d20d97134d8.jpg", + "text": "$$\n\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\ge \\operatorname* { m a x } _ { 0 \\le \\hat { \\alpha } \\le 1 , \\hat { \\beta } \\ge 0 } g ( \\hat { \\alpha } , \\hat { \\beta } ) .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 174, + 624, + 203 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "This formulation is non-convex and has a large number of variables. Since any $0 \\leq \\hat { \\alpha } \\leq 1$ , ${ \\hat { \\beta } } \\geq 0$ leads to a valid lower bound, the non-convexity does not affect soundness. When intermediate layer bounds are also allowed to be tightened during optimization, we can outperform the LP verifier for Eq. $\\textcircled { 1 0 }$ using fixed intermediate layer bounds. Typically, in many previous works $\\boxed { 8 } \\boxed { 2 3 } \\boxed { 6 }$ , when the LP formulation Eq. 10 is formed, intermediate layer bounds are pre-computed with bound propagation procedures $\\boxed { 8 } \\boxed { 2 3 }$ , which are far from optimal. To estimate the dimension of this problem, we denote the number of unstable neurons at layer $i$ as $s _ { i } : = \\operatorname { T r } ( | \\mathbf { S } ^ { ( i ) } | )$ . Each neuron in layer $i$ is associated with $2 \\times \\textstyle \\sum _ { k = 1 } ^ { i - 1 } s _ { k }$ variables $\\mathbf { { \\boldsymbol { \\alpha } } ^ { \\prime } }$ . Suppose each hidden layer has $d$ neurons $( s _ { i } = { O } ( d ) )$ , then ↵ˆ has 2 ⇥ PL\u00001i=1 d $\\begin{array} { r } { 2 \\times \\sum _ { i = 1 } ^ { L - 1 } d _ { i } \\sum _ { k = 1 } ^ { i - 1 } s _ { k } = O ( L ^ { 2 } d ^ { 2 } ) } \\end{array}$ variables in total. This can be too large for efficientintermediate neurons of the same layer, leading to a $\\mathbf { { \\boldsymbol { \\alpha } } ^ { \\prime } }$ $\\beta ^ { \\prime }$ total number of $O ( L ^ { 2 } d )$ variables to optimize. Note that a weaker form of joint optimization was also discussed in $\\lVert \\boldsymbol { \\mathsf { \\overline { { 4 5 } } } } \\rVert$ without $\\beta$ , and a detailed analysis can be found in Appendix B.2. ", + "bbox": [ + 173, + 209, + 826, + 386 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.4 $\\beta$ -CROWN with Branch and Bound ( $\\beta$ -CROWN BaB) ", + "text_level": 1, + "bbox": [ + 173, + 400, + 589, + 415 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We perform complete verification following BaB framework $\\pmb { \\bigtriangledown }$ using $\\beta$ -CROWN as the incomplete solver, and we use simple branching heuristics like BaBSR $\\mathbb { U }$ or FSB $\\mathbb { \\ m }$ . To efficiently utilize GPU, we also use batch splits to evaluate multiple subdomains in the same batch as in [44, 10]. We list our full algorithm $\\beta$ -CROWN BaB in Appendix B and we show it is sound and complete here: ", + "bbox": [ + 174, + 419, + 825, + 476 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 3.3. $\\beta$ -CROWN with Branch and Bound on splitting ReLUs is sound and complete. ", + "bbox": [ + 174, + 478, + 784, + 493 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Soundness is trivial because $\\beta$ -CROWN is a sound verifier. For completeness, it suffices to show that when all unstable ReLU neurons are split, $\\beta$ -CROWN gives the global minimum for Eq. 10. In contrast, combining CROWN $\\left[ \\left[ 4 6 \\right] \\right]$ with BaB does not yield a complete verifier, as it cannot detect infeasible splits and a slow LP solver is still needed to guarantee completeness $\\lVert \\rVert \\dot { \\boldsymbol { \\mathrm { E } } } \\rVert$ . Instead, $\\beta$ -CROWN can detect infeasible subdomains - according to duality theory, an infeasible primal problem leads to an unbounded dual objective, which can be detected (see Sec. B.3 for more details). ", + "bbox": [ + 174, + 502, + 825, + 587 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Additionally, we show the potential of early stopping a complete verifier as an incomplete verifier. BaB approaches the exact solution of Eq. $^ 1$ by splitting the problem into multiple subdomains, and more subdomains give a tighter lower bound for Eq. $^ { 1 . }$ Unlike traditional complete verifiers, $\\beta$ -CROWN is efficient to explore a large number of subdomains during a very short time, making $\\beta$ -CROWN BaB an attractive solution for efficient incomplete verification. ", + "bbox": [ + 174, + 593, + 826, + 662 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 Experimental Results ", + "text_level": 1, + "bbox": [ + 174, + 678, + 387, + 694 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.1 Comparison to Complete Verifiers ", + "text_level": 1, + "bbox": [ + 176, + 699, + 450, + 713 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate complete verification performance on dataset provided in $\\mathbb { \\left[ 2 3 , \\overline { { 1 0 } } \\right] }$ and used in VNNCOMP 2020 $\\vec { \\mathbb { B } 2 2 } \\rvert \\vec { | }$ . The benchmark contains three CIFAR-10 models (Base, Wide, and Deep) with 100 examples each. Each data example is associated with an $\\ell _ { \\infty }$ norm $\\epsilon$ and a target label for verification (referred to as a property to verify). The details of neural network structures and experimental setups can be found in Appendix $\\mathbf { C } .$ We compare against multiple baselines for complete verification: (1) BaBSR [7], a basic BaB and LP based verifier; (2) MIPplanet $\\mathbb { \\lVert \\rVert \\cdot \\rVert }$ , a customized MIP solver for NN verification where unstable ReLU neurons are randomly selected for splitting; (3) ERAN [35, 33, 36, 34], an abstract interpretation based verifier which performs well on this benchmark in VNN-COMP 2020; (4) GNN-Online $\\mathbb { \\left[ \\left. 2 3 \\right| \\right. }$ , a BaB and LP based verifiers using a learned Graph Neural Network (GNN) to guide the ReLU splits; (5) $\\mathrm { B D D + }$ BaBSR $\\pmb { \\Vert 6 \\Vert }$ , a verification framework based on Lagrangian decomposition on GPUs $\\mathrm { ( B D D + ) }$ ) with BaBSR branching strategy; (6) OVAL (BDD+ GNN) [6, 23], a strong verifier in VNN-COMP 2020 using $\\mathrm { B D D + }$ with GNN guiding the ReLU splits; (7) A.set BaBSR and (8) Big-M+A.set BaBSR $\\mathbb { m }$ , very recent dual-space verifiers on GPUs with a tighter linear relaxation than triangle LP relaxations; (9) Fast-and-Complete [45], which uses CROWN (LiRPA) on GPUs as the incomplete verifier in BaB without neuron split constraints; (10) BaDNB $\\mathrm { B D D + }$ FSB) $\\mathbb { \\ m }$ , a concurrent state-of-the-art complete verifier, using $\\mathrm { B D D + }$ on GPUs with FSB branching strategy. $\\beta$ -CROWN BaB can use either BaBSR or FSB branching heuristic, and we include both in evaluation. All methods use a 1 hour timeout threshold. ", + "bbox": [ + 173, + 717, + 826, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/477b6510373451cf02a1f5f4e97f98c796c4d61ab4254848eebc02c0df474832.jpg", + "table_caption": [ + "Table 1: Average runtime and average number of branches on three CIFAR-10 models over 100 properties. " + ], + "table_footnote": [], + "table_body": "
CIFAR-10 BaseCIFAR-10 WideCIFAR-10 Deep
Methodtime(s)branches%timeouttime(s)branches%timeouttime(s)branches%timeout
BaBSR2367.781020.5536.002871.14812.6549.002750.75401.2839.00
MIPplanet 152849.69-68.002417.53-46.002302.25140.00
ERAN*[35333634805.94=5.00632.20=9.00545.72=0.00
GNN-online [231794.85565.1333.001367.38372.7415.001055.33131.854.00
BDD+ BaBSR 6807.91195480.1420.00505.6574203.1110.00266.2812722.744.00
OVAL (BDD+ GNN)*[623]662.1767938.3816.00280.3817895.946.0094.691990.341.00
A.set BaBSR10381.7812004.607.00165.912233.103.00190.282491.552.00
BigM+A.setBaBSR[10390.4411938.757.00172.654050.593.00177.223275.252.00
Fast-and-Complete [45]695.01119522.6517.00495.8880519.859.00105.642455.111.00
BaDNB (BDD+ FSB)II309.2938239.047.00165.5311214.444.0010.50368.160.00
β-CROWNBaBSR226.06509608.506.00118.26217691.243.006.12204.660.00
β-CROWN FSB118.23208018.213.0078.32116912.572.005.6941.120.00
", + "bbox": [ + 169, + 104, + 828, + 258 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/e15f89e6f30fcd3a3be38f99210245d62207d65a1866c03408e05d561e72706d.jpg", + "image_caption": [ + "OVAL (BDD+ GNN) and ERAN results are from VNN-COMP 2020 report [22]. Other results were reported by their authors. ", + "Figure 1: Percentage of solved properties with growing running time. $\\beta$ -CROWN FSB (light green) and $\\beta$ -CROWN BaBSR (dark green) clearly lead in all 3 settings and solve over $90 \\%$ properties within 10 seconds. " + ], + "image_footnote": [], + "bbox": [ + 178, + 276, + 810, + 375 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 410, + 825, + 465 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We report the average verification time and branch numbers in Table $^ 1$ and plot the percentage of solved properties over time in Figure 1. $\\beta$ -CROWN FSB achieves the fastest average running time compared to all other baselines with minimal timeouts, and also clearly leads on the cactus plot. When using a weaker branching heuristic, $\\beta$ -CROWN BaBSR still outperforms all literature baselines, including very recent ones such as A.set BaBSR $\\mathbb { \\ m }$ , Fast-and-Complete $\\lVert \\boldsymbol { \\mathsf { A } } \\boldsymbol { \\mathsf { S } } \\rVert$ and BaDNB [11]. Our benefits are more clearly shown in Figure $\\bigstar \\bigstar$ where we solve over $90 \\%$ examples under 10s and most other verifiers can verify much less or none of the properties within 10s. We see a 2 to 3 orders of magitudes speedup in Figure 1 compared to CPU based verifiers such as MIPplanet and BaBSR. ", + "bbox": [ + 173, + 472, + 825, + 583 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 Comparison to Incomplete Verifiers ", + "text_level": 1, + "bbox": [ + 176, + 593, + 462, + 607 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Verified accuracy. In Table $\\lfloor 2 \\rfloor$ we compare against a few representative and strong incomplete verifiers on 5 convolutional networks and 4 MLP networks for MNIST and CIFAR-10 under the same set of 1000 images and perturbation $\\epsilon$ as reported in [35, 37, 26]. Among the baselines, kPoly [35], OptC2V $ { \\mathbb { I } } ^ { \\smash { \\sum } }$ and PRIMA $\\left[ \\left[ 2 6 \\right] \\right]$ utilize state-of-the-art multi-neuron linear relaxation for ReLUs and can bypass the single-neuron convex relaxation barrier $\\pmb { \\mathbb { B } } \\pmb { \\mathrm { O } } \\|$ , and are among the strongest incomplete verifiers. $\\beta$ -CROWN FSB achieves better verified accuracy on all models using a similar or less amount of time. Some models, such as MNIST ConvBig and CIFAR ConvBig, are quite challenging - the verified accuracy obtained by $\\beta$ -CROWN FSB is close to the upper bound found via PGD attack. ", + "bbox": [ + 173, + 611, + 826, + 723 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "To make more comprehensive evaluations, in Table $\\textcircled { 3 }$ we further compare against a state-of-the-art semidefinite programming (SDP) based verifier, SDP-FO $\\pmb { \\mathbb { Q } } \\|$ , on one MNIST and six CIFAR-10 models reported in their paper. The models were trained using adversarial training, which posed a challenge for verification $[ [ 2 8 ] ]$ . The SDP formulation can be tighter than linear relaxation based ones, but it is computationally expensive - SDP-FO takes 2 to 3 hours to converge on one GPU for verifying a single property, resulting 5,400 GPU hours to verify 200 testing images with 10 labels each. Due to resource limitations, we directly quote SDP-FO results from $\\blacktriangleleft$ on the same set of models. We evaluate verified accuracy on the same set of 200 test images for other baselines. We include a concurrent work PRIMA $\\pmb { \\mathbb { Z } } \\pmb { 6 } \\|$ , the strongest multi-neuron linear relaxation baseline in Table 2, which generally outperforms kPoly and OptC2V. Table $\\textcircled { 3 }$ shows that overall we are 3 orders of magnitude faster than SDP-FO while still achieving consistently higher verified accuracy on average. ", + "bbox": [ + 173, + 728, + 825, + 881 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Tightness of verification. In Figure $\\bigstar ,$ we compare the tightness of verification bounds against SDP-FO on two adversarially trained networks from $\\left[ \\left[ 9 \\right] \\right]$ . Specifically, we use the verification objective $f ( x ) : = z _ { y } ^ { ( L ) } ( x ) - z _ { y ^ { \\prime } } ^ { ( L ) } ( x ) ,$ , where $z ^ { ( L ) }$ is the logit layer output, $y$ and $y ^ { \\prime }$ are the true label and the runner-up label. For each test image, a 200-step PGD attack $\\mathbb { [ 2 4 ] }$ provides an adversarial upper bound $\\overline { { f } }$ of the optimal objective: $f ^ { * } \\leq { \\overline { { f } } }$ . Verifiers, on the other hand, can provide a verified lower bound $f \\leq f ^ { * }$ . Bounds from tighter verification methods lie closer to line $y = x$ in Figure $\\bigtriangledown$ Figure $\\left. 2 \\right.$ shows that on both PGD adversarially trained networks, $\\beta$ -CROWN FSB consistently outperforms SDP-FO for all 100 random test images. Importantly, for each point on the plots, $\\beta$ -CROWN FSB needs 3 minutes while SDP-FO needs 178 minutes on average. LP verifier with triangle relaxations produces much looser bounds than $\\beta$ -CROWN FSB and SDP-FO. Additional results are in Appendix C.2. ", + "bbox": [ + 174, + 882, + 821, + 911 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/1305a4b4ae42f032f1df94217c7b24a31fdd6452a5b982db29e85de1fc02a9b1.jpg", + "table_caption": [ + "Table 2: Verified accuracy $( \\% )$ and avg. time (s) of 1000 images evaluated on the ERAN models in [35, 37, 26]. kPoly, OptC2V and PRIMA are strong incomplete verifiers that can break the convex relaxation barrier [30]. The average time reported by us excludes examples that are classified incorrectly. " + ], + "table_footnote": [ + "\\* CROWN/DeepPoly evaluated on CPU. † PRIMA is a concurrent work and its results are from [26] (Oct 26, 2021 version), except that ResNet results are from personal communications with the authors due to a different input normalization used. ‡ Because these MLP models are fairly small, some of their intermediate layer bounds are computed by mixed integer programming (MIP) using $80 \\%$ time budget before branch and bound starts and $\\beta$ -CROWN FSB is used during the branch and bound process. We find that tighter intermediate bounds by MIP is beneficial for these small MLP models. " + ], + "table_body": "
DatasetModel (Same settings as 35.37261)CROWN/DeepPoly*1361 Verified%Time (s)kPoly 因 Ver.% Time(s)OptC2V Ver.%四 Time(s)PRIMA+ 园 Ver.% Time(s)β-CROWN FSB Ver.% Time(s)Upper bound
MNISTMLP5×100‡16.00.744.1307 42.913751.015969.9 10284.2
MLP8×10018.21.436.9 17138.475942.8301 62.010382.0
MLP5×20029.22.457.4 18760.140369.0 22477.48690.1
MLP8× 20025.95.650.6 46452.8345162.4 39573.59591.1
ConvSmall15.8334.7 47743.65559.872.77.073.2
ConvBig71.12173.6 4077.110277.542 11 79.33.180.4
CIFARConvSmall35.9439.986 39.810544.613 46.36.848.1
ConvBig42.14345.9 346No public code48.3176 51.615.355.0
ResNet24.1124.591cannot run24.81.724.8 1.624.8
", + "bbox": [ + 176, + 136, + 821, + 247 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/7be7195944088fdbcf2329920588fad8ef79ac3acefff7e1a563f6a8434c0eff.jpg", + "table_caption": [ + "Table 3: Verified accuracy $( \\% )$ and avg. per-example verification time (s) on 7 models from SDP-FO [9] CROWN/DeepPoly are fast but loose bound propagation based methods, and they cannot be improved with more running time. SDP-FO uses stronger semidefinite relaxations, which can be very slow and sometimes has convergence issues. PRIMA, a concurrent work, is the state-of-the-art relaxation barrier breaking method; we did not include kPoly and $\\mathrm { O p t C 2 V }$ because they are weaker than PRIMA (see Table 2). " + ], + "table_footnote": [ + "SDP-FO results are directly from their paper due to its very long running time (>20h per example). † PRIMA experiments were done using commi ${ \\widehat { 3 9 6 \\mathrm { d c } 7 \\mathrm { a } } } ,$ released on June 4, 2021. PRIMA and $\\beta$ -CROWN FSB results are on the same set of 200 examples (first 200 examples of CIFAR-10 dataset) and we don’t run verifiers on examples that are classified incorrectly or can be attacked by a 200-step PGD. $\\beta$ -CROWN uses 1 GPU and 1 CPU; PRIMA uses 1 GPU and 20 CPUs. " + ], + "table_body": "
DatasetModel ∈= 0.3 and ε= 2/255CROWN/DeepPoly Verified%SDP-FO*PRIMA [26]β-CROWN FSBUpper bound
MNISTTime (s)Ver.%Time(s)Ver.%Time(s)Ver.%Time(s)
CIFARCNN-A-Adv1.00.143.4>20h44.5135.970.521.176.5
CNN-B-Adv21.50.532.8>25h38.0343.646.532.265.0
CNN-B-Adv-443.50.946.0>25h53.543.854.011.663.5
CNN-A-Adv35.50.639.6>25h41.54.844.05.850.0
CNN-A-Adv-441.50.740.0>25h45.04.946.05.649.5
CNN-A-Mix CNN-A-Mix-423.5 38.00.4 0.539.6 47.8>25h >25h37.5 48.534.3 7.041.5 50.549.6 5.953.0 57.5
", + "bbox": [ + 191, + 354, + 805, + 460 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 508, + 825, + 626 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "VNN-COMP 2021 results. We encourage the readers to checkout the report of the Second International Verification of Neural Networks Competition (VNN-COMP 2021) $\\pmb { \\mathbb { B } } \\|$ with 9 additional benchmarks and 12 competing methods evaluated in a standardized testing environment on AWS. Our entry $\\alpha , \\beta$ -CROWN is based on the $\\beta$ -CROWN algorithm in this work and uses the same codebase. ", + "bbox": [ + 174, + 633, + 826, + 689 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 703, + 323, + 718 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Many early complete verifiers for neural networks relied on existing solvers such as MILP or SMT solvers [20, 15, 19, 12, 38] and were limited to very small problem instances. Branch and bound (BaB) ", + "bbox": [ + 174, + 722, + 826, + 750 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/f268324c95f4c9216eb4f87162c23c7018811152e56851544933396dfb737acb.jpg", + "image_caption": [ + "Figure 2: Verified lower bound v.s. PGD adversarial upper bound. A lower bound closer to the upper bound (closer to the line $y = x$ ) is better. $\\beta$ -CROWN FSB uses 3mins while SDP-FO needs 2 to 3 hours per point. " + ], + "image_footnote": [], + "bbox": [ + 210, + 757, + 782, + 878 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "based method was proposed to better exploit the network structure using LP-based incomplete verifier for bounding and ReLU splits for branching [8, 40, 23, 5]. Besides branching on ReLU neurons, input domain branching was also considered in [41, 29, 1] but limited by input dimensions $\\pmb { \\mathbb { B } } ] \\mathbf l$ . ", + "bbox": [ + 176, + 92, + 823, + 133 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Recently, a few approaches have been proposed to use efficient iterative solvers or bound propagation methods on GPUs without relying on LP solvers. Bunel et al. [6] decomposed the verification problem layer by layer, solved each layer in a closed form on GPUs, and used Lagrangian to enforce consistency between layers. However, their formulation only has the same power as LP and needs many iterations to converge. De Palma et al. $\\mathbb { \\ m }$ used a dual-space verifier with a linear relaxation [2, 37] tighter than triangle LP relaxation, but in most settings the extra computational costs and difficulties for an efficient implementation offset its benefits (more discussions in section $\\mathbf { B } . 2 )$ . A concurrent work BaDNB [11] proposed a new branching strategy, filtered smart branching (FSB), combined with Lagrangian decomposition to get better verification performance. Xu et al. $\\pm \\boxed { \\boxed { 4 4 } }$ used CROWN as a massively paralleled incomplete solver on GPUs for complete verification, but it cannot handle neuron split constraints, leading to suboptimal efficiency and high timeout rates. ", + "bbox": [ + 174, + 140, + 825, + 291 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "For incomplete verification, Salman et al. $\\textcircled { \\lvert 3 0 \\rvert }$ shows the inherent limitation of using per-neuron convex relaxations for verification problems. Singh et al. $\\pmb { \\Vert 3 5 \\Vert }$ and Müller et al. $[ [ 2 6 ] ]$ broke this barrier by considering constraints involving multiple ReLU neurons; Tjandraatmadja et al. $\\pmb { \\Vert 3 7 } \\Vert$ proposed to relax a linear layer with a ReLU neuron together using a strong mixed-integer programming formulation $\\mathbb { M }$ . SDP based relaxations $[ \\sqrt { 2 8 } , \\sqrt { 1 6 } , \\sqrt { 1 4 } ]$ typically produce tight bounds but with significantly higher cost. The most recent GPU based SDP verifier $\\bigstar \\bigstar$ is still relatively slow and can take 2 hours to verify a single image. In this work, we impose neuron split constraints using $\\beta$ -CROWN and combine it with branch and bound done in parallel on GPUs. Although for each subdomain in BaB, $\\beta$ -CROWN is still subject to the convex relaxation barrier, the efficiency of $\\beta$ -CROWN BaB allows it to quickly explore a very large number of subdomains and outperform existing convex barrier breaking incomplete verifiers under many scenarios in both runtime and tightness. ", + "bbox": [ + 174, + 297, + 825, + 450 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Additionally, another line of works train networks to enhance verified accuracy, typically using cheap incomplete verifiers at training time [42, 39, 25, 43, 18, 25, 47, 4, 32]. Traditionally only these verification-customized networks can have reasonable verified accuracy, while $\\beta$ -CROWN BaB can also give non-trivial verified accuracy on relatively large networks agnostic to verification. ", + "bbox": [ + 174, + 457, + 825, + 512 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 525, + 299, + 541 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We proposed $\\beta$ -CROWN, a new bound propagation method that can fully encode the neuron split constraints introduced in BaB, which clearly leads in both complete and incomplete verification settings. The success of $\\beta$ -CROWN comes from a few factors: (1) In Section $\\boxed { 3 . 1 }$ we show that $\\beta$ - CROWN is an GPU-friendly bound propagation algorithm significantly faster than $\\bar { L } P$ solvers. (2) In Section $^ { 3 . 2 , }$ we show that $\\beta$ -CROWN is solving an equivalent problem of the LP verifier with neuron split constraints. (3) In Section $^ { 3 . 3 , }$ we show that $\\beta$ -CROWN can jointly optimize intermediate layer bounds and achieve tighter bounds than typical $L P$ verifiers using fixed intermediate layer bounds. ", + "bbox": [ + 173, + 549, + 825, + 647 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Limitations. Our verifier has several limitations which are commonly shared by most existing BaB-based complete verifiers. First, we focused on ReLU which can be split into two linear cases. For other non-piecewise linear activation functions, although it is still possible to conduct branch and bound, it is difficult to guarantee completeness. Second, we discussed only the norm perturbations for input domains. In practice, the threat model may involve complicated and nonconvex perturbation specifications. Third, although our GPU accelerated verifier outperforms existing ones, all BaB based verifiers, including ours, are still limited to relatively small models far from the ImageNet scale. Finally, we have only demonstrated robustness verification of image classification tasks, and generalizing it to give verification guarantees for other tasks such as robust deep reinforcement learning $\\frac { 1 } { 1 2 7 }$ 48, 49] is an interesting direction for future work. ", + "bbox": [ + 174, + 661, + 825, + 799 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Societal Impact NNs have been used in an increasingly wide range of real-world applications and play an important role in artificial intelligence (AI). The trustworthiness and robustness of NNs have become crucial factors since AI plays an important role in modern society. $\\beta$ -CROWN is a strong neural network verifier which can be used to check certain properties of neural networks, which can be helpful for guaranteeing the robustness, correctness, and fairness of NNs in applications that can directly or indirectly impact human life. We believe our work has overall positive societal impacts, although it may potentially be misused to identify the weakness of NNs and guide attacks. ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgement ", + "text_level": 1, + "bbox": [ + 176, + 89, + 330, + 106 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "This work is supported by NSF grant CNS18-01426; an ARL Young Investigator (YIP) award; an NSF CAREER award; a Google Faculty Fellowship; a Capital One Research Grant; and a J.P. Morgan Faculty Award; Air Force Research Laboratory under FA8750-18-2-0058; NSF IIS-1901527, NSF IIS-2008173 and NSF CAREER-2048280; and NSF CNS-1932351. Huan Zhang is supported by funding from the Bosch Center for Artificial Intelligence. ", + "bbox": [ + 174, + 121, + 825, + 190 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 212, + 266, + 227 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "[1] G. Anderson, S. Pailoor, I. Dillig, and S. Chaudhuri. Optimization and abstraction: A synergistic approach for analyzing neural network robustness. In Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), 2019. [2] R. Anderson, J. Huchette, C. Tjandraatmadja, and J. P. Vielma. Strong convex relaxations and mixed-integer programming formulations for trained neural networks. Mathematical Programming, 2020. \n[3] S. Bak, C. Liu, and T. Johnson. 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When jointly optimized in interme-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 430, + 469, + 443 + ], + "spans": [ + { + "bbox": [ + 141, + 430, + 192, + 443 + ], + "score": 1.0, + "content": "diate layers,", + "type": "text" + }, + { + "bbox": [ + 192, + 431, + 200, + 442 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 430, + 469, + 443 + ], + "score": 1.0, + "content": "-CROWN generally produces better bounds than typical LP verifiers", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 442, + 469, + 453 + ], + "spans": [ + { + "bbox": [ + 141, + 442, + 469, + 453 + ], + "score": 1.0, + "content": "with neuron split constraints, while being as efficient and parallelizable as CROWN", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 452, + 469, + 464 + ], + "spans": [ + { + "bbox": [ + 141, + 452, + 420, + 464 + ], + "score": 1.0, + "content": "on GPUs. Applied to complete robustness verification benchmarks,", + "type": "text" + }, + { + "bbox": [ + 420, + 453, + 428, + 464 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 452, + 469, + 464 + ], + "score": 1.0, + "content": "-CROWN", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 464, + 469, + 475 + ], + "spans": [ + { + "bbox": [ + 141, + 464, + 469, + 475 + ], + "score": 1.0, + "content": "with BaB is up to three orders of magnitude faster than LP-based BaB methods, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 475, + 470, + 487 + ], + "spans": [ + { + "bbox": [ + 141, + 475, + 470, + 487 + ], + "score": 1.0, + "content": "is notably faster than all existing approaches while producing lower timeout rates.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 486, + 469, + 497 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 469, + 497 + ], + "score": 1.0, + "content": "By terminating BaB early, our method can also be used for efficient incomplete", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 496, + 469, + 509 + ], + "spans": [ + { + "bbox": [ + 141, + 496, + 469, + 509 + ], + "score": 1.0, + "content": "verification. We consistently achieve higher verified accuracy in many settings", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 508, + 470, + 519 + ], + "spans": [ + { + "bbox": [ + 141, + 508, + 470, + 519 + ], + "score": 1.0, + "content": "compared to powerful incomplete verifiers, including those based on convex barrier", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 141, + 519, + 469, + 531 + ], + "spans": [ + { + "bbox": [ + 141, + 519, + 469, + 531 + ], + "score": 1.0, + "content": "breaking techniques. Compared to the typically tightest but very costly semidefinite", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 529, + 470, + 542 + ], + "spans": [ + { + "bbox": [ + 141, + 529, + 470, + 542 + ], + "score": 1.0, + "content": "programming (SDP) based incomplete verifiers, we obtain higher verified accuracy", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 540, + 469, + 552 + ], + "spans": [ + { + "bbox": [ + 141, + 540, + 469, + 552 + ], + "score": 1.0, + "content": "with three orders of magnitudes less verification time. Our algorithm empowered", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 551, + 469, + 563 + ], + "spans": [ + { + "bbox": [ + 142, + 551, + 156, + 563 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 157, + 551, + 173, + 562 + ], + "score": 0.8, + "content": "\\alpha , \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 551, + 469, + 563 + ], + "score": 1.0, + "content": "-CROWN (alpha-beta-CROWN) verifier, the winning tool in VNN-COMP", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 142, + 562, + 419, + 574 + ], + "spans": [ + { + "bbox": [ + 142, + 562, + 419, + 574 + ], + "score": 1.0, + "content": "2021. Our code is available at http://PaperCode.cc/BetaCROWN.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 21, + "bbox_fs": [ + 141, + 344, + 470, + 574 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 597, + 190, + 610 + ], + "lines": [ + { + "bbox": [ + 105, + 596, + 192, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 596, + 192, + 613 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 637 + ], + "score": 1.0, + "content": "As neural networks (NNs) are being deployed in safety-critical applications, it becomes increasingly", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 634, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 507, + 648 + ], + "score": 1.0, + "content": "important to formally verify their behaviors under potentially malicious inputs. Broadly speaking,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "the neural network verification problem involves proving certain desired relationships between inputs", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "and outputs (often referred to as specifications), such as safety or robustness guarantees, for all inputs", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "inside some domain. Canonically, the problem can be cast as finding the global minima of some", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "functions on the network’s outputs (e.g., the difference between the predictions of the true label and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "score": 1.0, + "content": "another target label), within a bounded input set as constraints. This is a challenging problem due to", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 701, + 359, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 359, + 713 + ], + "score": 1.0, + "content": "the non-convexity and high dimensionality of neural networks.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 623, + 507, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 506, + 86 + ], + "score": 1.0, + "content": "We first focus on complete verification: the verifier should give a definite “yes/no” answer given", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 82, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 104, + 82, + 451, + 97 + ], + "score": 1.0, + "content": "sufficient time. Many complete verifiers rely on the branch and bound (BaB) method", + "type": "text" + }, + { + "bbox": [ + 451, + 83, + 463, + 95 + ], + "score": 0.67, + "content": "\\textcircled { 8 } \\textcircled { 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 82, + 506, + 97 + ], + "score": 1.0, + "content": "involving", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 108 + ], + "score": 1.0, + "content": "(1) branching by recursively splitting the original verification problem into subdomains (e.g., splitting", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 104, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 505, + 119 + ], + "score": 1.0, + "content": "a ReLU neuron into positive/negative linear regions by adding split constraints) and (2) bounding", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 129 + ], + "score": 1.0, + "content": "each subdomain with specialized incomplete verifiers. Traditional BaB-based verifiers use expensive", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 241, + 140 + ], + "score": 1.0, + "content": "linear programming (LP) solvers", + "type": "text" + }, + { + "bbox": [ + 241, + 127, + 284, + 138 + ], + "score": 0.35, + "content": "\\pm \\boxed { 1 } \\boxed { 2 3 } \\boxed { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "as incomplete verifiers which can fully encode neuron", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 381, + 151 + ], + "score": 1.0, + "content": "split constraints. Meanwhile, a recent verifier, Fast-and-Complete", + "type": "text" + }, + { + "bbox": [ + 381, + 137, + 399, + 149 + ], + "score": 0.5, + "content": "\\mathbb { \\lVert \\boldsymbol { 4 5 } \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 137, + 505, + 151 + ], + "score": 1.0, + "content": ", demonstrates that cheap", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "score": 1.0, + "content": "incomplete verifiers can significantly accelerate complete verification on GPUs over LP-based", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "ones thanks to their efficiency. Many cheap incomplete verifiers are based on bound propagation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "methods [46, 42, 41, 13, 17, 36, 44], i.e., maintaining and propagating tractable and sound bounds", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "score": 1.0, + "content": "through networks, and CROWN [46] is a representative which propagates a linear or quadratic bound.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 504, + 210 + ], + "score": 1.0, + "content": "However, unlike LP based verifiers, existing bound propagation methods lack the power to handle", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 383, + 222 + ], + "score": 1.0, + "content": "neuron split constraints introduced by BaB. For instance, given inputs", + "type": "text" + }, + { + "bbox": [ + 383, + 209, + 439, + 221 + ], + "score": 0.92, + "content": "x , y \\in [ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 209, + 506, + 222 + ], + "score": 1.0, + "content": ", they can bound", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 171, + 233 + ], + "score": 1.0, + "content": "a ReLU’s input", + "type": "text" + }, + { + "bbox": [ + 172, + 221, + 197, + 231 + ], + "score": 0.9, + "content": "x + y", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 220, + 209, + 233 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 209, + 220, + 238, + 232 + ], + "score": 0.89, + "content": "[ - 2 , 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "but they have no means to consider neuron split constraints such", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 117, + 243 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 231, + 162, + 243 + ], + "score": 0.92, + "content": "x - y \\ge 0", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "introduced by splitting another ReLU to the positive linear region. Such a problem", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "causes looser bounds and unnecessary branching, hurting the verification efficiency. Even worse,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "without considering these split constraints, bound propagation methods cannot detect many infeasible", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 262, + 463, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 187, + 277 + ], + "score": 1.0, + "content": "subdomains in BaB", + "type": "text" + }, + { + "bbox": [ + 187, + 263, + 205, + 275 + ], + "score": 0.61, + "content": "\\lVert \\boldsymbol { \\mathsf { \\Pi } } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 262, + 463, + 277 + ], + "score": 1.0, + "content": ", leading to incompleteness unless costly checking is performed.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 421, + 292 + ], + "score": 1.0, + "content": "In our work, we develop a new, fast bound propagation based incomplete verifier,", + "type": "text" + }, + { + "bbox": [ + 421, + 281, + 428, + 291 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "-CROWN. It solves", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "an optimization problem equivalent to the expensive LP based methods with neuron split constraints", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 370, + 315 + ], + "score": 1.0, + "content": "while still enjoying the efficiency of bound propagation methods.", + "type": "text" + }, + { + "bbox": [ + 371, + 302, + 378, + 313 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "-CROWN contains optimizable", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 151, + 326 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 152, + 313, + 160, + 324 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "which come from propagation of Lagrangian multipliers, and any valid settings of these", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "parameters yield sound bounds for verification. 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Traditional BaB-based verifiers use expensive", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 241, + 140 + ], + "score": 1.0, + "content": "linear programming (LP) solvers", + "type": "text" + }, + { + "bbox": [ + 241, + 127, + 284, + 138 + ], + "score": 0.35, + "content": "\\pm \\boxed { 1 } \\boxed { 2 3 } \\boxed { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "as incomplete verifiers which can fully encode neuron", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 381, + 151 + ], + "score": 1.0, + "content": "split constraints. Meanwhile, a recent verifier, Fast-and-Complete", + "type": "text" + }, + { + "bbox": [ + 381, + 137, + 399, + 149 + ], + "score": 0.5, + "content": "\\mathbb { \\lVert \\boldsymbol { 4 5 } \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 137, + 505, + 151 + ], + "score": 1.0, + "content": ", demonstrates that cheap", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 161 + ], + "score": 1.0, + "content": "incomplete verifiers can significantly accelerate complete verification on GPUs over LP-based", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "ones thanks to their efficiency. Many cheap incomplete verifiers are based on bound propagation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 183 + ], + "score": 1.0, + "content": "methods [46, 42, 41, 13, 17, 36, 44], i.e., maintaining and propagating tractable and sound bounds", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 182, + 506, + 194 + ], + "score": 1.0, + "content": "through networks, and CROWN [46] is a representative which propagates a linear or quadratic bound.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 71, + 506, + 194 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 504, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 504, + 210 + ], + "score": 1.0, + "content": "However, unlike LP based verifiers, existing bound propagation methods lack the power to handle", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 383, + 222 + ], + "score": 1.0, + "content": "neuron split constraints introduced by BaB. For instance, given inputs", + "type": "text" + }, + { + "bbox": [ + 383, + 209, + 439, + 221 + ], + "score": 0.92, + "content": "x , y \\in [ - 1 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 209, + 506, + 222 + ], + "score": 1.0, + "content": ", they can bound", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 171, + 233 + ], + "score": 1.0, + "content": "a ReLU’s input", + "type": "text" + }, + { + "bbox": [ + 172, + 221, + 197, + 231 + ], + "score": 0.9, + "content": "x + y", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 220, + 209, + 233 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 209, + 220, + 238, + 232 + ], + "score": 0.89, + "content": "[ - 2 , 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "but they have no means to consider neuron split constraints such", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 117, + 243 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 231, + 162, + 243 + ], + "score": 0.92, + "content": "x - y \\ge 0", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "introduced by splitting another ReLU to the positive linear region. Such a problem", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "causes looser bounds and unnecessary branching, hurting the verification efficiency. Even worse,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "without considering these split constraints, bound propagation methods cannot detect many infeasible", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 262, + 463, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 187, + 277 + ], + "score": 1.0, + "content": "subdomains in BaB", + "type": "text" + }, + { + "bbox": [ + 187, + 263, + 205, + 275 + ], + "score": 0.61, + "content": "\\lVert \\boldsymbol { \\mathsf { \\Pi } } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 262, + 463, + 277 + ], + "score": 1.0, + "content": ", leading to incompleteness unless costly checking is performed.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 198, + 506, + 277 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 281, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 421, + 292 + ], + "score": 1.0, + "content": "In our work, we develop a new, fast bound propagation based incomplete verifier,", + "type": "text" + }, + { + "bbox": [ + 421, + 281, + 428, + 291 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "-CROWN. It solves", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "an optimization problem equivalent to the expensive LP based methods with neuron split constraints", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 370, + 315 + ], + "score": 1.0, + "content": "while still enjoying the efficiency of bound propagation methods.", + "type": "text" + }, + { + "bbox": [ + 371, + 302, + 378, + 313 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "-CROWN contains optimizable", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 151, + 326 + ], + "score": 1.0, + "content": "parameters", + "type": "text" + }, + { + "bbox": [ + 152, + 313, + 160, + 324 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 312, + 505, + 326 + ], + "score": 1.0, + "content": "which come from propagation of Lagrangian multipliers, and any valid settings of these", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "parameters yield sound bounds for verification. These parameters are optimized using a few steps of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 335, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 397, + 348 + ], + "score": 1.0, + "content": "(super)gradient ascent to achieve bounds as tight as possible. Optimizing", + "type": "text" + }, + { + "bbox": [ + 397, + 335, + 406, + 346 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 335, + 505, + 348 + ], + "score": 1.0, + "content": "can also eliminate many", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "infeasible subdomains and avoid further useless branching. Furthermore, we can jointly optimize", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 259, + 369 + ], + "score": 1.0, + "content": "intermediate layer bounds similar to", + "type": "text" + }, + { + "bbox": [ + 259, + 356, + 277, + 368 + ], + "score": 0.77, + "content": "\\pm \\boxed { \\boxed { 4 4 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 356, + 443, + 369 + ], + "score": 1.0, + "content": "but also with the additional parameters", + "type": "text" + }, + { + "bbox": [ + 443, + 357, + 452, + 368 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 356, + 494, + 369 + ], + "score": 1.0, + "content": ", allowing", + "type": "text" + }, + { + "bbox": [ + 495, + 357, + 502, + 368 + ], + "score": 0.81, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 356, + 506, + 369 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "CROWN to tighten relaxations and outperform typical LP verifiers with fixed intermediate layer", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 312, + 390 + ], + "score": 1.0, + "content": "bounds. Unlike traditional LP-based BaB methods,", + "type": "text" + }, + { + "bbox": [ + 312, + 379, + 319, + 389 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "-CROWN can be efficiently implemented with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 390, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 401 + ], + "score": 1.0, + "content": "an automatic differentiation framework on GPUs to fully exploit the power of modern accelerators.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 401, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 187, + 412 + ], + "score": 1.0, + "content": "The combination of", + "type": "text" + }, + { + "bbox": [ + 187, + 401, + 195, + 411 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 401, + 276, + 412 + ], + "score": 1.0, + "content": "-CROWN and BaB (", + "type": "text" + }, + { + "bbox": [ + 276, + 401, + 283, + 411 + ], + "score": 0.76, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 401, + 505, + 412 + ], + "score": 1.0, + "content": "-CROWN BaB) produces a complete verifier with GPU", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 433, + 423 + ], + "score": 1.0, + "content": "acceleration, reducing the verification time of traditional LP based BaB verifiers", + "type": "text" + }, + { + "bbox": [ + 433, + 411, + 445, + 422 + ], + "score": 0.81, + "content": "\\textcircled { 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "by up to three", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "orders of magnitudes on a commonly used benchmark suite on CIFAR-10 [6, 10]. Compared to all", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "state-of-the-art GPU-based complete verifiers [7, 45, 10, 23, 6, 11], our approach is noticeably faster", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 443, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 352, + 456 + ], + "score": 1.0, + "content": "with lower timeout rates. Our algorithm empowered the tool", + "type": "text" + }, + { + "bbox": [ + 353, + 444, + 369, + 455 + ], + "score": 0.5, + "content": "\\alpha , \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 443, + 506, + 456 + ], + "score": 1.0, + "content": "-CROWN (alpha-beta-CROWN),", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 454, + 506, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 430, + 467 + ], + "score": 1.0, + "content": "which won the 2nd International Verification of Neural Networks Competition", + "type": "text" + }, + { + "bbox": [ + 430, + 454, + 443, + 466 + ], + "score": 0.49, + "content": "\\pmb { \\mathbb { B } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 454, + 506, + 467 + ], + "score": 1.0, + "content": "(VNN-COMP", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 466, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 506, + 477 + ], + "score": 1.0, + "content": "2021) with the highest total score and verified the most number of problem instances in 8 benchmarks.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 280, + 506, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 286, + 494 + ], + "score": 1.0, + "content": "Finally, by terminating our complete verifier", + "type": "text" + }, + { + "bbox": [ + 286, + 482, + 294, + 493 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "-CROWN BaB early, our approach can also function", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "as a more accurate incomplete verifier by returning an incomplete but sound lower bound of all", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "subdomains explored so far. 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Compared to the typically tightest but very costly incomplete verifier", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 143, + 550 + ], + "score": 1.0, + "content": "SDP-FO", + "type": "text" + }, + { + "bbox": [ + 144, + 536, + 156, + 548 + ], + "score": 0.61, + "content": "\\pmb { \\Vert }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 536, + 389, + 550 + ], + "score": 1.0, + "content": "based on the semidefinite programming (SDP) relaxations", + "type": "text" + }, + { + "bbox": [ + 390, + 536, + 422, + 547 + ], + "score": 0.7, + "content": "[ \\overbrace { 2 8 } , \\overbrace { 1 4 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 536, + 506, + 550 + ], + "score": 1.0, + "content": ", our method obtains", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "consistently higher verified accuracy while reducing verification time by three orders of magnitudes.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 482, + 506, + 560 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 570, + 189, + 584 + ], + "lines": [ + { + "bbox": [ + 103, + 568, + 191, + 588 + ], + "spans": [ + { + "bbox": [ + 103, + 568, + 191, + 588 + ], + "score": 1.0, + "content": "2 Background", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "title", + "bbox": [ + 108, + 589, + 396, + 601 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 397, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 397, + 603 + ], + "score": 1.0, + "content": "2.1 The neural network verification problem and its LP relaxation", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 506, + 707 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 288, + 623 + ], + "score": 1.0, + "content": "We define the input of a neural network as", + "type": "text" + }, + { + "bbox": [ + 288, + 609, + 332, + 620 + ], + "score": 0.91, + "content": "\\begin{array} { r l r } { x } & { { } \\in } & { \\mathbb { R } ^ { d _ { 0 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 608, + 506, + 623 + ], + "score": 1.0, + "content": ", and define the weights and biases of an", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 107, + 618, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 107, + 622, + 114, + 632 + ], + "score": 0.73, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 618, + 217, + 635 + ], + "score": 1.0, + "content": "-layer neural network as", + "type": "text" + }, + { + "bbox": [ + 217, + 621, + 291, + 632 + ], + "score": 0.91, + "content": "\\mathbf W ^ { ( i ) } \\in \\mathbb R ^ { d _ { i } \\times d _ { i - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 618, + 312, + 635 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 312, + 621, + 358, + 632 + ], + "score": 0.8, + "content": "\\mathbf { b } ^ { ( i ) } \\in \\mathbb { R } ^ { d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 618, + 362, + 635 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 362, + 621, + 430, + 634 + ], + "score": 0.85, + "content": "( i \\in \\{ 1 , \\cdots , L \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 618, + 506, + 635 + ], + "score": 1.0, + "content": "respectively. For", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 631, + 506, + 647 + ], + "spans": [ + { + "bbox": [ + 104, + 631, + 216, + 647 + ], + "score": 1.0, + "content": "simplicity we assume that", + "type": "text" + }, + { + "bbox": [ + 216, + 634, + 249, + 645 + ], + "score": 0.91, + "content": "d _ { L } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 631, + 262, + 647 + ], + "score": 1.0, + "content": "so", + "type": "text" + }, + { + "bbox": [ + 263, + 633, + 288, + 644 + ], + "score": 0.85, + "content": "\\mathbf { W } ^ { ( L ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 631, + 353, + 647 + ], + "score": 1.0, + "content": "is a vector and", + "type": "text" + }, + { + "bbox": [ + 353, + 633, + 372, + 644 + ], + "score": 0.89, + "content": "\\mathbf { b } ^ { ( L ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 631, + 506, + 647 + ], + "score": 1.0, + "content": "is a scalar. 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In practical settings, we typically have “specifications” to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "verify, which are (usually linear) functions of neural network outputs describing the desired behavior", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "of neural networks. For example, to guarantee robustness we typically investigate the margin between", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 484, + 140 + ], + "score": 1.0, + "content": "logits. Because the specification can also be seen as an output layer of NN and merged into", + "type": "text" + }, + { + "bbox": [ + 484, + 127, + 505, + 139 + ], + "score": 0.9, + "content": "f ( x )", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "under verification, we do not discuss it in detail in this work. We consider the canonical specification", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 149, + 406, + 162 + ], + "spans": [ + { + "bbox": [ + 107, + 149, + 146, + 162 + ], + "score": 0.94, + "content": "f ( x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 149, + 231, + 161 + ], + "score": 1.0, + "content": ": if we can prove that", + "type": "text" + }, + { + "bbox": [ + 232, + 149, + 270, + 161 + ], + "score": 0.86, + "content": "f ( x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 149, + 275, + 161 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 275, + 149, + 306, + 160 + ], + "score": 0.8, + "content": "\\forall x \\in { \\mathcal { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 149, + 340, + 161 + ], + "score": 1.0, + "content": ", we say", + "type": "text" + }, + { + "bbox": [ + 340, + 149, + 361, + 161 + ], + "score": 0.92, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 149, + 406, + 161 + ], + "score": 1.0, + "content": "is verified.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 505, + 233 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 131, + 180 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 132, + 167, + 139, + 177 + ], + "score": 0.75, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 164, + 432, + 180 + ], + "score": 1.0, + "content": "is a convex set, Eq. 1 is still a non-convex problem because the constraints", + "type": "text" + }, + { + "bbox": [ + 432, + 165, + 489, + 178 + ], + "score": 0.93, + "content": "\\hat { z } ^ { ( i ) } = \\sigma ( z ^ { ( i ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 164, + 506, + 180 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 104, + 177, + 414, + 191 + ], + "score": 1.0, + "content": "non-convex. Given unlimited time, complete verifiers can solve Eq. 1 exactly:", + "type": "text" + }, + { + "bbox": [ + 414, + 178, + 476, + 190 + ], + "score": 0.66, + "content": "f ^ { * } = \\operatorname* { m i n } { f ( x ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 177, + 481, + 191 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 481, + 178, + 505, + 189 + ], + "score": 0.59, + "content": "\\forall x \\in", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 113, + 199 + ], + "score": 0.74, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 188, + 506, + 201 + ], + "score": 1.0, + "content": ", so we can always conclude if the specification holds or not for any problem instance. On the other", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "hand, incomplete verifiers usually relax the non-convexity of neural networks to obtain a tractable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 217, + 223 + ], + "score": 1.0, + "content": "lower bound of the solution", + "type": "text" + }, + { + "bbox": [ + 218, + 210, + 248, + 223 + ], + "score": 0.91, + "content": "\\underline { { f } } \\le f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 210, + 262, + 223 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 262, + 210, + 288, + 223 + ], + "score": 0.91, + "content": "\\underline { { f } } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 210, + 310, + 223 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 311, + 210, + 341, + 222 + ], + "score": 0.92, + "content": "f ^ { * } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 210, + 353, + 223 + ], + "score": 1.0, + "content": "so", + "type": "text" + }, + { + "bbox": [ + 353, + 210, + 374, + 222 + ], + "score": 0.92, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 210, + 461, + 223 + ], + "score": 1.0, + "content": "can be verified; when", + "type": "text" + }, + { + "bbox": [ + 462, + 210, + 487, + 223 + ], + "score": 0.9, + "content": "\\underline { { f } } < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 210, + 506, + 223 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 221, + 452, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 231, + 234 + ], + "score": 1.0, + "content": "are not able to infer the sign of", + "type": "text" + }, + { + "bbox": [ + 232, + 222, + 243, + 233 + ], + "score": 0.89, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 221, + 452, + 234 + ], + "score": 1.0, + "content": "so cannot conclude if the specification holds or not.