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+[
+    {
+        "type": "text",
+        "text": "EMPIRICAL ANALYSIS OF UNLABELED ENTITY PROBLEM IN NAMED ENTITY RECOGNITION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Yangming Li, Lemao Liu, & Shuming Shi Tencent AI Lab {newmanli,redmondliu,shumingshi}@tencent.com ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In many scenarios, named entity recognition (NER) models severely suffer from unlabeled entity problem, where the entities of a sentence may not be fully annotated. Through empirical studies performed on synthetic datasets, we find two causes of performance degradation. One is the reduction of annotated entities and the other is treating unlabeled entities as negative instances. The first cause has less impact than the second one and can be mitigated by adopting pretraining language models. The second cause seriously misguides a model in training and greatly affects its performances. Based on the above observations, we propose a general approach, which can almost eliminate the misguidance brought by unlabeled entities. The key idea is to use negative sampling that, to a large extent, avoids training NER models with unlabeled entities. Experiments on synthetic datasets and real-world datasets show that our model is robust to unlabeled entity problem and surpasses prior baselines. On well-annotated datasets, our model is competitive with the state-of-the-art method1. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Named entity recognition (NER) is an important task in information extraction. Previous methods typically cast it into a sequence labeling problem by adopting IOB tagging scheme (Mesnil et al., 2015; Huang et al., 2015; Ma & Hovy, 2016; Akbik et al., 2018; Qin et al., 2019). A representative model is Bi-LSTM CRF (Lample et al., 2016). The great success achieved by these methods benefits from massive correctly labeled data. However, in some real scenarios, not all the entities in the training corpus are annotated. For example, in some NER tasks (Ling & Weld, 2012), the datasets contain too many entity types or a mention may be associated with multiple labels. Since manual annotation on this condition is too hard, some entities are inevitably neglected by human annotators. Situations in distantly supervised NER (Ren et al., 2015; Fries et al., 2017) are even more serious. To reduce handcraft annotation, distant supervision (Mintz et al., 2009) is applied to automatically produce labeled data. As a result, large amounts of entities in the corpus are missed due to the limited coverage of knowledge resources. We refer this to unlabeled entity problem, which largely degrades performances of NER models. ",
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+    },
+    {
+        "type": "text",
+        "text": "There are several approaches used in prior works to alleviate this problem. Fuzzy CRF and AutoNER (Shang et al., 2018b) allow models to learn from the phrases that may be potential entities. However, since these phrases are obtained through a distantly supervised phrase mining method (Shang et al., 2018a), many unlabeled entities in the training data may still not be recalled. In the context of only resorting to unlabeled corpora and an entity ontology, Mayhew et al. (2019); Peng et al. (2019) employ positive-unlabeled (PU) learning (Li & Liu, 2005) to unbiasedly and consistently estimate the task loss. In implementations, they build distinct binary classifiers for different labels. Nevertheless, the unlabeled entities still impact the classifiers of the corresponding entity types and, importantly, the model can’t disambiguate neighboring entities. Partial CRF (Tsuboi et al., 2008) is an extension of commonly used CRF (Lafferty et al., 2001) that supports learning from incomplete annotations. Yang et al. (2018); Nooralahzadeh et al. (2019); Jie et al. (2019) use it to circumvent training with false negatives. However, as fully annotated corpora are still required to get ground truth training negatives, this approach is not applicable to the situations where little or even no high-quality data is available. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, our goal is to study what are the impacts of unlabeled entity problem on the models and how to effectively eliminate them. Initially, we construct some synthetic datasets and introduce degradation rates. The datasets are constructed by randomly removing the annotated named entities in well-annotated datasets, e.g., CoNLL-2003 (Sang & De Meulder, 2003), with different probabilities. The degradation rates measure how severe an impact of unlabeled entity problem degrades the performances of models. Extensive studies are investigated on synthetic datasets. We find two causes: the reduction of annotated entities and treating unlabeled entities as negative instances. The first cause is obvious but has far fewer influences than the second one. Besides, it can be mitigated well by using a pretraining language model, like BERT (Devlin et al., 2019)), as the sentence encoder. The second cause seriously misleads the models in training and exerts a great negative impact on their performances. Even in less severe cases, it can sharply reduce the F1 score by about $2 0 \\%$ . Based on the above observations, we propose a novel method that is capable of eliminating the misguidance of unlabeled entities in training. The core idea is to apply negative sampling that avoids training NER models with unlabeled entities. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Extensive experiments have been conducted to verify the effectiveness of our approach. Studies on synthetic datasets and real-world datasets (e.g., EC) show that our model well handles unlabeled entities and notably surpasses prior baselines. On well-annotated datasets (e.g., CoNLL-2003), our model is competitive with the state-of-the-art method. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we formally define the unlabeled entity problem and briefly describe a strong baseline, BERT Tagging (Devlin et al., 2019), used in empirical studies. ",
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+    {
+        "type": "text",
+        "text": "2.1 UNLABELED ENTITY PROBLEM ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We denote an input sentence as $\\mathbf { x } = [ x _ { 1 } , x _ { 2 } , \\cdots , x _ { n } ]$ and the annotated named entity set as ${ \\bf y } =$ $\\{ y _ { 1 } , y _ { 2 } , \\cdots , y _ { m } \\}$ . $n$ is the sentence length and $m$ is the amount of entities. Each member $y _ { k }$ of set $\\mathbf { y }$ is a tuple $( i _ { k } , j _ { k } , l _ { k } )$ . $( i _ { k } , j _ { k } )$ is the span of an entity which corresponds to the phrase $\\mathbf { x } _ { i _ { k } , j _ { k } } = [ x _ { i _ { k } } , x _ { i _ { k } + 1 } , \\cdot \\cdot \\cdot , x _ { j _ { k } } ]$ and $l _ { k }$ is its label. The unlabeled entity problem is defined as, due to the limited coverage of machine annotator or the negligence of human annotator, some ground truth entities $\\widetilde { \\mathbf { y } }$ of the sentence $\\mathbf { x }$ are not covered by annotated entity set y. ",
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+    {
+        "type": "text",
+        "text": "For instance, given a sentence $\\mathbf { x } = [ \\mathrm { J a c k }$ , and, Mary, are, from, New, York] and a labeled entity set $\\mathbf { y } = \\{ ( 1 , 1 , \\bar { \\mathrm { P E R } } ) \\}$ , unlabeled entity problem is that some entities, like $( \\mathrm { 6 , \\dot { 7 } , L O C ) }$ , are neglected by annotators. These unlabeled entities are denoted as $\\widetilde { \\mathbf { y } } = \\{ ( 3 , 3 , \\mathrm { P E R } ) , ( 6 , 7 , \\mathrm { L O C } ) \\}$ . ",
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+    {
+        "type": "text",
+        "text": "2.2 BERT TAGGING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "BERT Tagging is present in Devlin et al. (2019), which adopts IOB tagging scheme, where each token $x _ { i }$ in a sentence $\\mathbf { x }$ is labeled with a fine-grained tag, such as B-ORG, I-LOC, or O. Formally, its output is a $n$ -length label sequence $\\mathbf { z } = [ z _ { 1 } , \\bar { z } _ { 2 } , \\cdot \\cdot \\cdot , z _ { n } ]$ . ",
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+    {
+        "type": "text",
+        "text": "Formally, BERT tagging firstly uses BERT to get the representation $\\mathbf { h } _ { i }$ for every token $x _ { i }$ ",
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/c961714e573dc7977ba0cd4ba2bb668969284ff8627e9b7bcbe40440ee450c02.jpg",
+        "text": "$$\n[ \\mathbf { h } _ { 1 } , \\mathbf { h } _ { 2 } , \\cdot \\cdot \\cdot \\mathbf { \\nabla } , \\mathbf { h } _ { n } ] = \\mathrm { B E R T } ( \\mathbf { x } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Then, the label distribution $\\mathbf { q } _ { i }$ is computed as $\\mathrm { S o f t m a x } ( \\mathbf { W } \\mathbf { h } _ { i } )$ . In training, the loss is induced as $\\begin{array} { r } { \\sum _ { 1 \\leq i \\leq n } - \\log { \\mathbf q } _ { i } [ z _ { i } ] } \\end{array}$ . At test time, it obtains the label for each token $x _ { i }$ by arg max $\\mathbf { q } _ { i }$ . ",
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+    {
+        "type": "text",
+        "text": "3 EMPIRICAL STUDIES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "To understand the impacts of unlabeled entity problem, we conduct empirical studies over multiple synthetic datasets, different methods, and various metrics. ",
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+        "type": "image",
+        "img_path": "images/12a49343baa487cabf1df8065948b51e0e3d6ad645b364ce37887dce590a4bd5.jpg",
+        "image_caption": [
+            "Figure 1: The empirical studies conducted on CoNLL-2003 dataset. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/08291b5c5ad5352b823484bdb00897ef3fcc47c354ce567ed299dfa0de2d61d6.jpg",
+        "image_caption": [
+            "Figure 2: The empirical studies investigated on OntoNotes 5.0 dataset. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3.1 PREPARATIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Synthetic Datasets. We use synthetic datasets to simulate poorly-annotated datasets that contain unlabeled entities. They are obtained by randomly removing the labeled entities of well-annotated datasets with different masking probabilities $p$ . The material datasets are CoNLL-2003 (Sang & De Meulder, 2003) and OntoNotes 5.0 (Pradhan et al., 2013). The probabilities $p$ are respectively set as $0 . 0 , 0 . 1 , 0 . 2 , \\cdots , 0 . 9$ . In this way, $2 \\times 1 0$ synthetic datasets are constructed. ",
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+    {
+        "type": "text",
+        "text": "Methods. We adopt two models. One of them is BERT Tagging, which has long been regarded as a strong baseline. The other is LSTM Tagging that replaces the original encoder (i.e., BERT) of BERT Tagging with LSTM (Hochreiter & Schmidhuber, 1997). We use it to study the effect of using pretraining language model. To explore the negative impact brought by unlabeled entities in training, we present an adjusted training loss for above two models: ",
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+        "type": "equation",
+        "img_path": "images/62142bdd79e7d730e26f1895594f377e931d2dc8d985eeb699e16ef06fc134d1.jpg",
+        "text": "$$\n\\Big ( \\sum _ { 1 \\leq i \\leq n } - \\log \\mathbf { q } _ { i } [ z _ { i } ] \\Big ) - \\Big ( \\sum _ { ( i ^ { \\prime } , j ^ { \\prime } , l ^ { \\prime } ) \\in \\widetilde { \\mathbf { y } } } \\sum _ { i ^ { \\prime } \\leq k \\leq j ^ { \\prime } } - \\log \\mathbf { q } _ { k } [ z _ { k } ] \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The idea here is to remove the incorrect loss incurred by unlabeled entities. Note that missed entity set $\\widetilde { \\mathbf { y } }$ is reachable in synthetic datasets but unknown in real-world datasets. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Metrics. Following prior works, the F1 scores of models are tested by using conlleval script2. We also design two degradation rates to measure the different impacts of unlabeled entity problem. One is erosion rate $\\alpha _ { p }$ and the other is misguidance rate $\\beta _ { p }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/941c1157c8e73414e0f0a7be7d6eab9b3c3ec566619d6ed410b4f34ee5a08568.jpg",
+        "text": "$$\n\\alpha _ { p } = \\frac { f _ { 0 } ^ { a } - f _ { p } ^ { a } } { f _ { 0 } ^ { a } } , \\beta _ { p } = \\frac { f _ { p } ^ { a } - f _ { p } } { f _ { p } ^ { a } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For a synthetic dataset with the masking probability being $p$ , $f _ { p }$ and $f _ { p } ^ { a }$ are the F1 scores of a model and its adjusted version, respectively. Note that $f _ { 0 } ^ { \\alpha }$ corresponds to $p \\bar { = } 0$ . Erosion rate $\\alpha _ { p }$ measures how severely the reduction of annotated entities degrades the F1 scores of a model. Misguidance rate $\\beta _ { p }$ measures how seriously unlabeled entities misguide the model in training. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/658021a70f774e2ead4cd1215d61c3ce7f148fa457e027ccd3049cf439fe8cf4.jpg",