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 506, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "A commonly used incomplete verification technique is to relax non-convex ReLU constraints with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "linear constraints and turn the verification problem into a linear programming (LP) problem, which", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 273 + ], + "score": 1.0, + "content": "can then be solved with linear solvers. We refer to it as the “LP verifier” in this paper. Specifically,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 102, + 267, + 509, + 290 + ], + "spans": [ + { + "bbox": [ + 102, + 267, + 131, + 290 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 131, + 270, + 246, + 287 + ], + "score": 0.93, + "content": "\\mathrm { R e L U } ( z _ { j } ^ { ( i ) } ) : = \\operatorname* { m a x } ( 0 , z _ { j } ^ { ( i ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 267, + 382, + 290 + ], + "score": 1.0, + "content": "and its intermediate layer bounds", + "type": "text" + }, + { + "bbox": [ + 383, + 270, + 453, + 287 + ], + "score": 0.93, + "content": "\\mathbf { l } _ { j } ^ { ( i ) } \\le z _ { j } ^ { ( i ) } \\overset { \\_ } { \\le } \\mathbf { u } _ { j } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 267, + 509, + 290 + ], + "score": 1.0, + "content": ", each ReLU", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 102, + 281, + 509, + 306 + ], + "spans": [ + { + "bbox": [ + 102, + 281, + 278, + 306 + ], + "score": 1.0, + "content": "can be categorized into three cases: (1) if", + "type": "text" + }, + { + "bbox": [ + 278, + 285, + 312, + 302 + ], + "score": 0.93, + "content": "\\mathbf { l } _ { j } ^ { ( i ) } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 281, + 432, + 306 + ], + "score": 1.0, + "content": "(ReLU in linear region) then", + "type": "text" + }, + { + "bbox": [ + 433, + 286, + 477, + 302 + ], + "score": 0.92, + "content": "\\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 281, + 509, + 306 + ], + "score": 1.0, + "content": "; (2) if", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 298, + 508, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 144, + 318 + ], + "score": 0.92, + "content": "\\mathbf { u } _ { j } ^ { ( i ) } \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 298, + 273, + 320 + ], + "score": 1.0, + "content": "(ReLU in inactive region) then", + "type": "text" + }, + { + "bbox": [ + 274, + 302, + 309, + 318 + ], + "score": 0.93, + "content": "\\hat { z } _ { j } ^ { ( i ) } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 298, + 338, + 320 + ], + "score": 1.0, + "content": "; (3) if", + "type": "text" + }, + { + "bbox": [ + 338, + 301, + 402, + 318 + ], + "score": 0.92, + "content": "\\mathbf { l } _ { j } ^ { ( i ) } \\le 0 \\le \\mathbf { u } _ { j } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 298, + 508, + 320 + ], + "score": 1.0, + "content": "(ReLU is unstable) then", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 100, + 314, + 508, + 343 + ], + "spans": [ + { + "bbox": [ + 100, + 314, + 227, + 343 + ], + "score": 1.0, + "content": "three linear bounds are used:", + "type": "text" + }, + { + "bbox": [ + 227, + 320, + 262, + 337 + ], + "score": 0.61, + "content": "\\hat { z } _ { j } ^ { ( i ) } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 314, + 266, + 343 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 266, + 320, + 309, + 337 + ], + "score": 0.62, + "content": "\\hat { z } _ { j } ^ { ( i ) } \\geq z _ { j } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 314, + 331, + 343 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 332, + 318, + 442, + 341 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\hat { z } _ { j } ^ { ( i ) } \\leq \\frac { \\mathbf { u } _ { j } ^ { ( i ) } } { \\mathbf { u } _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } } \\left( z _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 314, + 508, + 343 + ], + "score": 1.0, + "content": "; they are often", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 267, + 354 + ], + "score": 1.0, + "content": "referred to as the “triangle” relaxation", + "type": "text" + }, + { + "bbox": [ + 268, + 340, + 301, + 353 + ], + "score": 0.33, + "content": "\\mathbb { m } 5 \\mathbb { k } 2 \\mathbb { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 339, + 437, + 354 + ], + "score": 1.0, + "content": ". The intermediate layer bounds", + "type": "text" + }, + { + "bbox": [ + 437, + 340, + 451, + 352 + ], + "score": 0.84, + "content": "\\mathbf { l } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 339, + 471, + 354 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 340, + 488, + 352 + ], + "score": 0.89, + "content": "\\mathbf { u } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 339, + 506, + 354 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "usually obtained from a cheaper bound propagation method (see next subsection). LP verifiers can", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "provide relatively tight bounds but linear solvers are still expensive especially when the network is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 200, + 386 + ], + "score": 1.0, + "content": "large. Also, unlike our", + "type": "text" + }, + { + "bbox": [ + 200, + 374, + 208, + 385 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "-CROWN, they have to use fixed intermediate bounds and cannot use the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 384, + 437, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 331, + 398 + ], + "score": 1.0, + "content": "joint optimization of intermediate layer bounds (Section", + "type": "text" + }, + { + "bbox": [ + 331, + 384, + 349, + 397 + ], + "score": 0.75, + "content": "3 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 384, + 437, + 398 + ], + "score": 1.0, + "content": "to tighten relaxation.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 106, + 403, + 437, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 438, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 438, + 418 + ], + "score": 1.0, + "content": "2.2 CROWN: efficient incomplete verification by propagating linear bounds", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 418, + 506, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 393, + 432 + ], + "score": 1.0, + "content": "Another cheaper way to give a lower bound for the objective in Eq.", + "type": "text" + }, + { + "bbox": [ + 393, + 418, + 403, + 432 + ], + "score": 0.6, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "is through sound bound", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 429, + 507, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 200, + 443 + ], + "score": 1.0, + "content": "propagation. CROWN", + "type": "text" + }, + { + "bbox": [ + 200, + 429, + 218, + 441 + ], + "score": 0.85, + "content": "\\bar { \\lVert 4 6 rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 429, + 462, + 443 + ], + "score": 1.0, + "content": "is a representative method that propagates a linear bound of", + "type": "text" + }, + { + "bbox": [ + 463, + 430, + 483, + 442 + ], + "score": 0.92, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 429, + 507, + 443 + ], + "score": 1.0, + "content": "w.r.t.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 392, + 453 + ], + "score": 1.0, + "content": "every intermediate layer in a backward manner until reaching the input", + "type": "text" + }, + { + "bbox": [ + 393, + 442, + 399, + 450 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 440, + 506, + 453 + ], + "score": 1.0, + "content": ". CROWN uses two linear", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 101, + 451, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 101, + 452, + 371, + 479 + ], + "score": 1.0, + "content": "constraints to relax unstable ReLU neurons: a linear upper bound", + "type": "text" + }, + { + "bbox": [ + 371, + 451, + 479, + 475 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\hat { z } _ { j } ^ { ( i ) } \\leq \\frac { \\mathbf { u } _ { j } ^ { ( i ) } } { \\mathbf { u } _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } } \\left( z _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 452, + 506, + 475 + ], + "score": 1.0, + "content": "and a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 470, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 184, + 495 + ], + "score": 1.0, + "content": "linear lower bound", + "type": "text" + }, + { + "bbox": [ + 184, + 474, + 245, + 491 + ], + "score": 0.73, + "content": "\\hat { z } _ { j } ^ { ( i ) } \\geq \\alpha _ { j } ^ { ( i ) } z _ { j } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 474, + 305, + 491 + ], + "score": 0.83, + "content": "0 \\leq \\alpha _ { j } ^ { ( i ) } \\leq 1 ", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 470, + 506, + 495 + ], + "score": 1.0, + "content": "). 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Given", + "type": "text" + }, + { + "bbox": [ + 313, + 492, + 402, + 505 + ], + "score": 0.77, + "content": "w , v \\in \\mathbb { R } ^ { d } , 1 \\leq v \\leq \\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 491, + 501, + 506 + ], + "score": 1.0, + "content": "(element-wise), we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 506, + 362, + 520 + ], + "lines": [ + { + "bbox": [ + 246, + 506, + 362, + 520 + ], + "spans": [ + { + "bbox": [ + 246, + 506, + 362, + 520 + ], + "score": 0.9, + "content": "w ^ { \\top } \\mathrm { R e L U } ( v ) \\geq w ^ { \\top } \\mathbf { D } v + b ^ { \\prime } .", + "type": "interline_equation", + "image_path": "2068a804b82efa90a51fa0eb0b3bb6c8376efd5e229b79f91a053c2c7a168fda.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 246, + 506, + 362, + 520 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 133, + 535 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 523, + 144, + 532 + ], + "score": 0.39, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 522, + 335, + 535 + ], + "score": 1.0, + "content": "is a diagonal matrix containing free variables", + "type": "text" + }, + { + "bbox": [ + 336, + 523, + 387, + 534 + ], + "score": 0.92, + "content": "0 \\leq \\alpha _ { j } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 522, + 433, + 535 + ], + "score": 1.0, + "content": "only when", + "type": "text" + }, + { + "bbox": [ + 433, + 523, + 486, + 535 + ], + "score": 0.93, + "content": "\\mathbf { u } _ { j } > 0 > 1 _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 532, + 415, + 547 + ], + "spans": [ + { + "bbox": [ + 107, + 534, + 137, + 546 + ], + "score": 0.91, + "content": "w _ { j } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 532, + 300, + 547 + ], + "score": 1.0, + "content": ", while its rest values as well as constant", + "type": "text" + }, + { + "bbox": [ + 300, + 533, + 308, + 543 + ], + "score": 0.84, + "content": "b ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 532, + 384, + 547 + ], + "score": 1.0, + "content": "are determined by", + "type": "text" + }, + { + "bbox": [ + 384, + 534, + 411, + 545 + ], + "score": 0.85, + "content": "{ \\mathbf { l } } , { \\mathbf { u } } , w", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 532, + 415, + 547 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 504, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 305, + 566 + ], + "score": 1.0, + "content": "Detailed forms of each term are listed in Appendix", + "type": "text" + }, + { + "bbox": [ + 305, + 551, + 317, + 565 + ], + "score": 0.49, + "content": "\\mathbf { A } .", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 551, + 349, + 566 + ], + "score": 1.0, + "content": "Lemma", + "type": "text" + }, + { + "bbox": [ + 349, + 552, + 365, + 564 + ], + "score": 0.82, + "content": "\\boxed { 2 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 551, + 506, + 566 + ], + "score": 1.0, + "content": "can be repeatedly applied, resulting", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 562, + 487, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 476, + 576 + ], + "score": 1.0, + "content": "in an efficient back-substitution procedure to derive a linear lower bound of NN output w.r.t.", + "type": "text" + }, + { + "bbox": [ + 477, + 565, + 483, + 573 + ], + "score": 0.71, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 562, + 487, + 576 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 105, + 577, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 224, + 590 + ], + "score": 1.0, + "content": "Lemma 2.2 (CROWN bound", + "type": "text" + }, + { + "bbox": [ + 225, + 577, + 241, + 589 + ], + "score": 0.77, + "content": "[ \\overline { { | 4 6 | } }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 576, + 288, + 590 + ], + "score": 1.0, + "content": "). 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The main benefit of CROWN is its efficiency: CROWN can be efficiently implemented", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 298, + 681 + ], + "score": 1.0, + "content": "on machine learning accelerators such as GPUs", + "type": "text" + }, + { + "bbox": [ + 298, + 667, + 315, + 679 + ], + "score": 0.84, + "content": "\\pm \\mathbb { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 667, + 358, + 681 + ], + "score": 1.0, + "content": "and TPUs", + "type": "text" + }, + { + "bbox": [ + 359, + 667, + 376, + 679 + ], + "score": 0.81, + "content": "\\textcircled { \\scriptsize { 1 4 7 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 667, + 506, + 681 + ], + "score": 1.0, + "content": ", and it can be a few magnitudes", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "faster than an LP verifier which is hard to parallelize on GPUs. 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In practical settings, we typically have “specifications” to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "verify, which are (usually linear) functions of neural network outputs describing the desired behavior", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 506, + 129 + ], + "score": 1.0, + "content": "of neural networks. For example, to guarantee robustness we typically investigate the margin between", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 484, + 140 + ], + "score": 1.0, + "content": "logits. Because the specification can also be seen as an output layer of NN and merged into", + "type": "text" + }, + { + "bbox": [ + 484, + 127, + 505, + 139 + ], + "score": 0.9, + "content": "f ( x )", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 506, + 151 + ], + "score": 1.0, + "content": "under verification, we do not discuss it in detail in this work. We consider the canonical specification", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 149, + 406, + 162 + ], + "spans": [ + { + "bbox": [ + 107, + 149, + 146, + 162 + ], + "score": 0.94, + "content": "f ( x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 149, + 231, + 161 + ], + "score": 1.0, + "content": ": if we can prove that", + "type": "text" + }, + { + "bbox": [ + 232, + 149, + 270, + 161 + ], + "score": 0.86, + "content": "f ( x ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 149, + 275, + 161 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 275, + 149, + 306, + 160 + ], + "score": 0.8, + "content": "\\forall x \\in { \\mathcal { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 149, + 340, + 161 + ], + "score": 1.0, + "content": ", we say", + "type": "text" + }, + { + "bbox": [ + 340, + 149, + 361, + 161 + ], + "score": 0.92, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 149, + 406, + 161 + ], + "score": 1.0, + "content": "is verified.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 73, + 506, + 162 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 164, + 505, + 233 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 131, + 180 + ], + "score": 1.0, + "content": "When", + "type": "text" + }, + { + "bbox": [ + 132, + 167, + 139, + 177 + ], + "score": 0.75, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 164, + 432, + 180 + ], + "score": 1.0, + "content": "is a convex set, Eq. 1 is still a non-convex problem because the constraints", + "type": "text" + }, + { + "bbox": [ + 432, + 165, + 489, + 178 + ], + "score": 0.93, + "content": "\\hat { z } ^ { ( i ) } = \\sigma ( z ^ { ( i ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 164, + 506, + 180 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 177, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 104, + 177, + 414, + 191 + ], + "score": 1.0, + "content": "non-convex. Given unlimited time, complete verifiers can solve Eq. 1 exactly:", + "type": "text" + }, + { + "bbox": [ + 414, + 178, + 476, + 190 + ], + "score": 0.66, + "content": "f ^ { * } = \\operatorname* { m i n } { f ( x ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 177, + 481, + 191 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 481, + 178, + 505, + 189 + ], + "score": 0.59, + "content": "\\forall x \\in", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 113, + 199 + ], + "score": 0.74, + "content": "\\mathcal { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 188, + 506, + 201 + ], + "score": 1.0, + "content": ", so we can always conclude if the specification holds or not for any problem instance. On the other", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 505, + 212 + ], + "score": 1.0, + "content": "hand, incomplete verifiers usually relax the non-convexity of neural networks to obtain a tractable", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 217, + 223 + ], + "score": 1.0, + "content": "lower bound of the solution", + "type": "text" + }, + { + "bbox": [ + 218, + 210, + 248, + 223 + ], + "score": 0.91, + "content": "\\underline { { f } } \\le f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 210, + 262, + 223 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 262, + 210, + 288, + 223 + ], + "score": 0.91, + "content": "\\underline { { f } } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 210, + 310, + 223 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 311, + 210, + 341, + 222 + ], + "score": 0.92, + "content": "f ^ { * } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 210, + 353, + 223 + ], + "score": 1.0, + "content": "so", + "type": "text" + }, + { + "bbox": [ + 353, + 210, + 374, + 222 + ], + "score": 0.92, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 210, + 461, + 223 + ], + "score": 1.0, + "content": "can be verified; when", + "type": "text" + }, + { + "bbox": [ + 462, + 210, + 487, + 223 + ], + "score": 0.9, + "content": "\\underline { { f } } < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 210, + 506, + 223 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 221, + 452, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 231, + 234 + ], + "score": 1.0, + "content": "are not able to infer the sign of", + "type": "text" + }, + { + "bbox": [ + 232, + 222, + 243, + 233 + ], + "score": 0.89, + "content": "f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 221, + 452, + 234 + ], + "score": 1.0, + "content": "so cannot conclude if the specification holds or not.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 164, + 506, + 234 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 506, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "score": 1.0, + "content": "A commonly used incomplete verification technique is to relax non-convex ReLU constraints with", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "linear constraints and turn the verification problem into a linear programming (LP) problem, which", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 258, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 273 + ], + "score": 1.0, + "content": "can then be solved with linear solvers. We refer to it as the “LP verifier” in this paper. Specifically,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 102, + 267, + 509, + 290 + ], + "spans": [ + { + "bbox": [ + 102, + 267, + 131, + 290 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 131, + 270, + 246, + 287 + ], + "score": 0.93, + "content": "\\mathrm { R e L U } ( z _ { j } ^ { ( i ) } ) : = \\operatorname* { m a x } ( 0 , z _ { j } ^ { ( i ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 267, + 382, + 290 + ], + "score": 1.0, + "content": "and its intermediate layer bounds", + "type": "text" + }, + { + "bbox": [ + 383, + 270, + 453, + 287 + ], + "score": 0.93, + "content": "\\mathbf { l } _ { j } ^ { ( i ) } \\le z _ { j } ^ { ( i ) } \\overset { \\_ } { \\le } \\mathbf { u } _ { j } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 267, + 509, + 290 + ], + "score": 1.0, + "content": ", each ReLU", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 102, + 281, + 509, + 306 + ], + "spans": [ + { + "bbox": [ + 102, + 281, + 278, + 306 + ], + "score": 1.0, + "content": "can be categorized into three cases: (1) if", + "type": "text" + }, + { + "bbox": [ + 278, + 285, + 312, + 302 + ], + "score": 0.93, + "content": "\\mathbf { l } _ { j } ^ { ( i ) } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 281, + 432, + 306 + ], + "score": 1.0, + "content": "(ReLU in linear region) then", + "type": "text" + }, + { + "bbox": [ + 433, + 286, + 477, + 302 + ], + "score": 0.92, + "content": "\\hat { z } _ { j } ^ { ( i ) } = z _ { j } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 281, + 509, + 306 + ], + "score": 1.0, + "content": "; (2) if", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 298, + 508, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 144, + 318 + ], + "score": 0.92, + "content": "\\mathbf { u } _ { j } ^ { ( i ) } \\leq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 298, + 273, + 320 + ], + "score": 1.0, + "content": "(ReLU in inactive region) then", + "type": "text" + }, + { + "bbox": [ + 274, + 302, + 309, + 318 + ], + "score": 0.93, + "content": "\\hat { z } _ { j } ^ { ( i ) } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 298, + 338, + 320 + ], + "score": 1.0, + "content": "; (3) if", + "type": "text" + }, + { + "bbox": [ + 338, + 301, + 402, + 318 + ], + "score": 0.92, + "content": "\\mathbf { l } _ { j } ^ { ( i ) } \\le 0 \\le \\mathbf { u } _ { j } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 298, + 508, + 320 + ], + "score": 1.0, + "content": "(ReLU is unstable) then", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 100, + 314, + 508, + 343 + ], + "spans": [ + { + "bbox": [ + 100, + 314, + 227, + 343 + ], + "score": 1.0, + "content": "three linear bounds are used:", + "type": "text" + }, + { + "bbox": [ + 227, + 320, + 262, + 337 + ], + "score": 0.61, + "content": "\\hat { z } _ { j } ^ { ( i ) } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 314, + 266, + 343 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 266, + 320, + 309, + 337 + ], + "score": 0.62, + "content": "\\hat { z } _ { j } ^ { ( i ) } \\geq z _ { j } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 314, + 331, + 343 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 332, + 318, + 442, + 341 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\hat { z } _ { j } ^ { ( i ) } \\leq \\frac { \\mathbf { u } _ { j } ^ { ( i ) } } { \\mathbf { u } _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } } \\left( z _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 314, + 508, + 343 + ], + "score": 1.0, + "content": "; they are often", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 267, + 354 + ], + "score": 1.0, + "content": "referred to as the “triangle” relaxation", + "type": "text" + }, + { + "bbox": [ + 268, + 340, + 301, + 353 + ], + "score": 0.33, + "content": "\\mathbb { m } 5 \\mathbb { k } 2 \\mathbb { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 339, + 437, + 354 + ], + "score": 1.0, + "content": ". The intermediate layer bounds", + "type": "text" + }, + { + "bbox": [ + 437, + 340, + 451, + 352 + ], + "score": 0.84, + "content": "\\mathbf { l } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 339, + 471, + 354 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 471, + 340, + 488, + 352 + ], + "score": 0.89, + "content": "\\mathbf { u } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 339, + 506, + 354 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "usually obtained from a cheaper bound propagation method (see next subsection). LP verifiers can", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "provide relatively tight bounds but linear solvers are still expensive especially when the network is", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 373, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 200, + 386 + ], + "score": 1.0, + "content": "large. Also, unlike our", + "type": "text" + }, + { + "bbox": [ + 200, + 374, + 208, + 385 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 373, + 506, + 386 + ], + "score": 1.0, + "content": "-CROWN, they have to use fixed intermediate bounds and cannot use the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 384, + 437, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 384, + 331, + 398 + ], + "score": 1.0, + "content": "joint optimization of intermediate layer bounds (Section", + "type": "text" + }, + { + "bbox": [ + 331, + 384, + 349, + 397 + ], + "score": 0.75, + "content": "3 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 384, + 437, + 398 + ], + "score": 1.0, + "content": "to tighten relaxation.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 19.5, + "bbox_fs": [ + 100, + 236, + 509, + 398 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 403, + 437, + 416 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 438, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 438, + 418 + ], + "score": 1.0, + "content": "2.2 CROWN: efficient incomplete verification by propagating linear bounds", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 418, + 506, + 505 + ], + "lines": [ + { + "bbox": [ + 106, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 393, + 432 + ], + "score": 1.0, + "content": "Another cheaper way to give a lower bound for the objective in Eq.", + "type": "text" + }, + { + "bbox": [ + 393, + 418, + 403, + 432 + ], + "score": 0.6, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "is through sound bound", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 429, + 507, + 443 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 200, + 443 + ], + "score": 1.0, + "content": "propagation. CROWN", + "type": "text" + }, + { + "bbox": [ + 200, + 429, + 218, + 441 + ], + "score": 0.85, + "content": "\\bar { \\lVert 4 6 rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 429, + 462, + 443 + ], + "score": 1.0, + "content": "is a representative method that propagates a linear bound of", + "type": "text" + }, + { + "bbox": [ + 463, + 430, + 483, + 442 + ], + "score": 0.92, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 429, + 507, + 443 + ], + "score": 1.0, + "content": "w.r.t.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 392, + 453 + ], + "score": 1.0, + "content": "every intermediate layer in a backward manner until reaching the input", + "type": "text" + }, + { + "bbox": [ + 393, + 442, + 399, + 450 + ], + "score": 0.75, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 440, + 506, + 453 + ], + "score": 1.0, + "content": ". CROWN uses two linear", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 101, + 451, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 101, + 452, + 371, + 479 + ], + "score": 1.0, + "content": "constraints to relax unstable ReLU neurons: a linear upper bound", + "type": "text" + }, + { + "bbox": [ + 371, + 451, + 479, + 475 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\hat { z } _ { j } ^ { ( i ) } \\leq \\frac { \\mathbf { u } _ { j } ^ { ( i ) } } { \\mathbf { u } _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } } \\left( z _ { j } ^ { ( i ) } - \\mathbf { l } _ { j } ^ { ( i ) } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 452, + 506, + 475 + ], + "score": 1.0, + "content": "and a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 470, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 104, + 470, + 184, + 495 + ], + "score": 1.0, + "content": "linear lower bound", + "type": "text" + }, + { + "bbox": [ + 184, + 474, + 245, + 491 + ], + "score": 0.73, + "content": "\\hat { z } _ { j } ^ { ( i ) } \\geq \\alpha _ { j } ^ { ( i ) } z _ { j } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 474, + 305, + 491 + ], + "score": 0.83, + "content": "0 \\leq \\alpha _ { j } ^ { ( i ) } \\leq 1 ", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 470, + 506, + 495 + ], + "score": 1.0, + "content": "). We can then bound the output of a ReLU layer:", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 501, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 312, + 506 + ], + "score": 1.0, + "content": "Lemma 2.1 (ReLU relaxation in CROWN). Given", + "type": "text" + }, + { + "bbox": [ + 313, + 492, + 402, + 505 + ], + "score": 0.77, + "content": "w , v \\in \\mathbb { R } ^ { d } , 1 \\leq v \\leq \\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 491, + 501, + 506 + ], + "score": 1.0, + "content": "(element-wise), we have", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 29.5, + "bbox_fs": [ + 101, + 418, + 507, + 506 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 506, + 362, + 520 + ], + "lines": [ + { + "bbox": [ + 246, + 506, + 362, + 520 + ], + "spans": [ + { + "bbox": [ + 246, + 506, + 362, + 520 + ], + "score": 0.9, + "content": "w ^ { \\top } \\mathrm { R e L U } ( v ) \\geq w ^ { \\top } \\mathbf { D } v + b ^ { \\prime } .", + "type": "interline_equation", + "image_path": "2068a804b82efa90a51fa0eb0b3bb6c8376efd5e229b79f91a053c2c7a168fda.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 246, + 506, + 362, + 520 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 504, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 133, + 535 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 523, + 144, + 532 + ], + "score": 0.39, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 522, + 335, + 535 + ], + "score": 1.0, + "content": "is a diagonal matrix containing free variables", + "type": "text" + }, + { + "bbox": [ + 336, + 523, + 387, + 534 + ], + "score": 0.92, + "content": "0 \\leq \\alpha _ { j } \\leq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 522, + 433, + 535 + ], + "score": 1.0, + "content": "only when", + "type": "text" + }, + { + "bbox": [ + 433, + 523, + 486, + 535 + ], + "score": 0.93, + "content": "\\mathbf { u } _ { j } > 0 > 1 _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 532, + 415, + 547 + ], + "spans": [ + { + "bbox": [ + 107, + 534, + 137, + 546 + ], + "score": 0.91, + "content": "w _ { j } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 532, + 300, + 547 + ], + "score": 1.0, + "content": ", while its rest values as well as constant", + "type": "text" + }, + { + "bbox": [ + 300, + 533, + 308, + 543 + ], + "score": 0.84, + "content": "b ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 532, + 384, + 547 + ], + "score": 1.0, + "content": "are determined by", + "type": "text" + }, + { + "bbox": [ + 384, + 534, + 411, + 545 + ], + "score": 0.85, + "content": "{ \\mathbf { l } } , { \\mathbf { u } } , w", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 532, + 415, + 547 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 106, + 522, + 506, + 547 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 504, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 305, + 566 + ], + "score": 1.0, + "content": "Detailed forms of each term are listed in Appendix", + "type": "text" + }, + { + "bbox": [ + 305, + 551, + 317, + 565 + ], + "score": 0.49, + "content": "\\mathbf { A } .", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 551, + 349, + 566 + ], + "score": 1.0, + "content": "Lemma", + "type": "text" + }, + { + "bbox": [ + 349, + 552, + 365, + 564 + ], + "score": 0.82, + "content": "\\boxed { 2 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 551, + 506, + 566 + ], + "score": 1.0, + "content": "can be repeatedly applied, resulting", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 562, + 487, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 476, + 576 + ], + "score": 1.0, + "content": "in an efficient back-substitution procedure to derive a linear lower bound of NN output w.r.t.", + "type": "text" + }, + { + "bbox": [ + 477, + 565, + 483, + 573 + ], + "score": 0.71, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 562, + 487, + 576 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 551, + 506, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 577, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 224, + 590 + ], + "score": 1.0, + "content": "Lemma 2.2 (CROWN bound", + "type": "text" + }, + { + "bbox": [ + 225, + 577, + 241, + 589 + ], + "score": 0.77, + "content": "[ \\overline { { | 4 6 | } }", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 576, + 288, + 590 + ], + "score": 1.0, + "content": "). 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Incomplete", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 141 + ], + "score": 1.0, + "content": "verifiers can then estimate the lower bound of each subdomain with relaxations. If the lower bound", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 208, + 154 + ], + "score": 1.0, + "content": "produced for subdomain", + "type": "text" + }, + { + "bbox": [ + 208, + 141, + 218, + 152 + ], + "score": 0.86, + "content": "\\mathcal { C } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 141, + 270, + 154 + ], + "score": 1.0, + "content": "(denoted by", + "type": "text" + }, + { + "bbox": [ + 270, + 141, + 288, + 156 + ], + "score": 0.76, + "content": "\\underline { { f } } _ { \\mathcal { C } _ { i } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 141, + 359, + 154 + ], + "score": 1.0, + "content": "is greater than 0,", + "type": "text" + }, + { + "bbox": [ + 360, + 141, + 369, + 152 + ], + "score": 0.84, + "content": "\\mathcal { C } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 141, + 506, + 154 + ], + "score": 1.0, + "content": "is verified; otherwise, we further", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 188, + 166 + ], + "score": 1.0, + "content": "branch over domain", + "type": "text" + }, + { + "bbox": [ + 188, + 154, + 198, + 165 + ], + "score": 0.87, + "content": "\\mathcal { C } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "by splitting another unstable ReLU neuron. The process terminates when all", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "subdomains are verified. The completeness is guaranteed when all unstable ReLU neurons are split.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 506, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "LP verifier with neuron split constraints. A popular incomplete verifier used in BaB is the LP", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 99, + 197, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 99, + 197, + 259, + 223 + ], + "score": 1.0, + "content": "verifier. Essentially, when we split the", + "type": "text" + }, + { + "bbox": [ + 260, + 204, + 266, + 215 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 197, + 338, + 223 + ], + "score": 1.0, + "content": "-th ReLU in layer", + "type": "text" + }, + { + "bbox": [ + 339, + 204, + 343, + 213 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 197, + 423, + 223 + ], + "score": 1.0, + "content": ", we can simply add", + "type": "text" + }, + { + "bbox": [ + 424, + 200, + 458, + 217 + ], + "score": 0.93, + "content": "z _ { j } ^ { ( i ) } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 197, + 470, + 223 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 470, + 200, + 505, + 217 + ], + "score": 0.91, + "content": "z _ { j } ^ { ( i ) } < 0", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 131, + 229 + ], + "score": 1.0, + "content": "to Eq.", + "type": "text" + }, + { + "bbox": [ + 131, + 214, + 141, + 230 + ], + "score": 0.4, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 215, + 436, + 229 + ], + "score": 1.0, + "content": "and get a linearly relaxed lower bound to each subdomain. We denote the", + "type": "text" + }, + { + "bbox": [ + 437, + 216, + 461, + 227 + ], + "score": 0.9, + "content": "\\mathcal { Z } ^ { + ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 215, + 479, + 229 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 480, + 216, + 504, + 227 + ], + "score": 0.88, + "content": "\\mathcal { Z } ^ { - ( i ) }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 419, + 240 + ], + "score": 1.0, + "content": "as the set of neuron indices with positive and negative split constraints in layer", + "type": "text" + }, + { + "bbox": [ + 420, + 229, + 424, + 238 + ], + "score": 0.61, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 228, + 505, + 240 + ], + "score": 1.0, + "content": ". We define the split", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 239, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 186, + 254 + ], + "score": 1.0, + "content": "constraints at layer", + "type": "text" + }, + { + "bbox": [ + 186, + 242, + 191, + 252 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 240, + 203, + 254 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 203, + 239, + 454, + 255 + ], + "score": 0.91, + "content": "\\mathcal { Z } ^ { ( i ) } : = \\{ z ^ { ( i ) } \\mid z _ { j _ { 1 } } ^ { ( i ) } \\geq 0 , z _ { j _ { 2 } } ^ { ( i ) } < 0 , \\forall j _ { 1 } \\in \\mathcal { Z } ^ { + ( i ) } , \\forall j _ { 2 } \\in \\mathcal { Z } ^ { - ( i ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 239, + 506, + 255 + ], + "score": 1.0, + "content": ". We denote", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 258, + 266 + ], + "score": 1.0, + "content": "the vector of all pre-ReLU neurons as", + "type": "text" + }, + { + "bbox": [ + 258, + 256, + 264, + 263 + ], + "score": 0.74, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 253, + 345, + 266 + ], + "score": 1.0, + "content": ", and we define a set", + "type": "text" + }, + { + "bbox": [ + 345, + 254, + 355, + 263 + ], + "score": 0.79, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 253, + 496, + 266 + ], + "score": 1.0, + "content": "to represent the split constraints on", + "type": "text" + }, + { + "bbox": [ + 496, + 256, + 502, + 263 + ], + "score": 0.7, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 253, + 506, + 266 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 233, + 276 + ], + "score": 0.88, + "content": "\\mathcal { Z } = \\mathcal { Z } ^ { ( 1 ) } \\cap \\mathcal { Z } ^ { ( 2 ) } \\cap \\cdot \\cdot \\cdot \\cap \\mathcal { Z } ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 262, + 406, + 279 + ], + "score": 1.0, + "content": ". For convenience, we also use the shorthand", + "type": "text" + }, + { + "bbox": [ + 406, + 264, + 505, + 276 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { Z } } ^ { ( i ) } : = \\mathcal { Z } ^ { ( 1 ) } \\cap \\cdot \\cdot \\cdot \\cap \\mathcal { Z } ^ { ( i ) }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 273, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 104, + 273, + 123, + 290 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 276, + 237, + 289 + ], + "score": 0.88, + "content": "\\tilde { z } ^ { ( i ) } : = \\{ z ^ { ( 1 ) } , z ^ { ( 2 ) } , \\cdot \\cdot \\cdot , z ^ { ( i ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 273, + 506, + 290 + ], + "score": 1.0, + "content": ". LP verifiers can easily handle these neuron split constraints but are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 288, + 507, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 507, + 300 + ], + "score": 1.0, + "content": "more expensive than bound propagation methods like CROWN and cannot be accelerated on GPUs.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "Branching strategy. Branching strategies (selecting which ReLU neuron to split) are generally", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 503, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 503, + 335 + ], + "score": 1.0, + "content": "agnostic to the incomplete verifier used in BaB but do affect the overall BaB performance. BaBSR [7]", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "is a widely used strategy in complete verifiers, which is based on an fast estimates on objective", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "improvements after splitting each neuron. The neuron with highest estimated improvement is selected", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 332, + 369 + ], + "score": 1.0, + "content": "for branching. Recently, Filtered Smart Branching (FSB)", + "type": "text" + }, + { + "bbox": [ + 332, + 355, + 349, + 367 + ], + "score": 0.68, + "content": "\\bar { \\mathbb { M } } \\bar { \\mathbb { I } }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 354, + 505, + 369 + ], + "score": 1.0, + "content": "improves BaBSR by mimicking strong", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "branching - it utilizes bound propagation methods to evaluate the best a few candidates proposed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "by BaBSR and chooses the one with largest improvement. Graph neural network (GNN) based", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 222, + 402 + ], + "score": 1.0, + "content": "branching was also proposed", + "type": "text" + }, + { + "bbox": [ + 222, + 388, + 239, + 399 + ], + "score": 0.79, + "content": "\\mathbb { \\left[ \\left. 2 3 \\right] \\right. }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 387, + 261, + 402 + ], + "score": 1.0, + "content": ". Our", + "type": "text" + }, + { + "bbox": [ + 262, + 389, + 269, + 399 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 387, + 506, + 402 + ], + "score": 1.0, + "content": "-CROWN BaB is a general complete verification framework", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 398, + 483, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 483, + 412 + ], + "score": 1.0, + "content": "fit for any potential branching strategy, and we evaluate both BaBSR and FSB in experiments.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 106, + 426, + 398, + 441 + ], + "lines": [ + { + "bbox": [ + 104, + 425, + 399, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 124, + 444 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 124, + 428, + 133, + 441 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 425, + 399, + 444 + ], + "score": 1.0, + "content": "-CROWN for Complete and Incomplete Verification", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 293, + 460 + ], + "score": 1.0, + "content": "In this section, we first give intuitions on how", + "type": "text" + }, + { + "bbox": [ + 293, + 448, + 300, + 459 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "-CROWN handles neuron split constraints without", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 366, + 470 + ], + "score": 1.0, + "content": "costly LP solvers. Then we formally state the main theorem of", + "type": "text" + }, + { + "bbox": [ + 366, + 459, + 373, + 470 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "-CROWN from both primal and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 401, + 483 + ], + "score": 1.0, + "content": "dual spaces, and discuss how to tighten the bounds using free parameters", + "type": "text" + }, + { + "bbox": [ + 402, + 470, + 423, + 481 + ], + "score": 0.32, + "content": "\\alpha , \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 469, + 506, + 483 + ], + "score": 1.0, + "content": ". Lastly, we propose", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 480, + 493, + 493 + ], + "spans": [ + { + "bbox": [ + 107, + 480, + 114, + 492 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 480, + 493, + 493 + ], + "score": 1.0, + "content": "-CROWN BaB, a complete verifier that is also a strong incomplete verifier when stopped early.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "title", + "bbox": [ + 105, + 500, + 436, + 513 + ], + "lines": [ + { + "bbox": [ + 104, + 498, + 430, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 498, + 128, + 516 + ], + "score": 1.0, + "content": "3.1", + "type": "text" + }, + { + "bbox": [ + 129, + 501, + 136, + 513 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 498, + 430, + 516 + ], + "score": 1.0, + "content": "-CROWN: Linear Bound Propagation with Neuron Split Constraints", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 102, + 516, + 500, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 501, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 501, + 529 + ], + "score": 1.0, + "content": "The NN verification problem under neuron split constraints can be seen as an optimization problem:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 276, + 534, + 335, + 553 + ], + "lines": [ + { + "bbox": [ + 276, + 534, + 335, + 553 + ], + "spans": [ + { + "bbox": [ + 276, + 534, + 335, + 553 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } _ { x \\in { \\mathcal { C } } , z \\in { \\mathcal { Z } } } f ( x ) .", + "type": "interline_equation", + "image_path": "29b81d7a4774cfc4fc8d7a4d038ccb0ce94f5f906fe65723fe3528954e84b817.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 276, + 534, + 335, + 553 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 451, + 574 + ], + "score": 1.0, + "content": "Bound propagation methods like CROWN can give a relatively tight lower bound for", + "type": "text" + }, + { + "bbox": [ + 451, + 560, + 505, + 573 + ], + "score": 0.89, + "content": "\\scriptstyle \\operatorname* { m i n } _ { x \\in { \\mathcal { C } } } f ( x )", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 310, + 585 + ], + "score": 1.0, + "content": "but they cannot handle the neuron split constraints", + "type": "text" + }, + { + "bbox": [ + 310, + 572, + 337, + 582 + ], + "score": 0.92, + "content": "z \\in { \\mathcal { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 572, + 506, + 585 + ], + "score": 1.0, + "content": ". Before we present our main theorem, we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 469, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 469, + 596 + ], + "score": 1.0, + "content": "first show the intuition on how to apply split constraints to the bound propagation process.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 504, + 623 + ], + "lines": [ + { + "bbox": [ + 104, + 596, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 596, + 356, + 613 + ], + "score": 1.0, + "content": "To encode the neuron splits, we first define diagonal matrix", + "type": "text" + }, + { + "bbox": [ + 356, + 598, + 422, + 610 + ], + "score": 0.89, + "content": "\\mathbf { S } ^ { ( i ) } ~ \\in ~ \\mathbb { R } ^ { d _ { i } \\times d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 596, + 450, + 613 + ], + "score": 1.0, + "content": "in Eq.", + "type": "text" + }, + { + "bbox": [ + 450, + 597, + 461, + 613 + ], + "score": 0.79, + "content": "3", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 596, + 488, + 613 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 488, + 599, + 505, + 611 + ], + "score": 0.77, + "content": "i \\in", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 609, + 413, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 222, + 623 + ], + "score": 0.89, + "content": "[ 1 , \\cdot \\cdot \\cdot L - 1 ] , j \\in [ 1 , \\cdot \\cdot \\cdot , d _ { i } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 609, + 413, + 624 + ], + "score": 1.0, + "content": "are indices of layers and neurons, respectively:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "interline_equation", + "bbox": [ + 130, + 628, + 481, + 647 + ], + "lines": [ + { + "bbox": [ + 130, + 628, + 481, + 647 + ], + "spans": [ + { + "bbox": [ + 130, + 628, + 481, + 647 + ], + "score": 0.92, + "content": "{ \\bf S } _ { j , j } ^ { ( i ) } = - 1 ( \\mathrm { i f ~ s p l i t ~ } z _ { j } ^ { ( i ) } \\geq 0 ) ; \\quad { \\bf S } _ { j , j } ^ { ( i ) } = + 1 ( \\mathrm { i f ~ s p l i t ~ } z _ { j } ^ { ( i ) } < 0 ) ; \\quad { \\bf S } _ { j , j } ^ { ( i ) } = 0 ( \\mathrm { i f ~ n o ~ s p l i t ~ } z _ { j } ^ { ( i ) } )", + "type": "interline_equation", + "image_path": "42c29c4edf3482f89d2d60c251dd0147602ce69cfed2a01e36de47e396d6a1ae.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 130, + 628, + 481, + 647 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 104, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 104, + 653, + 433, + 666 + ], + "score": 1.0, + "content": "We start from the last layer and derive linear bounds for each intermediate layer", + "type": "text" + }, + { + "bbox": [ + 433, + 654, + 448, + 665 + ], + "score": 0.89, + "content": "z ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 653, + 467, + 666 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 468, + 654, + 483, + 665 + ], + "score": 0.89, + "content": "\\hat { z } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 666, + 171, + 678 + ], + "score": 1.0, + "content": "both constraints", + "type": "text" + }, + { + "bbox": [ + 171, + 667, + 196, + 677 + ], + "score": 0.9, + "content": "x \\in { \\mathcal { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 666, + 213, + 678 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 667, + 240, + 677 + ], + "score": 0.89, + "content": "z \\in { \\mathcal { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 666, + 398, + 678 + ], + "score": 1.0, + "content": ". 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Incomplete", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 131, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 141 + ], + "score": 1.0, + "content": "verifiers can then estimate the lower bound of each subdomain with relaxations. 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The process terminates when all", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "subdomains are verified. The completeness is guaranteed when all unstable ReLU neurons are split.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 102, + 89, + 507, + 177 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 506, + 299 + ], + "lines": [ + { + "bbox": [ + 106, + 189, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 189, + 506, + 202 + ], + "score": 1.0, + "content": "LP verifier with neuron split constraints. A popular incomplete verifier used in BaB is the LP", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 99, + 197, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 99, + 197, + 259, + 223 + ], + "score": 1.0, + "content": "verifier. Essentially, when we split the", + "type": "text" + }, + { + "bbox": [ + 260, + 204, + 266, + 215 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 197, + 338, + 223 + ], + "score": 1.0, + "content": "-th ReLU in layer", + "type": "text" + }, + { + "bbox": [ + 339, + 204, + 343, + 213 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 197, + 423, + 223 + ], + "score": 1.0, + "content": ", we can simply add", + "type": "text" + }, + { + "bbox": [ + 424, + 200, + 458, + 217 + ], + "score": 0.93, + "content": "z _ { j } ^ { ( i ) } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 197, + 470, + 223 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 470, + 200, + 505, + 217 + ], + "score": 0.91, + "content": "z _ { j } ^ { ( i ) } < 0", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 214, + 504, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 131, + 229 + ], + "score": 1.0, + "content": "to Eq.", + "type": "text" + }, + { + "bbox": [ + 131, + 214, + 141, + 230 + ], + "score": 0.4, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 215, + 436, + 229 + ], + "score": 1.0, + "content": "and get a linearly relaxed lower bound to each subdomain. We denote the", + "type": "text" + }, + { + "bbox": [ + 437, + 216, + 461, + 227 + ], + "score": 0.9, + "content": "\\mathcal { Z } ^ { + ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 215, + 479, + 229 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 480, + 216, + 504, + 227 + ], + "score": 0.88, + "content": "\\mathcal { Z } ^ { - ( i ) }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 419, + 240 + ], + "score": 1.0, + "content": "as the set of neuron indices with positive and negative split constraints in layer", + "type": "text" + }, + { + "bbox": [ + 420, + 229, + 424, + 238 + ], + "score": 0.61, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 228, + 505, + 240 + ], + "score": 1.0, + "content": ". We define the split", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 239, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 186, + 254 + ], + "score": 1.0, + "content": "constraints at layer", + "type": "text" + }, + { + "bbox": [ + 186, + 242, + 191, + 252 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 240, + 203, + 254 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 203, + 239, + 454, + 255 + ], + "score": 0.91, + "content": "\\mathcal { Z } ^ { ( i ) } : = \\{ z ^ { ( i ) } \\mid z _ { j _ { 1 } } ^ { ( i ) } \\geq 0 , z _ { j _ { 2 } } ^ { ( i ) } < 0 , \\forall j _ { 1 } \\in \\mathcal { Z } ^ { + ( i ) } , \\forall j _ { 2 } \\in \\mathcal { Z } ^ { - ( i ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 239, + 506, + 255 + ], + "score": 1.0, + "content": ". We denote", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 258, + 266 + ], + "score": 1.0, + "content": "the vector of all pre-ReLU neurons as", + "type": "text" + }, + { + "bbox": [ + 258, + 256, + 264, + 263 + ], + "score": 0.74, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 253, + 345, + 266 + ], + "score": 1.0, + "content": ", and we define a set", + "type": "text" + }, + { + "bbox": [ + 345, + 254, + 355, + 263 + ], + "score": 0.79, + "content": "\\mathcal { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 253, + 496, + 266 + ], + "score": 1.0, + "content": "to represent the split constraints on", + "type": "text" + }, + { + "bbox": [ + 496, + 256, + 502, + 263 + ], + "score": 0.7, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 253, + 506, + 266 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 262, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 233, + 276 + ], + "score": 0.88, + "content": "\\mathcal { Z } = \\mathcal { Z } ^ { ( 1 ) } \\cap \\mathcal { Z } ^ { ( 2 ) } \\cap \\cdot \\cdot \\cdot \\cap \\mathcal { Z } ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 262, + 406, + 279 + ], + "score": 1.0, + "content": ". For convenience, we also use the shorthand", + "type": "text" + }, + { + "bbox": [ + 406, + 264, + 505, + 276 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { Z } } ^ { ( i ) } : = \\mathcal { Z } ^ { ( 1 ) } \\cap \\cdot \\cdot \\cdot \\cap \\mathcal { Z } ^ { ( i ) }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 273, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 104, + 273, + 123, + 290 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 276, + 237, + 289 + ], + "score": 0.88, + "content": "\\tilde { z } ^ { ( i ) } : = \\{ z ^ { ( 1 ) } , z ^ { ( 2 ) } , \\cdot \\cdot \\cdot , z ^ { ( i ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 273, + 506, + 290 + ], + "score": 1.0, + "content": ". LP verifiers can easily handle these neuron split constraints but are", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 288, + 507, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 507, + 300 + ], + "score": 1.0, + "content": "more expensive than bound propagation methods like CROWN and cannot be accelerated on GPUs.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12, + "bbox_fs": [ + 99, + 189, + 507, + 300 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 311, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 311, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 505, + 325 + ], + "score": 1.0, + "content": "Branching strategy. Branching strategies (selecting which ReLU neuron to split) are generally", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 322, + 503, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 503, + 335 + ], + "score": 1.0, + "content": "agnostic to the incomplete verifier used in BaB but do affect the overall BaB performance. BaBSR [7]", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "is a widely used strategy in complete verifiers, which is based on an fast estimates on objective", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "improvements after splitting each neuron. The neuron with highest estimated improvement is selected", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 332, + 369 + ], + "score": 1.0, + "content": "for branching. Recently, Filtered Smart Branching (FSB)", + "type": "text" + }, + { + "bbox": [ + 332, + 355, + 349, + 367 + ], + "score": 0.68, + "content": "\\bar { \\mathbb { M } } \\bar { \\mathbb { I } }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 354, + 505, + 369 + ], + "score": 1.0, + "content": "improves BaBSR by mimicking strong", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "branching - it utilizes bound propagation methods to evaluate the best a few candidates proposed", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "by BaBSR and chooses the one with largest improvement. Graph neural network (GNN) based", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 222, + 402 + ], + "score": 1.0, + "content": "branching was also proposed", + "type": "text" + }, + { + "bbox": [ + 222, + 388, + 239, + 399 + ], + "score": 0.79, + "content": "\\mathbb { \\left[ \\left. 2 3 \\right] \\right. }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 387, + 261, + 402 + ], + "score": 1.0, + "content": ". Our", + "type": "text" + }, + { + "bbox": [ + 262, + 389, + 269, + 399 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 387, + 506, + 402 + ], + "score": 1.0, + "content": "-CROWN BaB is a general complete verification framework", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 398, + 483, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 483, + 412 + ], + "score": 1.0, + "content": "fit for any potential branching strategy, and we evaluate both BaBSR and FSB in experiments.