+        "image_caption": [
+            "Figure 3: This demonstrates how our model scores possible entities. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 OVERALL ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "The left parts of Fig. 1 and Fig. 2 show the results of empirical studies, where we evaluate the F1 scores of BERT Tagging and LSTM Tagging on 20 synthetic datasets. From them, we can draw the following observations. Firstly, the significant downward trends of solid lines confirm the fact that NER models severely suffer from unlabeled entity problem. For example, by setting the masking probability as 0.4, the performance of LSTM Tagging decreases by $3 3 . 0 1 \\%$ on CoNLL2003 and $1 9 . 5 8 \\%$ on OntoNotes 5.0. Secondly, in contrast, the dashed lines change very slowly, indicating that the models with adjusted training loss (see Eq. (2)) are much less influenced by the issue. For instance, when masking probability is 0.7, adopting adjusted loss preserves the F1 scores of BERT Tagging by $4 1 . 0 4 \\%$ on CoNLL-2003 and $5 7 . 3 8 \\%$ on OntoNotes 5.0. Lastly, for high masking probabilities, even though the negative impact of unlabeled entities is eliminated by adjusting the training loss, the performance still declines to a certain extent. For example, when masking probability is set as 0.8, the F1 scores of adjusted LSTM Tagging decrease by $3 1 . 6 4 \\%$ on CoNLL-2003 and $1 7 . 2 2 \\%$ on OntoNotes 5.0. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Reduction of Annotated Entities. From the last observation, we can infer that a cause of performance degradation is the reduction of annotated entities. In the middle parts of Fig. 1 and Fig. 2, we plot the change of erosion rates $\\alpha _ { p }$ (see Eq. (2)) with respect to masking probabilities. We can see that its impact is not very serious when in low masking probabilities but can’t be neglected when in high ones. Besides, using pre-training language models greatly mitigates the issue. As an example, when the probability is 0.8, on both CoNLL-2003 and OntoNotes 5.0, the erosion rates of adjusted BERT Tagging are only about half of those of adjusted LSTM Tagging. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Misguidance of Unlabeled Entities. From the last two observations, we can conclude that the primary cause is treating unlabeled entities as negative instances, which severely misleads the models during training. To better understand it, in the right parts of Fig. 1 and Fig. 2, we plot the change of misguidance rates $\\beta _ { p }$ (see Eq. (3)) with masking probabilities. These rates are essentially the percentage decreases of F1 scores. From them, we can see that the impact of misguidance is very much serious even when in low masking probabilities. ",
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+    {
+        "type": "text",
+        "text": "4 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Motivated by Sec. 3.2, we present a model that is robust to unlabeled entities. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "4.1 SCORING MODEL WITH BERT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Based on the findings in Sec. 3.2, we use BERT as the default encoder to mitigate the reduction of annotated entities. Specifically, given a sentence $\\mathbf { x }$ , we firstly obtain the token representations $\\mathbf { h } _ { i }$ with Eq. (1). Then, we get the representation for every phrase $\\mathbf { x } _ { i , j }$ as ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/6be0fc3fb3b2de94cd5342cc9e3649800a90bb30c0ecc564afea7088e5e70efb.jpg",
+        "text": "$$\n\\mathbf { s } _ { i , j } = \\mathbf { h } _ { i } \\oplus \\mathbf { h } _ { j } \\oplus \\left( \\mathbf { h } _ { i } - \\mathbf { h } _ { j } \\right) \\oplus \\big ( \\mathbf { h } _ { i } \\odot \\mathbf { h } _ { j } \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\oplus$ is column-wise vector concatenation and $\\odot$ is element-wise vector product. The design here is mainly inspired by Chen et al. (2017). ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Finally, multi-layer perceptron (MLP) computes the label distribution $\\mathbf { o } _ { i , j }$ for a span $( i , j )$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8e9c9bf8c8db94670a851faa0f3072ee5aad72350799370e6082c9047eb9aab2.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { o } _ { i , j } = \\mathrm { S o f t m a x } ( \\mathbf { U } \\operatorname { t a n h } ( \\mathbf { V } \\mathbf { s } _ { i , j } ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The term $\\mathbf { o } _ { i , j } [ l ]$ is the predicted score for an entity $( i , j , l )$ . ",
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+    {
+        "type": "text",
+        "text": "4.2 TRAINING VIA NEGATIVE SAMPLING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "From Sec. 3.2, we know that regarding all the unlabeled spans as negative instances certainly degrades the performances of models, since some of them may be missed entities. Our solution to this issue is negative sampling. Specifically, we randomly sample a small subset of unlabeled spans as the negative instances to induce the training loss. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given the annotated entity set $\\mathbf { y }$ , we firstly get all the negative instance candidates as ",
+        "bbox": [
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+            263,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/600228e853b0ebb745d08d689ac4ef22e839e9c90a915f4a286c8eb8c7857f27.jpg",
+        "text": "$$\n\\{ ( i , j , 0 ) \\mid ( i , j , l ) \\not \\in { \\bf y } , 1 \\leq i \\leq j \\leq n , l \\in \\mathcal { L } \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            281,
+            651,
+            299
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L }$ is the label space and O is the label for non-entity spans. ",
+        "bbox": [
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+            303,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Then, we uniformly sample a subset $\\widehat { \\mathbf { y } }$ from the whole candidate set. The size of sample set $\\widehat { \\mathbf { y } }$ is $\\lceil \\lambda * n \\rceil , 0 < \\lambda < 1$ , where $\\left\\lceil \\right\\rceil$ b is the ceiling function. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Ultimately, a span-level cross entropy loss used for training is incurred as ",