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 311, + 506, + 412 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 426, + 398, + 441 + ], + "lines": [ + { + "bbox": [ + 104, + 425, + 399, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 425, + 124, + 444 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 124, + 428, + 133, + 441 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 425, + 399, + 444 + ], + "score": 1.0, + "content": "-CROWN for Complete and Incomplete Verification", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 447, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 293, + 460 + ], + "score": 1.0, + "content": "In this section, we first give intuitions on how", + "type": "text" + }, + { + "bbox": [ + 293, + 448, + 300, + 459 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "-CROWN handles neuron split constraints without", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 366, + 470 + ], + "score": 1.0, + "content": "costly LP solvers. Then we formally state the main theorem of", + "type": "text" + }, + { + "bbox": [ + 366, + 459, + 373, + 470 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "-CROWN from both primal and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 401, + 483 + ], + "score": 1.0, + "content": "dual spaces, and discuss how to tighten the bounds using free parameters", + "type": "text" + }, + { + "bbox": [ + 402, + 470, + 423, + 481 + ], + "score": 0.32, + "content": "\\alpha , \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 469, + 506, + 483 + ], + "score": 1.0, + "content": ". Lastly, we propose", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 480, + 493, + 493 + ], + "spans": [ + { + "bbox": [ + 107, + 480, + 114, + 492 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 480, + 493, + 493 + ], + "score": 1.0, + "content": "-CROWN BaB, a complete verifier that is also a strong incomplete verifier when stopped early.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 448, + 506, + 493 + ] + }, + { + "type": "title", + "bbox": [ + 105, + 500, + 436, + 513 + ], + "lines": [ + { + "bbox": [ + 104, + 498, + 430, + 516 + ], + "spans": [ + { + "bbox": [ + 104, + 498, + 128, + 516 + ], + "score": 1.0, + "content": "3.1", + "type": "text" + }, + { + "bbox": [ + 129, + 501, + 136, + 513 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 498, + 430, + 516 + ], + "score": 1.0, + "content": "-CROWN: Linear Bound Propagation with Neuron Split Constraints", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 102, + 516, + 500, + 528 + ], + "lines": [ + { + "bbox": [ + 106, + 515, + 501, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 501, + 529 + ], + "score": 1.0, + "content": "The NN verification problem under neuron split constraints can be seen as an optimization problem:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 515, + 501, + 529 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 276, + 534, + 335, + 553 + ], + "lines": [ + { + "bbox": [ + 276, + 534, + 335, + 553 + ], + "spans": [ + { + "bbox": [ + 276, + 534, + 335, + 553 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } _ { x \\in { \\mathcal { C } } , z \\in { \\mathcal { Z } } } f ( x ) .", + "type": "interline_equation", + "image_path": "29b81d7a4774cfc4fc8d7a4d038ccb0ce94f5f906fe65723fe3528954e84b817.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 276, + 534, + 335, + 553 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 560, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 451, + 574 + ], + "score": 1.0, + "content": "Bound propagation methods like CROWN can give a relatively tight lower bound for", + "type": "text" + }, + { + "bbox": [ + 451, + 560, + 505, + 573 + ], + "score": 0.89, + "content": "\\scriptstyle \\operatorname* { m i n } _ { x \\in { \\mathcal { C } } } f ( x )", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 310, + 585 + ], + "score": 1.0, + "content": "but they cannot handle the neuron split constraints", + "type": "text" + }, + { + "bbox": [ + 310, + 572, + 337, + 582 + ], + "score": 0.92, + "content": "z \\in { \\mathcal { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 572, + 506, + 585 + ], + "score": 1.0, + "content": ". Before we present our main theorem, we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 582, + 469, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 469, + 596 + ], + "score": 1.0, + "content": "first show the intuition on how to apply split constraints to the bound propagation process.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 559, + 506, + 596 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 504, + 623 + ], + "lines": [ + { + "bbox": [ + 104, + 596, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 104, + 596, + 356, + 613 + ], + "score": 1.0, + "content": "To encode the neuron splits, we first define diagonal matrix", + "type": "text" + }, + { + "bbox": [ + 356, + 598, + 422, + 610 + ], + "score": 0.89, + "content": "\\mathbf { S } ^ { ( i ) } ~ \\in ~ \\mathbb { R } ^ { d _ { i } \\times d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 596, + 450, + 613 + ], + "score": 1.0, + "content": "in Eq.", + "type": "text" + }, + { + "bbox": [ + 450, + 597, + 461, + 613 + ], + "score": 0.79, + "content": "3", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 596, + 488, + 613 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 488, + 599, + 505, + 611 + ], + "score": 0.77, + "content": "i \\in", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 609, + 413, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 222, + 623 + ], + "score": 0.89, + "content": "[ 1 , \\cdot \\cdot \\cdot L - 1 ] , j \\in [ 1 , \\cdot \\cdot \\cdot , d _ { i } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 609, + 413, + 624 + ], + "score": 1.0, + "content": "are indices of layers and neurons, respectively:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 596, + 505, + 624 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 130, + 628, + 481, + 647 + ], + "lines": [ + { + "bbox": [ + 130, + 628, + 481, + 647 + ], + "spans": [ + { + "bbox": [ + 130, + 628, + 481, + 647 + ], + "score": 0.92, + "content": "{ \\bf S } _ { j , j } ^ { ( i ) } = - 1 ( \\mathrm { i f ~ s p l i t ~ } z _ { j } ^ { ( i ) } \\geq 0 ) ; \\quad { \\bf S } _ { j , j } ^ { ( i ) } = + 1 ( \\mathrm { i f ~ s p l i t ~ } z _ { j } ^ { ( i ) } < 0 ) ; \\quad { \\bf S } _ { j , j } ^ { ( i ) } = 0 ( \\mathrm { i f ~ n o ~ s p l i t ~ } z _ { j } ^ { ( i ) } )", + "type": "interline_equation", + "image_path": "42c29c4edf3482f89d2d60c251dd0147602ce69cfed2a01e36de47e396d6a1ae.jpg" + } + ] + } + ], + "index": 39, + "virtual_lines": [ + { + "bbox": [ + 130, + 628, + 481, + 647 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 104, + 653, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 104, + 653, + 433, + 666 + ], + "score": 1.0, + "content": "We start from the last layer and derive linear bounds for each intermediate layer", + "type": "text" + }, + { + "bbox": [ + 433, + 654, + 448, + 665 + ], + "score": 0.89, + "content": "z ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 653, + 467, + 666 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 468, + 654, + 483, + 665 + ], + "score": 0.89, + "content": "\\hat { z } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 653, + 505, + 666 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 104, + 666, + 171, + 678 + ], + "score": 1.0, + "content": "both constraints", + "type": "text" + }, + { + "bbox": [ + 171, + 667, + 196, + 677 + ], + "score": 0.9, + "content": "x \\in { \\mathcal { C } }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 666, + 213, + 678 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 214, + 667, + 240, + 677 + ], + "score": 0.89, + "content": "z \\in { \\mathcal { Z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 666, + 398, + 678 + ], + "score": 1.0, + "content": ". 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1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 81, + 504, + 99 + ], + "score": 1.0, + "content": "(we omit all constant terms to avoid clutter):", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 100, + 415, + 121 + ], + "lines": [ + { + "bbox": [ + 195, + 100, + 415, + 121 + ], + "spans": [ + { + "bbox": [ + 195, + 100, + 415, + 121 + ], + "score": 0.92, + "content": "\\operatorname* { m i n } _ { x \\in \\mathcal C , z \\in \\mathcal Z } f ( x ) \\ge \\operatorname* { m i n } _ { x \\in \\mathcal C , z \\in \\mathcal Z } \\mathbf W ^ { ( L ) } \\mathbf D ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \\mathrm { c o n s t . }", + "type": "interline_equation", + "image_path": "7f3e4767dc90b93cbbd550a079de48ee524afc6a2f7d63fe4c469b49059544ef.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 195, + 100, + 415, + 121 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 127, + 506, + 151 + ], + "lines": [ + { + "bbox": [ + 104, + 124, + 504, + 141 + ], + "spans": [ + { + "bbox": [ + 104, + 124, + 263, + 141 + ], + "score": 1.0, + "content": "To enforce the split neurons at layer", + "type": "text" + }, + { + "bbox": [ + 263, + 128, + 290, + 138 + ], + "score": 0.87, + "content": "L - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 124, + 438, + 141 + ], + "score": 1.0, + "content": ", we use a Lagrange function with", + "type": "text" + }, + { + "bbox": [ + 439, + 126, + 504, + 139 + ], + "score": 0.89, + "content": "\\beta ^ { ( L - 1 ) \\top } \\mathbf { S } ^ { ( L - 1 ) }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 196, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 196, + 151 + ], + "score": 1.0, + "content": "multiplied on z(L\u00001):", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 153, + 479, + 211 + ], + "lines": [ + { + "bbox": [ + 120, + 153, + 479, + 211 + ], + "spans": [ + { + "bbox": [ + 120, + 153, + 479, + 211 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle \\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\geq \\operatorname* { m i n } _ { \\bar { x } \\in \\mathcal { C } _ { \\mathcal { C } } ^ { ( L - 2 ) } \\in \\mathcal { L } ^ { ( L - 1 ) } \\geq 0 } \\operatorname* { m a x } _ { 0 ^ { ( L - 1 ) } \\geq 0 } \\mathbf { W } ^ { ( L ) } \\mathbf { D } ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) ^ { \\top } } \\mathbf { S } ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \\mathrm { c o n s t } } \\\\ { \\displaystyle \\qquad \\geq \\operatorname* { m a x } _ { \\beta ^ { ( L - 1 ) } \\geq 0 } \\operatorname* { m i n } _ { \\bar { x } \\in \\mathcal { Z } _ { \\mathcal { C } } ^ { ( L - 2 ) } \\in \\mathcal { Z } ( L - 2 ) } \\left( \\mathbf { W } ^ { ( L ) } \\mathbf { D } ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) ^ { \\top } } \\mathbf { S } ^ { ( L - 1 ) } \\right) z ^ { ( L - 1 ) } + \\mathrm { c o n s t } } \\end{array}", + "type": "interline_equation", + "image_path": "4881df06ffd2068cdaff4c089ecd7ab147f7dae2b8d0d373dae121415de3e9fa.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 120, + 153, + 479, + 172.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 120, + 172.33333333333334, + 479, + 191.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 120, + 191.66666666666669, + 479, + 211.00000000000003 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 506, + 292 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "The first inequality is due to the definition of the Lagrange function: we remove the constraint", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 99, + 223, + 512, + 262 + ], + "spans": [ + { + "bbox": [ + 99, + 229, + 512, + 262 + ], + "score": 1.0, + "content": "2 Zduality. Due to the design of S(L\u00001), neuron split z (L\u00001)j \u0000 0 has a negative multiplier \u0000\u0000(L\u00001)j", + "type": "text" + }, + { + "bbox": [ + 107, + 225, + 177, + 236 + ], + "score": 0.92, + "content": "z ^ { ( L - 1 ) } \\in \\mathcal { Z } ^ { ( \\bar { L } - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 223, + 506, + 240 + ], + "score": 1.0, + "content": "and use a multiplier to replace this constraint. The second inequality is due to weak", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 98, + 246, + 511, + 273 + ], + "spans": [ + { + "bbox": [ + 98, + 246, + 144, + 273 + ], + "score": 1.0, + "content": "and split", + "type": "text" + }, + { + "bbox": [ + 144, + 252, + 192, + 269 + ], + "score": 0.94, + "content": "z _ { j } ^ { ( L - 1 ) } < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 246, + 295, + 273 + ], + "score": 1.0, + "content": "has a positive multiplier", + "type": "text" + }, + { + "bbox": [ + 295, + 253, + 325, + 269 + ], + "score": 0.92, + "content": "\\beta _ { j } ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 246, + 351, + 273 + ], + "score": 1.0, + "content": ". Any", + "type": "text" + }, + { + "bbox": [ + 351, + 254, + 401, + 267 + ], + "score": 0.91, + "content": "\\beta ^ { ( L - 1 ) } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 246, + 511, + 273 + ], + "score": 1.0, + "content": "yields a lower bound for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 266, + 507, + 283 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 336, + 283 + ], + "score": 1.0, + "content": "the constrained optimization problem. Then we substitute", + "type": "text" + }, + { + "bbox": [ + 336, + 268, + 364, + 279 + ], + "score": 0.9, + "content": "z ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 266, + 386, + 283 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 386, + 267, + 489, + 280 + ], + "score": 0.91, + "content": "\\mathbf { W } ^ { ( L - 1 ) } \\hat { z } ^ { ( L - 2 ) } + \\mathbf { b } ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 266, + 507, + 283 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 278, + 152, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 278, + 152, + 294 + ], + "score": 1.0, + "content": "next layer:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 294, + 489, + 323 + ], + "lines": [ + { + "bbox": [ + 112, + 294, + 489, + 323 + ], + "spans": [ + { + "bbox": [ + 112, + 294, + 489, + 323 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\geq \\operatorname* { m a x } _ { \\beta ^ { ( L - 1 ) } \\geq 0 } \\operatorname* { m i n } _ { \\tilde { x } ^ { ( L - 2 ) } \\in \\mathcal { C } ( L - 2 ) } \\left( \\mathbf { W } ^ { ( L ) } \\mathbf { D } ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) ^ { \\top } } \\mathbf { S } ^ { ( L - 1 ) } \\right) \\mathbf { W } ^ { ( L - 1 ) } \\hat { z } ^ { ( L - 2 ) } + \\mathrm { c o n s t }", + "type": "interline_equation", + "image_path": "5458721cd6ce8eab141e678de6118d20e86f08ba767e7090d0b22e9ae7b2ef9a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 112, + 294, + 489, + 323 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 506, + 367 + ], + "lines": [ + { + "bbox": [ + 104, + 325, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 104, + 325, + 183, + 341 + ], + "score": 1.0, + "content": "We define a matrix", + "type": "text" + }, + { + "bbox": [ + 183, + 327, + 202, + 338 + ], + "score": 0.87, + "content": "\\mathbf { A } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 325, + 376, + 341 + ], + "score": 1.0, + "content": "to represent the linear relationship between", + "type": "text" + }, + { + "bbox": [ + 376, + 328, + 397, + 340 + ], + "score": 0.89, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 325, + 415, + 341 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 415, + 327, + 430, + 339 + ], + "score": 0.87, + "content": "\\hat { z } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 325, + 461, + 341 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 461, + 327, + 505, + 339 + ], + "score": 0.83, + "content": "\\mathbf { A } ^ { ( L - 1 ) } =", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 338, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 131, + 353 + ], + "score": 0.81, + "content": "\\mathbf { W } ^ { ( L ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 338, + 203, + 357 + ], + "score": 1.0, + "content": "according to Eq.", + "type": "text" + }, + { + "bbox": [ + 204, + 338, + 214, + 356 + ], + "score": 0.81, + "content": "_ 4", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 338, + 231, + 357 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 340, + 463, + 355 + ], + "score": 0.87, + "content": "\\mathbf { A } ^ { ( L - 2 ) } = \\bigl ( \\mathbf { A } ^ { ( L - 1 ) } \\mathbf { D } ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) } { } ^ { \\top } \\mathbf { S } ^ { ( L - 1 ) } \\bigr ) \\mathbf { W } ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 338, + 506, + 357 + ], + "score": 1.0, + "content": "by Eq. 6.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 351, + 469, + 370 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 238, + 370 + ], + "score": 1.0, + "content": "Considering 1-dimension output", + "type": "text" + }, + { + "bbox": [ + 238, + 354, + 281, + 367 + ], + "score": 0.36, + "content": "f ( \\boldsymbol { x } ) , \\mathbf { A } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 351, + 369, + 370 + ], + "score": 1.0, + "content": "has only 1 row. With", + "type": "text" + }, + { + "bbox": [ + 369, + 354, + 401, + 366 + ], + "score": 0.78, + "content": "\\mathbf { A } ^ { ( L - 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 351, + 469, + 370 + ], + "score": 1.0, + "content": ", Eq. 6 becomes:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 370, + 430, + 397 + ], + "lines": [ + { + "bbox": [ + 179, + 370, + 430, + 397 + ], + "spans": [ + { + "bbox": [ + 179, + 370, + 430, + 397 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\ge \\operatorname* { m a x } _ { \\beta ^ { ( L - 1 ) } \\ge 0 } \\operatorname* { m i n } _ { \\tilde { x } \\in \\mathcal { C } } \\quad \\mathbf { A } ^ { ( L - 2 ) } \\hat { z } ^ { ( L - 2 ) } + \\mathrm { c o n s t } ,", + "type": "interline_equation", + "image_path": "6ce4bd64f77b01732437fb947acb02645ff938fcf0d5deb3a877e08d82097e11.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 179, + 370, + 430, + 397 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 401, + 506, + 438 + ], + "lines": [ + { + "bbox": [ + 104, + 399, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 237, + 416 + ], + "score": 1.0, + "content": "which is in a form similar to Eq.", + "type": "text" + }, + { + "bbox": [ + 238, + 400, + 247, + 414 + ], + "score": 0.36, + "content": "\\boxed { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 399, + 406, + 416 + ], + "score": 1.0, + "content": "except for the outer maximization over", + "type": "text" + }, + { + "bbox": [ + 406, + 401, + 436, + 414 + ], + "score": 0.93, + "content": "\\beta ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 399, + 506, + 416 + ], + "score": 1.0, + "content": ". This allows the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 412, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 296, + 427 + ], + "score": 1.0, + "content": "back-substitution process (Eq. 4, Eq. 5, and Eq.", + "type": "text" + }, + { + "bbox": [ + 296, + 413, + 307, + 426 + ], + "score": 0.45, + "content": "\\textcircled { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 412, + 506, + 427 + ], + "score": 1.0, + "content": "to continue. In each step, we swap max and min", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 460, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 127, + 439 + ], + "score": 1.0, + "content": "as in", + "type": "text" + }, + { + "bbox": [ + 127, + 424, + 153, + 438 + ], + "score": 0.26, + "content": "\\mathrm { E q . } | \\overline { { 5 } } |", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 423, + 266, + 439 + ], + "score": 1.0, + "content": "so every maximization over", + "type": "text" + }, + { + "bbox": [ + 266, + 425, + 284, + 437 + ], + "score": 0.92, + "content": "\\beta ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 423, + 336, + 439 + ], + "score": 1.0, + "content": "is outside of", + "type": "text" + }, + { + "bbox": [ + 336, + 426, + 369, + 437 + ], + "score": 0.89, + "content": "\\operatorname* { m i n } _ { x \\in { \\mathcal { C } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 423, + 460, + 439 + ], + "score": 1.0, + "content": ". Eventually, we have:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 440, + 387, + 461 + ], + "lines": [ + { + "bbox": [ + 222, + 440, + 387, + 461 + ], + "spans": [ + { + "bbox": [ + 222, + 440, + 387, + 461 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\ge \\operatorname* { m a x } _ { \\beta \\ge 0 } \\operatorname* { m i n } _ { x \\in \\mathcal { C } } { \\mathbf { A } ^ { ( 0 ) } } x + \\mathrm { c o n s t } ,", + "type": "interline_equation", + "image_path": "b408a6b971acefbfee90b3ed9956949fec23c7985ad7a029e3b06dee82491cd5.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 222, + 440, + 387, + 461 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 275, + 480 + ], + "score": 1.0, + "content": "where \u0000 := ⇥ \u0000(1)> \u0000(2)> \u0000(L\u00001)>⇤>", + "type": "text" + }, + { + "bbox": [ + 280, + 466, + 347, + 482 + ], + "score": 1.0, + "content": "concatenates all", + "type": "text" + }, + { + "bbox": [ + 347, + 467, + 364, + 480 + ], + "score": 0.91, + "content": "\\beta ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 466, + 507, + 482 + ], + "score": 1.0, + "content": "vectors. 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2 ) } \\in \\mathcal { L } ^ { ( L - 1 ) } \\geq 0 } \\operatorname* { m a x } _ { 0 ^ { ( L - 1 ) } \\geq 0 } \\mathbf { W } ^ { ( L ) } \\mathbf { D } ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) ^ { \\top } } \\mathbf { S } ^ { ( L - 1 ) } z ^ { ( L - 1 ) } + \\mathrm { c o n s t } } \\\\ { \\displaystyle \\qquad \\geq \\operatorname* { m a x } _ { \\beta ^ { ( L - 1 ) } \\geq 0 } \\operatorname* { m i n } _ { \\bar { x } \\in \\mathcal { Z } _ { \\mathcal { C } } ^ { ( L - 2 ) } \\in \\mathcal { Z } ( L - 2 ) } \\left( \\mathbf { W } ^ { ( L ) } \\mathbf { D } ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) ^ { \\top } } \\mathbf { S } ^ { ( L - 1 ) } \\right) z ^ { ( L - 1 ) } + \\mathrm { c o n s t } } \\end{array}", + "type": "interline_equation", + "image_path": "4881df06ffd2068cdaff4c089ecd7ab147f7dae2b8d0d373dae121415de3e9fa.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 120, + 153, + 479, + 172.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 120, + 172.33333333333334, + 479, + 191.66666666666669 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 120, + 191.66666666666669, + 479, + 211.00000000000003 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 213, + 506, + 292 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 506, + 227 + ], + "score": 1.0, + "content": "The first inequality is due to the definition of the Lagrange function: we remove the constraint", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 99, + 223, + 512, + 262 + ], + "spans": [ + { + "bbox": [ + 99, + 229, + 512, + 262 + ], + "score": 1.0, + "content": "2 Zduality. Due to the design of S(L\u00001), neuron split z (L\u00001)j \u0000 0 has a negative multiplier \u0000\u0000(L\u00001)j", + "type": "text" + }, + { + "bbox": [ + 107, + 225, + 177, + 236 + ], + "score": 0.92, + "content": "z ^ { ( L - 1 ) } \\in \\mathcal { Z } ^ { ( \\bar { L } - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 223, + 506, + 240 + ], + "score": 1.0, + "content": "and use a multiplier to replace this constraint. The second inequality is due to weak", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 98, + 246, + 511, + 273 + ], + "spans": [ + { + "bbox": [ + 98, + 246, + 144, + 273 + ], + "score": 1.0, + "content": "and split", + "type": "text" + }, + { + "bbox": [ + 144, + 252, + 192, + 269 + ], + "score": 0.94, + "content": "z _ { j } ^ { ( L - 1 ) } < 0", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 246, + 295, + 273 + ], + "score": 1.0, + "content": "has a positive multiplier", + "type": "text" + }, + { + "bbox": [ + 295, + 253, + 325, + 269 + ], + "score": 0.92, + "content": "\\beta _ { j } ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 246, + 351, + 273 + ], + "score": 1.0, + "content": ". Any", + "type": "text" + }, + { + "bbox": [ + 351, + 254, + 401, + 267 + ], + "score": 0.91, + "content": "\\beta ^ { ( L - 1 ) } \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 246, + 511, + 273 + ], + "score": 1.0, + "content": "yields a lower bound for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 266, + 507, + 283 + ], + "spans": [ + { + "bbox": [ + 104, + 266, + 336, + 283 + ], + "score": 1.0, + "content": "the constrained optimization problem. Then we substitute", + "type": "text" + }, + { + "bbox": [ + 336, + 268, + 364, + 279 + ], + "score": 0.9, + "content": "z ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 266, + 386, + 283 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 386, + 267, + 489, + 280 + ], + "score": 0.91, + "content": "\\mathbf { W } ^ { ( L - 1 ) } \\hat { z } ^ { ( L - 2 ) } + \\mathbf { b } ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 266, + 507, + 283 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 278, + 152, + 294 + ], + "spans": [ + { + "bbox": [ + 104, + 278, + 152, + 294 + ], + "score": 1.0, + "content": "next layer:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 98, + 212, + 512, + 294 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 294, + 489, + 323 + ], + "lines": [ + { + "bbox": [ + 112, + 294, + 489, + 323 + ], + "spans": [ + { + "bbox": [ + 112, + 294, + 489, + 323 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\geq \\operatorname* { m a x } _ { \\beta ^ { ( L - 1 ) } \\geq 0 } \\operatorname* { m i n } _ { \\tilde { x } ^ { ( L - 2 ) } \\in \\mathcal { C } ( L - 2 ) } \\left( \\mathbf { W } ^ { ( L ) } \\mathbf { D } ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) ^ { \\top } } \\mathbf { S } ^ { ( L - 1 ) } \\right) \\mathbf { W } ^ { ( L - 1 ) } \\hat { z } ^ { ( L - 2 ) } + \\mathrm { c o n s t }", + "type": "interline_equation", + "image_path": "5458721cd6ce8eab141e678de6118d20e86f08ba767e7090d0b22e9ae7b2ef9a.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 112, + 294, + 489, + 323 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 506, + 367 + ], + "lines": [ + { + "bbox": [ + 104, + 325, + 505, + 341 + ], + "spans": [ + { + "bbox": [ + 104, + 325, + 183, + 341 + ], + "score": 1.0, + "content": "We define a matrix", + "type": "text" + }, + { + "bbox": [ + 183, + 327, + 202, + 338 + ], + "score": 0.87, + "content": "\\mathbf { A } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 325, + 376, + 341 + ], + "score": 1.0, + "content": "to represent the linear relationship between", + "type": "text" + }, + { + "bbox": [ + 376, + 328, + 397, + 340 + ], + "score": 0.89, + "content": "f ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 325, + 415, + 341 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 415, + 327, + 430, + 339 + ], + "score": 0.87, + "content": "\\hat { z } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 325, + 461, + 341 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 461, + 327, + 505, + 339 + ], + "score": 0.83, + "content": "\\mathbf { A } ^ { ( L - 1 ) } =", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 338, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 131, + 353 + ], + "score": 0.81, + "content": "\\mathbf { W } ^ { ( L ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 338, + 203, + 357 + ], + "score": 1.0, + "content": "according to Eq.", + "type": "text" + }, + { + "bbox": [ + 204, + 338, + 214, + 356 + ], + "score": 0.81, + "content": "_ 4", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 338, + 231, + 357 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 232, + 340, + 463, + 355 + ], + "score": 0.87, + "content": "\\mathbf { A } ^ { ( L - 2 ) } = \\bigl ( \\mathbf { A } ^ { ( L - 1 ) } \\mathbf { D } ^ { ( L - 1 ) } + \\beta ^ { ( L - 1 ) } { } ^ { \\top } \\mathbf { S } ^ { ( L - 1 ) } \\bigr ) \\mathbf { W } ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 338, + 506, + 357 + ], + "score": 1.0, + "content": "by Eq. 6.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 351, + 469, + 370 + ], + "spans": [ + { + "bbox": [ + 104, + 351, + 238, + 370 + ], + "score": 1.0, + "content": "Considering 1-dimension output", + "type": "text" + }, + { + "bbox": [ + 238, + 354, + 281, + 367 + ], + "score": 0.36, + "content": "f ( \\boldsymbol { x } ) , \\mathbf { A } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 351, + 369, + 370 + ], + "score": 1.0, + "content": "has only 1 row. With", + "type": "text" + }, + { + "bbox": [ + 369, + 354, + 401, + 366 + ], + "score": 0.78, + "content": "\\mathbf { A } ^ { ( L - 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 351, + 469, + 370 + ], + "score": 1.0, + "content": ", Eq. 6 becomes:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 325, + 506, + 370 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 370, + 430, + 397 + ], + "lines": [ + { + "bbox": [ + 179, + 370, + 430, + 397 + ], + "spans": [ + { + "bbox": [ + 179, + 370, + 430, + 397 + ], + "score": 0.91, + "content": "\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\ge \\operatorname* { m a x } _ { \\beta ^ { ( L - 1 ) } \\ge 0 } \\operatorname* { m i n } _ { \\tilde { x } \\in \\mathcal { C } } \\quad \\mathbf { A } ^ { ( L - 2 ) } \\hat { z } ^ { ( L - 2 ) } + \\mathrm { c o n s t } ,", + "type": "interline_equation", + "image_path": "6ce4bd64f77b01732437fb947acb02645ff938fcf0d5deb3a877e08d82097e11.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 179, + 370, + 430, + 397 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 401, + 506, + 438 + ], + "lines": [ + { + "bbox": [ + 104, + 399, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 237, + 416 + ], + "score": 1.0, + "content": "which is in a form similar to Eq.", + "type": "text" + }, + { + "bbox": [ + 238, + 400, + 247, + 414 + ], + "score": 0.36, + "content": "\\boxed { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 399, + 406, + 416 + ], + "score": 1.0, + "content": "except for the outer maximization over", + "type": "text" + }, + { + "bbox": [ + 406, + 401, + 436, + 414 + ], + "score": 0.93, + "content": "\\beta ^ { ( L - 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 399, + 506, + 416 + ], + "score": 1.0, + "content": ". This allows the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 412, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 296, + 427 + ], + "score": 1.0, + "content": "back-substitution process (Eq. 4, Eq. 5, and Eq.", + "type": "text" + }, + { + "bbox": [ + 296, + 413, + 307, + 426 + ], + "score": 0.45, + "content": "\\textcircled { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 412, + 506, + 427 + ], + "score": 1.0, + "content": "to continue. In each step, we swap max and min", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 460, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 127, + 439 + ], + "score": 1.0, + "content": "as in", + "type": "text" + }, + { + "bbox": [ + 127, + 424, + 153, + 438 + ], + "score": 0.26, + "content": "\\mathrm { E q . } | \\overline { { 5 } } |", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 423, + 266, + 439 + ], + "score": 1.0, + "content": "so every maximization over", + "type": "text" + }, + { + "bbox": [ + 266, + 425, + 284, + 437 + ], + "score": 0.92, + "content": "\\beta ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 423, + 336, + 439 + ], + "score": 1.0, + "content": "is outside of", + "type": "text" + }, + { + "bbox": [ + 336, + 426, + 369, + 437 + ], + "score": 0.89, + "content": "\\operatorname* { m i n } _ { x \\in { \\mathcal { C } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 423, + 460, + 439 + ], + "score": 1.0, + "content": ". Eventually, we have:", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 399, + 506, + 439 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 440, + 387, + 461 + ], + "lines": [ + { + "bbox": [ + 222, + 440, + 387, + 461 + ], + "spans": [ + { + "bbox": [ + 222, + 440, + 387, + 461 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { x \\in \\mathcal { C } , z \\in \\mathcal { Z } } f ( x ) \\ge \\operatorname* { m a x } _ { \\beta \\ge 0 } \\operatorname* { m i n } _ { x \\in \\mathcal { C } } { \\mathbf { A } ^ { ( 0 ) } } x + \\mathrm { c o n s t } ,", + "type": "interline_equation", + "image_path": "b408a6b971acefbfee90b3ed9956949fec23c7985ad7a029e3b06dee82491cd5.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 222, + 440, + 387, + 461 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 465, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 275, + 480 + ], + "score": 1.0, + "content": "where \u0000 := ⇥ \u0000(1)> \u0000(2)> \u0000(L\u00001)>⇤>", + "type": "text" + }, + { + "bbox": [ + 280, + 466, + 347, + 482 + ], + "score": 1.0, + "content": "concatenates all", + "type": "text" + }, + { + "bbox": [ + 347, + 467, + 364, + 480 + ], + "score": 0.91, + "content": "\\beta ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 466, + 507, + 482 + ], + "score": 1.0, + "content": "vectors. 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ˆz (i)j = 0, z(i)j < 0 (if j 2 Z \u0000(i) )", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 507, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 237, + 381 + ], + "score": 1.0, + "content": "Compared to the formulation in", + "type": "text" + }, + { + "bbox": [ + 237, + 369, + 254, + 380 + ], + "score": 0.82, + "content": "\\mathbb { \\lVert \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 367, + 505, + 381 + ], + "score": 1.0, + "content": ", we have neuron split constraints. 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ˆz (i)j = 0, z(i)j < 0 (if j 2 Z \u0000(i) )", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + } + ], + "index": 15, + "bbox_fs": [ + 102, + 256, + 506, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 507, + 403 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 237, + 381 + ], + "score": 1.0, + "content": "Compared to the formulation in", + "type": "text" + }, + { + "bbox": [ + 237, + 369, + 254, + 380 + ], + "score": 0.82, + "content": "\\mathbb { \\lVert \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 367, + 505, + 381 + ], + "score": 1.0, + "content": ", we have neuron split constraints. 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The objective", + "type": "text" + }, + { + "bbox": [ + 223, + 406, + 238, + 417 + ], + "score": 0.87, + "content": "d _ { L P }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 404, + 452, + 420 + ], + "score": 1.0, + "content": "for the dual problem of Eq. 10 can be represented as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 404, + 452, + 420 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 194, + 422, + 416, + 437 + ], + "lines": [ + { + "bbox": [ + 194, + 422, + 416, + 437 + ], + "spans": [ + { + "bbox": [ + 194, + 422, + 416, + 437 + ], + "score": 0.91, + "content": "d _ { L P } = - \\| \\pmb { a } + \\mathbf { P } \\pmb { \\beta } \\| _ { 1 } { \\cdot } \\epsilon + ( \\mathbf { P } ^ { \\top } x _ { 0 } + \\mathbf { q } ) ^ { \\top } \\pmb { \\beta } + \\pmb { a } ^ { \\top } x _ { 0 } + c ,", + "type": "interline_equation", + "image_path": "2a36bdf8f2ad41bafb23bfdf80952dffaa574b59fd39c50715d90166d950287b.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 194, + 422, + 416, + 437 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 442, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 133, + 455 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 443, + 154, + 453 + ], + "score": 0.34, + "content": "{ \\mathbf { } } _ { { \\mathbf { } } _ { a , { \\mathrm { ~ P ~ } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 442, + 363, + 455 + ], + "score": 1.0, + "content": ", q and c are defined in the same way as in Theorem", + "type": "text" + }, + { + "bbox": [ + 363, + 442, + 381, + 455 + ], + "score": 0.36, + "content": "3 . { { I } _ { : } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 442, + 400, + 455 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 400, + 443, + 426, + 454 + ], + "score": 0.89, + "content": "\\beta \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "corresponds to the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 313, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 298, + 468 + ], + "score": 1.0, + "content": "dual variables of neuron split constraints in Eq.", + "type": "text" + }, + { + "bbox": [ + 298, + 453, + 313, + 466 + ], + "score": 0.34, + "content": "\\boxed { I O }", + "type": "inline_equation" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 442, + 505, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 506, + 508 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 371, + 486 + ], + "score": 1.0, + "content": "A similar connection between CROWN and dual LP based verifier", + "type": "text" + }, + { + "bbox": [ + 372, + 474, + 389, + 485 + ], + "score": 0.85, + "content": "[ \\mathbb { A } 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 474, + 446, + 486 + ], + "score": 1.0, + "content": "was shown in", + "type": "text" + }, + { + "bbox": [ + 447, + 474, + 464, + 486 + ], + "score": 0.85, + "content": "\\pmb { \\mathbb { B } } 0 \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 474, + 506, + 486 + ], + "score": 1.0, + "content": ", and their", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 485, + 504, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 302, + 496 + ], + "score": 1.0, + "content": "results can be seen as a special case of ours when", + "type": "text" + }, + { + "bbox": [ + 302, + 485, + 329, + 496 + ], + "score": 0.91, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 485, + 504, + 496 + ], + "score": 1.0, + "content": "(none of the split constraints are active). 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For completeness, it suffices to show", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 300, + 423 + ], + "score": 1.0, + "content": "that when all unstable ReLU neurons are split,", + "type": "text" + }, + { + "bbox": [ + 301, + 411, + 308, + 421 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 409, + 505, + 423 + ], + "score": 1.0, + "content": "-CROWN gives the global minimum for Eq. 10.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 244, + 433 + ], + "score": 1.0, + "content": "In contrast, combining CROWN", + "type": "text" + }, + { + "bbox": [ + 245, + 420, + 262, + 432 + ], + "score": 0.62, + "content": "\\left[ \\left[ 4 6 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "with BaB does not yield a complete verifier, as it cannot", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 450, + 444 + ], + "score": 1.0, + "content": "detect infeasible splits and a slow LP solver is still needed to guarantee completeness", + "type": "text" + }, + { + "bbox": [ + 450, + 432, + 468, + 443 + ], + "score": 0.48, + "content": "\\lVert \\rVert \\dot { \\boldsymbol { \\mathrm { E } } } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 432, + 506, + 444 + ], + "score": 1.0, + "content": ". Instead,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 107, + 443, + 114, + 454 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "-CROWN can detect infeasible subdomains - according to duality theory, an infeasible primal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "problem leads to an unbounded dual objective, which can be detected (see Sec. B.3 for more details).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 506, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "score": 1.0, + "content": "Additionally, we show the potential of early stopping a complete verifier as an incomplete verifier.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 480, + 507, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 282, + 494 + ], + "score": 1.0, + "content": "BaB approaches the exact solution of Eq.", + "type": "text" + }, + { + "bbox": [ + 282, + 481, + 291, + 494 + ], + "score": 0.41, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 480, + 507, + 494 + ], + "score": 1.0, + "content": "by splitting the problem into multiple subdomains,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 507, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 339, + 506 + ], + "score": 1.0, + "content": "and more subdomains give a tighter lower bound for Eq.", + "type": "text" + }, + { + "bbox": [ + 340, + 492, + 350, + 505 + ], + "score": 0.25, + "content": "^ { 1 . }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 491, + 507, + 506 + ], + "score": 1.0, + "content": "Unlike traditional complete verifiers,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 501, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 107, + 503, + 114, + 514 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 501, + 506, + 517 + ], + "score": 1.0, + "content": "-CROWN is efficient to explore a large number of subdomains during a very short time, making", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 513, + 406, + 526 + ], + "spans": [ + { + "bbox": [ + 107, + 515, + 114, + 525 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 513, + 406, + 526 + ], + "score": 1.0, + "content": "-CROWN BaB an attractive solution for efficient incomplete verification.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 107, + 537, + 237, + 550 + ], + "lines": [ + { + "bbox": [ + 104, + 536, + 237, + 553 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 237, + 553 + ], + "score": 1.0, + "content": "4 Experimental Results", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 108, + 554, + 276, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 277, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 277, + 568 + ], + "score": 1.0, + "content": "4.1 Comparison to Complete Verifiers", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 507, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 394, + 581 + ], + "score": 1.0, + "content": "We evaluate complete verification performance on dataset provided in", + "type": "text" + }, + { + "bbox": [ + 395, + 568, + 428, + 580 + ], + "score": 0.77, + "content": "\\mathbb { \\left[ 2 3 , \\overline { { 1 0 } } \\right] }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 568, + 507, + 581 + ], + "score": 1.0, + "content": "and used in VNN-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 163, + 592 + ], + "score": 1.0, + "content": "COMP 2020", + "type": "text" + }, + { + "bbox": [ + 163, + 579, + 181, + 591 + ], + "score": 0.37, + "content": "\\vec { \\mathbb { B } 2 2 } \\rvert \\vec { | }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 579, + 506, + 592 + ], + "score": 1.0, + "content": ". The benchmark contains three CIFAR-10 models (Base, Wide, and Deep)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 383, + 603 + ], + "score": 1.0, + "content": "with 100 examples each. Each data example is associated with an", + "type": "text" + }, + { + "bbox": [ + 383, + 591, + 397, + 602 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 590, + 423, + 603 + ], + "score": 1.0, + "content": "norm", + "type": "text" + }, + { + "bbox": [ + 424, + 593, + 430, + 601 + ], + "score": 0.53, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "and a target label", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "for verification (referred to as a property to verify). The details of neural network structures and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 287, + 625 + ], + "score": 1.0, + "content": "experimental setups can be found in Appendix", + "type": "text" + }, + { + "bbox": [ + 288, + 612, + 299, + 625 + ], + "score": 0.53, + "content": "\\mathbf { C } .", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "We compare against multiple baselines for complete", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 427, + 635 + ], + "score": 1.0, + "content": "verification: (1) BaBSR [7], a basic BaB and LP based verifier; (2) MIPplanet", + "type": "text" + }, + { + "bbox": [ + 427, + 623, + 445, + 635 + ], + "score": 0.64, + "content": "\\mathbb { \\lVert \\rVert \\cdot \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 624, + 505, + 635 + ], + "score": 1.0, + "content": ", a customized", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 632, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 506, + 648 + ], + "score": 1.0, + "content": "MIP solver for NN verification where unstable ReLU neurons are randomly selected for splitting;", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "(3) ERAN [35, 33, 36, 34], an abstract interpretation based verifier which performs well on this", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 323, + 668 + ], + "score": 1.0, + "content": "benchmark in VNN-COMP 2020; 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Typically, in many previous works", + "type": "text" + }, + { + "bbox": [ + 463, + 199, + 502, + 211 + ], + "score": 0.38, + "content": "\\boxed { 8 } \\boxed { 2 3 } \\boxed { 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 199, + 506, + 213 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 223 + ], + "score": 1.0, + "content": "when the LP formulation Eq. 10 is formed, intermediate layer bounds are pre-computed with bound", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 221, + 507, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 200, + 235 + ], + "score": 1.0, + "content": "propagation procedures", + "type": "text" + }, + { + "bbox": [ + 200, + 222, + 227, + 233 + ], + "score": 0.81, + "content": "\\boxed { 8 } \\boxed { 2 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 221, + 507, + 235 + ], + "score": 1.0, + "content": ", which are far from optimal. To estimate the dimension of this problem,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 233, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 314, + 246 + ], + "score": 1.0, + "content": "we denote the number of unstable neurons at layer", + "type": "text" + }, + { + "bbox": [ + 315, + 234, + 320, + 244 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 233, + 331, + 246 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 332, + 233, + 397, + 246 + ], + "score": 0.93, + "content": "s _ { i } : = \\operatorname { T r } ( | \\mathbf { S } ^ { ( i ) } | )", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 233, + 489, + 246 + ], + "score": 1.0, + "content": ". Each neuron in layer", + "type": "text" + }, + { + "bbox": [ + 489, + 234, + 495, + 244 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 233, + 505, + 246 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 102, + 243, + 509, + 266 + ], + "spans": [ + { + "bbox": [ + 102, + 243, + 172, + 266 + ], + "score": 1.0, + "content": "associated with", + "type": "text" + }, + { + "bbox": [ + 172, + 245, + 226, + 259 + ], + "score": 0.91, + "content": "2 \\times \\textstyle \\sum _ { k = 1 } ^ { i - 1 } s _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 243, + 267, + 266 + ], + "score": 1.0, + "content": "variables", + "type": "text" + }, + { + "bbox": [ + 267, + 247, + 278, + 257 + ], + "score": 0.85, + "content": "\\mathbf { { \\boldsymbol { \\alpha } } ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 243, + 410, + 266 + ], + "score": 1.0, + "content": ". Suppose each hidden layer has", + "type": "text" + }, + { + "bbox": [ + 410, + 247, + 417, + 257 + ], + "score": 0.75, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 243, + 454, + 266 + ], + "score": 1.0, + "content": "neurons", + "type": "text" + }, + { + "bbox": [ + 455, + 246, + 501, + 259 + ], + "score": 0.89, + "content": "( s _ { i } = { O } ( d ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 243, + 509, + 266 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 102, + 252, + 510, + 284 + ], + "spans": [ + { + "bbox": [ + 102, + 255, + 204, + 278 + ], + "score": 1.0, + "content": "then ↵ˆ has 2 ⇥ PL\u00001i=1 d", + "type": "text" + }, + { + "bbox": [ + 153, + 259, + 295, + 273 + ], + "score": 0.82, + "content": "\\begin{array} { r } { 2 \\times \\sum _ { i = 1 } ^ { L - 1 } d _ { i } \\sum _ { k = 1 } ^ { i - 1 } s _ { k } = O ( L ^ { 2 } d ^ { 2 } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 252, + 510, + 284 + ], + "score": 1.0, + "content": "variables in total. 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Note that a weaker form of joint optimization was", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 456, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 176, + 306 + ], + "score": 1.0, + "content": "also discussed in", + "type": "text" + }, + { + "bbox": [ + 176, + 294, + 194, + 304 + ], + "score": 0.75, + "content": "\\lVert \\boldsymbol { \\mathsf { \\overline { { 4 5 } } } } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 293, + 227, + 306 + ], + "score": 1.0, + "content": "without", + "type": "text" + }, + { + "bbox": [ + 228, + 294, + 236, + 305 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 293, + 456, + 306 + ], + "score": 1.0, + "content": ", and a detailed analysis can be found in Appendix B.2.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11.5, + "bbox_fs": [ + 102, + 165, + 510, + 306 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 317, + 361, + 329 + ], + "lines": [ + { + "bbox": [ + 104, + 316, + 362, + 332 + ], + "spans": [ + { + "bbox": [ + 104, + 316, + 128, + 332 + ], + "score": 1.0, + "content": "3.4", + "type": "text" + }, + { + "bbox": [ + 129, + 318, + 136, + 329 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 316, + 288, + 332 + ], + "score": 1.0, + "content": "-CROWN with Branch and Bound (", + "type": "text" + }, + { + "bbox": [ + 288, + 318, + 295, + 329 + ], + "score": 0.75, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 316, + 362, + 332 + ], + "score": 1.0, + "content": "-CROWN BaB)", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 505, + 377 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 348, + 346 + ], + "score": 1.0, + "content": "We perform complete verification following BaB framework", + "type": "text" + }, + { + "bbox": [ + 348, + 332, + 361, + 344 + ], + "score": 0.82, + "content": "\\pmb { \\bigtriangledown }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 331, + 385, + 346 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 386, + 333, + 393, + 344 + ], + "score": 0.81, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 331, + 506, + 346 + ], + "score": 1.0, + "content": "-CROWN as the incomplete", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 343, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 351, + 356 + ], + "score": 1.0, + "content": "solver, and we use simple branching heuristics like BaBSR", + "type": "text" + }, + { + "bbox": [ + 351, + 344, + 363, + 355 + ], + "score": 0.58, + "content": "\\mathbb { U }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 343, + 397, + 356 + ], + "score": 1.0, + "content": "or FSB", + "type": "text" + }, + { + "bbox": [ + 397, + 343, + 415, + 354 + ], + "score": 0.37, + "content": "\\mathbb { \\ m }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 343, + 506, + 356 + ], + "score": 1.0, + "content": ". To efficiently utilize", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "GPU, we also use batch splits to evaluate multiple subdomains in the same batch as in [44, 10]. We", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 365, + 501, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 194, + 378 + ], + "score": 1.0, + "content": "list our full algorithm", + "type": "text" + }, + { + "bbox": [ + 195, + 366, + 202, + 377 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 365, + 501, + 378 + ], + "score": 1.0, + "content": "-CROWN BaB in Appendix B and we show it is sound and complete here:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 331, + 506, + 378 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 480, + 391 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 480, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 167, + 393 + ], + "score": 1.0, + "content": "Theorem 3.3.", + "type": "text" + }, + { + "bbox": [ + 167, + 379, + 174, + 391 + ], + "score": 0.79, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 378, + 480, + 393 + ], + "score": 1.0, + "content": "-CROWN with Branch and Bound on splitting ReLUs is sound and complete.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 378, + 480, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 398, + 505, + 465 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 224, + 411 + ], + "score": 1.0, + "content": "Soundness is trivial because", + "type": "text" + }, + { + "bbox": [ + 224, + 400, + 231, + 411 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "-CROWN is a sound verifier. For completeness, it suffices to show", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 409, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 300, + 423 + ], + "score": 1.0, + "content": "that when all unstable ReLU neurons are split,", + "type": "text" + }, + { + "bbox": [ + 301, + 411, + 308, + 421 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 409, + 505, + 423 + ], + "score": 1.0, + "content": "-CROWN gives the global minimum for Eq. 10.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 420, + 244, + 433 + ], + "score": 1.0, + "content": "In contrast, combining CROWN", + "type": "text" + }, + { + "bbox": [ + 245, + 420, + 262, + 432 + ], + "score": 0.62, + "content": "\\left[ \\left[ 4 6 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "with BaB does not yield a complete verifier, as it cannot", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 450, + 444 + ], + "score": 1.0, + "content": "detect infeasible splits and a slow LP solver is still needed to guarantee completeness", + "type": "text" + }, + { + "bbox": [ + 450, + 432, + 468, + 443 + ], + "score": 0.48, + "content": "\\lVert \\rVert \\dot { \\boldsymbol { \\mathrm { E } } } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 432, + 506, + 444 + ], + "score": 1.0, + "content": ". Instead,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 443, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 107, + 443, + 114, + 454 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 443, + 505, + 455 + ], + "score": 1.0, + "content": "-CROWN can detect infeasible subdomains - according to duality theory, an infeasible primal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 453, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 506, + 466 + ], + "score": 1.0, + "content": "problem leads to an unbounded dual objective, which can be detected (see Sec. B.3 for more details).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 399, + 506, + 466 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 506, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 506, + 483 + ], + "score": 1.0, + "content": "Additionally, we show the potential of early stopping a complete verifier as an incomplete verifier.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 480, + 507, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 282, + 494 + ], + "score": 1.0, + "content": "BaB approaches the exact solution of Eq.", + "type": "text" + }, + { + "bbox": [ + 282, + 481, + 291, + 494 + ], + "score": 0.41, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 480, + 507, + 494 + ], + "score": 1.0, + "content": "by splitting the problem into multiple subdomains,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 507, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 339, + 506 + ], + "score": 1.0, + "content": "and more subdomains give a tighter lower bound for Eq.", + "type": "text" + }, + { + "bbox": [ + 340, + 492, + 350, + 505 + ], + "score": 0.25, + "content": "^ { 1 . }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 491, + 507, + 506 + ], + "score": 1.0, + "content": "Unlike traditional complete verifiers,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 501, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 107, + 503, + 114, + 514 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 501, + 506, + 517 + ], + "score": 1.0, + "content": "-CROWN is efficient to explore a large number of subdomains during a very short time, making", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 513, + 406, + 526 + ], + "spans": [ + { + "bbox": [ + 107, + 515, + 114, + 525 + ], + "score": 0.82, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 513, + 406, + 526 + ], + "score": 1.0, + "content": "-CROWN BaB an attractive solution for efficient incomplete verification.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 469, + 507, + 526 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 537, + 237, + 550 + ], + "lines": [ + { + "bbox": [ + 104, + 536, + 237, + 553 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 237, + 553 + ], + "score": 1.0, + "content": "4 Experimental Results", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 108, + 554, + 276, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 277, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 277, + 568 + ], + "score": 1.0, + "content": "4.1 Comparison to Complete Verifiers", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 106, + 568, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 568, + 507, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 394, + 581 + ], + "score": 1.0, + "content": "We evaluate complete verification performance on dataset provided in", + "type": "text" + }, + { + "bbox": [ + 395, + 568, + 428, + 580 + ], + "score": 0.77, + "content": "\\mathbb { \\left[ 2 3 , \\overline { { 1 0 } } \\right] }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 568, + 507, + 581 + ], + "score": 1.0, + "content": "and used in VNN-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 579, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 163, + 592 + ], + "score": 1.0, + "content": "COMP 2020", + "type": "text" + }, + { + "bbox": [ + 163, + 579, + 181, + 591 + ], + "score": 0.37, + "content": "\\vec { \\mathbb { B } 2 2 } \\rvert \\vec { | }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 579, + 506, + 592 + ], + "score": 1.0, + "content": ". The benchmark contains three CIFAR-10 models (Base, Wide, and Deep)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 383, + 603 + ], + "score": 1.0, + "content": "with 100 examples each. Each data example is associated with an", + "type": "text" + }, + { + "bbox": [ + 383, + 591, + 397, + 602 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 590, + 423, + 603 + ], + "score": 1.0, + "content": "norm", + "type": "text" + }, + { + "bbox": [ + 424, + 593, + 430, + 601 + ], + "score": 0.53, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "and a target label", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "for verification (referred to as a property to verify). The details of neural network structures and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 612, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 612, + 287, + 625 + ], + "score": 1.0, + "content": "experimental setups can be found in Appendix", + "type": "text" + }, + { + "bbox": [ + 288, + 612, + 299, + 625 + ], + "score": 0.53, + "content": "\\mathbf { C } .", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 612, + 506, + 625 + ], + "score": 1.0, + "content": "We compare against multiple baselines for complete", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 427, + 635 + ], + "score": 1.0, + "content": "verification: (1) BaBSR [7], a basic BaB and LP based verifier; (2) MIPplanet", + "type": "text" + }, + { + "bbox": [ + 427, + 623, + 445, + 635 + ], + "score": 0.64, + "content": "\\mathbb { \\lVert \\rVert \\cdot \\rVert }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 624, + 505, + 635 + ], + "score": 1.0, + "content": ", a customized", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 632, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 506, + 648 + ], + "score": 1.0, + "content": "MIP solver for NN verification where unstable ReLU neurons are randomly selected for splitting;", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "(3) ERAN [35, 33, 36, 34], an abstract interpretation based verifier which performs well on this", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 323, + 668 + ], + "score": 1.0, + "content": "benchmark in VNN-COMP 2020; 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(6)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 686, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 400, + 703 + ], + "score": 1.0, + "content": "OVAL (BDD+ GNN) [6, 23], a strong verifier in VNN-COMP 2020 using", + "type": "text" + }, + { + "bbox": [ + 400, + 689, + 428, + 699 + ], + "score": 0.45, + "content": "\\mathrm { B D D + }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 686, + 506, + 703 + ], + "score": 1.0, + "content": "with GNN guiding", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 359, + 712 + ], + "score": 1.0, + "content": "the ReLU splits; (7) A.set BaBSR and (8) Big-M+A.set BaBSR", + "type": "text" + }, + { + "bbox": [ + 359, + 700, + 376, + 711 + ], + "score": 0.28, + "content": "\\mathbb { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 699, + 506, + 712 + ], + "score": 1.0, + "content": ", very recent dual-space verifiers", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "on GPUs with a tighter linear relaxation than triangle LP relaxations; (9) Fast-and-Complete [45],", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "which uses CROWN (LiRPA) on GPUs as the incomplete verifier in BaB without neuron split", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 214, + 349 + ], + "score": 1.0, + "content": "constraints; (10) BaDNB", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 214, + 336, + 244, + 347 + ], + "score": 0.34, + "content": "\\mathrm { B D D + }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 244, + 335, + 268, + 349 + ], + "score": 1.0, + "content": "FSB)", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 268, + 335, + 286, + 347 + ], + "score": 0.68, + "content": "\\mathbb { \\ m }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 286, + 335, + 506, + 349 + ], + "score": 1.0, + "content": ", a concurrent state-of-the-art complete verifier, using", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 136, + 358 + ], + "score": 0.62, + "content": "\\mathrm { B D D + }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 136, + 347, + 305, + 360 + ], + "score": 1.0, + "content": "on GPUs with FSB branching strategy.", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 306, + 347, + 313, + 359 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 313, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "-CROWN BaB can use either BaBSR or FSB", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 358, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 369 + ], + "score": 1.0, + "content": "branching heuristic, and we include both in evaluation. All methods use a 1 hour timeout threshold.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 43.5, + "bbox_fs": [ + 104, + 568, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 104, + 83, + 507, + 205 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 110, + 70, + 498, + 81 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 68, + 498, + 83 + ], + "spans": [ + { + "bbox": [ + 111, + 68, + 498, + 83 + ], + "score": 1.0, + "content": "Table 1: Average runtime and average number of branches on three CIFAR-10 models over 100 properties.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 104, + 83, + 507, + 205 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 83, + 507, + 205 + ], + "spans": [ + { + "bbox": [ + 104, + 83, + 507, + 205 + ], + "score": 0.968, + "html": "