+        "bbox": [
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+            359,
+            655,
+            375
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a669fe3e6ebf7be53fcbfbc94831dad27d7fb90341525c73ef1a70d1943f00ea.jpg",
+        "text": "$$\n\\Big ( \\sum _ { ( i , j , l ) \\in \\mathbf { y } } - \\log ( \\mathbf { o } _ { i , j } [ l ] ) \\Big ) + \\Big ( \\sum _ { ( i ^ { \\prime } , j ^ { \\prime } , l ^ { \\prime } ) \\in \\widehat { \\mathbf { y } } } - \\log \\bigl ( \\mathbf { o } _ { i ^ { \\prime } , j ^ { \\prime } } [ l ^ { \\prime } ] \\bigr ) \\Big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            377,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Negative sampling incorporates some randomness into the training loss, which reduces the risk of training a NER model with unlabeled entities. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 INFERENCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            464,
+            295,
+            478
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "At test time, firstly, the label for every span $( i , j )$ is obtained by $\\mathrm { \\ a r g m a x } _ { l } \\mathbf { o } _ { i , j } [ l ]$ . Then, we select the ones whose label $l$ is not $\\mathrm { O }$ as predicted entities. When the spans of inferred entities intersects, we preserve the one with the highest predicted score and discard the others. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The time complexity of inference is $\\mathcal { O } ( n ^ { 2 } )$ , which is majorly contributed by the span selection procedure. While this seems a bit higher compared with our counterparts, we find that, in practical use, its running time is far less than that of the forward computation of neural networks. The algorithm for inference is greedy yet effective. In experiments, we find that the probability of our heuristic selecting a wrong labeled span when resolving the span conflict is very low. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We show that, through negative sampling, the probability of not treating a specific missed entity in a $n$ -length sentence as the negative instance is larger than $\\textstyle 1 - { \\frac { 2 } { n - 3 } }$ : ",
+        "bbox": [
+            173,
+            660,
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+            690
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f52bee685d70067e3de2f4a3da913fa7b00cdf3c703b783356bed78a25d5beff.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\prod _ { 0 \\le i < \\lceil \\lambda n \\rceil } \\left( 1 - \\frac { 1 } { \\frac { n ( n + 1 ) } { 2 } - m - i } \\right) > \\left( 1 - \\frac { 1 } { \\frac { n ( n + 1 ) } { 2 } - m - \\lceil \\lambda n \\rceil } \\right) ^ { \\lceil \\lambda n \\rceil } } } \\\\ { { \\displaystyle \\qquad > \\left( 1 - \\frac { 1 } { \\frac { n ( n + 1 ) } { 2 } - n - n } \\right) ^ { n } \\ge \\left( 1 - n * \\frac { 1 } { \\frac { n ( n + 1 ) } { 2 } - n - n } \\right) = 1 - \\frac { 2 } { n - 3 } } } \\end{array} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            695,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Herand e n(n+1)2 − m is the amount of negative candidates. Besides, we use the facts, λ < 1, m ≤ n, $( 1 - \\bar { z } ) ^ { n } \\geq 1 - n z , 0 \\leq z \\leq 1$ , during the derivation. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that the above bound is only applicable to the special case where there is just one unlabeled entity in a sentence. We remain the strict proof for general cases to future work. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            864,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We have conducted extensive experiments on multiple datasets to verify the effectiveness of our method. Studies on synthetic datasets show that our model can almost eliminate the misguidance ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/896548e8366a57be72773dab37bbbf189660f66a0eeec4ef56fc0cc19f4efa3c.jpg",
+        "table_caption": [
+            "Table 1: The experiment results on two synthetic datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Masking Prob.</td><td colspan=\"2\">CoNLL-2003</td><td colspan=\"2\">OntoNotes 5.0</td></tr><tr><td>BERT Tagging</td><td>Our Model</td><td>BERT Tagging</td><td>Our Model</td></tr><tr><td>0.1</td><td>90.71</td><td>91.37</td><td>87.69</td><td>89.20</td></tr><tr><td>0.2</td><td>89.57</td><td>91.25</td><td>86.86</td><td>89.15</td></tr><tr><td>0.3</td><td>88.95</td><td>90.53</td><td>84.75</td><td>88.73</td></tr><tr><td>0.4</td><td>82.94</td><td>89.73</td><td>82.55</td><td>88.20</td></tr><tr><td>0.5</td><td>78.99</td><td>89.22</td><td>71.07</td><td>88.17</td></tr><tr><td>0.6</td><td>63.84</td><td>87.65</td><td>58.17</td><td>87.53</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b4b22a396a33c72df0cc060a52441c49a39b0dad956325e34b837146487fb67.jpg",
+        "image_caption": [
+            "Figure 4: The misguidance rates of BERT Tagging and our model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "brought by unlabeled entities in training. On real-world datasets, our model has notably outperformed prior baselines and achieved the state-of-the-art performances. On well-annotated datasets, our model is competitive to current state-of-the-art method. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6.1 SETTINGS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The setup of synthetic datasets and well-annotated datasets are the same as what we describe in Sec. 3.1. For real-world datasets, we use EC and NEWS, both of which are collected by Yang et al. (2018). EC is in e-commerce domain, which has 5 entity types: Brand, Product, Model, Material, and Specification. The data contains 2400 sentences tagged by human annotators and are divided into three parts: 1200 for training, 400 for dev, and 800 for testing. Yang et al. (2018) also construct an entity dictionary of size 927 and apply distant supervision on a raw corpus to obtain additional 2500 sentences for training. NEWS is from MSRA dataset (Levow, 2006). Yang et al. (2018) only adopt the PERSON entity type. Training data of size 3000, dev data of size 3328, and testing data of size 3186 are all sampled from MSRA. They collect an entity dictionary of size 71664 and perform distant supervision on the rest data to obtain extra 3722 training cases by using the dictionary. Both EC and NEWS contain a large amount of incompletely annotated sentences, and hence naturally suffer from the unlabeled entity problem. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We adopt the same hyper-parameter configurations of neural networks for all the datasets. L2 regularization and dropout ratio are respectively set as $1 \\times 1 0 ^ { - 5 }$ and 0.4 for reducing overfit. The dimension of scoring layers is 256. Ratio $\\lambda$ is set as 0.35. When the sentence encoder is LSTM, we set the hidden dimension as 512 and use pretrained word embeddings (Pennington et al., 2014; Song et al., 2018) to initialize word representations. We utilize Adam (Kingma & Ba, 2014) as the optimization algorithm and adopt the suggested hyper-parameters. At evaluation time, we convert the predictions of our models into IOB format and use conlleval script to compute the F1 score. In all the experiments, the improvements of our models over the baselines are statistically significant with rejection probabilities smaller than 0.05. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/304136abdfef2c8890feb498e8510b10440ce32ef5461b4934799993b46b47e7.jpg",
+        "table_caption": [
+            "Table 2: The experiment results on two well-annotated datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>CoNLL-2003</td><td>OntoNotes 5.0</td></tr><tr><td>FlairEmbedding (Akbik etal., 2018)</td><td>93.09</td><td>89.3</td></tr><tr><td>BERT-MRC(Li etal.,2020a)</td><td>93.04</td><td>91.11</td></tr><tr><td>HCR w/BERT (Luo et al., 2020)</td><td>93.37</td><td>90.30</td></tr><tr><td>BERT-Biaffine Model (Yu etal., 2020)</td><td>93.5</td><td>91.3</td></tr><tr><td>Our Model</td><td>93.42</td><td>90.59</td></tr></table>",
+        "bbox": [
+            250,
+            101,
+            748,
+            189
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/f3ad80bc2ae883bdb32a4b1decb095e7000f2acb295a789c2619fd4feb2dc0e0.jpg",
+        "table_caption": [
+            "Table 3: The experiment results on two real-world datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Method</td><td>EC</td><td>NEWS</td></tr><tr><td rowspan=\"6\">Yang et al. (2018)</td><td>String Matching via Ontology</td><td>44.02</td><td>47.75</td></tr><tr><td>BiLSTM+CRF</td><td>54.59</td><td>69.09</td></tr><tr><td>BiLSTM+CRFw/RL</td><td>56.23</td><td>73.19</td></tr><tr><td>BiLSTM+PartialCRF</td><td>60.08</td><td>78.38</td></tr><tr><td>BiLSTM+Partial CRF w/RL</td><td>61.45</td><td>79.22</td></tr><tr><td>WeightedPartial CRF</td><td>61.75</td><td>78.64</td></tr><tr><td>Jie et al. (2019) Nooralahzadeh et al. (2019)</td><td>BiLSTM+Partial CRFw/RL</td><td>63.56</td><td>80.04</td></tr><tr><td rowspan=\"2\">This Work</td><td>OurModel</td><td>66.17</td><td>85.39</td></tr><tr><td>OurModel w/o BERT,w/BiLSTM</td><td>64.68</td><td>82.11</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.2 RESULTS ON SYNTHETIC DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, our model is compared with BERT Tagging on the synthetic datasets of the masking probabilities being $0 . 1 , 0 . 2 , \\cdots , 0 . 6$ . From Table 1, we can get two conclusions. Firstly, our model significantly outperforms BERT Tagging, especially in high masking probabilities. For example, on CoNLL-2003, our F1 scores outnumber those of BERT Tagging by $1 . 8 8 \\%$ when the probability is 0.2 and $2 7 . 1 6 \\%$ when the probability is 0.6. Secondly, our model is very robust to the unlabeled entity problem. When increasing the masking probability from 0.1 to 0.5, the results of our model only decrease by $2 . 3 5 \\%$ on CoNLL-2003 and $1 . 9 1 \\%$ on OntoNotes 5.0. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Fig. 4 demonstrates the misguidance rate comparisons between BERT Tagging and our models. The way to adjust our model is reformulating Eq. (7) by defining the negative term via $\\{ ( i , j , \\mathrm { O } ) \\mid \\forall l :$ $( i , j , l ) \\not \\in { \\bf y } \\cup \\widetilde { { \\bf y } } \\}$ rather than the negatively sampled $\\hat { \\mathbf { y } }$ . The idea here is to avoid the unlabeled entities being sampled. From Fig. 4, we can discover that, in all masking probabilities, the misguidance rates of our model are far smaller than those of BERT Tagging and are consistently lower than $2 . 5 0 \\%$ . These indicate that, in training, our model indeed eliminates the misguidance brought by unlabeled entities to some extent. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.3 RESULTS ON FULLY ANNOTATED DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We additionally apply our model with negative sampling on the well-annotated datasets where the issue of incomplete entity annotation is not serious. As shown in Table 2, the F1 scores of our model are very close to current best results. Our model slightly underperforms BERT-Biaffine Model by only $0 . { \\dot { 0 } } 9 \\%$ on CoNLL-2003 and $0 . 7 8 \\%$ on OntoNotes 5.0. Besides, our model surpasses many other strong baselines. On OntoNotes 5.0, our model outperforms HCR w/ BERT by $0 . 3 2 \\%$ and Flair Embedding by $1 . 4 4 \\%$ . On CoNLL-2003, the improvements of F1 scores are $0 . 4 1 \\%$ over BERT-MRC and $0 . 3 5 \\%$ over Flair Embedding. All these results indicate that our model is still very effective when applied to high-quality data. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.4 RESULTS ON REAL-WORLD DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For two real-world datasets, a large portion of training data is obtained via distant supervision. As stated in Yang et al. (2018), the F1 scores of string matching through an entity dictionary are notably declined in terms of the low recall scores, although its precision scores are higher than those of other methods. Therefore, unlabeled entity problem is serious in the datasets. As shown in Table 3, the baselines come from three works (Yang et al., 2018; Nooralahzadeh et al., 2019; Jie et al., ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/510b8a385f05dd345a96722cf08c743e4b02d0cb87175dd544c60abfbfdfe26a.jpg",
+        "image_caption": [
+            "Figure 5: The results of our models with different ratio $\\lambda$ on synthetic datasets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            99,
+            774,
+            294
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/bd381a28080422592c6aa176663189bfd8963b610066e91d92997d55f113fcd3.jpg",
+        "table_caption": [