CIFAR-10 BaseCIFAR-10 WideCIFAR-10 Deep
Methodtime(s)branches%timeouttime(s)branches%timeouttime(s)branches%timeout
BaBSR2367.781020.5536.002871.14812.6549.002750.75401.2839.00
MIPplanet 152849.69-68.002417.53-46.002302.25140.00
ERAN*[35333634805.94=5.00632.20=9.00545.72=0.00
GNN-online [231794.85565.1333.001367.38372.7415.001055.33131.854.00
BDD+ BaBSR 6807.91195480.1420.00505.6574203.1110.00266.2812722.744.00
OVAL (BDD+ GNN)*[623]662.1767938.3816.00280.3817895.946.0094.691990.341.00
A.set BaBSR10381.7812004.607.00165.912233.103.00190.282491.552.00
BigM+A.setBaBSR[10390.4411938.757.00172.654050.593.00177.223275.252.00
Fast-and-Complete [45]695.01119522.6517.00495.8880519.859.00105.642455.111.00
BaDNB (BDD+ FSB)II309.2938239.047.00165.5311214.444.0010.50368.160.00
β-CROWNBaBSR226.06509608.506.00118.26217691.243.006.12204.660.00
β-CROWN FSB118.23208018.213.0078.32116912.572.005.6941.120.00
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Other results were reported by their authors.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "image_body", + "bbox": [ + 109, + 219, + 496, + 297 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 219, + 496, + 297 + ], + "spans": [ + { + "bbox": [ + 109, + 219, + 496, + 297 + ], + "score": 0.955, + "type": "image", + "image_path": "e15f89e6f30fcd3a3be38f99210245d62207d65a1866c03408e05d561e72706d.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 109, + 219, + 496, + 245.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 109, + 245.0, + 496, + 271.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 109, + 271.0, + 496, + 297.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 299, + 504, + 319 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 299, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 375, + 310 + ], + "score": 1.0, + "content": "Figure 1: Percentage of solved properties with growing running time.", + "type": "text" + }, + { + "bbox": [ + 375, + 299, + 382, + 309 + ], + "score": 0.81, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 299, + 505, + 310 + ], + "score": 1.0, + "content": "-CROWN FSB (light green) and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 107, + 309, + 113, + 319 + ], + "score": 0.8, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 308, + 379, + 321 + ], + "score": 1.0, + "content": "-CROWN BaBSR (dark green) clearly lead in all 3 settings and solve over", + "type": "text" + }, + { + "bbox": [ + 379, + 309, + 397, + 318 + ], + "score": 0.86, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "properties within 10 seconds.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 338 + ], + "score": 1.0, + "content": "which uses CROWN (LiRPA) on GPUs as the incomplete verifier in BaB without neuron split", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 214, + 349 + ], + "score": 1.0, + "content": "constraints; (10) BaDNB", + "type": "text" + }, + { + "bbox": [ + 214, + 336, + 244, + 347 + ], + "score": 0.34, + "content": "\\mathrm { B D D + }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 335, + 268, + 349 + ], + "score": 1.0, + "content": "FSB)", + "type": "text" + }, + { + "bbox": [ + 268, + 335, + 286, + 347 + ], + "score": 0.68, + "content": "\\mathbb { \\ m }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 335, + 506, + 349 + ], + "score": 1.0, + "content": ", a concurrent state-of-the-art complete verifier, using", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 136, + 358 + ], + "score": 0.62, + "content": "\\mathrm { B D D + }", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 347, + 305, + 360 + ], + "score": 1.0, + "content": "on GPUs with FSB branching strategy.", + "type": "text" + }, + { + "bbox": [ + 306, + 347, + 313, + 359 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "-CROWN BaB can use either BaBSR or FSB", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 358, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 369 + ], + "score": 1.0, + "content": "branching heuristic, and we include both in evaluation. All methods use a 1 hour timeout threshold.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 387, + 387 + ], + "score": 1.0, + "content": "We report the average verification time and branch numbers in Table", + "type": "text" + }, + { + "bbox": [ + 387, + 374, + 397, + 387 + ], + "score": 0.6, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 375, + 506, + 387 + ], + "score": 1.0, + "content": "and plot the percentage of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 268, + 397 + ], + "score": 1.0, + "content": "solved properties over time in Figure 1.", + "type": "text" + }, + { + "bbox": [ + 268, + 385, + 276, + 397 + ], + "score": 0.79, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "-CROWN FSB achieves the fastest average running time", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "compared to all other baselines with minimal timeouts, and also clearly leads on the cactus plot.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 271, + 419 + ], + "score": 1.0, + "content": "When using a weaker branching heuristic,", + "type": "text" + }, + { + "bbox": [ + 271, + 407, + 279, + 418 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "-CROWN BaBSR still outperforms all literature baselines,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 298, + 430 + ], + "score": 1.0, + "content": "including very recent ones such as A.set BaBSR", + "type": "text" + }, + { + "bbox": [ + 298, + 417, + 315, + 429 + ], + "score": 0.76, + "content": "\\mathbb { \\ m }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 418, + 396, + 430 + ], + "score": 1.0, + "content": ", Fast-and-Complete", + "type": "text" + }, + { + "bbox": [ + 396, + 417, + 414, + 429 + ], + "score": 0.72, + "content": "\\lVert \\boldsymbol { \\mathsf { A } } \\boldsymbol { \\mathsf { S } } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "and BaDNB [11]. Our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 271, + 441 + ], + "score": 1.0, + "content": "benefits are more clearly shown in Figure", + "type": "text" + }, + { + "bbox": [ + 271, + 429, + 281, + 442 + ], + "score": 0.78, + "content": "\\bigstar \\bigstar", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 428, + 365, + 441 + ], + "score": 1.0, + "content": "where we solve over", + "type": "text" + }, + { + "bbox": [ + 365, + 429, + 385, + 439 + ], + "score": 0.87, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "examples under 10s and most", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "score": 1.0, + "content": "other verifiers can verify much less or none of the properties within 10s. We see a 2 to 3 orders of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 450, + 493, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 493, + 463 + ], + "score": 1.0, + "content": "magitudes speedup in Figure 1 compared to CPU based verifiers such as MIPplanet and BaBSR.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 470, + 283, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 284, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 284, + 484 + ], + "score": 1.0, + "content": "4.2 Comparison to Incomplete Verifiers", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 506, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 228, + 498 + ], + "score": 1.0, + "content": "Verified accuracy. In Table", + "type": "text" + }, + { + "bbox": [ + 228, + 484, + 239, + 497 + ], + "score": 0.44, + "content": "\\lfloor 2 \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "we compare against a few representative and strong incomplete", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "verifiers on 5 convolutional networks and 4 MLP networks for MNIST and CIFAR-10 under the same", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 507, + 504, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 253, + 519 + ], + "score": 1.0, + "content": "set of 1000 images and perturbation", + "type": "text" + }, + { + "bbox": [ + 253, + 509, + 259, + 517 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 507, + 504, + 519 + ], + "score": 1.0, + "content": "as reported in [35, 37, 26]. Among the baselines, kPoly [35],", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 143, + 529 + ], + "score": 1.0, + "content": "OptC2V", + "type": "text" + }, + { + "bbox": [ + 143, + 518, + 161, + 529 + ], + "score": 0.42, + "content": " { \\mathbb { I } } ^ { \\smash { \\sum } }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 518, + 213, + 529 + ], + "score": 1.0, + "content": "and PRIMA", + "type": "text" + }, + { + "bbox": [ + 213, + 517, + 231, + 529 + ], + "score": 0.84, + "content": "\\left[ \\left[ 2 6 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "utilize state-of-the-art multi-neuron linear relaxation for ReLUs and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 324, + 541 + ], + "score": 1.0, + "content": "can bypass the single-neuron convex relaxation barrier", + "type": "text" + }, + { + "bbox": [ + 325, + 528, + 342, + 540 + ], + "score": 0.77, + "content": "\\pmb { \\mathbb { B } } \\pmb { \\mathrm { O } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 529, + 505, + 541 + ], + "score": 1.0, + "content": ", and are among the strongest incomplete", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 146, + 552 + ], + "score": 1.0, + "content": "verifiers.", + "type": "text" + }, + { + "bbox": [ + 146, + 540, + 154, + 551 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "-CROWN FSB achieves better verified accuracy on all models using a similar or less", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 550, + 507, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 550, + 507, + 564 + ], + "score": 1.0, + "content": "amount of time. Some models, such as MNIST ConvBig and CIFAR ConvBig, are quite challenging -", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 561, + 507, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 239, + 573 + ], + "score": 1.0, + "content": "the verified accuracy obtained by", + "type": "text" + }, + { + "bbox": [ + 240, + 562, + 247, + 573 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 561, + 507, + 573 + ], + "score": 1.0, + "content": "-CROWN FSB is close to the upper bound found via PGD attack.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 314, + 591 + ], + "score": 1.0, + "content": "To make more comprehensive evaluations, in Table", + "type": "text" + }, + { + "bbox": [ + 314, + 577, + 324, + 590 + ], + "score": 0.74, + "content": "\\textcircled { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "we further compare against a state-of-the-art", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 344, + 600 + ], + "score": 1.0, + "content": "semidefinite programming (SDP) based verifier, SDP-FO", + "type": "text" + }, + { + "bbox": [ + 345, + 588, + 357, + 600 + ], + "score": 0.31, + "content": "\\pmb { \\mathbb { Q } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 589, + 505, + 600 + ], + "score": 1.0, + "content": ", on one MNIST and six CIFAR-10", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "models reported in their paper. The models were trained using adversarial training, which posed a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 207, + 623 + ], + "score": 1.0, + "content": "challenge for verification", + "type": "text" + }, + { + "bbox": [ + 208, + 610, + 225, + 622 + ], + "score": 0.73, + "content": "[ [ 2 8 ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 610, + 506, + 623 + ], + "score": 1.0, + "content": ". The SDP formulation can be tighter than linear relaxation based ones,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "score": 1.0, + "content": "but it is computationally expensive - SDP-FO takes 2 to 3 hours to converge on one GPU for verifying", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "a single property, resulting 5,400 GPU hours to verify 200 testing images with 10 labels each. Due", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 364, + 655 + ], + "score": 1.0, + "content": "to resource limitations, we directly quote SDP-FO results from", + "type": "text" + }, + { + "bbox": [ + 365, + 643, + 377, + 654 + ], + "score": 0.52, + "content": "\\blacktriangleleft", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "on the same set of models. We", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "evaluate verified accuracy on the same set of 200 test images for other baselines. We include a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 664, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 207, + 677 + ], + "score": 1.0, + "content": "concurrent work PRIMA", + "type": "text" + }, + { + "bbox": [ + 207, + 665, + 224, + 676 + ], + "score": 0.74, + "content": "\\pmb { \\mathbb { Z } } \\pmb { 6 } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 664, + 504, + 677 + ], + "score": 1.0, + "content": ", the strongest multi-neuron linear relaxation baseline in Table 2, which", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 301, + 688 + ], + "score": 1.0, + "content": "generally outperforms kPoly and OptC2V. Table", + "type": "text" + }, + { + "bbox": [ + 302, + 676, + 311, + 689 + ], + "score": 0.62, + "content": "\\textcircled { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 676, + 505, + 688 + ], + "score": 1.0, + "content": "shows that overall we are 3 orders of magnitude", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 686, + 467, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 467, + 700 + ], + "score": 1.0, + "content": "faster than SDP-FO while still achieving consistently higher verified accuracy on average.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 261, + 713 + ], + "score": 1.0, + "content": "Tightness of verification. In Figure", + "type": "text" + }, + { + "bbox": [ + 262, + 699, + 271, + 713 + ], + "score": 0.28, + "content": "\\bigstar ,", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "we compare the tightness of verification bounds against", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 312, + 723 + ], + "score": 1.0, + "content": "SDP-FO on two adversarially trained networks from", + "type": "text" + }, + { + "bbox": [ + 312, + 712, + 324, + 722 + ], + "score": 0.73, + "content": "\\left[ \\left[ 9 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 711, + 505, + 723 + ], + "score": 1.0, + "content": ". 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CIFAR-10 BaseCIFAR-10 WideCIFAR-10 Deep
Methodtime(s)branches%timeouttime(s)branches%timeouttime(s)branches%timeout
BaBSR2367.781020.5536.002871.14812.6549.002750.75401.2839.00
MIPplanet 152849.69-68.002417.53-46.002302.25140.00
ERAN*[35333634805.94=5.00632.20=9.00545.72=0.00
GNN-online [231794.85565.1333.001367.38372.7415.001055.33131.854.00
BDD+ BaBSR 6807.91195480.1420.00505.6574203.1110.00266.2812722.744.00
OVAL (BDD+ GNN)*[623]662.1767938.3816.00280.3817895.946.0094.691990.341.00
A.set BaBSR10381.7812004.607.00165.912233.103.00190.282491.552.00
BigM+A.setBaBSR[10390.4411938.757.00172.654050.593.00177.223275.252.00
Fast-and-Complete [45]695.01119522.6517.00495.8880519.859.00105.642455.111.00
BaDNB (BDD+ FSB)II309.2938239.047.00165.5311214.444.0010.50368.160.00
β-CROWNBaBSR226.06509608.506.00118.26217691.243.006.12204.660.00
β-CROWN FSB118.23208018.213.0078.32116912.572.005.6941.120.00
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Other results were reported by their authors.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "image_body", + "bbox": [ + 109, + 219, + 496, + 297 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 219, + 496, + 297 + ], + "spans": [ + { + "bbox": [ + 109, + 219, + 496, + 297 + ], + "score": 0.955, + "type": "image", + "image_path": "e15f89e6f30fcd3a3be38f99210245d62207d65a1866c03408e05d561e72706d.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 109, + 219, + 496, + 245.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 109, + 245.0, + 496, + 271.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 109, + 271.0, + 496, + 297.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 299, + 504, + 319 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 299, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 375, + 310 + ], + "score": 1.0, + "content": "Figure 1: Percentage of solved properties with growing running time.", + "type": "text" + }, + { + "bbox": [ + 375, + 299, + 382, + 309 + ], + "score": 0.81, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 299, + 505, + 310 + ], + "score": 1.0, + "content": "-CROWN FSB (light green) and", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 107, + 309, + 113, + 319 + ], + "score": 0.8, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 308, + 379, + 321 + ], + "score": 1.0, + "content": "-CROWN BaBSR (dark green) clearly lead in all 3 settings and solve over", + "type": "text" + }, + { + "bbox": [ + 379, + 309, + 397, + 318 + ], + "score": 0.86, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "properties within 10 seconds.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 369 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 105, + 324, + 506, + 369 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 387, + 387 + ], + "score": 1.0, + "content": "We report the average verification time and branch numbers in Table", + "type": "text" + }, + { + "bbox": [ + 387, + 374, + 397, + 387 + ], + "score": 0.6, + "content": "^ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 375, + 506, + 387 + ], + "score": 1.0, + "content": "and plot the percentage of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 385, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 268, + 397 + ], + "score": 1.0, + "content": "solved properties over time in Figure 1.", + "type": "text" + }, + { + "bbox": [ + 268, + 385, + 276, + 397 + ], + "score": 0.79, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "-CROWN FSB achieves the fastest average running time", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "compared to all other baselines with minimal timeouts, and also clearly leads on the cactus plot.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 271, + 419 + ], + "score": 1.0, + "content": "When using a weaker branching heuristic,", + "type": "text" + }, + { + "bbox": [ + 271, + 407, + 279, + 418 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "-CROWN BaBSR still outperforms all literature baselines,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 418, + 298, + 430 + ], + "score": 1.0, + "content": "including very recent ones such as A.set BaBSR", + "type": "text" + }, + { + "bbox": [ + 298, + 417, + 315, + 429 + ], + "score": 0.76, + "content": "\\mathbb { \\ m }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 418, + 396, + 430 + ], + "score": 1.0, + "content": ", Fast-and-Complete", + "type": "text" + }, + { + "bbox": [ + 396, + 417, + 414, + 429 + ], + "score": 0.72, + "content": "\\lVert \\boldsymbol { \\mathsf { A } } \\boldsymbol { \\mathsf { S } } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "and BaDNB [11]. Our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 271, + 441 + ], + "score": 1.0, + "content": "benefits are more clearly shown in Figure", + "type": "text" + }, + { + "bbox": [ + 271, + 429, + 281, + 442 + ], + "score": 0.78, + "content": "\\bigstar \\bigstar", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 428, + 365, + 441 + ], + "score": 1.0, + "content": "where we solve over", + "type": "text" + }, + { + "bbox": [ + 365, + 429, + 385, + 439 + ], + "score": 0.87, + "content": "90 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "examples under 10s and most", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 451 + ], + "score": 1.0, + "content": "other verifiers can verify much less or none of the properties within 10s. We see a 2 to 3 orders of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 450, + 493, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 493, + 463 + ], + "score": 1.0, + "content": "magitudes speedup in Figure 1 compared to CPU based verifiers such as MIPplanet and BaBSR.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 374, + 506, + 463 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 470, + 283, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 468, + 284, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 284, + 484 + ], + "score": 1.0, + "content": "4.2 Comparison to Incomplete Verifiers", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 484, + 506, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 228, + 498 + ], + "score": 1.0, + "content": "Verified accuracy. In Table", + "type": "text" + }, + { + "bbox": [ + 228, + 484, + 239, + 497 + ], + "score": 0.44, + "content": "\\lfloor 2 \\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "we compare against a few representative and strong incomplete", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "verifiers on 5 convolutional networks and 4 MLP networks for MNIST and CIFAR-10 under the same", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 507, + 504, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 253, + 519 + ], + "score": 1.0, + "content": "set of 1000 images and perturbation", + "type": "text" + }, + { + "bbox": [ + 253, + 509, + 259, + 517 + ], + "score": 0.69, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 507, + 504, + 519 + ], + "score": 1.0, + "content": "as reported in [35, 37, 26]. Among the baselines, kPoly [35],", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 143, + 529 + ], + "score": 1.0, + "content": "OptC2V", + "type": "text" + }, + { + "bbox": [ + 143, + 518, + 161, + 529 + ], + "score": 0.42, + "content": " { \\mathbb { I } } ^ { \\smash { \\sum } }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 518, + 213, + 529 + ], + "score": 1.0, + "content": "and PRIMA", + "type": "text" + }, + { + "bbox": [ + 213, + 517, + 231, + 529 + ], + "score": 0.84, + "content": "\\left[ \\left[ 2 6 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "utilize state-of-the-art multi-neuron linear relaxation for ReLUs and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 324, + 541 + ], + "score": 1.0, + "content": "can bypass the single-neuron convex relaxation barrier", + "type": "text" + }, + { + "bbox": [ + 325, + 528, + 342, + 540 + ], + "score": 0.77, + "content": "\\pmb { \\mathbb { B } } \\pmb { \\mathrm { O } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 529, + 505, + 541 + ], + "score": 1.0, + "content": ", and are among the strongest incomplete", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 146, + 552 + ], + "score": 1.0, + "content": "verifiers.", + "type": "text" + }, + { + "bbox": [ + 146, + 540, + 154, + 551 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "-CROWN FSB achieves better verified accuracy on all models using a similar or less", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 550, + 507, + 564 + ], + "spans": [ + { + "bbox": [ + 104, + 550, + 507, + 564 + ], + "score": 1.0, + "content": "amount of time. Some models, such as MNIST ConvBig and CIFAR ConvBig, are quite challenging -", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 561, + 507, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 239, + 573 + ], + "score": 1.0, + "content": "the verified accuracy obtained by", + "type": "text" + }, + { + "bbox": [ + 240, + 562, + 247, + 573 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 561, + 507, + 573 + ], + "score": 1.0, + "content": "-CROWN FSB is close to the upper bound found via PGD attack.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 484, + 507, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 577, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 314, + 591 + ], + "score": 1.0, + "content": "To make more comprehensive evaluations, in Table", + "type": "text" + }, + { + "bbox": [ + 314, + 577, + 324, + 590 + ], + "score": 0.74, + "content": "\\textcircled { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "we further compare against a state-of-the-art", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 588, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 344, + 600 + ], + "score": 1.0, + "content": "semidefinite programming (SDP) based verifier, SDP-FO", + "type": "text" + }, + { + "bbox": [ + 345, + 588, + 357, + 600 + ], + "score": 0.31, + "content": "\\pmb { \\mathbb { Q } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 589, + 505, + 600 + ], + "score": 1.0, + "content": ", on one MNIST and six CIFAR-10", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 599, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 506, + 612 + ], + "score": 1.0, + "content": "models reported in their paper. The models were trained using adversarial training, which posed a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 610, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 207, + 623 + ], + "score": 1.0, + "content": "challenge for verification", + "type": "text" + }, + { + "bbox": [ + 208, + 610, + 225, + 622 + ], + "score": 0.73, + "content": "[ [ 2 8 ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 610, + 506, + 623 + ], + "score": 1.0, + "content": ". The SDP formulation can be tighter than linear relaxation based ones,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "score": 1.0, + "content": "but it is computationally expensive - SDP-FO takes 2 to 3 hours to converge on one GPU for verifying", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "a single property, resulting 5,400 GPU hours to verify 200 testing images with 10 labels each. Due", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 643, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 364, + 655 + ], + "score": 1.0, + "content": "to resource limitations, we directly quote SDP-FO results from", + "type": "text" + }, + { + "bbox": [ + 365, + 643, + 377, + 654 + ], + "score": 0.52, + "content": "\\blacktriangleleft", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "on the same set of models. We", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "evaluate verified accuracy on the same set of 200 test images for other baselines. We include a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 664, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 207, + 677 + ], + "score": 1.0, + "content": "concurrent work PRIMA", + "type": "text" + }, + { + "bbox": [ + 207, + 665, + 224, + 676 + ], + "score": 0.74, + "content": "\\pmb { \\mathbb { Z } } \\pmb { 6 } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 664, + 504, + 677 + ], + "score": 1.0, + "content": ", the strongest multi-neuron linear relaxation baseline in Table 2, which", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 301, + 688 + ], + "score": 1.0, + "content": "generally outperforms kPoly and OptC2V. Table", + "type": "text" + }, + { + "bbox": [ + 302, + 676, + 311, + 689 + ], + "score": 0.62, + "content": "\\textcircled { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 676, + 505, + 688 + ], + "score": 1.0, + "content": "shows that overall we are 3 orders of magnitude", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 686, + 467, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 467, + 700 + ], + "score": 1.0, + "content": "faster than SDP-FO while still achieving consistently higher verified accuracy on average.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 577, + 506, + 700 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 261, + 713 + ], + "score": 1.0, + "content": "Tightness of verification. In Figure", + "type": "text" + }, + { + "bbox": [ + 262, + 699, + 271, + 713 + ], + "score": 0.28, + "content": "\\bigstar ,", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "we compare the tightness of verification bounds against", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 312, + 723 + ], + "score": 1.0, + "content": "SDP-FO on two adversarially trained networks from", + "type": "text" + }, + { + "bbox": [ + 312, + 712, + 324, + 722 + ], + "score": 0.73, + "content": "\\left[ \\left[ 9 \\right] \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 711, + 505, + 723 + ], + "score": 1.0, + "content": ". Specifically, we use the verification objective", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 398, + 509, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 217, + 419 + ], + "score": 0.91, + "content": "f ( x ) : = z _ { y } ^ { ( L ) } ( x ) - z _ { y ^ { \\prime } } ^ { ( L ) } ( x ) ,", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 218, + 398, + 248, + 424 + ], + "score": 1.0, + "content": ", where", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 248, + 404, + 266, + 415 + ], + "score": 0.91, + "content": "z ^ { ( L ) }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 267, + 398, + 367, + 424 + ], + "score": 1.0, + "content": "is the logit layer output,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 367, + 407, + 374, + 417 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 374, + 398, + 392, + 424 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 392, + 406, + 401, + 417 + ], + "score": 0.88, + "content": "y ^ { \\prime }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 402, + 398, + 509, + 424 + ], + "score": 1.0, + "content": "are the true label and the", + "type": "text", + "cross_page": true + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 416, + 342, + 430 + ], + "score": 1.0, + "content": "runner-up label. For each test image, a 200-step PGD attack", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 343, + 417, + 360, + 429 + ], + "score": 0.75, + "content": "\\mathbb { [ 2 4 ] }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 361, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "provides an adversarial upper bound", + "type": "text", + "cross_page": true + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 107, + 429, + 114, + 441 + ], + "score": 0.86, + "content": "\\overline { { f } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 114, + 428, + 215, + 442 + ], + "score": 1.0, + "content": "of the optimal objective:", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 215, + 429, + 246, + 441 + ], + "score": 0.92, + "content": "f ^ { * } \\leq { \\overline { { f } } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 246, + 428, + 505, + 442 + ], + "score": 1.0, + "content": ". Verifiers, on the other hand, can provide a verified lower bound", + "type": "text", + "cross_page": true + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 440, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 137, + 453 + ], + "score": 0.88, + "content": "f \\leq f ^ { * }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 137, + 441, + 369, + 453 + ], + "score": 1.0, + "content": ". Bounds from tighter verification methods lie closer to line", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 369, + 442, + 394, + 452 + ], + "score": 0.93, + "content": "y = x", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 395, + 441, + 432, + 453 + ], + "score": 1.0, + "content": "in Figure", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 432, + 440, + 442, + 455 + ], + "score": 0.5, + "content": "\\bigtriangledown", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 442, + 441, + 470, + 453 + ], + "score": 1.0, + "content": "Figure", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 470, + 440, + 479, + 453 + ], + "score": 0.42, + "content": "\\left. 2 \\right.", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 479, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "shows", + "type": "text", + "cross_page": true + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 301, + 464 + ], + "score": 1.0, + "content": "that on both PGD adversarially trained networks,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 302, + 452, + 309, + 463 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 309, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "-CROWN FSB consistently outperforms SDP-FO", + "type": "text", + "cross_page": true + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 401, + 474 + ], + "score": 1.0, + "content": "for all 100 random test images. Importantly, for each point on the plots,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 402, + 462, + 409, + 474 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 410, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "-CROWN FSB needs 3", + "type": "text", + "cross_page": true + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "minutes while SDP-FO needs 178 minutes on average. LP verifier with triangle relaxations produces", + "type": "text", + "cross_page": true + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 483, + 492, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 209, + 497 + ], + "score": 1.0, + "content": "much looser bounds than", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 209, + 484, + 216, + 496 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 217, + 483, + 492, + 497 + ], + "score": 1.0, + "content": "-CROWN FSB and SDP-FO. Additional results are in Appendix C.2.", + "type": "text", + "cross_page": true + } + ], + "index": 29 + } + ], + "index": 42.5, + "bbox_fs": [ + 106, + 699, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 108, + 503, + 196 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 77, + 504, + 107 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 77, + 504, + 88 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 205, + 88 + ], + "score": 1.0, + "content": "Table 2: Verified accuracy", + "type": "text" + }, + { + "bbox": [ + 206, + 78, + 222, + 88 + ], + "score": 0.76, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 77, + 504, + 88 + ], + "score": 1.0, + "content": "and avg. time (s) of 1000 images evaluated on the ERAN models in [35, 37, 26].", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 87, + 504, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 504, + 99 + ], + "score": 1.0, + "content": "kPoly, OptC2V and PRIMA are strong incomplete verifiers that can break the convex relaxation barrier [30].", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 97, + 401, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 401, + 109 + ], + "score": 1.0, + "content": "The average time reported by us excludes examples that are classified incorrectly.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 108, + 108, + 503, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 108, + 503, + 196 + ], + "spans": [ + { + "bbox": [ + 108, + 108, + 503, + 196 + ], + "score": 0.975, + "html": "
DatasetModel (Same settings as 35.37261)CROWN/DeepPoly*1361 Verified%Time (s)kPoly 因 Ver.% Time(s)OptC2V Ver.%四 Time(s)PRIMA+ 园 Ver.% Time(s)β-CROWN FSB Ver.% Time(s)Upper bound
MNISTMLP5×100‡16.00.744.1307 42.913751.015969.9 10284.2
MLP8×10018.21.436.9 17138.475942.8301 62.010382.0
MLP5×20029.22.457.4 18760.140369.0 22477.48690.1
MLP8× 20025.95.650.6 46452.8345162.4 39573.59591.1
ConvSmall15.8334.7 47743.65559.872.77.073.2
ConvBig71.12173.6 4077.110277.542 11 79.33.180.4
CIFARConvSmall35.9439.986 39.810544.613 46.36.848.1
ConvBig42.14345.9 346No public code48.3176 51.615.355.0
ResNet24.1124.591cannot run24.81.724.8 1.624.8
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DatasetModel ∈= 0.3 and ε= 2/255CROWN/DeepPoly Verified%SDP-FO*PRIMA [26]β-CROWN FSBUpper bound
MNISTTime (s)Ver.%Time(s)Ver.%Time(s)Ver.%Time(s)
CIFARCNN-A-Adv1.00.143.4>20h44.5135.970.521.176.5
CNN-B-Adv21.50.532.8>25h38.0343.646.532.265.0
CNN-B-Adv-443.50.946.0>25h53.543.854.011.663.5
CNN-A-Adv35.50.639.6>25h41.54.844.05.850.0
CNN-A-Adv-441.50.740.0>25h45.04.946.05.649.5
CNN-A-Mix CNN-A-Mix-423.5 38.00.4 0.539.6 47.8>25h >25h37.5 48.534.3 7.041.5 50.549.6 5.953.0 57.5
", + "type": "table", + "image_path": "7be7195944088fdbcf2329920588fad8ef79ac3acefff7e1a563f6a8434c0eff.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 117, + 281, + 493, + 309.0 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 117, + 309.0, + 493, + 337.0 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 117, + 337.0, + 493, + 365.0 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 116, + 366, + 493, + 399 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 120, + 365, + 492, + 376 + ], + "spans": [ + { + "bbox": [ + 120, + 365, + 492, + 376 + ], + "score": 1.0, + "content": "SDP-FO results are directly from their paper due to its very long running time (>20h per example). † PRIMA experiments", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 117, + 374, + 492, + 384 + ], + "spans": [ + { + "bbox": [ + 117, + 374, + 191, + 384 + ], + "score": 1.0, + "content": "were done using commi", + "type": "text" + }, + { + "bbox": [ + 192, + 374, + 221, + 384 + ], + "score": 0.63, + "content": "{ \\widehat { 3 9 6 \\mathrm { d c } 7 \\mathrm { a } } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 374, + 338, + 384 + ], + "score": 1.0, + "content": "released on June 4, 2021. PRIMA and", + "type": "text" + }, + { + "bbox": [ + 339, + 375, + 344, + 383 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 374, + 492, + 384 + ], + "score": 1.0, + "content": "-CROWN FSB results are on the same set of 200", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 116, + 382, + 493, + 393 + ], + "spans": [ + { + "bbox": [ + 116, + 382, + 493, + 393 + ], + "score": 1.0, + "content": "examples (first 200 examples of CIFAR-10 dataset) and we don’t run verifiers on examples that are classified incorrectly or can", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 117, + 390, + 427, + 400 + ], + "spans": [ + { + "bbox": [ + 117, + 390, + 212, + 400 + ], + "score": 1.0, + "content": "be attacked by a 200-step PGD.", + "type": "text" + }, + { + "bbox": [ + 213, + 391, + 218, + 399 + ], + "score": 0.81, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 390, + 427, + 400 + ], + "score": 1.0, + "content": "-CROWN uses 1 GPU and 1 CPU; PRIMA uses 1 GPU and 20 CPUs.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 106, + 398, + 509, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 217, + 419 + ], + "score": 0.91, + "content": "f ( x ) : = z _ { y } ^ { ( L ) } ( x ) - z _ { y ^ { \\prime } } ^ { ( L ) } ( x ) ,", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 398, + 248, + 424 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 248, + 404, + 266, + 415 + ], + "score": 0.91, + "content": "z ^ { ( L ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 398, + 367, + 424 + ], + "score": 1.0, + "content": "is the logit layer output,", + "type": "text" + }, + { + "bbox": [ + 367, + 407, + 374, + 417 + ], + "score": 0.8, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 398, + 392, + 424 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 392, + 406, + 401, + 417 + ], + "score": 0.88, + "content": "y ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 398, + 509, + 424 + ], + "score": 1.0, + "content": "are the true label and the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 416, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 104, + 416, + 342, + 430 + ], + "score": 1.0, + "content": "runner-up label. For each test image, a 200-step PGD attack", + "type": "text" + }, + { + "bbox": [ + 343, + 417, + 360, + 429 + ], + "score": 0.75, + "content": "\\mathbb { [ 2 4 ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 416, + 505, + 430 + ], + "score": 1.0, + "content": "provides an adversarial upper bound", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 428, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 107, + 429, + 114, + 441 + ], + "score": 0.86, + "content": "\\overline { { f } }", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 428, + 215, + 442 + ], + "score": 1.0, + "content": "of the optimal objective:", + "type": "text" + }, + { + "bbox": [ + 215, + 429, + 246, + 441 + ], + "score": 0.92, + "content": "f ^ { * } \\leq { \\overline { { f } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 428, + 505, + 442 + ], + "score": 1.0, + "content": ". Verifiers, on the other hand, can provide a verified lower bound", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 440, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 137, + 453 + ], + "score": 0.88, + "content": "f \\leq f ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 441, + 369, + 453 + ], + "score": 1.0, + "content": ". Bounds from tighter verification methods lie closer to line", + "type": "text" + }, + { + "bbox": [ + 369, + 442, + 394, + 452 + ], + "score": 0.93, + "content": "y = x", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 441, + 432, + 453 + ], + "score": 1.0, + "content": "in Figure", + "type": "text" + }, + { + "bbox": [ + 432, + 440, + 442, + 455 + ], + "score": 0.5, + "content": "\\bigtriangledown", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 441, + 470, + 453 + ], + "score": 1.0, + "content": "Figure", + "type": "text" + }, + { + "bbox": [ + 470, + 440, + 479, + 453 + ], + "score": 0.42, + "content": "\\left. 2 \\right.", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "shows", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 451, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 301, + 464 + ], + "score": 1.0, + "content": "that on both PGD adversarially trained networks,", + "type": "text" + }, + { + "bbox": [ + 302, + 452, + 309, + 463 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 451, + 505, + 464 + ], + "score": 1.0, + "content": "-CROWN FSB consistently outperforms SDP-FO", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 401, + 474 + ], + "score": 1.0, + "content": "for all 100 random test images. Importantly, for each point on the plots,", + "type": "text" + }, + { + "bbox": [ + 402, + 462, + 409, + 474 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "-CROWN FSB needs 3", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "minutes while SDP-FO needs 178 minutes on average. LP verifier with triangle relaxations produces", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 483, + 492, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 209, + 497 + ], + "score": 1.0, + "content": "much looser bounds than", + "type": "text" + }, + { + "bbox": [ + 209, + 484, + 216, + 496 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 483, + 492, + 497 + ], + "score": 1.0, + "content": "-CROWN FSB and SDP-FO. Additional results are in Appendix C.2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 502, + 506, + 546 + ], + "lines": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 515 + ], + "score": 1.0, + "content": "VNN-COMP 2021 results. We encourage the readers to checkout the report of the Second Inter-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 513, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 418, + 525 + ], + "score": 1.0, + "content": "national Verification of Neural Networks Competition (VNN-COMP 2021)", + "type": "text" + }, + { + "bbox": [ + 419, + 513, + 431, + 524 + ], + "score": 0.75, + "content": "\\pmb { \\mathbb { B } } \\|", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 513, + 506, + 525 + ], + "score": 1.0, + "content": "with 9 additional", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 525, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 505, + 536 + ], + "score": 1.0, + "content": "benchmarks and 12 competing methods evaluated in a standardized testing environment on AWS. Our", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 535, + 505, + 548 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 129, + 548 + ], + "score": 1.0, + "content": "entry", + "type": "text" + }, + { + "bbox": [ + 129, + 536, + 146, + 546 + ], + "score": 0.52, + "content": "\\alpha , \\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 535, + 248, + 548 + ], + "score": 1.0, + "content": "-CROWN is based on the", + "type": "text" + }, + { + "bbox": [ + 248, + 536, + 255, + 547 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 535, + 505, + 548 + ], + "score": 1.0, + "content": "-CROWN algorithm in this work and uses the same codebase.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 557, + 198, + 569 + ], + "lines": [ + { + "bbox": [ + 104, + 555, + 199, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 199, + 572 + ], + "score": 1.0, + "content": "5 Related Work", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 572, + 506, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "Many early complete verifiers for neural networks relied on existing solvers such as MILP or SMT", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 506, + 595 + ], + "score": 1.0, + "content": "solvers [20, 15, 19, 12, 38] and were limited to very small problem instances. Branch and bound (BaB)", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35.5 + }, + { + "type": "image", + "bbox": [ + 129, + 600, + 479, + 696 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 129, + 600, + 479, + 696 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 129, + 600, + 479, + 696 + ], + "spans": [ + { + "bbox": [ + 129, + 600, + 479, + 696 + ], + "score": 0.827, + "type": "image", + "image_path": "f268324c95f4c9216eb4f87162c23c7018811152e56851544933396dfb737acb.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 129, + 600, + 479, + 632.0 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 129, + 632.0, + 479, + 664.0 + ], + "spans": [], + "index": 38 + }, + { + "bbox": [ + 129, + 664.0, + 479, + 696.0 + ], + "spans": [], + "index": 39 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 102, + 699, + 503, + 720 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 709 + ], + "score": 1.0, + "content": "Figure 2: Verified lower bound v.s. PGD adversarial upper bound. A lower bound closer to the upper bound", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 494, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 171, + 720 + ], + "score": 1.0, + "content": "(closer to the line", + "type": "text" + }, + { + "bbox": [ + 171, + 709, + 195, + 719 + ], + "score": 0.89, + "content": "y = x", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 709, + 231, + 720 + ], + "score": 1.0, + "content": ") is better.", + "type": "text" + }, + { + "bbox": [ + 232, + 709, + 239, + 719 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 709, + 494, + 720 + ], + "score": 1.0, + "content": "-CROWN FSB uses 3mins while SDP-FO needs 2 to 3 hours per point.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "index": 39.25 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 108, + 503, + 196 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 77, + 504, + 107 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 77, + 504, + 88 + ], + "spans": [ + { + "bbox": [ + 105, + 77, + 205, + 88 + ], + "score": 1.0, + "content": "Table 2: Verified accuracy", + "type": "text" + }, + { + "bbox": [ + 206, + 78, + 222, + 88 + ], + "score": 0.76, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 77, + 504, + 88 + ], + "score": 1.0, + "content": "and avg. time (s) of 1000 images evaluated on the ERAN models in [35, 37, 26].", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 87, + 504, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 504, + 99 + ], + "score": 1.0, + "content": "kPoly, OptC2V and PRIMA are strong incomplete verifiers that can break the convex relaxation barrier [30].", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 97, + 401, + 109 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 401, + 109 + ], + "score": 1.0, + "content": "The average time reported by us excludes examples that are classified incorrectly.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 108, + 108, + 503, + 196 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 108, + 503, + 196 + ], + "spans": [ + { + "bbox": [ + 108, + 108, + 503, + 196 + ], + "score": 0.975, + "html": "
DatasetModel (Same settings as 35.37261)CROWN/DeepPoly*1361 Verified%Time (s)kPoly 因 Ver.% Time(s)OptC2V Ver.%四 Time(s)PRIMA+ 园 Ver.% Time(s)β-CROWN FSB Ver.% Time(s)Upper bound
MNISTMLP5×100‡16.00.744.1307 42.913751.015969.9 10284.2
MLP8×10018.21.436.9 17138.475942.8301 62.010382.0
MLP5×20029.22.457.4 18760.140369.0 22477.48690.1
MLP8× 20025.95.650.6 46452.8345162.4 39573.59591.1
ConvSmall15.8334.7 47743.65559.872.77.073.2
ConvBig71.12173.6 4077.110277.542 11 79.33.180.4
CIFARConvSmall35.9439.986 39.810544.613 46.36.848.1
ConvBig42.14345.9 346No public code48.3176 51.615.355.0
ResNet24.1124.591cannot run24.81.724.8 1.624.8
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DatasetModel ∈= 0.3 and ε= 2/255CROWN/DeepPoly Verified%SDP-FO*PRIMA [26]β-CROWN FSBUpper bound
MNISTTime (s)Ver.%Time(s)Ver.%Time(s)Ver.%Time(s)
CIFARCNN-A-Adv1.00.143.4>20h44.5135.970.521.176.5
CNN-B-Adv21.50.532.8>25h38.0343.646.532.265.0
CNN-B-Adv-443.50.946.0>25h53.543.854.011.663.5
CNN-A-Adv35.50.639.6>25h41.54.844.05.850.0
CNN-A-Adv-441.50.740.0>25h45.04.946.05.649.5
CNN-A-Mix CNN-A-Mix-423.5 38.00.4 0.539.6 47.8>25h >25h37.5 48.534.3 7.041.5 50.549.6 5.953.0 57.5
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A lower bound closer to the upper bound", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 709, + 494, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 171, + 720 + ], + "score": 1.0, + "content": "(closer to the line", + "type": "text" + }, + { + "bbox": [ + 171, + 709, + 195, + 719 + ], + "score": 0.89, + "content": "y = x", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 709, + 231, + 720 + ], + "score": 1.0, + "content": ") is better.", + "type": "text" + }, + { + "bbox": [ + 232, + 709, + 239, + 719 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 709, + 494, + 720 + ], + "score": 1.0, + "content": "-CROWN FSB uses 3mins while SDP-FO needs 2 to 3 hours per point.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "index": 39.25 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 73, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "based method was proposed to better exploit the network structure using LP-based incomplete verifier", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "score": 1.0, + "content": "for bounding and ReLU splits for branching [8, 40, 23, 5]. 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SDP based relaxations", + "type": "text" + }, + { + "bbox": [ + 243, + 280, + 290, + 291 + ], + "score": 0.78, + "content": "[ \\sqrt { 2 8 } , \\sqrt { 1 6 } , \\sqrt { 1 4 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 279, + 505, + 293 + ], + "score": 1.0, + "content": "typically produce tight bounds but with significantly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 321, + 303 + ], + "score": 1.0, + "content": "higher cost. The most recent GPU based SDP verifier", + "type": "text" + }, + { + "bbox": [ + 321, + 291, + 333, + 302 + ], + "score": 0.67, + "content": "\\bigstar \\bigstar", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "is still relatively slow and can take 2 hours", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 438, + 315 + ], + "score": 1.0, + "content": "to verify a single image. In this work, we impose neuron split constraints using", + "type": "text" + }, + { + "bbox": [ + 438, + 302, + 446, + 313 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "-CROWN and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 313, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 506, + 324 + ], + "score": 1.0, + "content": "combine it with branch and bound done in parallel on GPUs. Although for each subdomain in BaB,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 107, + 324, + 114, + 335 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 324, + 408, + 336 + ], + "score": 1.0, + "content": "-CROWN is still subject to the convex relaxation barrier, the efficiency of", + "type": "text" + }, + { + "bbox": [ + 408, + 324, + 416, + 335 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "-CROWN BaB allows", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 335, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 346 + ], + "score": 1.0, + "content": "it to quickly explore a very large number of subdomains and outperform existing convex barrier", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 344, + 437, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 437, + 358 + ], + "score": 1.0, + "content": "breaking incomplete verifiers under many scenarios in both runtime and tightness.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 375 + ], + "score": 1.0, + "content": "Additionally, another line of works train networks to enhance verified accuracy, typically using cheap", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 371, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 386 + ], + "score": 1.0, + "content": "incomplete verifiers at training time [42, 39, 25, 43, 18, 25, 47, 4, 32]. Traditionally only these", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 420, + 397 + ], + "score": 1.0, + "content": "verification-customized networks can have reasonable verified accuracy, while", + "type": "text" + }, + { + "bbox": [ + 420, + 384, + 428, + 395 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "-CROWN BaB can", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 395, + 468, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 468, + 407 + ], + "score": 1.0, + "content": "also give non-trivial verified accuracy on relatively large networks agnostic to verification.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 416, + 183, + 429 + ], + "lines": [ + { + "bbox": [ + 104, + 414, + 185, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 185, + 432 + ], + "score": 1.0, + "content": "6 Conclusion", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 162, + 447 + ], + "score": 1.0, + "content": "We proposed", + "type": "text" + }, + { + "bbox": [ + 162, + 436, + 169, + 447 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 436, + 505, + 447 + ], + "score": 1.0, + "content": "-CROWN, a new bound propagation method that can fully encode the neuron split", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 447, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 459 + ], + "score": 1.0, + "content": "constraints introduced in BaB, which clearly leads in both complete and incomplete verification", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 205, + 470 + ], + "score": 1.0, + "content": "settings. The success of", + "type": "text" + }, + { + "bbox": [ + 205, + 458, + 213, + 469 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 457, + 420, + 470 + ], + "score": 1.0, + "content": "-CROWN comes from a few factors: (1) In Section", + "type": "text" + }, + { + "bbox": [ + 420, + 457, + 438, + 470 + ], + "score": 0.78, + "content": "\\boxed { 3 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 457, + 495, + 470 + ], + "score": 1.0, + "content": "we show that", + "type": "text" + }, + { + "bbox": [ + 495, + 458, + 502, + 469 + ], + "score": 0.81, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 433, + 481 + ], + "score": 1.0, + "content": "CROWN is an GPU-friendly bound propagation algorithm significantly faster than", + "type": "text" + }, + { + "bbox": [ + 433, + 469, + 446, + 479 + ], + "score": 0.41, + "content": "\\bar { L } P", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "solvers. 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(3) In Section", + "type": "text" + }, + { + "bbox": [ + 230, + 490, + 247, + 503 + ], + "score": 0.59, + "content": "^ { 3 . 3 , }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 491, + 303, + 502 + ], + "score": 1.0, + "content": "we show that", + "type": "text" + }, + { + "bbox": [ + 303, + 491, + 310, + 502 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "-CROWN can jointly optimize intermediate layer", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 501, + 502, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 297, + 514 + ], + "score": 1.0, + "content": "bounds and achieve tighter bounds than typical", + "type": "text" + }, + { + "bbox": [ + 297, + 502, + 311, + 511 + ], + "score": 0.51, + "content": "L P", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 501, + 502, + 514 + ], + "score": 1.0, + "content": "verifiers using fixed intermediate layer bounds.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "Limitations. Our verifier has several limitations which are commonly shared by most existing", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 534, + 507, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 507, + 547 + ], + "score": 1.0, + "content": "BaB-based complete verifiers. First, we focused on ReLU which can be split into two linear cases.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "For other non-piecewise linear activation functions, although it is still possible to conduct branch and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "bound, it is difficult to guarantee completeness. Second, we discussed only the norm perturbations", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "for input domains. In practice, the threat model may involve complicated and nonconvex perturbation", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "specifications. Third, although our GPU accelerated verifier outperforms existing ones, all BaB", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 590, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 506, + 601 + ], + "score": 1.0, + "content": "based verifiers, including ours, are still limited to relatively small models far from the ImageNet", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "score": 1.0, + "content": "scale. Finally, we have only demonstrated robustness verification of image classification tasks, and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "generalizing it to give verification guarantees for other tasks such as robust deep reinforcement", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 621, + 360, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 141, + 635 + ], + "score": 1.0, + "content": "learning", + "type": "text" + }, + { + "bbox": [ + 142, + 622, + 158, + 633 + ], + "score": 0.3, + "content": "\\frac { 1 } { 1 2 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 621, + 360, + 635 + ], + "score": 1.0, + "content": "48, 49] is an interesting direction for future work.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5 + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "Societal Impact NNs have been used in an increasingly wide range of real-world applications and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "play an important role in artificial intelligence (AI). The trustworthiness and robustness of NNs have", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 666, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 411, + 681 + ], + "score": 1.0, + "content": "become crucial factors since AI plays an important role in modern society.", + "type": "text" + }, + { + "bbox": [ + 412, + 667, + 419, + 678 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 666, + 506, + 681 + ], + "score": 1.0, + "content": "-CROWN is a strong", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "neural network verifier which can be used to check certain properties of neural networks, which can", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "be helpful for guaranteeing the robustness, correctness, and fairness of NNs in applications that can", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 699, + 507, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 507, + 713 + ], + "score": 1.0, + "content": "directly or indirectly impact human life. We believe our work has overall positive societal impacts,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 711, + 468, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 468, + 723 + ], + "score": 1.0, + "content": "although it may potentially be misused to identify the weakness of NNs and guide attacks.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 301, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 755 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 755 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 108, + 73, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "based method was proposed to better exploit the network structure using LP-based incomplete verifier", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 96 + ], + "score": 1.0, + "content": "for bounding and ReLU splits for branching [8, 40, 23, 5]. Besides branching on ReLU neurons,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 93, + 485, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 470, + 108 + ], + "score": 1.0, + "content": "input domain branching was also considered in [41, 29, 1] but limited by input dimensions", + "type": "text" + }, + { + "bbox": [ + 470, + 94, + 482, + 106 + ], + "score": 0.26, + "content": "\\pmb { \\mathbb { B } } ] \\mathbf l", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 93, + 485, + 108 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 72, + 507, + 108 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 111, + 505, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "Recently, a few approaches have been proposed to use efficient iterative solvers or bound propagation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "methods on GPUs without relying on LP solvers. Bunel et al. [6] decomposed the verification", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 106, + 133, + 505, + 145 + ], + "score": 1.0, + "content": "problem layer by layer, solved each layer in a closed form on GPUs, and used Lagrangian to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "enforce consistency between layers. However, their formulation only has the same power as LP", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 329, + 167 + ], + "score": 1.0, + "content": "and needs many iterations to converge. De Palma et al.", + "type": "text" + }, + { + "bbox": [ + 329, + 154, + 347, + 166 + ], + "score": 0.37, + "content": "\\mathbb { \\ m }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "used a dual-space verifier with a linear", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "relaxation [2, 37] tighter than triangle LP relaxation, but in most settings the extra computational costs", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 471, + 189 + ], + "score": 1.0, + "content": "and difficulties for an efficient implementation offset its benefits (more discussions in section", + "type": "text" + }, + { + "bbox": [ + 472, + 176, + 491, + 189 + ], + "score": 0.69, + "content": "\\mathbf { B } . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 176, + 506, + 189 + ], + "score": 1.0, + "content": ". A", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 507, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 507, + 200 + ], + "score": 1.0, + "content": "concurrent work BaDNB [11] proposed a new branching strategy, filtered smart branching (FSB),", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 465, + 210 + ], + "score": 1.0, + "content": "combined with Lagrangian decomposition to get better verification performance. Xu et al.", + "type": "text" + }, + { + "bbox": [ + 465, + 198, + 483, + 209 + ], + "score": 0.32, + "content": "\\pm \\boxed { \\boxed { 4 4 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 199, + 505, + 209 + ], + "score": 1.0, + "content": "used", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "CROWN as a massively paralleled incomplete solver on GPUs for complete verification, but it cannot", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 457, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 457, + 234 + ], + "score": 1.0, + "content": "handle neuron split constraints, leading to suboptimal efficiency and high timeout rates.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 111, + 507, + 234 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 236, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 106, + 236, + 505, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 280, + 248 + ], + "score": 1.0, + "content": "For incomplete verification, Salman et al.", + "type": "text" + }, + { + "bbox": [ + 280, + 236, + 298, + 248 + ], + "score": 0.75, + "content": "\\textcircled { \\lvert 3 0 \\rvert }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 236, + 505, + 248 + ], + "score": 1.0, + "content": "shows the inherent limitation of using per-neuron", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 330, + 259 + ], + "score": 1.0, + "content": "convex relaxations for verification problems. Singh et al.", + "type": "text" + }, + { + "bbox": [ + 331, + 246, + 349, + 259 + ], + "score": 0.62, + "content": "\\pmb { \\Vert 3 5 \\Vert }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 247, + 417, + 259 + ], + "score": 1.0, + "content": "and Müller et al.", + "type": "text" + }, + { + "bbox": [ + 417, + 246, + 434, + 258 + ], + "score": 0.71, + "content": "[ [ 2 6 ] ]", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 247, + 505, + 259 + ], + "score": 1.0, + "content": "broke this barrier", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 437, + 271 + ], + "score": 1.0, + "content": "by considering constraints involving multiple ReLU neurons; Tjandraatmadja et al.", + "type": "text" + }, + { + "bbox": [ + 437, + 258, + 455, + 270 + ], + "score": 0.56, + "content": "\\pmb { \\Vert 3 7 } \\Vert", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "proposed to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 282 + ], + "score": 1.0, + "content": "relax a linear layer with a ReLU neuron together using a strong mixed-integer programming formu-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 279, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 132, + 293 + ], + "score": 1.0, + "content": "lation", + "type": "text" + }, + { + "bbox": [ + 132, + 280, + 145, + 291 + ], + "score": 0.26, + "content": "\\mathbb { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 279, + 243, + 293 + ], + "score": 1.0, + "content": ". 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The most recent GPU based SDP verifier", + "type": "text" + }, + { + "bbox": [ + 321, + 291, + 333, + 302 + ], + "score": 0.67, + "content": "\\bigstar \\bigstar", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "is still relatively slow and can take 2 hours", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 506, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 438, + 315 + ], + "score": 1.0, + "content": "to verify a single image. In this work, we impose neuron split constraints using", + "type": "text" + }, + { + "bbox": [ + 438, + 302, + 446, + 313 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 302, + 506, + 315 + ], + "score": 1.0, + "content": "-CROWN and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 313, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 506, + 324 + ], + "score": 1.0, + "content": "combine it with branch and bound done in parallel on GPUs. Although for each subdomain in BaB,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 107, + 324, + 114, + 335 + ], + "score": 0.83, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 324, + 408, + 336 + ], + "score": 1.0, + "content": "-CROWN is still subject to the convex relaxation barrier, the efficiency of", + "type": "text" + }, + { + "bbox": [ + 408, + 324, + 416, + 335 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "-CROWN BaB allows", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 335, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 346 + ], + "score": 1.0, + "content": "it to quickly explore a very large number of subdomains and outperform existing convex barrier", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 344, + 437, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 437, + 358 + ], + "score": 1.0, + "content": "breaking incomplete verifiers under many scenarios in both runtime and tightness.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 236, + 506, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 406 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 375 + ], + "score": 1.0, + "content": "Additionally, another line of works train networks to enhance verified accuracy, typically using cheap", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 371, + 506, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 506, + 386 + ], + "score": 1.0, + "content": "incomplete verifiers at training time [42, 39, 25, 43, 18, 25, 47, 4, 32]. Traditionally only these", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 420, + 397 + ], + "score": 1.0, + "content": "verification-customized networks can have reasonable verified accuracy, while", + "type": "text" + }, + { + "bbox": [ + 420, + 384, + 428, + 395 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 383, + 506, + 397 + ], + "score": 1.0, + "content": "-CROWN BaB can", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 395, + 468, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 395, + 468, + 407 + ], + "score": 1.0, + "content": "also give non-trivial verified accuracy on relatively large networks agnostic to verification.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 360, + 506, + 407 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 416, + 183, + 429 + ], + "lines": [ + { + "bbox": [ + 104, + 414, + 185, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 414, + 185, + 432 + ], + "score": 1.0, + "content": "6 Conclusion", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 435, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 162, + 447 + ], + "score": 1.0, + "content": "We proposed", + "type": "text" + }, + { + "bbox": [ + 162, + 436, + 169, + 447 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 436, + 505, + 447 + ], + "score": 1.0, + "content": "-CROWN, a new bound propagation method that can fully encode the neuron split", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 447, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 506, + 459 + ], + "score": 1.0, + "content": "constraints introduced in BaB, which clearly leads in both complete and incomplete verification", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 205, + 470 + ], + "score": 1.0, + "content": "settings. The success of", + "type": "text" + }, + { + "bbox": [ + 205, + 458, + 213, + 469 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 457, + 420, + 470 + ], + "score": 1.0, + "content": "-CROWN comes from a few factors: (1) In Section", + "type": "text" + }, + { + "bbox": [ + 420, + 457, + 438, + 470 + ], + "score": 0.78, + "content": "\\boxed { 3 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 457, + 495, + 470 + ], + "score": 1.0, + "content": "we show that", + "type": "text" + }, + { + "bbox": [ + 495, + 458, + 502, + 469 + ], + "score": 0.81, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 433, + 481 + ], + "score": 1.0, + "content": "CROWN is an GPU-friendly bound propagation algorithm significantly faster than", + "type": "text" + }, + { + "bbox": [ + 433, + 469, + 446, + 479 + ], + "score": 0.41, + "content": "\\bar { L } P", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "solvers. 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(3) In Section", + "type": "text" + }, + { + "bbox": [ + 230, + 490, + 247, + 503 + ], + "score": 0.59, + "content": "^ { 3 . 3 , }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 491, + 303, + 502 + ], + "score": 1.0, + "content": "we show that", + "type": "text" + }, + { + "bbox": [ + 303, + 491, + 310, + 502 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 491, + 505, + 502 + ], + "score": 1.0, + "content": "-CROWN can jointly optimize intermediate layer", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 501, + 502, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 297, + 514 + ], + "score": 1.0, + "content": "bounds and achieve tighter bounds than typical", + "type": "text" + }, + { + "bbox": [ + 297, + 502, + 311, + 511 + ], + "score": 0.51, + "content": "L P", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 501, + 502, + 514 + ], + "score": 1.0, + "content": "verifiers using fixed intermediate layer bounds.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 436, + 506, + 514 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "Limitations. Our verifier has several limitations which are commonly shared by most existing", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 534, + 507, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 507, + 547 + ], + "score": 1.0, + "content": "BaB-based complete verifiers. First, we focused on ReLU which can be split into two linear cases.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 546, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 546, + 505, + 558 + ], + "score": 1.0, + "content": "For other non-piecewise linear activation functions, although it is still possible to conduct branch and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "bound, it is difficult to guarantee completeness. Second, we discussed only the norm perturbations", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "for input domains. In practice, the threat model may involve complicated and nonconvex perturbation", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "specifications. Third, although our GPU accelerated verifier outperforms existing ones, all BaB", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 590, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 506, + 601 + ], + "score": 1.0, + "content": "based verifiers, including ours, are still limited to relatively small models far from the ImageNet", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 505, + 612 + ], + "score": 1.0, + "content": "scale. Finally, we have only demonstrated robustness verification of image classification tasks, and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "generalizing it to give verification guarantees for other tasks such as robust deep reinforcement", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 621, + 360, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 141, + 635 + ], + "score": 1.0, + "content": "learning", + "type": "text" + }, + { + "bbox": [ + 142, + 622, + 158, + 633 + ], + "score": 0.3, + "content": "\\frac { 1 } { 1 2 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 621, + 360, + 635 + ], + "score": 1.0, + "content": "48, 49] is an interesting direction for future work.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 523, + 507, + 635 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 645, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "Societal Impact NNs have been used in an increasingly wide range of real-world applications and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "play an important role in artificial intelligence (AI). 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We believe our work has overall positive societal impacts,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 711, + 468, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 468, + 723 + ], + "score": 1.0, + "content": "although it may potentially be misused to identify the weakness of NNs and guide attacks.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 645, + 507, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 71, + 202, + 84 + ], + "lines": [ + { + "bbox": [ + 106, + 70, + 204, + 87 + ], + "spans": [ + { + "bbox": [ + 106, + 70, + 204, + 87 + ], + "score": 1.0, + "content": "Acknowledgement", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 96, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 106, + 96, + 505, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 505, + 109 + ], + "score": 1.0, + "content": "This work is supported by NSF grant CNS18-01426; an ARL Young Investigator (YIP) award; an", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 505, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 505, + 120 + ], + "score": 1.0, + "content": "NSF CAREER award; a Google Faculty Fellowship; a Capital One Research Grant; and a J.P. 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CIFAR-10 BaseCIFAR-10 WideCIFAR-10 Deep
Methodtime(s)branches%timeouttime(s)branches%timeouttime(s)branches%timeout
BaBSR2367.781020.5536.002871.14812.6549.002750.75401.2839.00
MIPplanet 152849.69-68.002417.53-46.002302.25140.00
ERAN*[35333634805.94=5.00632.20=9.00545.72=0.00
GNN-online [231794.85565.1333.001367.38372.7415.001055.33131.854.00
BDD+ BaBSR 6807.91195480.1420.00505.6574203.1110.00266.2812722.744.00
OVAL (BDD+ GNN)*[623]662.1767938.3816.00280.3817895.946.0094.691990.341.00
A.set BaBSR10381.7812004.607.00165.912233.103.00190.282491.552.00
BigM+A.setBaBSR[10390.4411938.757.00172.654050.593.00177.223275.252.00
Fast-and-Complete [45]695.01119522.6517.00495.8880519.859.00105.642455.111.00
BaDNB (BDD+ FSB)II309.2938239.047.00165.5311214.444.0010.50368.160.00
β-CROWNBaBSR226.06509608.506.00118.26217691.243.006.12204.660.00
β-CROWN FSB118.23208018.213.0078.32116912.572.005.6941.120.00
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DatasetModel ∈= 0.3 and ε= 2/255CROWN/DeepPoly Verified%SDP-FO*PRIMA [26]β-CROWN FSBUpper bound
MNISTTime (s)Ver.%Time(s)Ver.%Time(s)Ver.%Time(s)
CIFARCNN-A-Adv1.00.143.4>20h44.5135.970.521.176.5
CNN-B-Adv21.50.532.8>25h38.0343.646.532.265.0
CNN-B-Adv-443.50.946.0>25h53.543.854.011.663.5
CNN-A-Adv35.50.639.6>25h41.54.844.05.850.0
CNN-A-Adv-441.50.740.0>25h45.04.946.05.649.5
CNN-A-Mix CNN-A-Mix-423.5 38.00.4 0.539.6 47.8>25h >25h37.5 48.534.3 7.041.5 50.549.6 5.953.0 57.5
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DatasetModel (Same settings as 35.37261)CROWN/DeepPoly*1361 Verified%Time (s)kPoly 因 Ver.% Time(s)OptC2V Ver.%四 Time(s)PRIMA+ 园 Ver.% Time(s)β-CROWN FSB Ver.% Time(s)Upper bound
MNISTMLP5×100‡16.00.744.1307 42.913751.015969.9 10284.2
MLP8×10018.21.436.9 17138.475942.8301 62.010382.0
MLP5×20029.22.457.4 18760.140369.0 22477.48690.1
MLP8× 20025.95.650.6 46452.8345162.4 39573.59591.1
ConvSmall15.8334.7 47743.65559.872.77.073.2
ConvBig71.12173.6 4077.110277.542 11 79.33.180.4
CIFARConvSmall35.9439.986 39.810544.613 46.36.848.1
ConvBig42.14345.9 346No public code48.3176 51.615.355.0
ResNet24.1124.591cannot run24.81.724.8 1.624.8
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"" + } + ], + "page_info": { + "page_no": 12, + "width": 1700, + "height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/qpsl2dR9twy/qpsl2dR9twy.md b/parse/train/qpsl2dR9twy/qpsl2dR9twy.md new file mode 100644 index 0000000000000000000000000000000000000000..42c2532cc9fa98073f40f32bb145c91cce009501 --- /dev/null +++ b/parse/train/qpsl2dR9twy/qpsl2dR9twy.md @@ -0,0 +1,250 @@ +# COMMUNICATION IN MULTI-AGENT REINFORCEMENT LEARNING: INTENTION SHARING + +Woojun Kim, Jongeui Park, Youngchul Sung∗ +School of Electrical Engineering, KAIST +Daejeon, South Korea +{woojun.kim, jongeui.park, ycsung}@kaist.ac.kr + +# ABSTRACT + +Communication is one of the core components for learning coordinated behavior in multi-agent systems. In this paper, we propose a new communication scheme named Intention Sharing (IS) for multi-agent reinforcement learning in order to enhance the coordination among agents. In the proposed IS scheme, each agent generates an imagined trajectory by modeling the environment dynamics and other agents’ actions. The imagined trajectory is a simulated future trajectory of each agent based on the learned model of the environment dynamics and other agents and represents each agent’s future action plan. Each agent compresses this imagined trajectory capturing its future action plan to generate its intention message for communication by applying an attention mechanism to learn the relative importance of the components in the imagined trajectory based on the received message from other agents. Numeral results show that the proposed IS scheme significantly outperforms other communication schemes in multi-agent reinforcement learning. + +# 1 INTRODUCTION + +Reinforcement learning (RL) has achieved remarkable success in various complex control problems such as robotics and games (Gu et al. (2017); Mnih et al. (2013); Silver et al. (2017)). Multi-agent reinforcement learning (MARL) extends RL to multi-agent systems, which model many practical real-world problems such as connected cars and smart cities (Roscia et al. (2013)). There exist several distinct problems in MARL inherent to the nature of multi-agent learning (Gupta et al. (2017); Lowe et al. (2017)). One such problem is how to learn coordinated behavior among multiple agents and various approaches to tackling this problem have been proposed (Jaques et al. (2018); Pesce & Montana (2019); Kim et al. (2020)). One promising approach to learning coordinated behavior is learning communication protocol among multiple agents (Foerster et al. (2016); Sukhbaatar et al. (2016); Jiang & Lu (2018); Das et al. (2019)). The line of recent researches on communication for MARL adopts end-to-end training based on differential communication channel (Foerster et al. (2016); Jiang & Lu (2018); Das et al. (2019)). That is, a message-generation network is defined at each agent and connected to other agents’ policies or critic networks through communication channels. Then, the message-generation network is trained by using the gradient of other agents’ policy or critic losses. Typically, the message-generation network is conditioned on the current observation or the hidden state of a recurrent network with observations as input. Thus, the trained message encodes the past and current observation information to minimize other agents’ policy or critic loss. It has been shown that due to the capability of sharing observation information, this kind of communication scheme has good performance as compared to communication-free MARL algorithms such as independent learning, which is widely used in MARL, in partially observable environments. + +In this paper, we consider the following further question for communication in MARL: + +”How to harness the benefit of communication beyond sharing partial observation.” + +We propose intention of each agent as the content of message to address the above question. Sharing intention using communication has been used in natural multi-agent systems like human society. + +For example, drivers use signal light to inform other drivers of their intentions. A car driver may slow down if a driver in his or her left lane turns the right signal light on. In this case, the signal light encodes the driver’s intention, which indicates the driver’s future behavior, not current or past observation such as the field view. By sharing intention using signal light, drivers coordinate their drive with each other. In this paper, we formalize and propose a new communication scheme for MARL named Intention sharing (IS) in order to go beyond existing observation-sharing schemes for communication in MARL. The proposed IS scheme allows each agent to share its intention with other agents in the form of encoded imagined trajectory. That is, each agent generates an imagined trajectory by modeling the environment dynamics and other agents’ actions. Then, each agent learns the relative importance of the components in the imagined trajectory based on the received messages from other agents by using an attention model. The output of the attention model is an encoded imagined trajectory capturing the intention of the agent and used as the communication message. We evaluate the proposed IS scheme in several multi-agent environments requiring coordination among agents. Numerical result shows that the proposed IS scheme significantly outperforms other existing communication schemes for MARL including the state-of-the-art algorithms such as ATOC and TarMAC. + +# 2 RELATED WORKS + +Under the asymmetry in learning resources between the training and execution phases, the framework of centralized training and decentralized execution (CTDE), which assumes the availability of all system information in the training phase and distributed policy in the execution phase, has been adopted in most recent MARL researches (Lowe et al. (2017); Foerster et al. (2018); Iqbal & Sha (2018); Kim et al. (2020)). Under the framework of CTDE, learning communication protocol has been considered to enhance performance in the decentralized execution phase for various multi-agent tasks (Foerster et al. (2016); Jiang & Lu (2018); Das et al. (2019)). For this purpose, Foerster et al. (2016) proposed Differentiable Inter-Agent Learning (DIAL). DIAL trains a message-generation network by connecting it to other agents’ Q-networks and allowing gradient flow through communication channels in the training phase. Then, in the execution phase the messages are generated and passed to other agents through communication channels. Jiang & Lu (2018) proposed an attentional communication model named ATOC to learn when to communicate and how to combine information received from other agents through communication based on attention mechanism. Das et al. (2019) proposed Targeted Multi-Agent Communication (TarMAC) to learn the message-generation network in order to produce different messages for different agents based on a signature-based attention model. The message-generation networks in the aforementioned algorithms are conditioned on the current observation or a hidden state of LSTM. Under partially observable environments, such messages which encode past and current observations are useful but do not capture any future information. In our approach, we use not only the current information but also future information to generate messages and the weight between the current and future information is adaptively learned according to the environment. This yields further performance enhancement, as we will see in Section 5. + +In our approach, the encoded imagined trajectory capturing the intention of each agent is used as the communication message in MARL. Imagined trajectory was used in other problems too. Racaniere \` et al. (2017) used imagined trajectory to augment it into the policy and critic for combining modelbased and model-free approaches in single-agent RL. It is shown that arbitrary imagined trajectory (rolled-out trajectory by using a random policy or own policy) is useful for single-agent RL in terms of performance and data efficiency. Strouse et al. (2018) introduced information-regularizer to share or hide agent’s intention to other agents for a multi-goal MARL setting in which some agents know the goal and other agents do not know the goal. By maximizing (or minimizing) the mutual information between the goal and action, an agent knowing the goal learns to share (or hide) its intention to other agents not knowing the goal in cooperative (or competitive) tasks. They showed that sharing intention is effective in the cooperative case. + +In addition to our approach, Theory of Mind (ToM) and Opponent Modeling (OM) use the notion of intention. Rabinowitz et al. (2018) proposed the Theory of Mind network (ToM-net) to predict other agents’ behaviors by using meta-learning. Raileanu et al. (2018) proposed Self Other-Modeling (SOM) to infer other agents’ goal in an online manner. Both ToM and OM take advantage of predicting other agents’ behaviors capturing the intention. One difference between our approach and the aforementioned two methods is that we use communication to share the intention instead of inference. That is, the agents in our approach allow other agents to know their intention directly through communication, whereas the agents in ToM and OM should figure out other agents’ intention by themselves. Furthermore, the messages in our approach include future information by rolling out the policy, whereas ToM and CM predict only the current or just next time-step information. + +# 3 SYSTEM MODEL + +We consider a partially observable $N$ -agent Markov game (Littman (1994)) and assume that communication among agents is available. At time step $t$ , Agent $i$ observes its own observation $o _ { t } ^ { i }$ , which is a part of the global environment state $s _ { t }$ , and selects action $a _ { t } ^ { i } \in { \mathcal { A } } _ { i }$ and message $m _ { t } ^ { i } \in \mathcal { M } _ { i }$ based on its own observation $o _ { t } ^ { i }$ and its own previous time step message $m _ { t - 1 } ^ { i }$ plus the received messages from other agents, i.e., $m _ { t - 1 } = ( m _ { t - 1 } ^ { 1 } , \cdot \cdot \cdot , m _ { t - 1 } ^ { N } )$ . We assume that the message $m _ { t } ^ { i }$ of Agent $i$ is sent to all other agents and available at other agents at the next time step, i.e., time $t + 1$ . The joint actions ccording to the tra $a _ { t } \overset { \cdot } { = } ( a _ { t } ^ { 1 } , \cdots , a _ { t } ^ { N } )$ nt state and th $s _ { t + 1 }$ and rewardsard function $\{ r _ { t } ^ { i } \} _ { i = 1 } ^ { N }$ $\mathcal { \bar { T } } : \mathcal { S } \times \mathcal { A } \times \mathcal { S } [ 0 , 1 ]$ $R ^ { i } : \mathcal { S } \times \mathcal { A } \mathbb { R }$ , respectively, where $s$ and $\begin{array} { r } { A = \prod _ { i = 1 } ^ { N } A ^ { i } } \end{array}$ are the environment state space and the joint action space, respectively. The goal of Agent $i$ is to find the policy $\pi ^ { i }$ that maximizes its discounted return $\begin{array} { r } { R _ { t } ^ { i } = \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } } r _ { t ^ { \prime } } ^ { i } } \end{array}$ . Hence, the objective function of Agent $i$ is defined as $J _ { i } ( \pi ^ { i } ) = \mathbb { E } _ { \pi } \left[ R _ { 0 } ^ { i } \right]$ , where $\pi = ( \pi ^ { 1 } , \cdots , \pi ^ { N } )$ and $\gamma \in [ 0 , 1 ]$ are the joint policy and the discounting factor, respectively. + +# 4 THE PROPOSED INTENTION SHARING SCHEME + +The key idea behind the IS scheme is that multiple agents communicate with other agents by sending their implicit future plans, which carry their intention. The received messages capturing the intention of other agents enable the agent to coordinate its action with those of other agents. We now describe the architecture of the proposed IS scheme. At time step $t$ , Agent $i$ selects an action $a _ { t } ^ { i } \sim \pi ^ { i } ( \cdot | o _ { t } ^ { i } , m _ { t - 1 } )$ and a message $\stackrel { \cdot } { m } _ { t } ^ { i } = M G N ^ { i } ( o _ { t } ^ { i } , m _ { t - 1 } , \pi ^ { i } )$ based on its own observation $o _ { t } ^ { i }$ and received messages $m _ { t - 1 }$ , where $M G N ^ { i }$ is the message-generation network (MGN) of Agent $i$ . The MGN consists of two components: Imagined trajectory generation module (ITGM) and attention module (AM). Each agent generates an imagined trajectory by using ITGM and learns the importance of each imagined step in the imagined trajectory by using AM. The output of AM is an encoded imagined trajectory reflecting the importance of imagined steps and is used as the communication message. The overall architecture of the proposed IS scheme is shown in Fig. 1. In the following we describe the detail of each module. + +# 4.1 IMAGINED TRAJECTORY GENERATION MODULE (ITGM) + +The role of ITGM is to produce the next imagined step. ITGM takes the received messages, observation, and action as input and yields the predicted next observation and predicted action as output. By stacking ITGMs, we generate an imagined trajectory, as shown in Fig. 1. For Agent $i$ at time step $t$ , we define an $H$ -length imagined trajectory as + +$$ +\tau ^ { i } = ( \tau _ { t } ^ { i } , \hat { \tau } _ { t + 1 } ^ { i } , \cdot \cdot \cdot , \hat { \tau } _ { t + H - 1 } ^ { i } ) , +$$ + +where $\hat { \tau } _ { t + k } ^ { i } = ( \hat { o } _ { t + k } ^ { i } , \hat { a } _ { t + k } ^ { i } )$ is the imagined step at time step $t + k$ . Note that $\tau _ { t } ^ { i } = ( o _ { t } ^ { i } , a _ { t } ^ { i } )$ is the true values of observation and action, but the imagined steps except $\tau _ { t } ^ { i }$ are predicted values. + +ITGM consists of a roll-out policy and two predictors: Other agents’ action predictor $f _ { a } ^ { i } ( o _ { t } ^ { i } )$ (we will call this predictor simply action predictor) and observation predictor $f _ { o } ^ { i } ( o _ { t } ^ { i } , a _ { t } ^ { i } , a _ { t } ^ { - i } )$ . First, we model the action predictor which takes the observation as input and produces other agents’ predicted actions. The output of the action predictor is given by + +$$ +\begin{array} { r } { f _ { a } ^ { i } ( o _ { t } ^ { i } ) = ( \hat { a } _ { t } ^ { 1 } , \cdot \cdot \cdot , \hat { a } _ { t } ^ { i - 1 } , \hat { a } _ { t } ^ { i + 1 } , \cdot \cdot \cdot , \hat { a } ^ { N } ) = : \hat { a } _ { t } ^ { - i } } \end{array} +$$ + +Note that the action predictor can be trained by the previously proposed opponent modeling method (Rabinowitz et al. (2018); Raileanu et al. (2018)) and can take the received messages as input. Next, we model the observation predictor $f _ { o } ^ { i } ( o _ { t } ^ { i } , a _ { t } ^ { i } , \hat { a } _ { t } ^ { - i } )$ which is conditioned on the observation $o _ { t } ^ { i }$ , own action $a _ { t } ^ { i }$ , and the output of the action predictor $\hat { a } _ { t } ^ { - i }$ . Here, we adopt the dynamics function that predicts the difference between the next observation and the current observation, i.e., $o _ { t + 1 } ^ { i } ~ - ~ o _ { t } ^ { i }$ instead of the next observation $o _ { t + 1 } ^ { i }$ proposed in (Nagabandi et al. (2018)) in order to reduce model bias in the early stage of learning. Hence, the next observation can be written as + +![](images/c9b874186cbaf9921bdc0929bdc73a3541afd60d23c39ea1d0cb95194fbd81c8.jpg) +Figure 1: Overall architecture of the IS scheme from the perspective of Agent $i$ + +$$ +\hat { o } _ { t + 1 } ^ { i } = o _ { t } ^ { i } + f _ { o } ^ { i } ( o _ { t } ^ { i } , a _ { t } ^ { i } , \hat { a } _ { t } ^ { - i } ) . +$$ + +By injecting the predicted next observation and the received messages into the roll-out policy in ITGM, we obtain the predicted next action $\hat { a } _ { t + 1 } ^ { i } = \pi ^ { i } ( \hat { o } _ { t + 1 } ^ { i } , m _ { t - 1 } )$ . Here, we use the current policy as the roll-out policy. Combining $\hat { o } _ { t + 1 } ^ { i }$ and $\hat { a } _ { t + 1 } ^ { i }$ , we obtain next imagined step at time step $t + 1$ , $\tau _ { t + 1 } ^ { i } = ( \widehat { o } _ { t + 1 } ^ { i } , \widehat { a } _ { t + 1 } ^ { i } )$ . In order to produce an $H$ -length imagined trajectory, we inject the output of ITGM and the received messages $m _ { t - 1 }$ into the input of ITGM recursively. Note that we use the received messages at time step $t$ , $m _ { t - 1 }$ , in every recursion of ITGM.1 + +# 4.2 ATTENTION MODULE (AM) + +Instead of the naive approach that uses the imagined trajectory $[ \tau _ { t } , \cdot \cdot \cdot , \tau _ { t + H - 1 } ]$ directly as the message, we apply an attention mechanism in order to learn the relative importance of imagined steps and encode the imagined trajectory according to the relative importance. We adopt the scaledot product attention proposed in (Vaswani et al. (2017)) as our AM. Our AM consists of three components: query, key, and values. The output of AM is the weighted sum of values, where the weight of values is determined by the dot product of the query and the corresponding key. In our model, the query consists of the received messages, and the key and value consist of the imagined trajectory. For Agent $i$ at time step $t$ , the query, key and value are defined as + +$$ +\begin{array} { r l } & { q _ { t } ^ { i } = W _ { Q } ^ { i } m _ { t - 1 } = W _ { Q } ^ { i } \Bigl [ m _ { t - 1 } ^ { 1 } \lVert m _ { t - 1 } ^ { 2 } \rVert \cdots \lVert m _ { t - 1 } ^ { N - 1 } \lVert m _ { t - 1 } ^ { N } \Bigr ] \in \mathbb { R } ^ { d _ { k } } } \\ & { k _ { t } ^ { i } = \Bigl [ W _ { K } ^ { i } \tau _ { t } , \cdots , \underbrace { W _ { K } ^ { i } \tau _ { t + h - 1 } } _ { = : k _ { t } ^ { i , h } } , \cdots , W _ { K } ^ { i } \tau _ { t + H - 1 } \Bigr ] \in \mathbb { R } ^ { H \times d _ { k } } } \\ & { v _ { t } ^ { i } = \Bigl [ W _ { V } ^ { i } \tau _ { t } , \cdots , \underbrace { W _ { V } ^ { i } \tau _ { t + h - 1 } } _ { = : v _ { t } ^ { i , h } } , \cdots , W _ { V } ^ { i } \tau _ { t + H - 1 } \Bigr ] \in \mathbb { R } ^ { H \times d _ { m } } , } \end{array} +$$ + +where $W _ { Q } ^ { i } \in \mathbb { R } ^ { d _ { k } \times N d _ { m } }$ , $W _ { K } ^ { i } \in \mathbb { R } ^ { d _ { k } \times d _ { \tau } }$ and $W _ { V } ^ { i } \in \mathbb { R } ^ { d _ { m } \times d _ { \tau } }$ are learnable parameters and operation $\parallel$ denotes the concatenation of vectors. The output $m _ { t } ^ { i }$ of the attention model, which is used for message, is the weighted sum of the values: + +$$ +m _ { t } ^ { i } = \sum _ { h = 1 } ^ { H } \alpha _ { h } ^ { i } v _ { t } ^ { i , h } , +$$ + +where the weight vector $\alpha ^ { i } = ( \alpha _ { 1 } ^ { i } , \cdot \cdot \cdot , \alpha _ { H } ^ { i } )$ is computed as + +$$ +\alpha ^ { i } = \mathrm { s o f t m a x } \left[ \frac { q _ { t } ^ { i ^ { 2 } } k _ { t } ^ { i , 0 } } { \sqrt { d _ { k } } } , \cdots , \underbrace { \frac { q _ { t } ^ { i ^ { T } } k _ { t } ^ { i , h } } { \sqrt { d _ { k } } } } _ { = : \alpha _ { h } ^ { i } } , \cdots , \frac { q _ { t } ^ { i ^ { T } } k _ { t } ^ { i , H } } { \sqrt { d _ { k } } } \right] . +$$ + +The weight of each value is computed by the dot product of the corresponding key and query. Since the projections of the imagined trajectory and the received messages are used for key and query, respectively, the weight can be interpreted as the relative importance of imagined step given the received messages. Note that $W _ { Q }$ , $W _ { K }$ and $W _ { V }$ are updated through the gradients from the other agents. + +# 4.3 TRAINING + +We implement the proposed IS scheme on the top of MADDPG (Lowe et al. (2017)), but it can be applied to other MARL algorithms. MADDPG is a well-known MARL algorithm and is briefly explained in Appendix A. In order to handle continuous state-action spaces, the actor, critic, observation predictor, and action predictor are parameterized by deep neural networks. For Agent $i$ , let ${ \theta } _ { \mu } ^ { i } , { \theta } _ { Q } ^ { i } , \bar { \theta } _ { o } ^ { i }$ , and $\theta _ { a } ^ { i }$ be the deep neural network parameters of actor, critic, observation predictor, and action predictor, respectively. Let $W ^ { i } = ( W _ { Q } ^ { i } , W _ { K } ^ { i } , W _ { V } ^ { i } )$ be the trainable parameters in the attention module of Agent $i$ . The centralized critic for Agent $i$ , $Q ^ { i }$ , is updated to minimize the following loss: + +$$ +L _ { Q } ( \theta _ { Q } ^ { i } ) = \mathbb { E } _ { x , a , r ^ { i } , x ^ { \prime } } \Big [ ( y ^ { i } - Q ^ { i } ( x , a ) ) ^ { 2 } \Big ] , \quad y ^ { i } = r ^ { i } + \gamma Q ^ { i - } ( x ^ { \prime } , a ^ { \prime } ) | _ { a ^ { j ^ { \prime } } = \mu ^ { i - } ( \sigma ^ { \prime } { } ^ { j } , m ) } , +$$ + +where $Q ^ { i - }$ and $\mu ^ { i - }$ are the target Q-function and the target policy of Agent $i$ and parameterized by $\theta _ { \mu } ^ { i - } , \theta _ { Q } ^ { i - }$ , respectively. The policy is updated to minimize the policy gradient loss: + +$$ +\nabla _ { \theta _ { \mu } ^ { i } } J ( \theta _ { \mu } ^ { i } ) = \mathbb { E } _ { x , a } \Big [ \nabla _ { \theta _ { \mu } ^ { i } } \mu ^ { i } ( o ^ { i } , m ) \nabla _ { a ^ { i } } Q ^ { i } ( x , a ) | _ { a ^ { i } = \mu ^ { i } ( o ^ { i } , m ) } \Big ] +$$ + +Since the MGN is connected to the agent’s own policy and other agents’ policies, the attention module parameters $W ^ { i }$ are trained by gradient flow from all agents. The gradient of Agent $i$ ’s attention module parameters is given by $\mathsf { \bar { V } } _ { W ^ { i } } J ( W ^ { i } ) =$ + +$$ +\frac { 1 } { N } \sum _ { j = 1 } ^ { N } \left[ \mathbb { E } _ { \bar { x } , \overline { { m } } , \bar { x } , a } \Big [ \nabla _ { W ^ { i } } M G N ( \tilde { m } ^ { i } | \bar { \sigma } ^ { i } , \overline { { m } } ) \nabla _ { \tilde { m } ^ { i } } \mu ^ { j } ( \sigma ^ { j } , \tilde { m } ^ { i } , \tilde { m } ^ { - i } ) \nabla _ { a ^ { j } } Q ^ { j } ( x , a ) | _ { a ^ { j } = \mu ^ { j } ( \sigma ^ { j } , m ) } \Big ] \right] , +$$ + +where $\overline { { O } } ^ { i }$ and $\overline { { m } }$ are the previous observation and received messages, respectively. The gradient of the attention module parameters are obtained by applying the chain rule to policy gradient. + +Both the action predictor and the observation predictor are trained based on supervised learning and the loss functions for agent $i$ are given by + +$$ +\begin{array} { r l } & { L ( \theta _ { a } ^ { i } ) = \mathbb { E } _ { o ^ { i } , a } \Big [ \big ( f _ { \theta _ { a } ^ { i } } ^ { i } ( o ^ { i } ) - a ^ { - i } \big ) ^ { 2 } \Big ] } \\ & { L ( \theta _ { o } ^ { i } ) = \mathbb { E } _ { o ^ { i } , a , o ^ { \prime } ^ { i } } \Big [ \big ( \big ( o ^ { \prime ^ { i } } - o ^ { i } \big ) - f _ { \theta _ { o } ^ { i } } ^ { i } \big ( o ^ { i } , a ^ { i } , \hat { a } ^ { - i } \big ) \big ) ^ { 2 } \Big ] . } \end{array} +$$ + +# 5 EXPERIMENT + +In order to evaluate the proposed algorithm and compare it with other communication schemes fairly, we implemented existing baselines on the top of the same MADDPG used for the proposed scheme. + +The considered baselines are as follows. 1) MADDPG (Lowe et al. (2017)): we can assess the gain of introducing communication from this baseline. 2) DIAL (Foerster et al. (2016)): we modified DIAL, which is based on Q-learning, to our setting by connecting the message-generation network to other agents’ policies and allowing the gradient flow through communication channel. 3) TarMAC (Das et al. (2019)): we adopted the key concept of TarMAC in which the agent sends targeted messages using a signature-based attention model. 4) Comm-OA: the message consists of its own observation and action. 5) ATOC (Jiang & Lu (2018)): an attentional communication model which learns when communication is needed and how to combine the information of agents. We considered three multi-agent environments: predator-prey, cooperative navigation, and traffic junction, and we slightly modified the conventional environments to require more coordination among agents. + +![](images/65e2e8a9585fe2006b682abcc9d4060731b5c342e805424fd1e2bd25bde7fbfa.jpg) +Figure 2: Considered environments: (a) predator-prey (PP), (b) cooperative-navigation (CN), and (c) traffic-junction (TJ) + +# 5.1 ENVIRONMENTS + +Predator-prey (PP) The predator-prey environment is a standard task in multi-agent systems. We used a PP environment that consists of $N$ predators and fixed $M$ preys in a continuous state-action domain. We control the actions of predators and the goal is to capture as many preys as possible in a given time. Each agent observes the positions of predators and preys. When $C$ predators catch a prey simultaneously, the prey is captured and all predators get shared reward $R _ { 1 }$ . At every time when all the preys are captured, the preys are respawn and the shared reward value $R _ { 1 }$ increases by one with initial value one to accelerate the capture speed for the given time. We simulated three cases: $( N = 2 , C = 1 )$ ), $( N = 3 , C = 1 )$ , and $\mathit { N } = 4 , C = 2 )$ with all $M = 9$ preys, where the fixed positions of the preys are shown in Fig.2(a). In the cases of $( N = 2 , C = 1$ ) and $( N = 3 , C = 1 )$ ), the initial positions of all predators are the same and randomly determined. Thus, the predators should learn not only how to capture preys but also how to spread out. In the case of $( N = 4 , C = 2 )$ , the initial positions of all predators are randomly determined independently. Thus, the predators should learn to capture preys in group of two. + +Cooperative-navigation (CN) The goal of cooperative navigation introduced in (Lowe et al. (2017)) is for $N$ agents to cover $L$ landmarks while avoiding collisions among the agents. We modified the original environment so that collisions occur more easily. We set $L = N$ , increased the size of agent, and assigned a specific landmark to cover to each agent (i.e., each agent should cover the landmark of the same color in Fig.2(b)). Each agent observes the positions of predators and landmarks. The agent receives shared reward $R _ { 1 }$ which is the sum of the distance between each agent and the corresponding landmark at each time step and success reward $N ^ { \prime } \times R _ { 2 }$ where $N ^ { \prime }$ is the number of the covered landmark. Agents who collide with other agents receive negative reward $R _ { 3 }$ . We simulated the environment with $N = L = 3$ , $R _ { 1 } = 1 / 3$ , $R _ { 2 } = 1$ and $R _ { 3 } = - 5$ . + +Traffic-junction (TJ) We modified the traffic-junction introduced in Sukhbaatar et al. (2016) to continuous state-action domain. In the beginning of an episode, each agent is randomly located in a predefined initial position and assigned one of three routes: left, right or straight, as seen in Fig.2(c). The observation of each agent consists of the positions of all agents (no route information of other agents) and 2 one-hot vectors which encodes the initial position and assigned route of the agent. The action of each agent is a real value in $( 0 , 1 )$ , which indicates the distance to go along the assigned route from the current position. The goal is to go to the destination as fast as possible while avoiding collision with other agents. To achieve the goal, we design reward with three components. Each agent receives success reward $R _ { 1 }$ if it arrives at the destination without any collision with other agents, collision negative reward $R _ { 2 }$ if its position is overlapped with that of other agent, and time negative reward $R _ { 3 }$ to avoid traffic jam. When an agent arrives at the destination, the agent is assigned a new initial position and the route. An episode ends when $T$ time steps elapse. We set $R _ { 1 } = 2 0$ , $R _ { 2 } = - 1 0$ , and $R _ { 3 } = - 0 . 0 1 \tau$ , where $\tau$ is the total time step after agent is initialized. + +![](images/8387baf51bb80be1110568befb74b35cde463335a1b82d273a70194c0d32c2a4.jpg) +Figure 3: Performance for MADDPG (blue), DIAL (green), TarMAC (red), Comm-OA (purple), ATOC (cyan) and the proposed IS method (black). + +# 5.2 RESULTS + +Fig. 3 shows the performance of the proposed IS scheme and the considered baselines on the PP, CN, and TJ environments. Figs.3(a)-(d) shows the learning curves of the algorithms on PP and CN and Figs. (e)-(f) show the average return using deterministic policy over 100 episodes every 250000 time steps. All performance is averaged over 10 different seeds. It is seen that Comm-OA performs similarly to MADDPG in the considered environments. Since the received messages come from other agents at the previous time step, Comm-OA in which the communication message consists of agent’s observation and action performs similarly to MADDPG. Unlike Comm-OA, DIAL, TarMAC, and ATOC outperform MADDPG and the performance gain comes from the benefit of learning communication protocol in the considered environments except PP with $N = 4$ . In PP with $N = 4$ , four agents need to coordinate to spread out in group of two to capture preys. In this complicated coordination requirement, simply learning communication protocol based on past and current information did not obtain benefit from communication. On the contrary, the proposed IS scheme sharing intention with other agents achieved the required coordination even in this complicated environment. + +# 5.3 ANALYSIS + +Imagined trajectory The proposed IS scheme uses the encoded imagined trajectory as the message content. Each agent rolls out an imagined trajectory based on its own policy and trained models including action predictor and observation predictor. Since the access to other agents’ policies is not available, the true trajectory and the imagined trajectory can mismatch. Especially, the mismatch is large in the beginning of an episode because each agent does not receive any messages from other agents (In this case, we inject zero vector instead of the received messages into the policy). We expect that the mismatch will gradually decrease as the episode progresses and this can be interpreted as the procedure of coordination among agents. Fig.5 shows the positions of all agents and each agent’s imagined trajectory over time step in one episode for predator-prey with $N = 3$ predators, where the initial positions of the agents after the end of training $( t = 0$ ) is bottom right on the map. Note that each agent estimates the future positions of other agents as well as their own future position due to the assumption of full observability. The first, second, and third row of Fig.5 show the imagined trajectories of all agents at Agent 1 (red), Agent 2 (green) and Agent 3 (blue), respectively. Note that the imagined trajectory of each agent represents its future plan for the environment. As seen in Fig.5, at $t = 0$ the intention of both Agent 1 and Agent 3 is to move to the left to catch preys. At $t = 1$ , all agents receive the messages from other agents. It is observed that Agent 3 changes its future plan to catch preys around the center while Agent 1 maintains its future plan. This procedure shows that coordination between Agent 1 and Agent 3 starts to occur. It is seen that as time goes, each agent roughly predicts other agents’ future actions. + +![](images/7b3b19c5a9045ea3e1116c74ee3a2c72b0b990f2a8adf5552c566ad452bf7ac2.jpg) +Figure 4: Performance for our proposed method with different length of imagined trajectory $H$ and without attention module. + +We conducted experiments to examine the impact of the length of the imagined trajectory $H$ . Fig.4 shows the performances of the proposed method for different values of $H$ . It is seen that the training speed is reduced when $H = 7$ as compared to $H = 3$ or $H = 5$ . However, the final performance all outperforms the baseline. + +Attention In the proposed IS scheme, the imagined trajectory is encoded based on the attention module to capture the importance of components in the imagined trajectory. Recall that the message of Agent $i$ is expressed as $\begin{array} { r } { m _ { t } ^ { i } = \sum _ { h = 1 } ^ { H } \alpha _ { h } ^ { i } v _ { t } ^ { i , h } } \end{array}$ , as seen in (7), where $\alpha _ { h } ^ { i }$ denotes the importance of $v _ { t } ^ { i , h }$ , which is the encoded imagined step. Note that the previously proposed communication schemes are the special case corresponding to $\alpha ^ { i } = ( 1 , 0 , \cdots , 0 )$ . In Fig.5, the brightness of each circle is proportional to the attention weight. At time step $t \ = \ K$ , where $K = 3 7$ , $\alpha _ { 2 } ^ { 1 }$ , which 2indicates when Agent 1 moves to the prey in the bottom middle, is the highest. In addition, $\alpha _ { 4 } ^ { 3 }$ which indicates when Agent 3 moves to the prey in the left middle, is the highest. Hence, the agent tends to send future information when it is near a prey. Similar attention weight tendency is also captured in the time step $t = K + 1$ and $t = K + 2$ . + +As aforementioned, the aim of the IS scheme is to communicate with other agents based on their own future plans. How far future is important depends on the environment and on the tasks. In order to analyze the tendency in the importance of future plans, we averaged the attention weight over the trajectories on the fully observable PP environment with 3 agents and a partially observable PP environment with 3 agents in which each agent knows the locations of other agents within a certain range from the agent. The result is summarized in Table 1. It is observed that the current information (time $k$ ) and the farthest future information (time $k + 4 )$ are mainly used as the message content in the fully observable case, whereas the current information and the information next to the present (time $k$ and $k + 1 )$ ) are mainly used in the partially observable environment. This is because sharing observation information is more critical in the partially observable case than the fully observable case. A key aspect of the proposed IS scheme is that it adaptively selects most important steps as the message content depending on the environment by using the attention module. + +We conducted an ablation study for the attention module and the result is shown in Fig. 4. We compared the proposed IS scheme with and without the attention module. We replace the attention module with an averaging layer, which is the special case corresponding to $\alpha ^ { i } = ( \textstyle { \frac { 1 } { H } } , \cdot \cdot \cdot , \textstyle { \frac { 1 } { H } } )$ . Fig. 4 shows that the proposed IS scheme with the attention module yields better performance than the one without the attention module. This shows the necessity of the attention module. In the PP environment with 4 agents, the imagined trajectory alone without the attention module improves the training speed while the final performance is similar to that of MADDPG. In the TJ environment with 3 agents, the imagined trajectory alone without the attention module improves both the final performance and the training speed. + +![](images/2d65899b9ef8f947fa967ca00c1b95f2451e28838ca23027d3e02cd83f73438d.jpg) +Figure 5: Imagined trajectories and attention weights of each agent on PP $\left( \mathrm { N } { = } 3 \right)$ : 1st row - agent1 (red), 2nd row - agent2 (green), and 3rd row - agent3 (blue). Black squares, circle inside the timesicon, and other circles denote the prey, current position, and estimated future positions, respectively. The brightness of the circle is proportional to the attention weight. $K = 3 7$ . + +Table 1: Averaged attention weight over the trajectory at time step $k$ + +