+            "Table 4: The PCCs between F1 score and degradation rates. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Metric</td><td colspan=\"2\">CoNLL-2003</td><td colspan=\"2\">OntoNotes 5.0</td></tr><tr><td>BERTTagging</td><td>LSTMTagging</td><td>BERTTagging</td><td>LSTM Tagging</td></tr><tr><td>Erosion Rate αp</td><td>-0.94</td><td>-0.90</td><td>-0.85</td><td>-0.82</td></tr><tr><td>Misguidance Rate βp</td><td>-1.00</td><td>-0.96</td><td>-1.00</td><td>-0.98</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "2019). Yang et al. (2018) use Partial CRF to circumvent all possible unlabeled entities and utilize reinforcement learning (RL) to adaptively skip noisy annotation. Jie et al. (2019) and Nooralahzadeh et al. (2019) respectively improve Partial CRF and the policy of RL. All the F1 scores of baselines are copied from Yang et al. (2018); Nooralahzadeh et al. (2019), except for that of Weighted Partial CRF, which is obtained by rerunning its open-source code3. ",
+        "bbox": [
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+            452,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our model has significantly outperformed prior baselines and obtained new state-of-the-art results. Compared with prior best model (Nooralahzadeh et al., 2019), we achieve the improvements of $3 . 9 4 \\%$ on EC and $6 . 2 7 \\%$ on NEWS. Compared with strong baseline, BiLSTM $^ +$ Partial CRF, the increases of F1 scores are $9 . 2 0 \\%$ and $8 . 2 \\bar { 1 } \\%$ . To make fair comparisons, we replace BERT with LSTM. Even so, we still outperform (Nooralahzadeh et al., 2019) by $1 . 7 6 \\%$ on EC and $2 . 5 2 \\%$ on NEWS. All these strongly confirm the effectiveness of our model. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.5 RATIO $\\lambda$ IN NEGATIVE SAMPLING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, setting ratio $\\lambda$ (see Sec. 4.2) as too large values or too small values both are inappropriate. Large ratios increase the risks of training negatives containing unlabeled entities. Small ratios reduce the number of negative instances used for training, leading to underfitting. Fig. 5 shows the experiments on some synthetic datasets with the ratio $\\lambda$ of our method being $0 . 1 , 0 . 2 , \\cdots , 0 . 9$ . From it, we can see that all the score curves are roughly arched, which verifies our intuition. Besides, we find that $0 . 3 < \\lambda < 0 . 4$ performs well in all the cases. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.6 VALIDITY OF DEGRADATION RATES ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As Table 4 shows, we use Pearson’s correlation coefficient (PCC) to measure the statistical correlations between degradation rates (e.g., misguidance rate $\\beta _ { p }$ ) and the F1 score. We can see that the correlation scores are generally close to $- 1$ . For example, for LSTM Tagging, on the synthetic datasets built from CoNLL-2003, the correlation score of erosion rate $\\alpha _ { p }$ is $- 0 . 9 0$ and that of misguidance rate $\\beta _ { p }$ is $- 0 . 9 6$ . The results indicate that not only degradation rates quantify specific impacts of unlabeled entities but also their negative values change synchronously with the F1 score. We conclude that degradation rates are appropriate metrics for evaluation. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            344,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "NER is a classical task in information extraction. Previous works commonly treat it as a sequence labeling problem by using IOB tagging scheme (Huang et al., 2015; Akbik et al., 2018; Luo et al., 2020; Li et al., 2020b;c). Each word in the sentence is labeled as B-tag if it is the beginning of an entity, I-tag if it’s inside but not the first one within the entity, or O otherwise. This approach is extensively studied in prior works. For example, Akbik et al. (2018) propose Flair Embedding that pretrains character embedding in large corpora and uses it rather than token representations to represent a sentence. Recently, there is a growing interest in span-based models (Li et al., 2020a; Yu et al., 2020). They treat the spans, instead of single words, as the basic units for labeling. For example, Li et al. (2020a) present BERT-MRC that regards NER as a MRC task, where named entities are extracted as retrieving answer spans. Span-based models are also prevalent in language modeling (Li et al., 2020d), syntactic analysis (Stern et al., 2017), etc. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In some practical applications (e.g., fine-grained NER (Zhang et al., 2020)), NER models are faced with unlabeled entity problem, where the unlabeled entities seriously degrade the performances of models. Several approaches to this issue have been proposed. Fuzzy CRF and AutoNER (Shang et al., 2018b) allow learning from high-quality phrases. However, since these phrases are obtained through distant supervision, the unlabeled entities in the corpora may still be missed. PU learning (Peng et al., 2019; Mayhew et al., 2019) unbiasedly and consistently estimates the training loss. Nevertheless, the unlabeled entities still impact the classifiers of the corresponding entity types and, importantly, the model can’t disambiguate neighboring entities. Partial CRF (Yang et al., 2018; Jie et al., 2019) supports learning from incomplete annotations. However, because fully annotated corpora are still needed to training models with true negative instances, this type of approach is not applicable to the situations where no high-quality data is available. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "8 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            469,
+            318,
+            486
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we study what are the impacts of unlabeled entities on NER models and how to effectively eliminate them. Through empirical studies performed on synthetic datasets, we find two causes: the reduction of annotated entities and treating unlabeled entities as training negatives. The first cause has fewer influences than the second one and can be mitigated by adopting pretraining language models. The second cause seriously misleads the models in training and greatly affects their performances. Based on the above observations, we propose a novel method that is capable of eliminating the misguidance of unlabeled entities during training. The core idea is to apply negative sampling that avoids training NER models with unlabeled entities. Experiments on synthetic datasets and real-world datasets demonstrate that our model handles unlabeled entities well and significantly outperforms previous baselines. On well-annotated datasets, our model is competitive with the existing state-of-the-art approach. ",
+        "bbox": [
+            174,
+            502,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            678,
+            285,
+            693
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alan Akbik, Duncan Blythe, and Roland Vollgraf. Contextual string embeddings for sequence labeling. In Proceedings of the 27th International Conference on Computational Linguistics, pp. 1638–1649, 2018. ",
+        "bbox": [
+            176,
+            702,
+            823,
+            744
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Qian Chen, Xiaodan Zhu, Zhen-Hua Ling, Si Wei, Hui Jiang, and Diana Inkpen. Enhanced LSTM for natural language inference. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 1657–1668, Vancouver, Canada, July 2017. Association for Computational Linguistics. doi: 10.18653/v1/P17-1152. URL https: //www.aclweb.org/anthology/P17-1152. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            827
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pp. 4171–4186, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics. doi: 10.18653/v1/N19-1423. URL https: //www.aclweb.org/anthology/N19-1423. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jason Fries, Sen Wu, Alex Ratner, and Christopher Re. Swellshark: A generative model for biomed-´ ical named entity recognition without labeled data. arXiv preprint arXiv:1704.06360, 2017. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8): 1735–1780, 1997. ",
+        "bbox": [
+            173,
+            143,
+            823,
+            171
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhiheng Huang, Wei Xu, and Kai Yu. Bidirectional lstm-crf models for sequence tagging. arXiv preprint arXiv:1508.01991, 2015. ",
+        "bbox": [
+            173,
+            183,
+            821,
+            212
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhanming Jie, Pengjun Xie, Wei Lu, Ruixue Ding, and Linlin Li. Better modeling of incomplete annotations for named entity recognition. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pp. 729–734, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics. doi: 10.18653/v1/N19-1079. URL https://www.aclweb.org/anthology/N19-1079. ",
+        "bbox": [
+            173,
+            222,
+            825,
+            308
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. ",
+        "bbox": [
+            173,
+            318,
+            823,
+            347
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "John Lafferty, Andrew McCallum, and Fernando CN Pereira. Conditional random fields: Probabilistic models for segmenting and labeling sequence data. 2001. ",
+        "bbox": [
+            173,
+            357,
+            821,
+            387
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Guillaume Lample, Miguel Ballesteros, Sandeep Subramanian, Kazuya Kawakami, and Chris Dyer. Neural architectures for named entity recognition. In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 260–270, San Diego, California, June 2016. Association for Computational Linguistics. doi: 10.18653/v1/N16-1030. URL https://www.aclweb.org/anthology/ N16-1030. ",
+        "bbox": [
+            173,
+            397,
+            825,
+            481
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gina-Anne Levow. The third international chinese language processing bakeoff: Word segmentation and named entity recognition. In Proceedings of the Fifth SIGHAN Workshop on Chinese Language Processing, pp. 108–117, 2006. ",
+        "bbox": [
+            174,
+            492,
+            821,
+            535
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xiao-Li Li and Bing Liu. Learning from positive and unlabeled examples with different data distributions. In European conference on machine learning, pp. 218–229. Springer, 2005. ",
+        "bbox": [
+            173,
+            546,
+            821,
+            575
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xiaoya Li, Jingrong Feng, Yuxian Meng, Qinghong Han, Fei Wu, and Jiwei Li. A unified MRC framework for named entity recognition. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pp. 5849–5859, Online, July 2020a. Association for Computational Linguistics. doi: 10.18653/v1/2020.acl-main.519. URL https: //www.aclweb.org/anthology/2020.acl-main.519. ",
+        "bbox": [
+            173,
+            585,
+            825,
+            656
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yangming Li, Han Li, Kaisheng Yao, and Xiaolong Li. Handling rare entities for neural sequence labeling. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pp. 6441–6451, Online, July 2020b. Association for Computational Linguistics. doi: 10.18653/v1/2020.acl-main.574. URL https://www.aclweb.org/anthology/2020. acl-main.574. ",
+        "bbox": [
+            174,
+            666,
+            825,
+            737
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yangming Li, Lemao Liu, and Shuming Shi. Segmenting natural language sentences via lexical unit analysis. arXiv preprint arXiv:2012.05418, 2020c. ",
+        "bbox": [
+            174,
+            748,
+            823,
+            777
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yangming Li, Kaisheng Yao, Libo Qin, Shuang Peng, Yijia Liu, and Xiaolong Li. Span-based neural buffer: Towards efficient and effective utilization of long-distance context for neural sequence models. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pp. 8277– 8284, 2020d. ",
+        "bbox": [
+            174,
+            787,
+            825,
+            844
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xiao Ling and Daniel S Weld. Fine-grained entity recognition. In AAAI, volume 12, pp. 94–100, 2012. ",
+        "bbox": [
+            171,
+            854,
+            823,
+            885
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ying Luo, Fengshun Xiao, and Hai Zhao. Hierarchical contextualized representation for named entity recognition. In AAAI, pp. 8441–8448, 2020. ",
+        "bbox": [
+            176,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xuezhe Ma and Eduard Hovy. End-to-end sequence labeling via bi-directional LSTM-CNNs-CRF. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 1064–1074, Berlin, Germany, August 2016. Association for Computational Linguistics. doi: 10.18653/v1/P16-1101. URL https://www.aclweb.org/ anthology/P16-1101. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Stephen Mayhew, Snigdha Chaturvedi, Chen-Tse Tsai, and Dan Roth. Named entity recognition with partially annotated training data. In Proceedings of the 23rd Conference on Computational Natural Language Learning (CoNLL), pp. 645–655, Hong Kong, China, November 2019. Association for Computational Linguistics. doi: 10.18653/v1/K19-1060. URL https://www.aclweb.org/anthology/K19-1060. ",