Imagined stepskk+1k+2k+3k+4
Fully observable PP (N=3)Partially observable PP (N=3)0.330.320.180.220.150.170.140.150.200.14
+ +# 6 CONCLUSION + +In this paper, we proposed the IS scheme, a new communication protocol, based on sharing intention among multiple agents for MARL. The message-generation network in the proposed IS scheme consists of ITGM, which is used for producing predicted future trajectories, and AM, which learns the importance of imagined steps based on the received messages. The message in the proposed scheme is encoded imagined trajectory capturing the agent’s intention so that the communication message includes the future information as well as the current information, and their weights are adaptively determined depending on the environment. We studied examples of imagined trajectories and attention weights. It is observed that the proposed IS scheme generates meaningful imagined trajectories and attention weights. Numerical results show that the proposed IS scheme outperforms other communication algorithms including state-of-the-art algorithms. Furthermore, we expect that the key idea of the proposed IS scheme combining with other communication algorithms such as ATOC and TarMAC would yield even better performance. + +# 7 ACKNOWLEDGMENTS + +This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning(NRF2017R1E1A1A03070788). + +# REFERENCES + +Abhishek Das, Theophile Gervet, Joshua Romoff, Dhruv Batra, Devi Parikh, Mike Rabbat, and ´ Joelle Pineau. Tarmac: Targeted multi-agent communication. In International Conference on Machine Learning, pp. 1538–1546, 2019. + +Jakob Foerster, Ioannis Alexandros Assael, Nando De Freitas, and Shimon Whiteson. Learning to communicate with deep multi-agent reinforcement learning. In Advances in neural information processing systems, pp. 2137–2145, 2016. + +Jakob N Foerster, Gregory Farquhar, Triantafyllos Afouras, Nantas Nardelli, and Shimon Whiteson. 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Playing atari with deep reinforcement learning. arXiv preprint arXiv:1312.5602, 2013. + +Anusha Nagabandi, Gregory Kahn, Ronald S Fearing, and Sergey Levine. Neural network dynamics for model-based deep reinforcement learning with model-free fine-tuning. In 2018 IEEE International Conference on Robotics and Automation (ICRA), pp. 7559–7566. IEEE, 2018. + +Emanuele Pesce and Giovanni Montana. Improving coordination in small-scale multi-agent deep reinforcement learning through memory-driven communication. arXiv preprint arXiv:1901.03887, 2019. + +Neil C Rabinowitz, Frank Perbet, H Francis Song, Chiyuan Zhang, SM Eslami, and Matthew Botvinick. Machine theory of mind. arXiv preprint arXiv:1802.07740, 2018. + +Sebastien Racani ´ ere, Th \` eophane Weber, David Reichert, Lars Buesing, Arthur Guez, ´ Danilo Jimenez Rezende, Adria Puigdomenech Badia, Oriol Vinyals, Nicolas Heess, Yujia Li, et al. Imagination-augmented agents for deep reinforcement learning. In Advances in neural information processing systems, pp. 5690–5701, 2017. + +Roberta Raileanu, Emily Denton, Arthur Szlam, and Rob Fergus. Modeling others using oneself in multi-agent reinforcement learning. arXiv preprint arXiv:1802.09640, 2018. + +Mariacristina Roscia, Michela Longo, and George Cristian Lazaroiu. Smart city by multi-agent systems. In 2013 International Conference on Renewable Energy Research and Applications (ICRERA), pp. 371–376. IEEE, 2013. + +David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. nature, 550(7676):354–359, 2017. + +DJ Strouse, Max Kleiman-Weiner, Josh Tenenbaum, Matt Botvinick, and David J Schwab. Learning to share and hide intentions using information regularization. In Advances in Neural Information Processing Systems, pp. 10249–10259, 2018. + +Sainbayar Sukhbaatar, Rob Fergus, et al. Learning multiagent communication with backpropagation. In Advances in neural information processing systems, pp. 2244–2252, 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. + +# A MULTI-AGENT DEEP DETERMINISTIC POLICY GRADIENT (MADDPG) + +MADDPG is an extended version of DDPG to multi-agent systems under the framework of CTDE (Lowe et al. (2017)). Each agent has a deterministic policy $a _ { t } = \mu _ { \theta _ { \mu } } ^ { i } ( o _ { t } )$ conditioned on its own observation $o _ { t }$ and a centralized critic $Q _ { \theta _ { Q } } ^ { i } ( x , a ) = E \big [ R _ { t } ^ { i } \big | x _ { t } = x , a _ { t } = a \big ]$ conditioned on the joint actions $a _ { t }$ and state information $x _ { t }$ . Here, $x _ { t }$ can be state $s _ { t }$ or the set of observations $( o _ { t } ^ { 1 } , \cdots , o _ { t } ^ { N } )$ . The centralized critic is trained by minimizing the following loss: + +$$ +L _ { Q } ( \theta _ { Q } ) = \mathbb { E } _ { x , a , r ^ { i } , x ^ { \prime } } \Big [ ( y ^ { i } - Q _ { \theta _ { Q } } ^ { i } ( x , a ) ) ^ { 2 } \Big ] , ~ y ^ { i } = r ^ { i } + \gamma Q _ { \theta _ { Q } ^ { - } } ^ { i } ( x ^ { \prime } , a ^ { \prime } ) | _ { a ^ { j ^ { \prime } } = \mu ^ { i - } ( o ^ { j } ) } , +$$ + +where $\theta _ { Q } ^ { - }$ is the parameter of target Q-function and $\mu ^ { i - }$ is the target policy of Agent $i$ . The policy is trained by Deterministic Policy Gradient (DPG), and the gradient of the objective with respect to the policy parameter $\theta _ { \mu ^ { i } }$ is given by + +$$ +\nabla _ { \theta _ { \mu ^ { i } } } J ( \mu ^ { i } ) = \mathbb { E } _ { x , a } \Big [ \nabla _ { \theta _ { \mu ^ { i } } } \mu ^ { i } ( o ^ { i } ) \nabla _ { a ^ { i } } Q _ { \theta _ { Q } } ^ { i } ( x , a ) | _ { a ^ { i } = \mu ^ { i } ( o ^ { i } ) } \Big ] . +$$ + +# B TRAINING DETAILS AND HYPERPARAMETERS + +Table 2: Hyperparameters of all algorithms + +
MADDPGTARMACDIALATOCIS
REPLAY BUFFER SIZE2×1052×1052×1052×1052×105
DISCOUNT FACTOR0.990.990.990.990.99
MINI-BATCH SIZE128128128128128
OPTIMIZERADAMADAMADAMADAMADAM
LEARNING RATE0.00050.00050.00050.00050.0005
NUMBER OF HIDDEN LAYERS (ALL NETWORKS)22222
NUMBER OF HIDDENUNITSPERLAYER128128128128128
ACTIVATION FUNCTION FOR HIDDEN LAYERRELURELURELURELURELU
MESSAGE DIMENSION ON PP15555
MESSAGE DIMENSION ON CN3333
MESSAGE DIMENSION ON TJ3333
ATTENTION DIMENSION ON PP5-55
ATTENTION DIMENSION ON CJ333
ATTENTION DIMENSION ON TJ333
IMAGINED TRAJECTORY LENGTH H-5
+ +# C ADDITIONAL ABLATION STUDY + +We conducted an additional experiment to examine whether performance improvement is gained from sharing intention or having a prediction of the future. We compared the proposed IS scheme with MADPPG- $\boldsymbol { \cdot } \mathbf { p }$ in which the agent does not use communication, but uses their own imagined trajectory as additional input. Fig. 6 shows that the proposed IS scheme outperforms MADDPG-p. Thus, sharing intention, which is a core idea of this paper, is more important than having a prediction of the future. + +![](images/855716cc9dda436e0c9a3111e7d6d848e9a8f2fcdae2392503dbf27cdacc0c44.jpg) +Figure 6: Performance for MADDPG (blue), MADDPG-p (orange), and the proposed IS method (black). + +# D PSEUDO CODE + +# Algorithm 1 Intention Sharing (IS) Communication Scheme + +Initialize parameter $\theta _ { \mu } ^ { i } , \theta _ { Q } ^ { i } , \theta _ { \mu } ^ { i - } , \theta _ { Q } ^ { i - } , \theta _ { o } ^ { i } , \theta _ { a } ^ { i } , W ^ { i } , \forall i \in \{ 1 , \cdots , N \}$ +for episode $: = 1 , 2 , \cdots$ do Initialize state $s _ { 1 }$ , messages $m _ { 0 } = ( \overrightarrow { 0 } , \cdot \cdot \cdot , \overrightarrow { 0 } )$ and each agent observes $o _ { 1 } ^ { i }$ for $t < = T$ and $s _ { t } \neq$ terminal do Each agent receives the messages $m _ { t - 1 } = ( m _ { t - 1 } ^ { 1 } , \cdot \cdot \cdot , m _ { t - 1 } ^ { N } )$ Each agent selects action $a _ { t } ^ { i } \sim \pi ^ { i } ( \cdot | o _ { t } ^ { i } , m _ { t - 1 } )$ for each agent $i$ Execute $\mathbf { \delta } \mathbf { a } _ { t }$ and each agent $i$ receives $r _ { t }$ and $o _ { t + 1 } ^ { i }$ for $h = 1 , 2 , \cdots , H$ do Predict other agents’ actions $\hat { a } _ { t + h - 1 } ^ { - i }$ from the action predictor $f _ { a } ^ { i }$ Generate $\hat { o } _ { t + h } ^ { i }$ from observation predictor $f _ { o } ^ { i } ( o _ { t + h - 1 } ^ { i } , \hat { a } _ { t + h - 1 } ^ { i } , \hat { a } _ { t + h - 1 } ^ { - i } )$ Generate $\hat { a } _ { t + h } ^ { i } \sim \pi ^ { i } ( \cdot | \hat { o } _ { t + h } ^ { i } , m _ { t - 1 } )$ Each agent generates the messages $m _ { t } ^ { i }$ by injecting $\tau ^ { i } = ( \tau _ { t } ^ { i } , \hat { \tau } _ { t + 1 } ^ { i } , \cdot \cdot \cdot , \hat { \tau } _ { t + H - 1 } ^ { i } )$ into Attention Module (AM) Store transitions in $D$ end for for each gradient step do Update $\theta _ { Q } ^ { i }$ and $( \theta _ { o } ^ { i } , \theta _ { a } ^ { i } )$ by minimizing the loss (9) and the loss (12) Update $\theta _ { \mu } ^ { i }$ and $W ^ { i }$ based on the gradient (10) and the gradient (11) end for Update $\theta _ { \mu } ^ { i - } , \theta _ { Q } ^ { i - }$ using the moving average method +end for \ No newline at end of file diff --git a/parse/train/qpsl2dR9twy/qpsl2dR9twy_content_list.json b/parse/train/qpsl2dR9twy/qpsl2dR9twy_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..6e03b94b17b0dfe40de8097e1356723a0e8f9442 --- /dev/null +++ b/parse/train/qpsl2dR9twy/qpsl2dR9twy_content_list.json @@ -0,0 +1,1302 @@ +[ + { + "type": "text", + "text": "COMMUNICATION IN MULTI-AGENT REINFORCEMENT LEARNING: INTENTION SHARING ", + "text_level": 1, + "bbox": [ + 174, + 98, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Woojun Kim, Jongeui Park, Youngchul Sung∗ \nSchool of Electrical Engineering, KAIST \nDaejeon, South Korea \n{woojun.kim, jongeui.park, ycsung}@kaist.ac.kr ", + "bbox": [ + 184, + 170, + 632, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 262, + 544, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Communication is one of the core components for learning coordinated behavior in multi-agent systems. In this paper, we propose a new communication scheme named Intention Sharing (IS) for multi-agent reinforcement learning in order to enhance the coordination among agents. In the proposed IS scheme, each agent generates an imagined trajectory by modeling the environment dynamics and other agents’ actions. The imagined trajectory is a simulated future trajectory of each agent based on the learned model of the environment dynamics and other agents and represents each agent’s future action plan. Each agent compresses this imagined trajectory capturing its future action plan to generate its intention message for communication by applying an attention mechanism to learn the relative importance of the components in the imagined trajectory based on the received message from other agents. Numeral results show that the proposed IS scheme significantly outperforms other communication schemes in multi-agent reinforcement learning. ", + "bbox": [ + 233, + 295, + 764, + 474 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 502, + 336, + 518 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Reinforcement learning (RL) has achieved remarkable success in various complex control problems such as robotics and games (Gu et al. (2017); Mnih et al. (2013); Silver et al. (2017)). Multi-agent reinforcement learning (MARL) extends RL to multi-agent systems, which model many practical real-world problems such as connected cars and smart cities (Roscia et al. (2013)). There exist several distinct problems in MARL inherent to the nature of multi-agent learning (Gupta et al. (2017); Lowe et al. (2017)). One such problem is how to learn coordinated behavior among multiple agents and various approaches to tackling this problem have been proposed (Jaques et al. (2018); Pesce & Montana (2019); Kim et al. (2020)). One promising approach to learning coordinated behavior is learning communication protocol among multiple agents (Foerster et al. (2016); Sukhbaatar et al. (2016); Jiang & Lu (2018); Das et al. (2019)). The line of recent researches on communication for MARL adopts end-to-end training based on differential communication channel (Foerster et al. (2016); Jiang & Lu (2018); Das et al. (2019)). That is, a message-generation network is defined at each agent and connected to other agents’ policies or critic networks through communication channels. Then, the message-generation network is trained by using the gradient of other agents’ policy or critic losses. Typically, the message-generation network is conditioned on the current observation or the hidden state of a recurrent network with observations as input. Thus, the trained message encodes the past and current observation information to minimize other agents’ policy or critic loss. It has been shown that due to the capability of sharing observation information, this kind of communication scheme has good performance as compared to communication-free MARL algorithms such as independent learning, which is widely used in MARL, in partially observable environments. ", + "bbox": [ + 174, + 534, + 825, + 811 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we consider the following further question for communication in MARL: ", + "bbox": [ + 176, + 819, + 741, + 833 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "”How to harness the benefit of communication beyond sharing partial observation.” ", + "bbox": [ + 223, + 845, + 767, + 859 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose intention of each agent as the content of message to address the above question. Sharing intention using communication has been used in natural multi-agent systems like human society. ", + "bbox": [ + 174, + 872, + 825, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "For example, drivers use signal light to inform other drivers of their intentions. A car driver may slow down if a driver in his or her left lane turns the right signal light on. In this case, the signal light encodes the driver’s intention, which indicates the driver’s future behavior, not current or past observation such as the field view. By sharing intention using signal light, drivers coordinate their drive with each other. In this paper, we formalize and propose a new communication scheme for MARL named Intention sharing (IS) in order to go beyond existing observation-sharing schemes for communication in MARL. The proposed IS scheme allows each agent to share its intention with other agents in the form of encoded imagined trajectory. That is, each agent generates an imagined trajectory by modeling the environment dynamics and other agents’ actions. Then, each agent learns the relative importance of the components in the imagined trajectory based on the received messages from other agents by using an attention model. The output of the attention model is an encoded imagined trajectory capturing the intention of the agent and used as the communication message. We evaluate the proposed IS scheme in several multi-agent environments requiring coordination among agents. Numerical result shows that the proposed IS scheme significantly outperforms other existing communication schemes for MARL including the state-of-the-art algorithms such as ATOC and TarMAC. ", + "bbox": [ + 174, + 104, + 825, + 325 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORKS ", + "text_level": 1, + "bbox": [ + 176, + 348, + 352, + 364 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Under the asymmetry in learning resources between the training and execution phases, the framework of centralized training and decentralized execution (CTDE), which assumes the availability of all system information in the training phase and distributed policy in the execution phase, has been adopted in most recent MARL researches (Lowe et al. (2017); Foerster et al. (2018); Iqbal & Sha (2018); Kim et al. (2020)). Under the framework of CTDE, learning communication protocol has been considered to enhance performance in the decentralized execution phase for various multi-agent tasks (Foerster et al. (2016); Jiang & Lu (2018); Das et al. (2019)). For this purpose, Foerster et al. (2016) proposed Differentiable Inter-Agent Learning (DIAL). DIAL trains a message-generation network by connecting it to other agents’ Q-networks and allowing gradient flow through communication channels in the training phase. Then, in the execution phase the messages are generated and passed to other agents through communication channels. Jiang & Lu (2018) proposed an attentional communication model named ATOC to learn when to communicate and how to combine information received from other agents through communication based on attention mechanism. Das et al. (2019) proposed Targeted Multi-Agent Communication (TarMAC) to learn the message-generation network in order to produce different messages for different agents based on a signature-based attention model. The message-generation networks in the aforementioned algorithms are conditioned on the current observation or a hidden state of LSTM. Under partially observable environments, such messages which encode past and current observations are useful but do not capture any future information. In our approach, we use not only the current information but also future information to generate messages and the weight between the current and future information is adaptively learned according to the environment. This yields further performance enhancement, as we will see in Section 5. ", + "bbox": [ + 174, + 382, + 825, + 686 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In our approach, the encoded imagined trajectory capturing the intention of each agent is used as the communication message in MARL. Imagined trajectory was used in other problems too. Racaniere \\` et al. (2017) used imagined trajectory to augment it into the policy and critic for combining modelbased and model-free approaches in single-agent RL. It is shown that arbitrary imagined trajectory (rolled-out trajectory by using a random policy or own policy) is useful for single-agent RL in terms of performance and data efficiency. Strouse et al. (2018) introduced information-regularizer to share or hide agent’s intention to other agents for a multi-goal MARL setting in which some agents know the goal and other agents do not know the goal. By maximizing (or minimizing) the mutual information between the goal and action, an agent knowing the goal learns to share (or hide) its intention to other agents not knowing the goal in cooperative (or competitive) tasks. They showed that sharing intention is effective in the cooperative case. ", + "bbox": [ + 174, + 694, + 825, + 847 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In addition to our approach, Theory of Mind (ToM) and Opponent Modeling (OM) use the notion of intention. Rabinowitz et al. (2018) proposed the Theory of Mind network (ToM-net) to predict other agents’ behaviors by using meta-learning. Raileanu et al. (2018) proposed Self Other-Modeling (SOM) to infer other agents’ goal in an online manner. Both ToM and OM take advantage of predicting other agents’ behaviors capturing the intention. One difference between our approach and the aforementioned two methods is that we use communication to share the intention instead of inference. That is, the agents in our approach allow other agents to know their intention directly through communication, whereas the agents in ToM and OM should figure out other agents’ intention by themselves. Furthermore, the messages in our approach include future information by rolling out the policy, whereas ToM and CM predict only the current or just next time-step information. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 SYSTEM MODEL ", + "text_level": 1, + "bbox": [ + 176, + 193, + 343, + 209 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We consider a partially observable $N$ -agent Markov game (Littman (1994)) and assume that communication among agents is available. At time step $t$ , Agent $i$ observes its own observation $o _ { t } ^ { i }$ , which is a part of the global environment state $s _ { t }$ , and selects action $a _ { t } ^ { i } \\in { \\mathcal { A } } _ { i }$ and message $m _ { t } ^ { i } \\in \\mathcal { M } _ { i }$ based on its own observation $o _ { t } ^ { i }$ and its own previous time step message $m _ { t - 1 } ^ { i }$ plus the received messages from other agents, i.e., $m _ { t - 1 } = ( m _ { t - 1 } ^ { 1 } , \\cdot \\cdot \\cdot , m _ { t - 1 } ^ { N } )$ . We assume that the message $m _ { t } ^ { i }$ of Agent $i$ is sent to all other agents and available at other agents at the next time step, i.e., time $t + 1$ . The joint actions ccording to the tra $a _ { t } \\overset { \\cdot } { = } ( a _ { t } ^ { 1 } , \\cdots , a _ { t } ^ { N } )$ nt state and th $s _ { t + 1 }$ and rewardsard function $\\{ r _ { t } ^ { i } \\} _ { i = 1 } ^ { N }$ $\\mathcal { \\bar { T } } : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S } [ 0 , 1 ]$ $R ^ { i } : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R }$ , respectively, where $s$ and $\\begin{array} { r } { A = \\prod _ { i = 1 } ^ { N } A ^ { i } } \\end{array}$ are the environment state space and the joint action space, respectively. The goal of Agent $i$ is to find the policy $\\pi ^ { i }$ that maximizes its discounted return $\\begin{array} { r } { R _ { t } ^ { i } = \\sum _ { t ^ { \\prime } = t } ^ { \\infty } \\gamma ^ { t ^ { \\prime } } r _ { t ^ { \\prime } } ^ { i } } \\end{array}$ . Hence, the objective function of Agent $i$ is defined as $J _ { i } ( \\pi ^ { i } ) = \\mathbb { E } _ { \\pi } \\left[ R _ { 0 } ^ { i } \\right]$ , where $\\pi = ( \\pi ^ { 1 } , \\cdots , \\pi ^ { N } )$ and $\\gamma \\in [ 0 , 1 ]$ are the joint policy and the discounting factor, respectively. ", + "bbox": [ + 173, + 224, + 825, + 417 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 THE PROPOSED INTENTION SHARING SCHEME ", + "text_level": 1, + "bbox": [ + 174, + 436, + 594, + 454 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The key idea behind the IS scheme is that multiple agents communicate with other agents by sending their implicit future plans, which carry their intention. The received messages capturing the intention of other agents enable the agent to coordinate its action with those of other agents. We now describe the architecture of the proposed IS scheme. At time step $t$ , Agent $i$ selects an action $a _ { t } ^ { i } \\sim \\pi ^ { i } ( \\cdot | o _ { t } ^ { i } , m _ { t - 1 } )$ and a message $\\stackrel { \\cdot } { m } _ { t } ^ { i } = M G N ^ { i } ( o _ { t } ^ { i } , m _ { t - 1 } , \\pi ^ { i } )$ based on its own observation $o _ { t } ^ { i }$ and received messages $m _ { t - 1 }$ , where $M G N ^ { i }$ is the message-generation network (MGN) of Agent $i$ . The MGN consists of two components: Imagined trajectory generation module (ITGM) and attention module (AM). Each agent generates an imagined trajectory by using ITGM and learns the importance of each imagined step in the imagined trajectory by using AM. The output of AM is an encoded imagined trajectory reflecting the importance of imagined steps and is used as the communication message. The overall architecture of the proposed IS scheme is shown in Fig. 1. In the following we describe the detail of each module. ", + "bbox": [ + 173, + 468, + 825, + 636 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4.1 IMAGINED TRAJECTORY GENERATION MODULE (ITGM) ", + "text_level": 1, + "bbox": [ + 173, + 652, + 611, + 667 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The role of ITGM is to produce the next imagined step. ITGM takes the received messages, observation, and action as input and yields the predicted next observation and predicted action as output. By stacking ITGMs, we generate an imagined trajectory, as shown in Fig. 1. For Agent $i$ at time step $t$ , we define an $H$ -length imagined trajectory as ", + "bbox": [ + 173, + 678, + 823, + 734 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/ec38485abcb04143b194ac2580c9f34bc7b97c940287cce4bdf705849fb5ba5b.jpg", + "text": "$$\n\\tau ^ { i } = ( \\tau _ { t } ^ { i } , \\hat { \\tau } _ { t + 1 } ^ { i } , \\cdot \\cdot \\cdot , \\hat { \\tau } _ { t + H - 1 } ^ { i } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 400, + 741, + 598, + 760 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\hat { \\tau } _ { t + k } ^ { i } = ( \\hat { o } _ { t + k } ^ { i } , \\hat { a } _ { t + k } ^ { i } )$ is the imagined step at time step $t + k$ . Note that $\\tau _ { t } ^ { i } = ( o _ { t } ^ { i } , a _ { t } ^ { i } )$ is the true values of observation and action, but the imagined steps except $\\tau _ { t } ^ { i }$ are predicted values. ", + "bbox": [ + 176, + 767, + 825, + 799 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "ITGM consists of a roll-out policy and two predictors: Other agents’ action predictor $f _ { a } ^ { i } ( o _ { t } ^ { i } )$ (we will call this predictor simply action predictor) and observation predictor $f _ { o } ^ { i } ( o _ { t } ^ { i } , a _ { t } ^ { i } , a _ { t } ^ { - i } )$ . First, we model the action predictor which takes the observation as input and produces other agents’ predicted actions. The output of the action predictor is given by ", + "bbox": [ + 174, + 805, + 825, + 863 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/076ed65955d84fe46fa1366fd5016c41cd4585508056e1dedd8f0e6fa1347f43.jpg", + "text": "$$\n\\begin{array} { r } { f _ { a } ^ { i } ( o _ { t } ^ { i } ) = ( \\hat { a } _ { t } ^ { 1 } , \\cdot \\cdot \\cdot , \\hat { a } _ { t } ^ { i - 1 } , \\hat { a } _ { t } ^ { i + 1 } , \\cdot \\cdot \\cdot , \\hat { a } ^ { N } ) = : \\hat { a } _ { t } ^ { - i } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 869, + 656, + 888 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Note that the action predictor can be trained by the previously proposed opponent modeling method (Rabinowitz et al. (2018); Raileanu et al. (2018)) and can take the received messages as input. Next, we model the observation predictor $f _ { o } ^ { i } ( o _ { t } ^ { i } , a _ { t } ^ { i } , \\hat { a } _ { t } ^ { - i } )$ which is conditioned on the observation $o _ { t } ^ { i }$ , own action $a _ { t } ^ { i }$ , and the output of the action predictor $\\hat { a } _ { t } ^ { - i }$ . Here, we adopt the dynamics function that predicts the difference between the next observation and the current observation, i.e., $o _ { t + 1 } ^ { i } ~ - ~ o _ { t } ^ { i }$ instead of the next observation $o _ { t + 1 } ^ { i }$ proposed in (Nagabandi et al. (2018)) in order to reduce model bias in the early stage of learning. Hence, the next observation can be written as ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/c9b874186cbaf9921bdc0929bdc73a3541afd60d23c39ea1d0cb95194fbd81c8.jpg", + "image_caption": [ + "Figure 1: Overall architecture of the IS scheme from the perspective of Agent $i$ " + ], + "image_footnote": [], + "bbox": [ + 251, + 102, + 748, + 352 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 410, + 825, + 486 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2d9a4e670cba7f02d7792d2fabedc97f1629067b4edc252769d6a46fcf1fd769.jpg", + "text": "$$\n\\hat { o } _ { t + 1 } ^ { i } = o _ { t } ^ { i } + f _ { o } ^ { i } ( o _ { t } ^ { i } , a _ { t } ^ { i } , \\hat { a } _ { t } ^ { - i } ) .\n$$", + "text_format": "latex", + "bbox": [ + 405, + 491, + 593, + 511 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "By injecting the predicted next observation and the received messages into the roll-out policy in ITGM, we obtain the predicted next action $\\hat { a } _ { t + 1 } ^ { i } = \\pi ^ { i } ( \\hat { o } _ { t + 1 } ^ { i } , m _ { t - 1 } )$ . Here, we use the current policy as the roll-out policy. Combining $\\hat { o } _ { t + 1 } ^ { i }$ and $\\hat { a } _ { t + 1 } ^ { i }$ , we obtain next imagined step at time step $t + 1$ , $\\tau _ { t + 1 } ^ { i } = ( \\widehat { o } _ { t + 1 } ^ { i } , \\widehat { a } _ { t + 1 } ^ { i } )$ . In order to produce an $H$ -length imagined trajectory, we inject the output of ITGM and the received messages $m _ { t - 1 }$ into the input of ITGM recursively. Note that we use the received messages at time step $t$ , $m _ { t - 1 }$ , in every recursion of ITGM.1 ", + "bbox": [ + 173, + 522, + 825, + 611 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.2 ATTENTION MODULE (AM) ", + "text_level": 1, + "bbox": [ + 176, + 626, + 408, + 641 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Instead of the naive approach that uses the imagined trajectory $[ \\tau _ { t } , \\cdot \\cdot \\cdot , \\tau _ { t + H - 1 } ]$ directly as the message, we apply an attention mechanism in order to learn the relative importance of imagined steps and encode the imagined trajectory according to the relative importance. We adopt the scaledot product attention proposed in (Vaswani et al. (2017)) as our AM. Our AM consists of three components: query, key, and values. The output of AM is the weighted sum of values, where the weight of values is determined by the dot product of the query and the corresponding key. In our model, the query consists of the received messages, and the key and value consist of the imagined trajectory. For Agent $i$ at time step $t$ , the query, key and value are defined as ", + "bbox": [ + 173, + 651, + 826, + 765 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5d6a99a55bde962d9c51d7c95949c61a7184499e504824e723d731ddf8635002.jpg", + "text": "$$\n\\begin{array} { r l } & { q _ { t } ^ { i } = W _ { Q } ^ { i } m _ { t - 1 } = W _ { Q } ^ { i } \\Bigl [ m _ { t - 1 } ^ { 1 } \\lVert m _ { t - 1 } ^ { 2 } \\rVert \\cdots \\lVert m _ { t - 1 } ^ { N - 1 } \\lVert m _ { t - 1 } ^ { N } \\Bigr ] \\in \\mathbb { R } ^ { d _ { k } } } \\\\ & { k _ { t } ^ { i } = \\Bigl [ W _ { K } ^ { i } \\tau _ { t } , \\cdots , \\underbrace { W _ { K } ^ { i } \\tau _ { t + h - 1 } } _ { = : k _ { t } ^ { i , h } } , \\cdots , W _ { K } ^ { i } \\tau _ { t + H - 1 } \\Bigr ] \\in \\mathbb { R } ^ { H \\times d _ { k } } } \\\\ & { v _ { t } ^ { i } = \\Bigl [ W _ { V } ^ { i } \\tau _ { t } , \\cdots , \\underbrace { W _ { V } ^ { i } \\tau _ { t + h - 1 } } _ { = : v _ { t } ^ { i , h } } , \\cdots , W _ { V } ^ { i } \\tau _ { t + H - 1 } \\Bigr ] \\in \\mathbb { R } ^ { H \\times d _ { m } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 770, + 709, + 888 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $W _ { Q } ^ { i } \\in \\mathbb { R } ^ { d _ { k } \\times N d _ { m } }$ , $W _ { K } ^ { i } \\in \\mathbb { R } ^ { d _ { k } \\times d _ { \\tau } }$ and $W _ { V } ^ { i } \\in \\mathbb { R } ^ { d _ { m } \\times d _ { \\tau } }$ are learnable parameters and operation $\\parallel$ denotes the concatenation of vectors. The output $m _ { t } ^ { i }$ of the attention model, which is used for message, is the weighted sum of the values: ", + "bbox": [ + 174, + 101, + 825, + 148 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0ff4daececb0a9c5ab491108ef763c3d874ef1d3b446a4ada7966633acd4ef65.jpg", + "text": "$$\nm _ { t } ^ { i } = \\sum _ { h = 1 } ^ { H } \\alpha _ { h } ^ { i } v _ { t } ^ { i , h } ,\n$$", + "text_format": "latex", + "bbox": [ + 437, + 154, + 558, + 196 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where the weight vector $\\alpha ^ { i } = ( \\alpha _ { 1 } ^ { i } , \\cdot \\cdot \\cdot , \\alpha _ { H } ^ { i } )$ is computed as ", + "bbox": [ + 173, + 202, + 568, + 218 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/dcc0a95792c4821928dff21ea63a151937abfa6dc2e0733492c3787234d12140.jpg", + "text": "$$\n\\alpha ^ { i } = \\mathrm { s o f t m a x } \\left[ \\frac { q _ { t } ^ { i ^ { 2 } } k _ { t } ^ { i , 0 } } { \\sqrt { d _ { k } } } , \\cdots , \\underbrace { \\frac { q _ { t } ^ { i ^ { T } } k _ { t } ^ { i , h } } { \\sqrt { d _ { k } } } } _ { = : \\alpha _ { h } ^ { i } } , \\cdots , \\frac { q _ { t } ^ { i ^ { T } } k _ { t } ^ { i , H } } { \\sqrt { d _ { k } } } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 222, + 678, + 284 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The weight of each value is computed by the dot product of the corresponding key and query. Since the projections of the imagined trajectory and the received messages are used for key and query, respectively, the weight can be interpreted as the relative importance of imagined step given the received messages. Note that $W _ { Q }$ , $W _ { K }$ and $W _ { V }$ are updated through the gradients from the other agents. ", + "bbox": [ + 173, + 286, + 825, + 358 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.3 TRAINING ", + "text_level": 1, + "bbox": [ + 174, + 373, + 287, + 388 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We implement the proposed IS scheme on the top of MADDPG (Lowe et al. (2017)), but it can be applied to other MARL algorithms. MADDPG is a well-known MARL algorithm and is briefly explained in Appendix A. In order to handle continuous state-action spaces, the actor, critic, observation predictor, and action predictor are parameterized by deep neural networks. For Agent $i$ , let ${ \\theta } _ { \\mu } ^ { i } , { \\theta } _ { Q } ^ { i } , \\bar { \\theta } _ { o } ^ { i }$ , and $\\theta _ { a } ^ { i }$ be the deep neural network parameters of actor, critic, observation predictor, and action predictor, respectively. Let $W ^ { i } = ( W _ { Q } ^ { i } , W _ { K } ^ { i } , W _ { V } ^ { i } )$ be the trainable parameters in the attention module of Agent $i$ . The centralized critic for Agent $i$ , $Q ^ { i }$ , is updated to minimize the following loss: ", + "bbox": [ + 173, + 398, + 826, + 517 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/eaac6d75e2f6e9a125877a265c10500fabdaca7eff985a74b610595cf2e30291.jpg", + "text": "$$\nL _ { Q } ( \\theta _ { Q } ^ { i } ) = \\mathbb { E } _ { x , a , r ^ { i } , x ^ { \\prime } } \\Big [ ( y ^ { i } - Q ^ { i } ( x , a ) ) ^ { 2 } \\Big ] , \\quad y ^ { i } = r ^ { i } + \\gamma Q ^ { i - } ( x ^ { \\prime } , a ^ { \\prime } ) | _ { a ^ { j ^ { \\prime } } = \\mu ^ { i - } ( \\sigma ^ { \\prime } { } ^ { j } , m ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 223, + 518, + 772, + 546 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $Q ^ { i - }$ and $\\mu ^ { i - }$ are the target Q-function and the target policy of Agent $i$ and parameterized by $\\theta _ { \\mu } ^ { i - } , \\theta _ { Q } ^ { i - }$ , respectively. The policy is updated to minimize the policy gradient loss: ", + "bbox": [ + 169, + 551, + 823, + 583 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/86d88c0573b9dcc3b1efb226da1155381541ec230ccbea22bd6a9ccb27ba07ac.jpg", + "text": "$$\n\\nabla _ { \\theta _ { \\mu } ^ { i } } J ( \\theta _ { \\mu } ^ { i } ) = \\mathbb { E } _ { x , a } \\Big [ \\nabla _ { \\theta _ { \\mu } ^ { i } } \\mu ^ { i } ( o ^ { i } , m ) \\nabla _ { a ^ { i } } Q ^ { i } ( x , a ) | _ { a ^ { i } = \\mu ^ { i } ( o ^ { i } , m ) } \\Big ]\n$$", + "text_format": "latex", + "bbox": [ + 302, + 588, + 692, + 616 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Since the MGN is connected to the agent’s own policy and other agents’ policies, the attention module parameters $W ^ { i }$ are trained by gradient flow from all agents. The gradient of Agent $i$ ’s attention module parameters is given by $\\mathsf { \\bar { V } } _ { W ^ { i } } J ( W ^ { i } ) =$ ", + "bbox": [ + 176, + 627, + 825, + 670 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7fd262c939d78843e70804e0c0be0c30ee7134da3115154acde763fae6f3d2a8.jpg", + "text": "$$\n\\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\left[ \\mathbb { E } _ { \\bar { x } , \\overline { { m } } , \\bar { x } , a } \\Big [ \\nabla _ { W ^ { i } } M G N ( \\tilde { m } ^ { i } | \\bar { \\sigma } ^ { i } , \\overline { { m } } ) \\nabla _ { \\tilde { m } ^ { i } } \\mu ^ { j } ( \\sigma ^ { j } , \\tilde { m } ^ { i } , \\tilde { m } ^ { - i } ) \\nabla _ { a ^ { j } } Q ^ { j } ( x , a ) | _ { a ^ { j } = \\mu ^ { j } ( \\sigma ^ { j } , m ) } \\Big ] \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 183, + 675, + 785, + 719 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\overline { { O } } ^ { i }$ and $\\overline { { m } }$ are the previous observation and received messages, respectively. The gradient of the attention module parameters are obtained by applying the chain rule to policy gradient. ", + "bbox": [ + 174, + 724, + 826, + 755 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Both the action predictor and the observation predictor are trained based on supervised learning and the loss functions for agent $i$ are given by ", + "bbox": [ + 174, + 761, + 823, + 790 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/158ea335aa690e79dd105448147c8f6d2b2e21a6166904a89ea9b0fba2935bc5.jpg", + "text": "$$\n\\begin{array} { r l } & { L ( \\theta _ { a } ^ { i } ) = \\mathbb { E } _ { o ^ { i } , a } \\Big [ \\big ( f _ { \\theta _ { a } ^ { i } } ^ { i } ( o ^ { i } ) - a ^ { - i } \\big ) ^ { 2 } \\Big ] } \\\\ & { L ( \\theta _ { o } ^ { i } ) = \\mathbb { E } _ { o ^ { i } , a , o ^ { \\prime } ^ { i } } \\Big [ \\big ( \\big ( o ^ { \\prime ^ { i } } - o ^ { i } \\big ) - f _ { \\theta _ { o } ^ { i } } ^ { i } \\big ( o ^ { i } , a ^ { i } , \\hat { a } ^ { - i } \\big ) \\big ) ^ { 2 } \\Big ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 792, + 671, + 851 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 EXPERIMENT ", + "text_level": 1, + "bbox": [ + 174, + 864, + 318, + 881 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In order to evaluate the proposed algorithm and compare it with other communication schemes fairly, we implemented existing baselines on the top of the same MADDPG used for the proposed scheme. ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The considered baselines are as follows. 1) MADDPG (Lowe et al. (2017)): we can assess the gain of introducing communication from this baseline. 2) DIAL (Foerster et al. (2016)): we modified DIAL, which is based on Q-learning, to our setting by connecting the message-generation network to other agents’ policies and allowing the gradient flow through communication channel. 3) TarMAC (Das et al. (2019)): we adopted the key concept of TarMAC in which the agent sends targeted messages using a signature-based attention model. 4) Comm-OA: the message consists of its own observation and action. 5) ATOC (Jiang & Lu (2018)): an attentional communication model which learns when communication is needed and how to combine the information of agents. We considered three multi-agent environments: predator-prey, cooperative navigation, and traffic junction, and we slightly modified the conventional environments to require more coordination among agents. ", + "bbox": [ + 173, + 103, + 825, + 242 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/65e2e8a9585fe2006b682abcc9d4060731b5c342e805424fd1e2bd25bde7fbfa.jpg", + "image_caption": [ + "Figure 2: Considered environments: (a) predator-prey (PP), (b) cooperative-navigation (CN), and (c) traffic-junction (TJ) " + ], + "image_footnote": [], + "bbox": [ + 250, + 256, + 748, + 376 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 ENVIRONMENTS ", + "text_level": 1, + "bbox": [ + 176, + 452, + 328, + 467 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Predator-prey (PP) The predator-prey environment is a standard task in multi-agent systems. We used a PP environment that consists of $N$ predators and fixed $M$ preys in a continuous state-action domain. We control the actions of predators and the goal is to capture as many preys as possible in a given time. Each agent observes the positions of predators and preys. When $C$ predators catch a prey simultaneously, the prey is captured and all predators get shared reward $R _ { 1 }$ . At every time when all the preys are captured, the preys are respawn and the shared reward value $R _ { 1 }$ increases by one with initial value one to accelerate the capture speed for the given time. We simulated three cases: $( N = 2 , C = 1 )$ ), $( N = 3 , C = 1 )$ , and $\\mathit { N } = 4 , C = 2 )$ with all $M = 9$ preys, where the fixed positions of the preys are shown in Fig.2(a). In the cases of $( N = 2 , C = 1$ ) and $( N = 3 , C = 1 )$ ), the initial positions of all predators are the same and randomly determined. Thus, the predators should learn not only how to capture preys but also how to spread out. In the case of $( N = 4 , C = 2 )$ , the initial positions of all predators are randomly determined independently. Thus, the predators should learn to capture preys in group of two. ", + "bbox": [ + 173, + 478, + 825, + 660 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Cooperative-navigation (CN) The goal of cooperative navigation introduced in (Lowe et al. (2017)) is for $N$ agents to cover $L$ landmarks while avoiding collisions among the agents. We modified the original environment so that collisions occur more easily. We set $L = N$ , increased the size of agent, and assigned a specific landmark to cover to each agent (i.e., each agent should cover the landmark of the same color in Fig.2(b)). Each agent observes the positions of predators and landmarks. The agent receives shared reward $R _ { 1 }$ which is the sum of the distance between each agent and the corresponding landmark at each time step and success reward $N ^ { \\prime } \\times R _ { 2 }$ where $N ^ { \\prime }$ is the number of the covered landmark. Agents who collide with other agents receive negative reward $R _ { 3 }$ . We simulated the environment with $N = L = 3$ , $R _ { 1 } = 1 / 3$ , $R _ { 2 } = 1$ and $R _ { 3 } = - 5$ . ", + "bbox": [ + 173, + 666, + 825, + 792 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Traffic-junction (TJ) We modified the traffic-junction introduced in Sukhbaatar et al. (2016) to continuous state-action domain. In the beginning of an episode, each agent is randomly located in a predefined initial position and assigned one of three routes: left, right or straight, as seen in Fig.2(c). The observation of each agent consists of the positions of all agents (no route information of other agents) and 2 one-hot vectors which encodes the initial position and assigned route of the agent. The action of each agent is a real value in $( 0 , 1 )$ , which indicates the distance to go along the assigned route from the current position. The goal is to go to the destination as fast as possible while avoiding collision with other agents. To achieve the goal, we design reward with three components. Each agent receives success reward $R _ { 1 }$ if it arrives at the destination without any collision with other agents, collision negative reward $R _ { 2 }$ if its position is overlapped with that of other agent, and time negative reward $R _ { 3 }$ to avoid traffic jam. When an agent arrives at the destination, the agent is assigned a new initial position and the route. An episode ends when $T$ time steps elapse. We set $R _ { 1 } = 2 0$ , $R _ { 2 } = - 1 0$ , and $R _ { 3 } = - 0 . 0 1 \\tau$ , where $\\tau$ is the total time step after agent is initialized. ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/8387baf51bb80be1110568befb74b35cde463335a1b82d273a70194c0d32c2a4.jpg", + "image_caption": [ + "Figure 3: Performance for MADDPG (blue), DIAL (green), TarMAC (red), Comm-OA (purple), ATOC (cyan) and the proposed IS method (black). " + ], + "image_footnote": [], + "bbox": [ + 189, + 109, + 794, + 363 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 434, + 825, + 491 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 507, + 277, + 522 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Fig. 3 shows the performance of the proposed IS scheme and the considered baselines on the PP, CN, and TJ environments. Figs.3(a)-(d) shows the learning curves of the algorithms on PP and CN and Figs. (e)-(f) show the average return using deterministic policy over 100 episodes every 250000 time steps. All performance is averaged over 10 different seeds. It is seen that Comm-OA performs similarly to MADDPG in the considered environments. Since the received messages come from other agents at the previous time step, Comm-OA in which the communication message consists of agent’s observation and action performs similarly to MADDPG. Unlike Comm-OA, DIAL, TarMAC, and ATOC outperform MADDPG and the performance gain comes from the benefit of learning communication protocol in the considered environments except PP with $N = 4$ . In PP with $N = 4$ , four agents need to coordinate to spread out in group of two to capture preys. In this complicated coordination requirement, simply learning communication protocol based on past and current information did not obtain benefit from communication. On the contrary, the proposed IS scheme sharing intention with other agents achieved the required coordination even in this complicated environment. ", + "bbox": [ + 173, + 534, + 825, + 727 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.3 ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 744, + 287, + 758 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Imagined trajectory The proposed IS scheme uses the encoded imagined trajectory as the message content. Each agent rolls out an imagined trajectory based on its own policy and trained models including action predictor and observation predictor. Since the access to other agents’ policies is not available, the true trajectory and the imagined trajectory can mismatch. Especially, the mismatch is large in the beginning of an episode because each agent does not receive any messages from other agents (In this case, we inject zero vector instead of the received messages into the policy). We expect that the mismatch will gradually decrease as the episode progresses and this can be interpreted as the procedure of coordination among agents. Fig.5 shows the positions of all agents and each agent’s imagined trajectory over time step in one episode for predator-prey with $N = 3$ predators, where the initial positions of the agents after the end of training $( t = 0$ ) is bottom right on the map. Note that each agent estimates the future positions of other agents as well as their own future position due to the assumption of full observability. The first, second, and third row of Fig.5 show the imagined trajectories of all agents at Agent 1 (red), Agent 2 (green) and Agent 3 (blue), respectively. Note that the imagined trajectory of each agent represents its future plan for the environment. As seen in Fig.5, at $t = 0$ the intention of both Agent 1 and Agent 3 is to move to the left to catch preys. At $t = 1$ , all agents receive the messages from other agents. It is observed that Agent 3 changes its future plan to catch preys around the center while Agent 1 maintains its future plan. This procedure shows that coordination between Agent 1 and Agent 3 starts to occur. It is seen that as time goes, each agent roughly predicts other agents’ future actions. ", + "bbox": [ + 174, + 770, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/7b3b19c5a9045ea3e1116c74ee3a2c72b0b990f2a8adf5552c566ad452bf7ac2.jpg", + "image_caption": [ + "Figure 4: Performance for our proposed method with different length of imagined trajectory $H$ and without attention module. " + ], + "image_footnote": [], + "bbox": [ + 194, + 123, + 823, + 308 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 388, + 825, + 501 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We conducted experiments to examine the impact of the length of the imagined trajectory $H$ . Fig.4 shows the performances of the proposed method for different values of $H$ . It is seen that the training speed is reduced when $H = 7$ as compared to $H = 3$ or $H = 5$ . However, the final performance all outperforms the baseline. ", + "bbox": [ + 174, + 507, + 825, + 563 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Attention In the proposed IS scheme, the imagined trajectory is encoded based on the attention module to capture the importance of components in the imagined trajectory. Recall that the message of Agent $i$ is expressed as $\\begin{array} { r } { m _ { t } ^ { i } = \\sum _ { h = 1 } ^ { H } \\alpha _ { h } ^ { i } v _ { t } ^ { i , h } } \\end{array}$ , as seen in (7), where $\\alpha _ { h } ^ { i }$ denotes the importance of $v _ { t } ^ { i , h }$ , which is the encoded imagined step. Note that the previously proposed communication schemes are the special case corresponding to $\\alpha ^ { i } = ( 1 , 0 , \\cdots , 0 )$ . In Fig.5, the brightness of each circle is proportional to the attention weight. At time step $t \\ = \\ K$ , where $K = 3 7$ , $\\alpha _ { 2 } ^ { 1 }$ , which 2indicates when Agent 1 moves to the prey in the bottom middle, is the highest. In addition, $\\alpha _ { 4 } ^ { 3 }$ which indicates when Agent 3 moves to the prey in the left middle, is the highest. Hence, the agent tends to send future information when it is near a prey. Similar attention weight tendency is also captured in the time step $t = K + 1$ and $t = K + 2$ . ", + "bbox": [ + 174, + 569, + 825, + 715 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As aforementioned, the aim of the IS scheme is to communicate with other agents based on their own future plans. How far future is important depends on the environment and on the tasks. In order to analyze the tendency in the importance of future plans, we averaged the attention weight over the trajectories on the fully observable PP environment with 3 agents and a partially observable PP environment with 3 agents in which each agent knows the locations of other agents within a certain range from the agent. The result is summarized in Table 1. It is observed that the current information (time $k$ ) and the farthest future information (time $k + 4 )$ are mainly used as the message content in the fully observable case, whereas the current information and the information next to the present (time $k$ and $k + 1 )$ ) are mainly used in the partially observable environment. This is because sharing observation information is more critical in the partially observable case than the fully observable case. A key aspect of the proposed IS scheme is that it adaptively selects most important steps as the message content depending on the environment by using the attention module. ", + "bbox": [ + 173, + 722, + 825, + 888 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We conducted an ablation study for the attention module and the result is shown in Fig. 4. We compared the proposed IS scheme with and without the attention module. We replace the attention module with an averaging layer, which is the special case corresponding to $\\alpha ^ { i } = ( \\textstyle { \\frac { 1 } { H } } , \\cdot \\cdot \\cdot , \\textstyle { \\frac { 1 } { H } } )$ . Fig. 4 shows that the proposed IS scheme with the attention module yields better performance than the one without the attention module. This shows the necessity of the attention module. In the PP environment with 4 agents, the imagined trajectory alone without the attention module improves the training speed while the final performance is similar to that of MADDPG. In the TJ environment with 3 agents, the imagined trajectory alone without the attention module improves both the final performance and the training speed. ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/2d65899b9ef8f947fa967ca00c1b95f2451e28838ca23027d3e02cd83f73438d.jpg", + "image_caption": [ + "Figure 5: Imagined trajectories and attention weights of each agent on PP $\\left( \\mathrm { N } { = } 3 \\right)$ : 1st row - agent1 (red), 2nd row - agent2 (green), and 3rd row - agent3 (blue). Black squares, circle inside the timesicon, and other circles denote the prey, current position, and estimated future positions, respectively. The brightness of the circle is proportional to the attention weight. $K = 3 7$ . " + ], + "image_footnote": [], + "bbox": [ + 200, + 113, + 782, + 304 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 414, + 825, + 513 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/2d5ae03fd7187820f538d90d110497490060f685d187f74e3019bd182951e92a.jpg", + "table_caption": [ + "Table 1: Averaged attention weight over the trajectory at time step $k$ " + ], + "table_footnote": [], + "table_body": "
Imagined stepskk+1k+2k+3k+4
Fully observable PP (N=3)Partially observable PP (N=3)0.330.320.180.220.150.170.140.150.200.14
", + "bbox": [ + 266, + 526, + 728, + 574 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 630, + 318, + 645 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we proposed the IS scheme, a new communication protocol, based on sharing intention among multiple agents for MARL. The message-generation network in the proposed IS scheme consists of ITGM, which is used for producing predicted future trajectories, and AM, which learns the importance of imagined steps based on the received messages. The message in the proposed scheme is encoded imagined trajectory capturing the agent’s intention so that the communication message includes the future information as well as the current information, and their weights are adaptively determined depending on the environment. We studied examples of imagined trajectories and attention weights. It is observed that the proposed IS scheme generates meaningful imagined trajectories and attention weights. Numerical results show that the proposed IS scheme outperforms other communication algorithms including state-of-the-art algorithms. Furthermore, we expect that the key idea of the proposed IS scheme combining with other communication algorithms such as ATOC and TarMAC would yield even better performance. ", + "bbox": [ + 174, + 661, + 825, + 828 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 178, + 849, + 424, + 866 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning(NRF2017R1E1A1A03070788). 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Each agent has a deterministic policy $a _ { t } = \\mu _ { \\theta _ { \\mu } } ^ { i } ( o _ { t } )$ conditioned on its own observation $o _ { t }$ and a centralized critic $Q _ { \\theta _ { Q } } ^ { i } ( x , a ) = E \\big [ R _ { t } ^ { i } \\big | x _ { t } = x , a _ { t } = a \\big ]$ conditioned on the joint actions $a _ { t }$ and state information $x _ { t }$ . Here, $x _ { t }$ can be state $s _ { t }$ or the set of observations $( o _ { t } ^ { 1 } , \\cdots , o _ { t } ^ { N } )$ . The centralized critic is trained by minimizing the following loss: ", + "bbox": [ + 173, + 132, + 825, + 213 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/99bb1d16c882d48c96de914142f9fc31d167595061041f7d94bb6bf8583f128d.jpg", + "text": "$$\nL _ { Q } ( \\theta _ { Q } ) = \\mathbb { E } _ { x , a , r ^ { i } , x ^ { \\prime } } \\Big [ ( y ^ { i } - Q _ { \\theta _ { Q } } ^ { i } ( x , a ) ) ^ { 2 } \\Big ] , ~ y ^ { i } = r ^ { i } + \\gamma Q _ { \\theta _ { Q } ^ { - } } ^ { i } ( x ^ { \\prime } , a ^ { \\prime } ) | _ { a ^ { j ^ { \\prime } } = \\mu ^ { i - } ( o ^ { j } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 227, + 218, + 767, + 246 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "where $\\theta _ { Q } ^ { - }$ is the parameter of target Q-function and $\\mu ^ { i - }$ is the target policy of Agent $i$ . The policy is trained by Deterministic Policy Gradient (DPG), and the gradient of the objective with respect to the policy parameter $\\theta _ { \\mu ^ { i } }$ is given by ", + "bbox": [ + 174, + 253, + 825, + 299 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/ae52996ba851ace79af9913002c75eaa64fdce69f5629d13451bcb4960868017.jpg", + "text": "$$\n\\nabla _ { \\theta _ { \\mu ^ { i } } } J ( \\mu ^ { i } ) = \\mathbb { E } _ { x , a } \\Big [ \\nabla _ { \\theta _ { \\mu ^ { i } } } \\mu ^ { i } ( o ^ { i } ) \\nabla _ { a ^ { i } } Q _ { \\theta _ { Q } } ^ { i } ( x , a ) | _ { a ^ { i } = \\mu ^ { i } ( o ^ { i } ) } \\Big ] .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 306, + 689, + 333 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B TRAINING DETAILS AND HYPERPARAMETERS ", + "text_level": 1, + "bbox": [ + 173, + 102, + 588, + 118 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/5aa4ed79f3c8329becc2c884cd0bf7d61238a47ad4091df54151cfc8982f3f3c.jpg", + "table_caption": [ + "Table 2: Hyperparameters of all algorithms " + ], + "table_footnote": [], + "table_body": "
MADDPGTARMACDIALATOCIS
REPLAY BUFFER SIZE2×1052×1052×1052×1052×105
DISCOUNT FACTOR0.990.990.990.990.99
MINI-BATCH SIZE128128128128128
OPTIMIZERADAMADAMADAMADAMADAM
LEARNING RATE0.00050.00050.00050.00050.0005
NUMBER OF HIDDEN LAYERS (ALL NETWORKS)22222
NUMBER OF HIDDENUNITSPERLAYER128128128128128
ACTIVATION FUNCTION FOR HIDDEN LAYERRELURELURELURELURELU
MESSAGE DIMENSION ON PP15555
MESSAGE DIMENSION ON CN3333
MESSAGE DIMENSION ON TJ3333
ATTENTION DIMENSION ON PP5-55
ATTENTION DIMENSION ON CJ333
ATTENTION DIMENSION ON TJ333
IMAGINED TRAJECTORY LENGTH H-5
", + "bbox": [ + 173, + 179, + 861, + 396 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C ADDITIONAL ABLATION STUDY ", + "text_level": 1, + "bbox": [ + 176, + 102, + 473, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We conducted an additional experiment to examine whether performance improvement is gained from sharing intention or having a prediction of the future. We compared the proposed IS scheme with MADPPG- $\\boldsymbol { \\cdot } \\mathbf { p }$ in which the agent does not use communication, but uses their own imagined trajectory as additional input. Fig. 6 shows that the proposed IS scheme outperforms MADDPG-p. Thus, sharing intention, which is a core idea of this paper, is more important than having a prediction of the future. ", + "bbox": [ + 173, + 133, + 825, + 217 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/855716cc9dda436e0c9a3111e7d6d848e9a8f2fcdae2392503dbf27cdacc0c44.jpg", + "image_caption": [ + "Figure 6: Performance for MADDPG (blue), MADDPG-p (orange), and the proposed IS method (black). " + ], + "image_footnote": [], + "bbox": [ + 235, + 243, + 746, + 397 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "D PSEUDO CODE ", + "text_level": 1, + "bbox": [ + 174, + 102, + 334, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Algorithm 1 Intention Sharing (IS) Communication Scheme ", + "text_level": 1, + "bbox": [ + 173, + 138, + 573, + 155 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Initialize parameter $\\theta _ { \\mu } ^ { i } , \\theta _ { Q } ^ { i } , \\theta _ { \\mu } ^ { i - } , \\theta _ { Q } ^ { i - } , \\theta _ { o } ^ { i } , \\theta _ { a } ^ { i } , W ^ { i } , \\forall i \\in \\{ 1 , \\cdots , N \\}$ \nfor episode $: = 1 , 2 , \\cdots$ do Initialize state $s _ { 1 }$ , messages $m _ { 0 } = ( \\overrightarrow { 0 } , \\cdot \\cdot \\cdot , \\overrightarrow { 0 } )$ and each agent observes $o _ { 1 } ^ { i }$ for $t < = T$ and $s _ { t } \\neq$ terminal do Each agent receives the messages $m _ { t - 1 } = ( m _ { t - 1 } ^ { 1 } , \\cdot \\cdot \\cdot , m _ { t - 1 } ^ { N } )$ Each agent selects action $a _ { t } ^ { i } \\sim \\pi ^ { i } ( \\cdot | o _ { t } ^ { i } , m _ { t - 1 } )$ for each agent $i$ Execute $\\mathbf { \\delta } \\mathbf { a } _ { t }$ and each agent $i$ receives $r _ { t }$ and $o _ { t + 1 } ^ { i }$ for $h = 1 , 2 , \\cdots , H$ do Predict other agents’ actions $\\hat { a } _ { t + h - 1 } ^ { - i }$ from the action predictor $f _ { a } ^ { i }$ Generate $\\hat { o } _ { t + h } ^ { i }$ from observation predictor $f _ { o } ^ { i } ( o _ { t + h - 1 } ^ { i } , \\hat { a } _ { t + h - 1 } ^ { i } , \\hat { a } _ { t + h - 1 } ^ { - i } )$ Generate $\\hat { a } _ { t + h } ^ { i } \\sim \\pi ^ { i } ( \\cdot | \\hat { o } _ { t + h } ^ { i } , m _ { t - 1 } )$ Each agent generates the messages $m _ { t } ^ { i }$ by injecting $\\tau ^ { i } = ( \\tau _ { t } ^ { i } , \\hat { \\tau } _ { t + 1 } ^ { i } , \\cdot \\cdot \\cdot , \\hat { \\tau } _ { t + H - 1 } ^ { i } )$ into Attention Module (AM) Store transitions in $D$ end for for each gradient step do Update $\\theta _ { Q } ^ { i }$ and $( \\theta _ { o } ^ { i } , \\theta _ { a } ^ { i } )$ by minimizing the loss (9) and the loss (12) Update $\\theta _ { \\mu } ^ { i }$ and $W ^ { i }$ based on the gradient (10) and the gradient (11) end for Update $\\theta _ { \\mu } ^ { i - } , \\theta _ { Q } ^ { i - }$ using the moving average method \nend for ", + "bbox": [ + 173, + 147, + 825, + 484 + ], + "page_idx": 14 + } +] \ No newline at end of file diff --git a/parse/train/qpsl2dR9twy/qpsl2dR9twy_middle.json b/parse/train/qpsl2dR9twy/qpsl2dR9twy_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..35a0563750ce1c0e3f86865455e50050e4dee268 --- /dev/null +++ b/parse/train/qpsl2dR9twy/qpsl2dR9twy_middle.json @@ -0,0 +1,34006 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 502, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 98 + ], + "score": 1.0, + "content": "COMMUNICATION IN MULTI-AGENT REINFORCEMENT", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 100, + 355, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 355, + 117 + ], + "score": 1.0, + "content": "LEARNING: INTENTION SHARING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 113, + 135, + 387, + 180 + ], + "lines": [ + { + "bbox": [ + 112, + 133, + 308, + 149 + ], + "spans": [ + { + "bbox": [ + 112, + 133, + 308, + 149 + ], + "score": 1.0, + "content": "Woojun Kim, Jongeui Park, Youngchul Sung∗", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 145, + 280, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 145, + 280, + 159 + ], + "score": 1.0, + "content": "School of Electrical Engineering, KAIST", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 111, + 156, + 204, + 170 + ], + "spans": [ + { + "bbox": [ + 111, + 156, + 204, + 170 + ], + "score": 1.0, + "content": "Daejeon, South Korea", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 112, + 167, + 388, + 182 + ], + "spans": [ + { + "bbox": [ + 112, + 167, + 388, + 182 + ], + "score": 1.0, + "content": "{woojun.kim, jongeui.park, ycsung}@kaist.ac.kr", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 278, + 208, + 333, + 220 + ], + "lines": [ + { + "bbox": [ + 277, + 208, + 335, + 221 + ], + "spans": [ + { + "bbox": [ + 277, + 208, + 335, + 221 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 143, + 234, + 468, + 376 + ], + "lines": [ + { + "bbox": [ + 142, + 234, + 469, + 246 + ], + "spans": [ + { + "bbox": [ + 142, + 234, + 469, + 246 + ], + "score": 1.0, + "content": "Communication is one of the core components for learning coordinated behavior", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 245, + 470, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 245, + 470, + 258 + ], + "score": 1.0, + "content": "in multi-agent systems. 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Numeral results show that the proposed IS scheme significantly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 365, + 469, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 469, + 379 + ], + "score": 1.0, + "content": "outperforms other communication schemes in multi-agent reinforcement learning.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 108, + 398, + 206, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "Reinforcement learning (RL) has achieved remarkable success in various complex control problems", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "such as robotics and games (Gu et al. (2017); Mnih et al. (2013); Silver et al. (2017)). Multi-agent", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "reinforcement learning (MARL) extends RL to multi-agent systems, which model many practical", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "real-world problems such as connected cars and smart cities (Roscia et al. (2013)). There exist sev-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "eral distinct problems in MARL inherent to the nature of multi-agent learning (Gupta et al. (2017);", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "Lowe et al. (2017)). One such problem is how to learn coordinated behavior among multiple agents", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "and various approaches to tackling this problem have been proposed (Jaques et al. (2018); Pesce &", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 513 + ], + "score": 1.0, + "content": "Montana (2019); Kim et al. (2020)). One promising approach to learning coordinated behavior is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "learning communication protocol among multiple agents (Foerster et al. (2016); Sukhbaatar et al.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "(2016); Jiang & Lu (2018); Das et al. (2019)). The line of recent researches on communication", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "for MARL adopts end-to-end training based on differential communication channel (Foerster et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "(2016); Jiang & Lu (2018); Das et al. (2019)). That is, a message-generation network is defined at", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "each agent and connected to other agents’ policies or critic networks through communication chan-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "nels. Then, the message-generation network is trained by using the gradient of other agents’ policy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "or critic losses. Typically, the message-generation network is conditioned on the current observation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 602 + ], + "score": 1.0, + "content": "or the hidden state of a recurrent network with observations as input. Thus, the trained message", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "encodes the past and current observation information to minimize other agents’ policy or critic loss.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "It has been shown that due to the capability of sharing observation information, this kind of commu-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "nication scheme has good performance as compared to communication-free MARL algorithms such", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 633, + 486, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 486, + 644 + ], + "score": 1.0, + "content": "as independent learning, which is widely used in MARL, in partially observable environments.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 108, + 649, + 454, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 456, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 456, + 662 + ], + "score": 1.0, + "content": "In this paper, we consider the following further question for communication in MARL:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 137, + 670, + 470, + 681 + ], + "lines": [ + { + "bbox": [ + 138, + 668, + 471, + 683 + ], + "spans": [ + { + "bbox": [ + 138, + 668, + 471, + 683 + ], + "score": 1.0, + "content": "”How to harness the benefit of communication beyond sharing partial observation.”", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 691, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 690, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 705 + ], + "score": 1.0, + "content": "We propose intention of each agent as the content of message to address the above question. Sharing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 701, + 505, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 715 + ], + "score": 1.0, + "content": "intention using communication has been used in natural multi-agent systems like human society.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + } + ], + "page_idx": 0, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 119, + 722, + 202, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 720, + 204, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 720, + 204, + 734 + ], + "score": 1.0, + "content": "∗Corresponding author", + "type": "text" + } + ] + } + ] + }, + { + "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, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "1", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 502, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 505, + 98 + ], + "score": 1.0, + "content": "COMMUNICATION IN MULTI-AGENT REINFORCEMENT", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 100, + 355, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 100, + 355, + 117 + ], + "score": 1.0, + "content": "LEARNING: INTENTION SHARING", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "list", + "bbox": [ + 113, + 135, + 387, + 180 + ], + "lines": [ + { + "bbox": [ + 112, + 133, + 308, + 149 + ], + "spans": [ + { + "bbox": [ + 112, + 133, + 308, + 149 + ], + "score": 1.0, + "content": "Woojun Kim, Jongeui Park, Youngchul Sung∗", + "type": "text" + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 112, + 145, + 280, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 145, + 280, + 159 + ], + "score": 1.0, + "content": "School of Electrical Engineering, KAIST", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 111, + 156, + 204, + 170 + ], + "spans": [ + { + "bbox": [ + 111, + 156, + 204, + 170 + ], + "score": 1.0, + "content": "Daejeon, South Korea", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 112, + 167, + 388, + 182 + ], + "spans": [ + { + "bbox": [ + 112, + 167, + 388, + 182 + ], + "score": 1.0, + "content": "{woojun.kim, jongeui.park, ycsung}@kaist.ac.kr", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + } + ], + "index": 3.5, + "bbox_fs": [ + 111, + 133, + 388, + 182 + ] + }, + { + "type": "title", + "bbox": [ + 278, + 208, + 333, + 220 + ], + "lines": [ + { + "bbox": [ + 277, + 208, + 335, + 221 + ], + "spans": [ + { + "bbox": [ + 277, + 208, + 335, + 221 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 143, + 234, + 468, + 376 + ], + "lines": [ + { + "bbox": [ + 142, + 234, + 469, + 246 + ], + "spans": [ + { + "bbox": [ + 142, + 234, + 469, + 246 + ], + "score": 1.0, + "content": "Communication is one of the core components for learning coordinated behavior", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 245, + 470, + 258 + ], + "spans": [ + { + "bbox": [ + 141, + 245, + 470, + 258 + ], + "score": 1.0, + "content": "in multi-agent systems. In this paper, we propose a new communication scheme", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 256, + 469, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 469, + 268 + ], + "score": 1.0, + "content": "named Intention Sharing (IS) for multi-agent reinforcement learning in order to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 267, + 470, + 280 + ], + "spans": [ + { + "bbox": [ + 142, + 267, + 470, + 280 + ], + "score": 1.0, + "content": "enhance the coordination among agents. In the proposed IS scheme, each agent", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 279, + 469, + 290 + ], + "spans": [ + { + "bbox": [ + 141, + 279, + 469, + 290 + ], + "score": 1.0, + "content": "generates an imagined trajectory by modeling the environment dynamics and other", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 301 + ], + "score": 1.0, + "content": "agents’ actions. The imagined trajectory is a simulated future trajectory of each", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "score": 1.0, + "content": "agent based on the learned model of the environment dynamics and other agents", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 310, + 469, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 310, + 469, + 324 + ], + "score": 1.0, + "content": "and represents each agent’s future action plan. Each agent compresses this imag-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 469, + 334 + ], + "score": 1.0, + "content": "ined trajectory capturing its future action plan to generate its intention message for", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 333, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 469, + 344 + ], + "score": 1.0, + "content": "communication by applying an attention mechanism to learn the relative impor-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 470, + 357 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 470, + 357 + ], + "score": 1.0, + "content": "tance of the components in the imagined trajectory based on the received message", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 469, + 366 + ], + "score": 1.0, + "content": "from other agents. Numeral results show that the proposed IS scheme significantly", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 365, + 469, + 379 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 469, + 379 + ], + "score": 1.0, + "content": "outperforms other communication schemes in multi-agent reinforcement learning.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13, + "bbox_fs": [ + 141, + 234, + 470, + 379 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 398, + 206, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 208, + 414 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 643 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "Reinforcement learning (RL) has achieved remarkable success in various complex control problems", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "such as robotics and games (Gu et al. (2017); Mnih et al. (2013); Silver et al. (2017)). Multi-agent", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "reinforcement learning (MARL) extends RL to multi-agent systems, which model many practical", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "real-world problems such as connected cars and smart cities (Roscia et al. (2013)). There exist sev-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 505, + 480 + ], + "score": 1.0, + "content": "eral distinct problems in MARL inherent to the nature of multi-agent learning (Gupta et al. (2017);", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "Lowe et al. (2017)). One such problem is how to learn coordinated behavior among multiple agents", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "and various approaches to tackling this problem have been proposed (Jaques et al. (2018); Pesce &", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 513 + ], + "score": 1.0, + "content": "Montana (2019); Kim et al. (2020)). One promising approach to learning coordinated behavior is", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "learning communication protocol among multiple agents (Foerster et al. (2016); Sukhbaatar et al.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "(2016); Jiang & Lu (2018); Das et al. (2019)). The line of recent researches on communication", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "for MARL adopts end-to-end training based on differential communication channel (Foerster et al.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 506, + 557 + ], + "score": 1.0, + "content": "(2016); Jiang & Lu (2018); Das et al. (2019)). That is, a message-generation network is defined at", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "each agent and connected to other agents’ policies or critic networks through communication chan-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "nels. Then, the message-generation network is trained by using the gradient of other agents’ policy", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "or critic losses. Typically, the message-generation network is conditioned on the current observation", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 602 + ], + "score": 1.0, + "content": "or the hidden state of a recurrent network with observations as input. Thus, the trained message", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "encodes the past and current observation information to minimize other agents’ policy or critic loss.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 623 + ], + "score": 1.0, + "content": "It has been shown that due to the capability of sharing observation information, this kind of commu-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 634 + ], + "score": 1.0, + "content": "nication scheme has good performance as compared to communication-free MARL algorithms such", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 633, + 486, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 486, + 644 + ], + "score": 1.0, + "content": "as independent learning, which is widely used in MARL, in partially observable environments.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 424, + 506, + 644 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 649, + 454, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 648, + 456, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 456, + 662 + ], + "score": 1.0, + "content": "In this paper, we consider the following further question for communication in MARL:", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 648, + 456, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 137, + 670, + 470, + 681 + ], + "lines": [ + { + "bbox": [ + 138, + 668, + 471, + 683 + ], + "spans": [ + { + "bbox": [ + 138, + 668, + 471, + 683 + ], + "score": 1.0, + "content": "”How to harness the benefit of communication beyond sharing partial observation.”", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 138, + 668, + 471, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 691, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 690, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 705 + ], + "score": 1.0, + "content": "We propose intention of each agent as the content of message to address the above question. Sharing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 701, + 505, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 715 + ], + "score": 1.0, + "content": "intention using communication has been used in natural multi-agent systems like human society.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 690, + 505, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "For example, drivers use signal light to inform other drivers of their intentions. A car driver may", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "slow down if a driver in his or her left lane turns the right signal light on. In this case, the signal", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "light encodes the driver’s intention, which indicates the driver’s future behavior, not current or past", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "observation such as the field view. By sharing intention using signal light, drivers coordinate their", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "drive with each other. In this paper, we formalize and propose a new communication scheme for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "MARL named Intention sharing (IS) in order to go beyond existing observation-sharing schemes", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "for communication in MARL. The proposed IS scheme allows each agent to share its intention with", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "other agents in the form of encoded imagined trajectory. That is, each agent generates an imagined", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "trajectory by modeling the environment dynamics and other agents’ actions. Then, each agent learns", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "the relative importance of the components in the imagined trajectory based on the received messages", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 206 + ], + "score": 1.0, + "content": "from other agents by using an attention model. The output of the attention model is an encoded", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "score": 1.0, + "content": "imagined trajectory capturing the intention of the agent and used as the communication message.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "We evaluate the proposed IS scheme in several multi-agent environments requiring coordination", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "among agents. Numerical result shows that the proposed IS scheme significantly outperforms other", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "existing communication schemes for MARL including the state-of-the-art algorithms such as ATOC", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 163, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 163, + 258 + ], + "score": 1.0, + "content": "and TarMAC.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 276, + 216, + 289 + ], + "lines": [ + { + "bbox": [ + 104, + 275, + 219, + 291 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 219, + 291 + ], + "score": 1.0, + "content": "2 RELATED WORKS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 504, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 504, + 315 + ], + "score": 1.0, + "content": "Under the asymmetry in learning resources between the training and execution phases, the frame-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 504, + 326 + ], + "score": 1.0, + "content": "work of centralized training and decentralized execution (CTDE), which assumes the availability", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 337 + ], + "score": 1.0, + "content": "of all system information in the training phase and distributed policy in the execution phase, has", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "been adopted in most recent MARL researches (Lowe et al. (2017); Foerster et al. (2018); Iqbal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "& Sha (2018); Kim et al. (2020)). Under the framework of CTDE, learning communication proto-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "col has been considered to enhance performance in the decentralized execution phase for various", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "multi-agent tasks (Foerster et al. (2016); Jiang & Lu (2018); Das et al. (2019)). For this pur-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 380, + 507, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 507, + 393 + ], + "score": 1.0, + "content": "pose, Foerster et al. (2016) proposed Differentiable Inter-Agent Learning (DIAL). DIAL trains a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "score": 1.0, + "content": "message-generation network by connecting it to other agents’ Q-networks and allowing gradient", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "flow through communication channels in the training phase. Then, in the execution phase the mes-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "sages are generated and passed to other agents through communication channels. Jiang & Lu (2018)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "score": 1.0, + "content": "proposed an attentional communication model named ATOC to learn when to communicate and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "how to combine information received from other agents through communication based on attention", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "mechanism. Das et al. (2019) proposed Targeted Multi-Agent Communication (TarMAC) to learn", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "the message-generation network in order to produce different messages for different agents based", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "on a signature-based attention model. The message-generation networks in the aforementioned al-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "gorithms are conditioned on the current observation or a hidden state of LSTM. Under partially", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "observable environments, such messages which encode past and current observations are useful but", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "do not capture any future information. In our approach, we use not only the current information", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "score": 1.0, + "content": "but also future information to generate messages and the weight between the current and future", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "information is adaptively learned according to the environment. This yields further performance", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 534, + 274, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 274, + 544 + ], + "score": 1.0, + "content": "enhancement, as we will see in Section 5.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "In our approach, the encoded imagined trajectory capturing the intention of each agent is used as the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "score": 1.0, + "content": "communication message in MARL. Imagined trajectory was used in other problems too. Racaniere `", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "et al. (2017) used imagined trajectory to augment it into the policy and critic for combining model-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 581, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 597 + ], + "score": 1.0, + "content": "based and model-free approaches in single-agent RL. It is shown that arbitrary imagined trajectory", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "(rolled-out trajectory by using a random policy or own policy) is useful for single-agent RL in", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "terms of performance and data efficiency. Strouse et al. (2018) introduced information-regularizer", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "to share or hide agent’s intention to other agents for a multi-goal MARL setting in which some", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "agents know the goal and other agents do not know the goal. By maximizing (or minimizing) the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "mutual information between the goal and action, an agent knowing the goal learns to share (or hide)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "its intention to other agents not knowing the goal in cooperative (or competitive) tasks. They showed", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 659, + 335, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 335, + 673 + ], + "score": 1.0, + "content": "that sharing intention is effective in the cooperative case.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "In addition to our approach, Theory of Mind (ToM) and Opponent Modeling (OM) use the notion of", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "intention. Rabinowitz et al. (2018) proposed the Theory of Mind network (ToM-net) to predict other", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "agents’ behaviors by using meta-learning. Raileanu et al. (2018) proposed Self Other-Modeling", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "(SOM) to infer other agents’ goal in an online manner. Both ToM and OM take advantage of pre-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "dicting other agents’ behaviors capturing the intention. One difference between our approach and", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52 + } + ], + "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, + 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, + 83, + 505, + 258 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "For example, drivers use signal light to inform other drivers of their intentions. A car driver may", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "slow down if a driver in his or her left lane turns the right signal light on. In this case, the signal", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "light encodes the driver’s intention, which indicates the driver’s future behavior, not current or past", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "observation such as the field view. By sharing intention using signal light, drivers coordinate their", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "drive with each other. In this paper, we formalize and propose a new communication scheme for", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "MARL named Intention sharing (IS) in order to go beyond existing observation-sharing schemes", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "for communication in MARL. The proposed IS scheme allows each agent to share its intention with", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "other agents in the form of encoded imagined trajectory. That is, each agent generates an imagined", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "trajectory by modeling the environment dynamics and other agents’ actions. Then, each agent learns", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "the relative importance of the components in the imagined trajectory based on the received messages", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 206 + ], + "score": 1.0, + "content": "from other agents by using an attention model. The output of the attention model is an encoded", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 217 + ], + "score": 1.0, + "content": "imagined trajectory capturing the intention of the agent and used as the communication message.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "We evaluate the proposed IS scheme in several multi-agent environments requiring coordination", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "among agents. Numerical result shows that the proposed IS scheme significantly outperforms other", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "existing communication schemes for MARL including the state-of-the-art algorithms such as ATOC", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 247, + 163, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 163, + 258 + ], + "score": 1.0, + "content": "and TarMAC.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 81, + 506, + 258 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 276, + 216, + 289 + ], + "lines": [ + { + "bbox": [ + 104, + 275, + 219, + 291 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 219, + 291 + ], + "score": 1.0, + "content": "2 RELATED WORKS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 303, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 106, + 304, + 504, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 304, + 504, + 315 + ], + "score": 1.0, + "content": "Under the asymmetry in learning resources between the training and execution phases, the frame-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 313, + 504, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 504, + 326 + ], + "score": 1.0, + "content": "work of centralized training and decentralized execution (CTDE), which assumes the availability", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 337 + ], + "score": 1.0, + "content": "of all system information in the training phase and distributed policy in the execution phase, has", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "been adopted in most recent MARL researches (Lowe et al. (2017); Foerster et al. (2018); Iqbal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "& Sha (2018); Kim et al. (2020)). Under the framework of CTDE, learning communication proto-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 370 + ], + "score": 1.0, + "content": "col has been considered to enhance performance in the decentralized execution phase for various", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "multi-agent tasks (Foerster et al. (2016); Jiang & Lu (2018); Das et al. (2019)). For this pur-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 380, + 507, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 507, + 393 + ], + "score": 1.0, + "content": "pose, Foerster et al. (2016) proposed Differentiable Inter-Agent Learning (DIAL). DIAL trains a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "score": 1.0, + "content": "message-generation network by connecting it to other agents’ Q-networks and allowing gradient", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "flow through communication channels in the training phase. Then, in the execution phase the mes-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "sages are generated and passed to other agents through communication channels. Jiang & Lu (2018)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "score": 1.0, + "content": "proposed an attentional communication model named ATOC to learn when to communicate and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "how to combine information received from other agents through communication based on attention", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "mechanism. Das et al. (2019) proposed Targeted Multi-Agent Communication (TarMAC) to learn", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "the message-generation network in order to produce different messages for different agents based", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 481 + ], + "score": 1.0, + "content": "on a signature-based attention model. The message-generation networks in the aforementioned al-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "gorithms are conditioned on the current observation or a hidden state of LSTM. Under partially", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "observable environments, such messages which encode past and current observations are useful but", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "do not capture any future information. In our approach, we use not only the current information", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 525 + ], + "score": 1.0, + "content": "but also future information to generate messages and the weight between the current and future", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 535 + ], + "score": 1.0, + "content": "information is adaptively learned according to the environment. This yields further performance", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 534, + 274, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 274, + 544 + ], + "score": 1.0, + "content": "enhancement, as we will see in Section 5.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 304, + 507, + 544 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "In our approach, the encoded imagined trajectory capturing the intention of each agent is used as the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "score": 1.0, + "content": "communication message in MARL. Imagined trajectory was used in other problems too. Racaniere `", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "et al. (2017) used imagined trajectory to augment it into the policy and critic for combining model-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 581, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 597 + ], + "score": 1.0, + "content": "based and model-free approaches in single-agent RL. It is shown that arbitrary imagined trajectory", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "(rolled-out trajectory by using a random policy or own policy) is useful for single-agent RL in", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "terms of performance and data efficiency. Strouse et al. (2018) introduced information-regularizer", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "to share or hide agent’s intention to other agents for a multi-goal MARL setting in which some", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "agents know the goal and other agents do not know the goal. By maximizing (or minimizing) the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "mutual information between the goal and action, an agent knowing the goal learns to share (or hide)", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "its intention to other agents not knowing the goal in cooperative (or competitive) tasks. They showed", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 659, + 335, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 335, + 673 + ], + "score": 1.0, + "content": "that sharing intention is effective in the cooperative case.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 550, + 506, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "In addition to our approach, Theory of Mind (ToM) and Opponent Modeling (OM) use the notion of", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "intention. Rabinowitz et al. (2018) proposed the Theory of Mind network (ToM-net) to predict other", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "agents’ behaviors by using meta-learning. Raileanu et al. (2018) proposed Self Other-Modeling", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "(SOM) to infer other agents’ goal in an online manner. Both ToM and OM take advantage of pre-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "dicting other agents’ behaviors capturing the intention. One difference between our approach and", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "the aforementioned two methods is that we use communication to share the intention instead of", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "inference. That is, the agents in our approach allow other agents to know their intention directly", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "score": 1.0, + "content": "through communication, whereas the agents in ToM and OM should figure out other agents’ inten-", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "tion by themselves. Furthermore, the messages in our approach include future information by rolling", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 492, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 492, + 140 + ], + "score": 1.0, + "content": "out the policy, whereas ToM and CM predict only the current or just next time-step information.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 52, + "bbox_fs": [ + 105, + 676, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "the aforementioned two methods is that we use communication to share the intention instead of", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "inference. 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In the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 492, + 303, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 303, + 504 + ], + "score": 1.0, + "content": "following we describe the detail of each module.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 106, + 517, + 374, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 374, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 374, + 531 + ], + "score": 1.0, + "content": "4.1 IMAGINED TRAJECTORY GENERATION MODULE (ITGM)", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 504, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "The role of ITGM is to produce the next imagined step. 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The MGN consists of two components: Imagined trajectory generation module (ITGM) and at-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "tention module (AM). Each agent generates an imagined trajectory by using ITGM and learns the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 472 + ], + "score": 1.0, + "content": "importance of each imagined step in the imagined trajectory by using AM. The output of AM is an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "encoded imagined trajectory reflecting the importance of imagined steps and is used as the commu-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "nication message. The overall architecture of the proposed IS scheme is shown in Fig. 1. In the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 492, + 303, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 303, + 504 + ], + "score": 1.0, + "content": "following we describe the detail of each module.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 370, + 506, + 504 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 517, + 374, + 529 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 374, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 374, + 531 + ], + "score": 1.0, + "content": "4.1 IMAGINED TRAJECTORY GENERATION MODULE (ITGM)", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 504, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 550 + ], + "score": 1.0, + "content": "The role of ITGM is to produce the next imagined step. ITGM takes the received messages, obser-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "vation, and action as input and yields the predicted next observation and predicted action as output.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 468, + 573 + ], + "score": 1.0, + "content": "By stacking ITGMs, we generate an imagined trajectory, as shown in Fig. 1. For Agent", + "type": "text" + }, + { + "bbox": [ + 468, + 561, + 473, + 570 + ], + "score": 0.75, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "at time", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 570, + 318, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 125, + 584 + ], + "score": 1.0, + "content": "step", + "type": "text" + }, + { + "bbox": [ + 125, + 572, + 130, + 581 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 570, + 186, + 584 + ], + "score": 1.0, + "content": ", we define an", + "type": "text" + }, + { + "bbox": [ + 187, + 571, + 197, + 581 + ], + "score": 0.84, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 570, + 318, + 584 + ], + "score": 1.0, + "content": "-length imagined trajectory as", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 538, + 506, + 584 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 587, + 366, + 602 + ], + "lines": [ + { + "bbox": [ + 245, + 587, + 366, + 602 + ], + "spans": [ + { + "bbox": [ + 245, + 587, + 366, + 602 + ], + "score": 0.93, + "content": "\\tau ^ { i } = ( \\tau _ { t } ^ { i } , \\hat { \\tau } _ { t + 1 } ^ { i } , \\cdot \\cdot \\cdot , \\hat { \\tau } _ { t + H - 1 } ^ { i } ) ,", + "type": "interline_equation", + "image_path": "ec38485abcb04143b194ac2580c9f34bc7b97c940287cce4bdf705849fb5ba5b.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 245, + 587, + 366, + 602 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 608, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 133, + 624 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 608, + 219, + 622 + ], + "score": 0.93, + "content": "\\hat { \\tau } _ { t + k } ^ { i } = ( \\hat { o } _ { t + k } ^ { i } , \\hat { a } _ { t + k } ^ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 606, + 354, + 624 + ], + "score": 1.0, + "content": "is the imagined step at time step", + "type": "text" + }, + { + "bbox": [ + 355, + 609, + 378, + 619 + ], + "score": 0.9, + "content": "t + k", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 606, + 424, + 624 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 425, + 608, + 479, + 621 + ], + "score": 0.93, + "content": "\\tau _ { t } ^ { i } = ( o _ { t } ^ { i } , a _ { t } ^ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 606, + 506, + 624 + ], + "score": 1.0, + "content": "is the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 621, + 473, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 378, + 635 + ], + "score": 1.0, + "content": "true values of observation and action, but the imagined steps except", + "type": "text" + }, + { + "bbox": [ + 378, + 621, + 388, + 633 + ], + "score": 0.89, + "content": "\\tau _ { t } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 621, + 473, + 635 + ], + "score": 1.0, + "content": "are predicted values.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 606, + 506, + 635 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 638, + 505, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 505, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 459, + 652 + ], + "score": 1.0, + "content": "ITGM consists of a roll-out policy and two predictors: Other agents’ action predictor", + "type": "text" + }, + { + "bbox": [ + 459, + 637, + 486, + 650 + ], + "score": 0.92, + "content": "f _ { a } ^ { i } ( o _ { t } ^ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 637, + 505, + 652 + ], + "score": 1.0, + "content": "(we", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 650, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 403, + 664 + ], + "score": 1.0, + "content": "will call this predictor simply action predictor) and observation predictor", + "type": "text" + }, + { + "bbox": [ + 403, + 650, + 461, + 663 + ], + "score": 0.95, + "content": "f _ { o } ^ { i } ( o _ { t } ^ { i } , a _ { t } ^ { i } , a _ { t } ^ { - i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 650, + 505, + 664 + ], + "score": 1.0, + "content": ". 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(2018); Raileanu et al. (2018)) and can take the received messages as input. Next,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 326, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 250, + 340 + ], + "score": 1.0, + "content": "we model the observation predictor", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 250, + 326, + 309, + 339 + ], + "score": 0.93, + "content": "f _ { o } ^ { i } ( o _ { t } ^ { i } , a _ { t } ^ { i } , \\hat { a } _ { t } ^ { - i } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 309, + 326, + 472, + 340 + ], + "score": 1.0, + "content": "which is conditioned on the observation", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 472, + 326, + 482, + 339 + ], + "score": 0.89, + "content": "o _ { t } ^ { i }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 482, + 326, + 506, + 340 + ], + "score": 1.0, + "content": ", own", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 134, + 352 + ], + "score": 1.0, + "content": "action", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 134, + 339, + 144, + 351 + ], + "score": 0.88, + "content": "a _ { t } ^ { i }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 145, + 339, + 305, + 352 + ], + "score": 1.0, + "content": ", and the output of the action predictor", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 305, + 338, + 321, + 351 + ], + "score": 0.91, + "content": "\\hat { a } _ { t } ^ { - i }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 321, + 339, + 506, + 352 + ], + "score": 1.0, + "content": ". 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In the cases of", + "type": "text" + }, + { + "bbox": [ + 419, + 468, + 483, + 479 + ], + "score": 0.89, + "content": "( N = 2 , C = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 467, + 506, + 480 + ], + "score": 1.0, + "content": ") and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 109, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 109, + 478, + 171, + 490 + ], + "score": 0.89, + "content": "( N = 3 , C = 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "), the initial positions of all predators are the same and randomly determined. Thus,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "the predators should learn not only how to capture preys but also how to spread out. In the case of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 109, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 109, + 501, + 171, + 512 + ], + "score": 0.89, + "content": "( N = 4 , C = 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 499, + 506, + 514 + ], + "score": 1.0, + "content": ", the initial positions of all predators are randomly determined independently. Thus,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 511, + 345, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 345, + 525 + ], + "score": 1.0, + "content": "the predators should learn to capture preys in group of two.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "Cooperative-navigation (CN) The goal of cooperative navigation introduced in (Lowe et al. (2017))", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 132, + 552 + ], + "score": 1.0, + "content": "is for", + "type": "text" + }, + { + "bbox": [ + 132, + 540, + 142, + 549 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 539, + 209, + 552 + ], + "score": 1.0, + "content": "agents to cover", + "type": "text" + }, + { + "bbox": [ + 210, + 540, + 218, + 549 + ], + "score": 0.75, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "landmarks while avoiding collisions among the agents. We modified", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 392, + 563 + ], + "score": 1.0, + "content": "the original environment so that collisions occur more easily. We set", + "type": "text" + }, + { + "bbox": [ + 392, + 550, + 426, + 560 + ], + "score": 0.91, + "content": "L = N", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 550, + 506, + 563 + ], + "score": 1.0, + "content": ", increased the size", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "of agent, and assigned a specific landmark to cover to each agent (i.e., each agent should cover", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "the landmark of the same color in Fig.2(b)). Each agent observes the positions of predators and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 295, + 595 + ], + "score": 1.0, + "content": "landmarks. The agent receives shared reward", + "type": "text" + }, + { + "bbox": [ + 296, + 583, + 309, + 594 + ], + "score": 0.89, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "which is the sum of the distance between each", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 414, + 606 + ], + "score": 1.0, + "content": "agent and the corresponding landmark at each time step and success reward", + "type": "text" + }, + { + "bbox": [ + 414, + 594, + 452, + 605 + ], + "score": 0.92, + "content": "N ^ { \\prime } \\times R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 594, + 481, + 606 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 481, + 594, + 494, + 604 + ], + "score": 0.89, + "content": "N ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "the number of the covered landmark. Agents who collide with other agents receive negative reward", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 616, + 458, + 628 + ], + "spans": [ + { + "bbox": [ + 107, + 616, + 119, + 627 + ], + "score": 0.85, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 616, + 268, + 628 + ], + "score": 1.0, + "content": ". We simulated the environment with", + "type": "text" + }, + { + "bbox": [ + 268, + 616, + 317, + 627 + ], + "score": 0.88, + "content": "N = L = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 616, + 320, + 628 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 321, + 616, + 362, + 628 + ], + "score": 0.87, + "content": "R _ { 1 } = 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 616, + 366, + 628 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 366, + 616, + 398, + 627 + ], + "score": 0.86, + "content": "R _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 616, + 415, + 628 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 416, + 616, + 455, + 627 + ], + "score": 0.91, + "content": "R _ { 3 } = - 5", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 616, + 458, + 628 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Traffic-junction (TJ) We modified the traffic-junction introduced in Sukhbaatar et al. (2016) to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "continuous state-action domain. In the beginning of an episode, each agent is randomly located", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "in a predefined initial position and assigned one of three routes: left, right or straight, as seen in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 666, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 678 + ], + "score": 1.0, + "content": "Fig.2(c). The observation of each agent consists of the positions of all agents (no route information", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "of other agents) and 2 one-hot vectors which encodes the initial position and assigned route of the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 302, + 702 + ], + "score": 1.0, + "content": "agent. The action of each agent is a real value in", + "type": "text" + }, + { + "bbox": [ + 302, + 688, + 324, + 700 + ], + "score": 0.9, + "content": "( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 687, + 505, + 702 + ], + "score": 1.0, + "content": ", which indicates the distance to go along the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "assigned route from the current position. The goal is to go to the destination as fast as possible while", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "avoiding collision with other agents. To achieve the goal, we design reward with three components.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 255, + 732 + ], + "score": 1.0, + "content": "Each agent receives success reward", + "type": "text" + }, + { + "bbox": [ + 256, + 721, + 269, + 732 + ], + "score": 0.88, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "if it arrives at the destination without any collision with", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "The considered baselines are as follows. 1) MADDPG (Lowe et al. (2017)): we can assess the gain", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "of introducing communication from this baseline. 2) DIAL (Foerster et al. (2016)): we modified", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "DIAL, which is based on Q-learning, to our setting by connecting the message-generation network", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "to other agents’ policies and allowing the gradient flow through communication channel. 3) TarMAC", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 127, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 506, + 138 + ], + "score": 1.0, + "content": "(Das et al. 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We considered", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 184 + ], + "score": 1.0, + "content": "three multi-agent environments: predator-prey, cooperative navigation, and traffic junction, and we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 477, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 477, + 194 + ], + "score": 1.0, + "content": "slightly modified the conventional environments to require more coordination among agents.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 81, + 506, + 194 + ] + }, + { + "type": "image", + "bbox": [ + 153, + 203, + 458, + 298 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 153, + 203, + 458, + 298 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 153, + 203, + 458, + 298 + ], + "spans": [ + { + "bbox": [ + 153, + 203, + 458, + 298 + ], + "score": 0.97, + "type": "image", + "image_path": "65e2e8a9585fe2006b682abcc9d4060731b5c342e805424fd1e2bd25bde7fbfa.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 153, + 203, + 458, + 234.66666666666666 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 153, + 234.66666666666666, + 458, + 266.3333333333333 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 153, + 266.3333333333333, + 458, + 298.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 307, + 505, + 330 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "score": 1.0, + "content": "Figure 2: Considered environments: (a) predator-prey (PP), (b) cooperative-navigation (CN), and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 318, + 201, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 201, + 332 + ], + "score": 1.0, + "content": "(c) traffic-junction (TJ)", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + } + ], + "index": 12.25 + }, + { + "type": "title", + "bbox": [ + 108, + 358, + 201, + 370 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 203, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 203, + 371 + ], + "score": 1.0, + "content": "5.1 ENVIRONMENTS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 379, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 505, + 393 + ], + "score": 1.0, + "content": "Predator-prey (PP) The predator-prey environment is a standard task in multi-agent systems. We", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 264, + 403 + ], + "score": 1.0, + "content": "used a PP environment that consists of", + "type": "text" + }, + { + "bbox": [ + 264, + 391, + 274, + 401 + ], + "score": 0.81, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 390, + 355, + 403 + ], + "score": 1.0, + "content": "predators and fixed", + "type": "text" + }, + { + "bbox": [ + 355, + 391, + 367, + 401 + ], + "score": 0.71, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "preys in a continuous state-action", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "score": 1.0, + "content": "domain. We control the actions of predators and the goal is to capture as many preys as possible in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 430, + 425 + ], + "score": 1.0, + "content": "a given time. Each agent observes the positions of predators and preys. 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At every time", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 435, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 450, + 448 + ], + "score": 1.0, + "content": "when all the preys are captured, the preys are respawn and the shared reward value", + "type": "text" + }, + { + "bbox": [ + 451, + 435, + 464, + 446 + ], + "score": 0.89, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 435, + 506, + 448 + ], + "score": 1.0, + "content": "increases", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 445, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 458 + ], + "score": 1.0, + "content": "by one with initial value one to accelerate the capture speed for the given time. We simulated", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 455, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 161, + 470 + ], + "score": 1.0, + "content": "three cases:", + "type": "text" + }, + { + "bbox": [ + 162, + 457, + 230, + 468 + ], + "score": 0.88, + "content": "( N = 2 , C = 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 455, + 239, + 470 + ], + "score": 1.0, + "content": "),", + "type": "text" + }, + { + "bbox": [ + 239, + 457, + 309, + 468 + ], + "score": 0.9, + "content": "( N = 3 , C = 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 455, + 336, + 470 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 336, + 457, + 405, + 468 + ], + "score": 0.89, + "content": "\\mathit { N } = 4 , C = 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 455, + 443, + 470 + ], + "score": 1.0, + "content": "with all", + "type": "text" + }, + { + "bbox": [ + 443, + 457, + 477, + 467 + ], + "score": 0.89, + "content": "M = 9", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 455, + 506, + 470 + ], + "score": 1.0, + "content": "preys,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 419, + 480 + ], + "score": 1.0, + "content": "where the fixed positions of the preys are shown in Fig.2(a). In the cases of", + "type": "text" + }, + { + "bbox": [ + 419, + 468, + 483, + 479 + ], + "score": 0.89, + "content": "( N = 2 , C = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 467, + 506, + 480 + ], + "score": 1.0, + "content": ") and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 109, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 109, + 478, + 171, + 490 + ], + "score": 0.89, + "content": "( N = 3 , C = 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "), the initial positions of all predators are the same and randomly determined. Thus,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "the predators should learn not only how to capture preys but also how to spread out. In the case of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 109, + 499, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 109, + 501, + 171, + 512 + ], + "score": 0.89, + "content": "( N = 4 , C = 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 499, + 506, + 514 + ], + "score": 1.0, + "content": ", the initial positions of all predators are randomly determined independently. Thus,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 511, + 345, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 345, + 525 + ], + "score": 1.0, + "content": "the predators should learn to capture preys in group of two.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 380, + 506, + 525 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "Cooperative-navigation (CN) The goal of cooperative navigation introduced in (Lowe et al. (2017))", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 132, + 552 + ], + "score": 1.0, + "content": "is for", + "type": "text" + }, + { + "bbox": [ + 132, + 540, + 142, + 549 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 539, + 209, + 552 + ], + "score": 1.0, + "content": "agents to cover", + "type": "text" + }, + { + "bbox": [ + 210, + 540, + 218, + 549 + ], + "score": 0.75, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "landmarks while avoiding collisions among the agents. We modified", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 392, + 563 + ], + "score": 1.0, + "content": "the original environment so that collisions occur more easily. We set", + "type": "text" + }, + { + "bbox": [ + 392, + 550, + 426, + 560 + ], + "score": 0.91, + "content": "L = N", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 550, + 506, + 563 + ], + "score": 1.0, + "content": ", increased the size", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "of agent, and assigned a specific landmark to cover to each agent (i.e., each agent should cover", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 506, + 586 + ], + "score": 1.0, + "content": "the landmark of the same color in Fig.2(b)). Each agent observes the positions of predators and", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 295, + 595 + ], + "score": 1.0, + "content": "landmarks. The agent receives shared reward", + "type": "text" + }, + { + "bbox": [ + 296, + 583, + 309, + 594 + ], + "score": 0.89, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "which is the sum of the distance between each", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 414, + 606 + ], + "score": 1.0, + "content": "agent and the corresponding landmark at each time step and success reward", + "type": "text" + }, + { + "bbox": [ + 414, + 594, + 452, + 605 + ], + "score": 0.92, + "content": "N ^ { \\prime } \\times R _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 594, + 481, + 606 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 481, + 594, + 494, + 604 + ], + "score": 0.89, + "content": "N ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 594, + 506, + 606 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "the number of the covered landmark. Agents who collide with other agents receive negative reward", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 616, + 458, + 628 + ], + "spans": [ + { + "bbox": [ + 107, + 616, + 119, + 627 + ], + "score": 0.85, + "content": "R _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 616, + 268, + 628 + ], + "score": 1.0, + "content": ". We simulated the environment with", + "type": "text" + }, + { + "bbox": [ + 268, + 616, + 317, + 627 + ], + "score": 0.88, + "content": "N = L = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 616, + 320, + 628 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 321, + 616, + 362, + 628 + ], + "score": 0.87, + "content": "R _ { 1 } = 1 / 3", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 616, + 366, + 628 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 366, + 616, + 398, + 627 + ], + "score": 0.86, + "content": "R _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 616, + 415, + 628 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 416, + 616, + 455, + 627 + ], + "score": 0.91, + "content": "R _ { 3 } = - 5", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 616, + 458, + 628 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 528, + 506, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Traffic-junction (TJ) We modified the traffic-junction introduced in Sukhbaatar et al. (2016) to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "continuous state-action domain. In the beginning of an episode, each agent is randomly located", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "in a predefined initial position and assigned one of three routes: left, right or straight, as seen in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 666, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 504, + 678 + ], + "score": 1.0, + "content": "Fig.2(c). The observation of each agent consists of the positions of all agents (no route information", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "of other agents) and 2 one-hot vectors which encodes the initial position and assigned route of the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 687, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 302, + 702 + ], + "score": 1.0, + "content": "agent. The action of each agent is a real value in", + "type": "text" + }, + { + "bbox": [ + 302, + 688, + 324, + 700 + ], + "score": 0.9, + "content": "( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 687, + 505, + 702 + ], + "score": 1.0, + "content": ", which indicates the distance to go along the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "assigned route from the current position. The goal is to go to the destination as fast as possible while", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "avoiding collision with other agents. To achieve the goal, we design reward with three components.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 255, + 732 + ], + "score": 1.0, + "content": "Each agent receives success reward", + "type": "text" + }, + { + "bbox": [ + 256, + 721, + 269, + 732 + ], + "score": 0.88, + "content": "R _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "if it arrives at the destination without any collision with", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 263, + 357 + ], + "score": 1.0, + "content": "other agents, collision negative reward", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 263, + 345, + 276, + 356 + ], + "score": 0.89, + "content": "R _ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 277, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "if its position is overlapped with that of other agent, and", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 193, + 369 + ], + "score": 1.0, + "content": "time negative reward", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 193, + 357, + 206, + 367 + ], + "score": 0.89, + "content": "R _ { 3 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 207, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "to avoid traffic jam. When an agent arrives at the destination, the agent is", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 388, + 379 + ], + "score": 1.0, + "content": "assigned a new initial position and the route. An episode ends when", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 389, + 367, + 397, + 377 + ], + "score": 0.75, + "content": "T", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 398, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "time steps elapse. 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Note that each agent estimates the future positions of other agents as well as their", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "own future position due to the assumption of full observability. The first, second, and third row of", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 320, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 332 + ], + "score": 1.0, + "content": "Fig.5 show the imagined trajectories of all agents at Agent 1 (red), Agent 2 (green) and Agent 3", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "(blue), respectively. Note that the imagined trajectory of each agent represents its future plan for the", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 241, + 354 + ], + "score": 1.0, + "content": "environment. As seen in Fig.5, at", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 242, + 342, + 266, + 352 + ], + "score": 0.89, + "content": "t = 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 266, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "the intention of both Agent 1 and Agent 3 is to move to the", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 352, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 197, + 364 + ], + "score": 1.0, + "content": "left to catch preys. At", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 198, + 353, + 222, + 362 + ], + "score": 0.9, + "content": "t = 1", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 222, + 352, + 506, + 364 + ], + "score": 1.0, + "content": ", all agents receive the messages from other agents. It is observed that", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "score": 1.0, + "content": "Agent 3 changes its future plan to catch preys around the center while Agent 1 maintains its future", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 375, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 387 + ], + "score": 1.0, + "content": "plan. This procedure shows that coordination between Agent 1 and Agent 3 starts to occur. It is seen", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 385, + 405, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 405, + 398 + ], + "score": 1.0, + "content": "that as time goes, each agent roughly predicts other agents’ future actions.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 610, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 119, + 98, + 504, + 244 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 119, + 98, + 504, + 244 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 119, + 98, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 119, + 98, + 504, + 244 + ], + "score": 0.969, + "type": "image", + "image_path": "7b3b19c5a9045ea3e1116c74ee3a2c72b0b990f2a8adf5552c566ad452bf7ac2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 119, + 98, + 504, + 146.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 119, + 146.66666666666666, + 504, + 195.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 119, + 195.33333333333331, + 504, + 243.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 254, + 506, + 276 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 253, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 476, + 267 + ], + "score": 1.0, + "content": "Figure 4: Performance for our proposed method with different length of imagined trajectory", + "type": "text" + }, + { + "bbox": [ + 477, + 254, + 487, + 264 + ], + "score": 0.73, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 253, + 505, + 267 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 265, + 212, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 212, + 277 + ], + "score": 1.0, + "content": "without attention module.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 505, + 397 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 505, + 320 + ], + "score": 1.0, + "content": "own future position due to the assumption of full observability. The first, second, and third row of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 320, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 332 + ], + "score": 1.0, + "content": "Fig.5 show the imagined trajectories of all agents at Agent 1 (red), Agent 2 (green) and Agent 3", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "(blue), respectively. Note that the imagined trajectory of each agent represents its future plan for the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 241, + 354 + ], + "score": 1.0, + "content": "environment. As seen in Fig.5, at", + "type": "text" + }, + { + "bbox": [ + 242, + 342, + 266, + 352 + ], + "score": 0.89, + "content": "t = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "the intention of both Agent 1 and Agent 3 is to move to the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 352, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 197, + 364 + ], + "score": 1.0, + "content": "left to catch preys. At", + "type": "text" + }, + { + "bbox": [ + 198, + 353, + 222, + 362 + ], + "score": 0.9, + "content": "t = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 352, + 506, + 364 + ], + "score": 1.0, + "content": ", all agents receive the messages from other agents. It is observed that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 504, + 375 + ], + "score": 1.0, + "content": "Agent 3 changes its future plan to catch preys around the center while Agent 1 maintains its future", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 375, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 387 + ], + "score": 1.0, + "content": "plan. This procedure shows that coordination between Agent 1 and Agent 3 starts to occur. It is seen", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 385, + 405, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 405, + 398 + ], + "score": 1.0, + "content": "that as time goes, each agent roughly predicts other agents’ future actions.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 401, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 466, + 415 + ], + "score": 1.0, + "content": "We conducted experiments to examine the impact of the length of the imagined trajectory", + "type": "text" + }, + { + "bbox": [ + 467, + 403, + 477, + 412 + ], + "score": 0.67, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 401, + 505, + 415 + ], + "score": 1.0, + "content": ". Fig.4", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 388, + 426 + ], + "score": 1.0, + "content": "shows the performances of the proposed method for different values of", + "type": "text" + }, + { + "bbox": [ + 389, + 414, + 398, + 423 + ], + "score": 0.79, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 412, + 505, + 426 + ], + "score": 1.0, + "content": ". It is seen that the training", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 198, + 437 + ], + "score": 1.0, + "content": "speed is reduced when", + "type": "text" + }, + { + "bbox": [ + 199, + 424, + 227, + 434 + ], + "score": 0.9, + "content": "H = 7", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 424, + 291, + 437 + ], + "score": 1.0, + "content": "as compared to", + "type": "text" + }, + { + "bbox": [ + 291, + 424, + 320, + 434 + ], + "score": 0.9, + "content": "H = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 424, + 331, + 437 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 332, + 424, + 360, + 434 + ], + "score": 0.89, + "content": "H = 5", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 424, + 505, + 437 + ], + "score": 1.0, + "content": ". However, the final performance all", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 435, + 210, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 210, + 447 + ], + "score": 1.0, + "content": "outperforms the baseline.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 451, + 505, + 567 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 505, + 464 + ], + "score": 1.0, + "content": "Attention In the proposed IS scheme, the imagined trajectory is encoded based on the attention", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "module to capture the importance of components in the imagined trajectory. Recall that the message", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 103, + 473, + 507, + 493 + ], + "spans": [ + { + "bbox": [ + 103, + 473, + 145, + 493 + ], + "score": 1.0, + "content": "of Agent", + "type": "text" + }, + { + "bbox": [ + 146, + 477, + 151, + 486 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 473, + 216, + 493 + ], + "score": 1.0, + "content": "is expressed as", + "type": "text" + }, + { + "bbox": [ + 216, + 474, + 300, + 489 + ], + "score": 0.94, + "content": "\\begin{array} { r } { m _ { t } ^ { i } = \\sum _ { h = 1 } ^ { H } \\alpha _ { h } ^ { i } v _ { t } ^ { i , h } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 473, + 393, + 493 + ], + "score": 1.0, + "content": ", as seen in (7), where", + "type": "text" + }, + { + "bbox": [ + 394, + 475, + 407, + 488 + ], + "score": 0.9, + "content": "\\alpha _ { h } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 473, + 507, + 493 + ], + "score": 1.0, + "content": "denotes the importance", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 102, + 486, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 102, + 486, + 118, + 504 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 487, + 135, + 501 + ], + "score": 0.91, + "content": "v _ { t } ^ { i , h }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 486, + 505, + 504 + ], + "score": 1.0, + "content": ", which is the encoded imagined step. Note that the previously proposed communication", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 294, + 513 + ], + "score": 1.0, + "content": "schemes are the special case corresponding to", + "type": "text" + }, + { + "bbox": [ + 295, + 500, + 371, + 512 + ], + "score": 0.95, + "content": "\\alpha ^ { i } = ( 1 , 0 , \\cdots , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 500, + 505, + 513 + ], + "score": 1.0, + "content": ". In Fig.5, the brightness of each", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 352, + 524 + ], + "score": 1.0, + "content": "circle is proportional to the attention weight. At time step", + "type": "text" + }, + { + "bbox": [ + 353, + 512, + 384, + 522 + ], + "score": 0.88, + "content": "t \\ = \\ K", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 511, + 417, + 524 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 417, + 511, + 455, + 522 + ], + "score": 0.85, + "content": "K = 3 7", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 511, + 460, + 524 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 461, + 511, + 473, + 523 + ], + "score": 0.86, + "content": "\\alpha _ { 2 } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 511, + 506, + 524 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 102, + 516, + 501, + 540 + ], + "spans": [ + { + "bbox": [ + 102, + 516, + 489, + 540 + ], + "score": 1.0, + "content": "2indicates when Agent 1 moves to the prey in the bottom middle, is the highest. In addition,", + "type": "text" + }, + { + "bbox": [ + 489, + 522, + 501, + 534 + ], + "score": 0.88, + "content": "\\alpha _ { 4 } ^ { 3 }", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "which indicates when Agent 3 moves to the prey in the left middle, is the highest. Hence, the agent", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "tends to send future information when it is near a prey. Similar attention weight tendency is also", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 555, + 318, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 207, + 567 + ], + "score": 1.0, + "content": "captured in the time step", + "type": "text" + }, + { + "bbox": [ + 207, + 555, + 252, + 566 + ], + "score": 0.92, + "content": "t = K + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 555, + 269, + 567 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 270, + 555, + 314, + 566 + ], + "score": 0.92, + "content": "t = K + 2", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 555, + 318, + 567 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 505, + 584 + ], + "score": 1.0, + "content": "As aforementioned, the aim of the IS scheme is to communicate with other agents based on their", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 583, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 594 + ], + "score": 1.0, + "content": "own future plans. How far future is important depends on the environment and on the tasks. In order", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "to analyze the tendency in the importance of future plans, we averaged the attention weight over", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "the trajectories on the fully observable PP environment with 3 agents and a partially observable PP", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "environment with 3 agents in which each agent knows the locations of other agents within a certain", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 628, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 505, + 639 + ], + "score": 1.0, + "content": "range from the agent. The result is summarized in Table 1. 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This is because sharing", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "observation information is more critical in the partially observable case than the fully observable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 682, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 506, + 694 + ], + "score": 1.0, + "content": "case. 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At time step", + "type": "text" + }, + { + "bbox": [ + 353, + 512, + 384, + 522 + ], + "score": 0.88, + "content": "t \\ = \\ K", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 511, + 417, + 524 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 417, + 511, + 455, + 522 + ], + "score": 0.85, + "content": "K = 3 7", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 511, + 460, + 524 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 461, + 511, + 473, + 523 + ], + "score": 0.86, + "content": "\\alpha _ { 2 } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 511, + 506, + 524 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 102, + 516, + 501, + 540 + ], + "spans": [ + { + "bbox": [ + 102, + 516, + 489, + 540 + ], + "score": 1.0, + "content": "2indicates when Agent 1 moves to the prey in the bottom middle, is the highest. In addition,", + "type": "text" + }, + { + "bbox": [ + 489, + 522, + 501, + 534 + ], + "score": 0.88, + "content": "\\alpha _ { 4 } ^ { 3 }", + "type": "inline_equation" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "which indicates when Agent 3 moves to the prey in the left middle, is the highest. Hence, the agent", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 505, + 557 + ], + "score": 1.0, + "content": "tends to send future information when it is near a prey. 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How far future is important depends on the environment and on the tasks. 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The result is summarized in Table 1. 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This is because sharing", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "observation information is more critical in the partially observable case than the fully observable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 682, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 506, + 694 + ], + "score": 1.0, + "content": "case. 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We", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "compared the proposed IS scheme with and without the attention module. We replace the attention", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 103, + 326, + 507, + 346 + ], + "spans": [ + { + "bbox": [ + 103, + 326, + 407, + 346 + ], + "score": 1.0, + "content": "module with an averaging layer, which is the special case corresponding to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 408, + 328, + 482, + 342 + ], + "score": 0.93, + "content": "\\alpha ^ { i } = ( \\textstyle { \\frac { 1 } { H } } , \\cdot \\cdot \\cdot , \\textstyle { \\frac { 1 } { H } } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 482, + 326, + 507, + 346 + ], + "score": 1.0, + "content": ". 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Initialize parameter 0μ,,0i-,0,,0,Wi,∀i∈ {1,.,N} for episode = 1,2,.. do
Initialize state s1,messages mo = (,. , ) and each agent observes 0i for t<= Tand st ≠ terminal do Each agent receives the messages mt-1 = (mt-1, ,mt-1)
Each agent selects action ai ~ πi(-loε,mt-1) for each agent i
Execute at and each agent i receives rt and Ot+1
for h=1,2,.,H do Predict other agents' actions at+h-1 from the action predictor fi
Generate Ot+n from observation predictor fi(Ot+h-1,@t+h-1,@t+h-1)
Generate at+h ~π²（-1O+h,mt-1) end for
Eachagent generates the messages mbyinjecting Ti = (𝑖,+1, ,+H-1) intoAtten-
tion Module (AM) Store transitions in D
end for
for each gradient step do Update 0 and (0i,0) by minimizing the loss (9) and the loss (12)
Update 0, and Wi based on the gradient (1O) and the gradient (11) end for Update 0i- , 0- using the moving average method
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0000000000000000000000000000000000000000..b552dd1ebc68526687d468e76b7a19f82b6a1a15 --- /dev/null +++ b/parse/train/ryMxXPFex/ryMxXPFex.md @@ -0,0 +1,851 @@ +# DISCRETE VARIATIONAL AUTOENCODERS + +Jason Tyler Rolfe +D-Wave Systems +Burnaby, BC V5G-4M9, Canada +jrolfe@dwavesys.com + +# ABSTRACT + +Probabilistic models with discrete latent variables naturally capture datasets composed of discrete classes. However, they are difficult to train efficiently, since backpropagation through discrete variables is generally not possible. We present a novel method to train a class of probabilistic models with discrete latent variables using the variational autoencoder framework, including backpropagation through the discrete latent variables. The associated class of probabilistic models comprises an undirected discrete component and a directed hierarchical continuous component. The discrete component captures the distribution over the disconnected smooth manifolds induced by the continuous component. As a result, this class of models efficiently learns both the class of objects in an image, and their specific realization in pixels, from unsupervised data; and outperforms state-ofthe-art methods on the permutation-invariant MNIST, Omniglot, and Caltech-101 Silhouettes datasets. + +# 1 INTRODUCTION + +Unsupervised learning of probabilistic models is a powerful technique, facilitating tasks such as denoising and inpainting, and regularizing supervised tasks such as classification (Hinton et al., 2006; Salakhutdinov & Hinton, 2009; Rasmus et al., 2015). Many datasets of practical interest are projections of underlying distributions over real-world objects into an observation space; the pixels of an image, for example. When the real-world objects are of discrete types subject to continuous transformations, these datasets comprise multiple disconnected smooth manifolds. For instance, natural images change smoothly with respect to the position and pose of objects, as well as scene lighting. At the same time, it is extremely difficult to directly transform the image of a person to one of a car while remaining on the manifold of natural images. + +It would be natural to represent the space within each disconnected component with continuous variables, and the selection amongst these components with discrete variables. In contrast, most stateof-the-art probabilistic models use exclusively discrete variables — as do DBMs (Salakhutdinov & Hinton, 2009), NADEs (Larochelle & Murray, 2011), sigmoid belief networks (Spiegelhalter & Lauritzen, 1990; Bornschein et al., 2016), and DARNs (Gregor et al., 2014) — or exclusively continuous variables — as do VAEs (Kingma & Welling, 2014; Rezende et al., 2014) and GANs (Goodfellow et al., 2014).1 Moreover, it would be desirable to apply the efficient variational autoencoder framework to models with discrete values, but this has proven difficult, since backpropagation through discrete variables is generally not possible (Bengio et al., 2013; Raiko et al., 2015). + +We introduce a novel class of probabilistic models, comprising an undirected graphical model defined over binary latent variables, followed by multiple directed layers of continuous latent variables. This class of models captures both the discrete class of the object in an image, and its specific continuously deformable realization. Moreover, we show how these models can be trained efficiently using the variational autoencoder framework, including backpropagation through the binary latent variables. We ensure that the evidence lower bound remains tight by incorporating a hierarchical approximation to the posterior distribution of the latent variables, which can model strong correlations. Since these models efficiently marry the variational autoencoder framework with discrete latent variables, we call them discrete variational autoencoders (discrete VAEs). + +# 1.1 VARIATIONAL AUTOENCODERS ARE INCOMPATIBLE WITH DISCRETE DISTRIBUTIONS + +Conventionally, unsupervised learning algorithms maximize the log-likelihood of an observed dataset under a probabilistic model. Even stochastic approximations to the gradient of the loglikelihood generally require samples from the posterior and prior of the model. However, sampling from undirected graphical models is generally intractable (Long & Servedio, 2010), as is sampling from the posterior of a directed graphical model conditioned on its leaf variables (Dagum $\&$ Luby, 1993). + +In contrast to the exact log-likelihood, it can be computationally efficient to optimize a lower bound on the log-likelihood (Jordan et al., 1999), such as the evidence lower bound (ELBO, $\mathcal { L } ( x , \theta , \phi )$ ; Hinton & Zemel, 1994): + +$$ +\mathcal { L } ( x , \theta , \phi ) = \log p ( x | \theta ) - \mathrm { K L } [ q ( z | x , \phi ) | | p ( z | x , \theta ) ] , +$$ + +where $q ( z | x , \phi )$ is a computationally tractable approximation to the posterior distribution $p ( z | x , \theta )$ . We denote the observed random variables by $x$ , the latent random variables by $z$ , the parameters of the generative model by $\theta$ , and the parameters of the approximating posterior by $\phi$ . The variational autoencoder (VAE; Kingma & Welling, 2014; Rezende et al., 2014; Kingma et al., 2014) regroups the evidence lower bound of Equation 1 as: + +$$ +\begin{array} { r } { \mathcal { L } ( x , \theta , \phi ) = - \underbrace { \mathrm { K L } \left[ q ( z | x , \phi ) | | p ( z | \theta ) \right] } _ { \mathrm { K L } \mathrm { t e r m } } + \underbrace { \mathbb { E } _ { q } \left[ \log p ( x | z , \theta ) \right] } _ { \mathrm { a u t o e n c o d i n g ~ t e r m } } . } \end{array} +$$ + +In many cases of practical interest, such as Gaussian $q ( z | x )$ and $p ( z )$ , the KL term of Equation 2 can be computed analytically. Moreover, a low-variance stochastic approximation to the gradient of the autoencoding term can be obtained using backpropagation and the reparameterization trick, so long as samples from the approximating posterior $q ( \bar { z } | \bar { x } )$ can be drawn using a differentiable, deterministic function $f ( x , \phi , \rho )$ of the combination of the inputs, the parameters, and a set of inputand parameter-independent random variables $\rho \sim D$ . For instance, samples can be drawn from a Gaussian distribution with mean and variance determined by the input, $\mathcal { N } ( m ( x , \phi ) , v ( x , \phi ) )$ , using $f ( x , \phi , \rho ) = m ( x , \phi ) + \sqrt { v ( x , \phi ) } \cdot \rho$ , where $\rho \sim \mathcal { N } ( 0 , 1 )$ . When such an $f ( x , \phi , \rho )$ exists, + +$$ +\frac { \partial } { \partial \phi } \mathbb { E } _ { q ( z | x , \phi ) } \left[ \log p ( x | z , \theta ) \right] \approx \frac { 1 } { N } \sum _ { \rho \sim D } \frac { \partial } { \partial \phi } \log p ( x | f ( x , \rho , \phi ) , \theta ) . +$$ + +The reparameterization trick can be generalized to a large set of distributions, including nonfactorial approximating posteriors. We address this issue carefully in Appendix A, where we find that an analog of Equation 3 holds. Specifically, $\mathcal { D } _ { i }$ is the uniform distribution between 0 and 1, and + +$$ +f ( x ) = \mathbf { F } ^ { - 1 } ( x ) , +$$ + +where $\mathbf { F }$ is the conditional-marginal cumulative distribution function (CDF) defined by: + +$$ +F _ { i } ( \mathbf { x } ) = \int _ { x _ { i } ^ { \prime } = - \infty } ^ { x } p \left( x _ { i } ^ { \prime } | x _ { 1 } , \dots , x _ { i - 1 } \right) . +$$ + +However, this generalization is only possible if the inverse of the conditional-marginal CDF exists and is differentiable. + +A formulation comparable to Equation 3 is not possible for discrete distributions, such as restricted Boltzmann machines (RBMs) (Smolensky, 1986): + +$$ +p ( z ) = \frac { 1 } { \mathcal { Z } _ { p } } e ^ { - E _ { p } ( z ) } = \frac { 1 } { \mathcal { Z } _ { p } } \cdot e ^ { \left( z ^ { \top } W z + b ^ { \top } z \right) } , +$$ + +where $z \in \{ 0 , 1 \} ^ { n }$ , ${ \mathcal { Z } } _ { p }$ is the partition function of $p ( z )$ , and the lateral connection matrix $W$ is triangular. Any approximating posterior that only assigns nonzero probability to a discrete domain corresponds to a CDF that is piecewise-contant. That is, the range of the CDF is a proper subset of the interval $[ 0 , 1 ]$ . The domain of the inverse CDF is thus also a proper subset of $[ 0 , 1 ]$ , and its derivative is not defined, as required in Equations 3 and 4.2 + +In the following sections, we present the discrete variational autoencoder (discrete VAE), a hierarchical probabilistic model consising of an RBM,3 followed by multiple directed layers of continuous latent variables. This model is efficiently trainable using the variational autoencoder formalism, as in Equation 3, including backpropagation through its discrete latent variables. + +# 1.2 RELATED WORK + +Recently, there have been many efforts to develop effective unsupervised learning techniques by building upon variational autoencoders. Importance weighted autoencoders (Burda et al., 2016), Hamiltonian variational inference (Salimans et al., 2015), normalizing flows (Rezende & Mohamed, 2015), and variational Gaussian processes (Tran et al., 2016) improve the approximation to the posterior distribution. Ladder variational autoencoders (Sønderby et al., 2016) increase the power of the architecture of both approximating posterior and prior. Neural adaptive importance sampling (Du et al., 2015) and reweighted wake-sleep (Bornschein & Bengio, 2015) use sophisticated approximations to the gradient of the log-likelihood that do not admit direct backpropagation. Structured variational autoencoders use conjugate priors to construct powerful approximating posterior distributions (Johnson et al., 2016). + +It is easy to construct a stochastic approximation to the gradient of the ELBO that admits both discrete and continuous latent variables, and only requires computationally tractable samples. Unfortunately, this naive estimate is impractically high-variance, leading to slow training and poor performance (Paisley et al., 2012). The variance of the gradient can be reduced somewhat using the baseline technique, originally called REINFORCE in the reinforcement learning literature (Mnih & Gregor, 2014; Williams, 1992; Mnih & Rezende, 2016), which we discuss in greater detail in Appendix B. + +Prior efforts by Makhzani et al. (2015) to use multimodal priors with implicit discrete variables governing the modes did not successfully align the modes of the prior with the intrinsic clusters of the dataset. Rectified Gaussian units allow spike-and-slab sparsity in a VAE, but the discrete variables are also implicit, and their prior factorial and thus unimodal (Salimans, 2016). Graves (2016) computes VAE-like gradient approximations for mixture models, but the component models are assumed to be simple factorial distributions. In contrast, discrete VAEs generalize to powerful multimodal priors on the discrete variables, and a wider set of mappings to the continuous units. + +The generative model underlying the discrete variational autoencoder resembles a deep belief network (DBN; Hinton et al., 2006). A DBN comprises a sigmoid belief network, the top layer of which is conditioned on the visible units of an RBM. In contrast to a DBN, we use a bipartite Boltzmann machine, with both sides of the bipartite split connected to the rest of the model. Moreover, all hidden layers below the bipartite Boltzmann machine are composed of continuous latent variables with a fully autoregressive layer-wise connection architecture. Each layer $j$ receives connections from all previous layers $i < j$ , with connections from the bipartite Boltzmann machine mediated by a set of smoothing variables. However, these architectural differences are secondary to those in the gradient estimation technique. Whereas DBNs are traditionally trained by unrolling a succession of RBMs, discrete variational autoencoders use the reparameterization trick to backpropagate through the evidence lower bound. + +# 2 BACKPROPAGATING THROUGH DISCRETE LATENT VARIABLES BY ADDINGCONTINUOUS LATENT VARIABLES + +When working with an approximating posterior over discrete latent variables, we can effectively smooth the conditional-marginal CDF (defined by Equation 5 and Appendix A) by augmenting the latent representation with a set of continous random variables. The conditional-marginal CDF over the new continuous variables is invertible and its inverse is differentiable, as required in Equations 3 and 4. We redefine the generative model so that the conditional distribution of the observed variables given the latent variables only depends on the new continuous latent space. This does not alter the fundamental form of the model, or the KL term of Equation 2; rather, it can be interpreted as adding a noisy nonlinearity, like dropout (Srivastava et al., 2014) or batch normalization with a small minibatch (Ioffe & Szegedy, 2015), to each latent variable in the approximating posterior and the prior. The conceptual motivation for this approach is discussed in Appendix C. + +![](images/0e229301772a577b94a9bbe88faeff7ac8e5f98e0ae2924455834014a8b1ceff.jpg) +Figure 1: Graphical models of the smoothed approximating posterior (a) and prior (b), and the network realizing the autoencoding term of the ELBO from Equation 2 (c). Continuous latent variables $\zeta _ { i }$ are smoothed analogs of discrete latent variables $z _ { i }$ , and insulate $z$ from the observed variables $x$ in the prior (b). This facilitates the marginalization of the discrete $z$ in the autoencoding term of the ELBO, resulting in a network (c) in which all operations are deterministic and differentiable given independent stochastic input $\rho \sim U [ 0 , 1 ]$ . + +Specifically, as shown in Figure 1a, we augment the latent representation in the approximating posterior with continuous random variables $\bar { \zeta } , { } ^ { 4 }$ conditioned on the discrete latent variables $z$ of the RBM: + +$$ +\begin{array} { r l } { q ( \zeta , z | x , \phi ) = { \displaystyle r ( \zeta | z ) \cdot q ( z | x , \phi ) } , } & { \quad \mathrm { w h e r e } } \\ { r ( \zeta | z ) = \displaystyle \prod _ { i } r ( \zeta _ { i } | z _ { i } ) . } \end{array} +$$ + +The support of $r ( \zeta | z )$ for all values of $z$ must be connected, so the marginal distribution $\begin{array} { r } { q ( \zeta | x , \phi ) = \sum _ { z } r ( \zeta | z ) \cdot q ( z | x , \phi ) } \end{array}$ has a constant, connected support so long as $0 < q ( z | x , \phi ) < 1$ . We further require that $r ( \zeta | z )$ is continuous and differentiable except at the endpoints of its support, so the inverse conditional-marginal CDF of $q ( \zeta | x , \phi )$ is differentiable in Equations 3 and 4, as we discuss in Appendix A. + +As shown in Figure 1b, we correspondingly augment the prior with $\zeta$ : + +$$ +p ( \zeta , z | \theta ) = r ( \zeta | z ) \cdot p ( z | \theta ) , +$$ + +where $r ( \zeta | z )$ is the same as for the approximating posterior. Finally, we require that the conditional distribution over the observed variables only depends on $\zeta$ : + +$$ +\begin{array} { r } { p ( x | \zeta , z , \theta ) = p ( x | \zeta , \theta ) . } \end{array} +$$ + +The smoothing distribution $r ( \zeta | z )$ transforms the model into a continuous function of the distribution over $z$ , and allows us to use Equations 2 and 3 directly to obtain low-variance stochastic approximations to the gradient. + +Given this expansion, we can simplify Equations 3 and 4 by dropping the dependence on $z$ and applying Equation 16 of Appendix A, which generalizes Equation 3: + +$$ +\frac { \partial } { \partial \phi } \mathbb { E } _ { q ( \zeta , z | x , \phi ) } \left[ \log p ( x | \zeta , z , \theta ) \right] \approx \frac { 1 } { N } \sum _ { \rho \sim U ( 0 , 1 ) ^ { n } } \frac { \partial } { \partial \phi } \log p \left( x | \mathbf { F } _ { q ( \zeta | x , \phi ) } ^ { - 1 } ( \rho ) , \theta \right) . +$$ + +If the approximating posterior is factorial, then each $F _ { i }$ is an independent CDF, without conditioning or marginalization. + +As we shall demonstrate in Section 2.1, $\mathbf { F } _ { q ( \zeta | x , \phi ) } ^ { - 1 } ( \rho )$ is a function of $q ( z \ = \ 1 | x , \phi )$ , where $q ( z = 1 | x , \phi )$ is a deterministic probability value calculated by a parameterized function, such as a neural network. The autoencoder implicit in Equation 8 is shown in Figure 1c. Initially, input $x$ is passed into a deterministic feedforward network $q ( z = 1 | x , \phi )$ , for which the final nonlinearity is the logistic function. Its outpuinto the deterministic function $q$ h an independent randoto produce a sample of varia. This e , $\rho \sim U [ 0 , 1 ]$ , is passedhe original $\mathbf { F } _ { q ( \zeta | x , \phi ) } ^ { - 1 } ( \rho )$ $\zeta$ $\zeta$ input $x$ , is finally passed to $\log p \left( x | \zeta , \theta \right)$ . The expectation of this log probability with respect to $\rho$ is the autoencoding term of the VAE formalism, as in Equation 2. Moreover, conditioned on the input and the independent $\rho$ , this autoencoder is deterministic and differentiable, so backpropagation can be used to produce a low-variance, computationally-efficient approximation to the gradient. + +# 2.1 SPIKE-AND-EXPONENTIAL SMOOTHING TRANSFORMATION + +As a concrete example consistent with sparse coding, consider the spike-and-exponential transformation from binary $z$ to continuous $\zeta$ : + +$$ +\begin{array} { r l } & { r ( \zeta _ { i } | z _ { i } = 0 ) = \left\{ \begin{array} { l l } { \infty , } & { \mathrm { i f ~ } \zeta _ { i } = 0 } \\ { 0 , } & { \mathrm { o t h e r w i s e } } \end{array} \right. \quad \quad \quad F _ { r ( \zeta _ { i } | z _ { i } = 0 ) } ( \zeta ^ { \prime } ) = 1 } \\ & { r ( \zeta _ { i } | z _ { i } = 1 ) = \left\{ \begin{array} { l l } { \frac { \beta e ^ { \beta \zeta } } { e ^ { \beta } - 1 } , } & { \mathrm { i f ~ } 0 \leq \zeta _ { i } \leq 1 } \\ { 0 , } & { \mathrm { o t h e r w i s e } } \end{array} \right. \quad \quad \quad F _ { r ( \zeta _ { i } | z _ { i } = 1 ) } ( \zeta ^ { \prime } ) = \frac { e ^ { \beta \zeta } } { e ^ { \beta } - 1 } \bigg | _ { 0 } ^ { \zeta ^ { \prime } } = \frac { e ^ { \beta \zeta ^ { \prime } } - 1 } { e ^ { \beta } - 1 } } \end{array} +$$ + +where $\begin{array} { r } { F _ { p } ( \zeta ^ { \prime } ) = \int _ { - \infty } ^ { \zeta ^ { \prime } } p ( \zeta ) \cdot d \zeta } \end{array}$ is the CDF of probability distribution $p$ in the domain $[ 0 , 1 ]$ . This transformation from $z _ { i }$ to $\zeta _ { i }$ is invertible: $\zeta _ { i } = 0 \Leftrightarrow z _ { i } = 0$ , and $\zeta _ { i } > 0 \Leftrightarrow z _ { i } = 1$ almost surely.5 + +We can now find the CDF for $q ( \zeta | x , \phi )$ as a function of $q ( z = 1 | x , \phi )$ in the domain $[ 0 , 1 ]$ , marginalizing out the discrete $z$ : + +$$ +\begin{array} { r l } & { F _ { q ( \zeta | x , \phi ) } ( \zeta ^ { \prime } ) = ( 1 - q ( z = 1 | x , \phi ) ) \cdot F _ { r ( \zeta _ { i } | z _ { i } = 0 ) } ( \zeta ^ { \prime } ) + q ( z = 1 | x , \phi ) \cdot F _ { r ( \zeta _ { i } | z _ { i } = 1 ) } ( \zeta ^ { \prime } ) } \\ & { \qquad = q ( z = 1 | x , \phi ) \cdot \left( \cfrac { e ^ { \beta \zeta ^ { \prime } } - 1 } { e ^ { \beta } - 1 } - 1 \right) + 1 . } \end{array} +$$ + +To evaluate the autoencoder of Figure 1c, and through it the gradient approximation of Equation 8, we must invert the conditional-marginal CDF $F _ { q ( \zeta | x , \phi ) }$ : + +$$ +F _ { q ( \zeta \left. x , \phi \right) } ^ { - 1 } ( \rho ) = \left\{ \begin{array} { l l } { \frac { 1 } { \beta } \cdot \log \left[ \left( \frac { \rho + q - 1 } { q } \right) \cdot \left( e ^ { \beta } - 1 \right) + 1 \right] , } & { \mathrm { i f ~ } \rho \geq 1 - q } \\ { 0 , } & { \mathrm { o t h e r w i s e } } \end{array} \right. +$$ + +where we use the substitution $q ( z = 1 | x , \phi ) \to q$ to simplify notation. For all values of the independent random variable $\rho \sim U [ 0 , 1 ]$ , the function $F _ { q ( \zeta | x , \phi ) } ^ { - 1 } ( \rho )$ rectifies the input $q ( z = 1 | x , \phi )$ if $q \le 1 - \rho$ in a manner analogous to a rectified linear unit (ReLU), as shown in Figure 2a. It is also quasi-sigmoidal, in that $\bar { F ^ { - 1 } }$ is increasing but concave-down if $q > 1 - \rho$ . The effect of $\rho$ on $F ^ { - 1 }$ is qualitatively similar to that of dropout (Srivastava et al., 2014), depicted in Figure 2b, or the noise injected by batch normalization (Ioffe & Szegedy, 2015) using small minibatches, shown in Figure 2c. + +Other expansions to the continuous space are possible. In Appendix D.1, we consider the case where both $r ( \zeta _ { i } | z _ { i } = 0 )$ and $r ( \zeta _ { i } | z _ { i } = 1 )$ are linear functions of $\zeta$ ; in Appendix D.2, we develop a spikeand-slab transformation; and in Appendix $\mathrm { E }$ , we explore a spike-and-Gaussian transformation where the continuous $\zeta$ is directly dependent on the input $x$ in addition to the discrete $z$ . + +![](images/80695ecff259b01c4903d90399e9f5448ca13d773121a847a8dfc9d966acbdc2.jpg) +Figure 2: Inverse CDF of the spike-and-exponential smoothing transformation for $\rho \in \{ 0 . 2 , 0 . 5 , 0 . 8 \}$ ; $\beta = 1$ (dotted), $\beta = 3$ (solid), and $\beta = 5$ (dashed) (a). Rectified linear unit with dropout rate 0.5 (b). Shift (red) and scale (green) noise from batch normalization; with magnitude 0.3 (dashed), $- 0 . 3$ (dotted), or 0 (solid blue); before a rectified linear unit (c). In all cases, the abcissa is the input and the ordinate is the output of the effective transfer function. The novel stochastic nonlinearity $F _ { q ( \zeta | x , \phi ) } ^ { - 1 } ( \rho )$ from Figure 1c, of which (a) is an example, is qualitatively similar to the familiar stochastic nonlinearities induced by dropout (b) or batch normalization (c). + +# 3 ACCOMMODATING EXPLAINING-AWAY WITH A HIERARCHICALAPPROXIMATING POSTERIOR + +When a probabilistic model is defined in terms of a prior distribution $p ( z )$ and a conditional distribution $p ( x | z )$ , the observation of $x$ often induces strong correlations in the posterior $p ( z | x )$ due to phenomena such as explaining-away (Pearl, 1988). Moreover, we wish to use an RBM as the prior distribution (Equation 6), which itself may have strong correlations. In contrast, to maintain tractability, many variational approximations use a product of independent approximating posterior distributions (e.g., mean-field methods, but also Kingma & Welling (2014); Rezende et al. (2014)). + +To accommodate strong correlations in the posterior distribution while maintaining tractability, we introduce a hierarchy into the approximating posterior $q ( z | x )$ over the discrete latent variables. Specifically, we divide the latent variables $z$ of the RBM into disjoint groups, $z _ { 1 } , \dots , z _ { k }$ ,6 and define the approximating posterior via a directed acyclic graphical model over these groups: + +$$ +{ \begin{array} { r l } { q ( z _ { 1 } , \zeta _ { 1 } , \dots , z _ { k } , \zeta _ { k } | x , \phi ) = \displaystyle \prod _ { 1 \leq j \leq k } r ( \zeta _ { j } | z _ { j } ) \cdot q \left( z _ { j } | \zeta _ { i < j } , x , \phi \right) } & { \quad { \mathrm { w h e r e } } } \\ { q ( z _ { j } | \zeta _ { i < j } , x , \phi ) = \displaystyle { \frac { e ^ { g _ { j } \left( \zeta _ { i < j } , x , \phi \right) ^ { \top } \cdot z _ { j } } } { \prod _ { z _ { i } \in z _ { j } } \left( 1 + e ^ { g _ { z _ { i } } \left( \zeta _ { i < j } , x , \phi \right) } \right) } } , } \end{array} } +$$ + +$z _ { j } \in \{ 0 , 1 \} ^ { n }$ , and $g _ { j } ( \zeta _ { i < j } , x , \phi )$ is a parameterized function of the inputs and preceding $\zeta _ { i }$ , such as a neural network. The corresponding graphical model is depicted in Figure 3a, and the integration of such hierarchical approximating posteriors into the reparameterization trick is discussed in Appendix A. If each group $z _ { j }$ contains a single variable, this dependence structure is analogous to that of a deep autoregressive network (DARN; Gregor et al., 2014), and can represent any distribution. However, the dependence of $z _ { j }$ on the preceding discrete variables $z _ { i < j }$ is always mediated by the continuous variables $\zeta _ { i < j }$ . + +This hierarchical approximating posterior does not affect the form of the autoencoding term in Equation 8, except to increase the depth of the autoencoder, as shown in Figure 3b. The deterministic probability value $q ( z _ { j } = 1 | \zeta _ { i < j } , \bar { x } , \phi )$ of Equation 10 is parameterized, generally by a neural network, in a manner analogous to Section 2. However, the final logistic function is made explicit in Equation 10 to simplify Equation 12. For each successive layer $j$ of the autoencoder, input $x$ and all previous $\zeta _ { i < j }$ are passed into the network computing $q ( z = 1 | \zeta _ { i < j } , x , \phi )$ . Its output $q _ { j }$ , along with an independent random variable $\rho \sim U [ 0 , 1 ]$ , is passed to the deterministic function F− q(ζj |ζi