+        "bbox": [
+            173,
+            186,
+            825,
+            257
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "G. Mesnil, Y. Dauphin, K. Yao, Y. Bengio, L. Deng, D. Hakkani-Tur, X. He, L. Heck, G. Tur, D. Yu, and G. Zweig. Using recurrent neural networks for slot filling in spoken language understanding. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 23(3):530–539, 2015. doi: 10.1109/TASLP.2014.2383614. ",
+        "bbox": [
+            173,
+            270,
+            825,
+            325
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Mike Mintz, Steven Bills, Rion Snow, and Dan Jurafsky. Distant supervision for relation extraction without labeled data. In Proceedings of the Joint Conference of the 47th Annual Meeting of the ACL and the 4th International Joint Conference on Natural Language Processing of the AFNLP, pp. 1003–1011, 2009. ",
+        "bbox": [
+            173,
+            338,
+            825,
+            395
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Farhad Nooralahzadeh, Jan Tore Lønning, and Lilja Øvrelid. Reinforcement-based denoising of distantly supervised ner with partial annotation. In Proceedings of the 2nd Workshop on Deep Learning Approaches for Low-Resource NLP (DeepLo 2019), pp. 225–233, 2019. ",
+        "bbox": [
+            174,
+            409,
+            825,
+            452
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Minlong Peng, Xiaoyu Xing, Qi Zhang, Jinlan Fu, and Xuanjing Huang. Distantly supervised named entity recognition using positive-unlabeled learning. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pp. 2409–2419, Florence, Italy, July 2019. Association for Computational Linguistics. doi: 10.18653/v1/P19-1231. URL https: //www.aclweb.org/anthology/P19-1231. ",
+        "bbox": [
+            173,
+            464,
+            825,
+            535
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jeffrey Pennington, Richard Socher, and Christopher D Manning. Glove: Global vectors for word representation. In Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP), pp. 1532–1543, 2014. ",
+        "bbox": [
+            173,
+            547,
+            825,
+            590
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sameer Pradhan, Alessandro Moschitti, Nianwen Xue, Hwee Tou Ng, Anders Bjorkelund, Olga ¨ Uryupina, Yuchen Zhang, and Zhi Zhong. Towards robust linguistic analysis using ontonotes. In Proceedings of the Seventeenth Conference on Computational Natural Language Learning, pp. 143–152, 2013. ",
+        "bbox": [
+            174,
+            603,
+            825,
+            660
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Libo Qin, Wanxiang Che, Yangming Li, Haoyang Wen, and Ting Liu. A stack-propagation framework with token-level intent detection for spoken language understanding. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP), pp. 2078– 2087, Hong Kong, China, November 2019. Association for Computational Linguistics. doi: 10.18653/v1/D19-1214. URL https://www.aclweb.org/anthology/D19-1214. ",
+        "bbox": [
+            173,
+            672,
+            825,
+            757
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Xiang Ren, Ahmed El-Kishky, Chi Wang, Fangbo Tao, Clare R Voss, and Jiawei Han. Clustype: Effective entity recognition and typing by relation phrase-based clustering. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 995–1004, 2015. ",
+        "bbox": [
+            173,
+            770,
+            825,
+            827
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Erik F Sang and Fien De Meulder. Introduction to the conll-2003 shared task: Languageindependent named entity recognition. arXiv preprint cs/0306050, 2003. ",
+        "bbox": [
+            171,
+            839,
+            821,
+            868
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jingbo Shang, Jialu Liu, Meng Jiang, Xiang Ren, Clare R Voss, and Jiawei Han. Automated phrase mining from massive text corpora. IEEE Transactions on Knowledge and Data Engineering, 30 (10):1825–1837, 2018a. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            922
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jingbo Shang, Liyuan Liu, Xiaotao Gu, Xiang Ren, Teng Ren, and Jiawei Han. Learning named entity tagger using domain-specific dictionary. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pp. 2054–2064, Brussels, Belgium, OctoberNovember 2018b. Association for Computational Linguistics. doi: 10.18653/v1/D18-1230. URL https://www.aclweb.org/anthology/D18-1230. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yan Song, Shuming Shi, Jing Li, and Haisong Zhang. Directional skip-gram: Explicitly distinguishing left and right context for word embeddings. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers), pp. 175–180, New Orleans, Louisiana, June 2018. Association for Computational Linguistics. doi: 10.18653/v1/N18-2028. URL https://www.aclweb.org/anthology/N18-2028. ",
+        "bbox": [
+            174,
+            183,
+            825,
+            267
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Mitchell Stern, Jacob Andreas, and Dan Klein. A minimal span-based neural constituency parser. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 818–827, Vancouver, Canada, July 2017. Association for Computational Linguistics. doi: 10.18653/v1/P17-1076. URL https://www.aclweb.org/anthology/ P17-1076. ",
+        "bbox": [
+            173,
+            275,
+            825,
+            345
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yuta Tsuboi, Hisashi Kashima, Shinsuke Mori, Hiroki Oda, and Yuji Matsumoto. Training conditional random fields using incomplete annotations. In Proceedings of the 22nd International Conference on Computational Linguistics (Coling 2008), pp. 897–904, 2008. ",
+        "bbox": [
+            176,
+            354,
+            821,
+            397
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yaosheng Yang, Wenliang Chen, Zhenghua Li, Zhengqiu He, and Min Zhang. Distantly supervised ner with partial annotation learning and reinforcement learning. In Proceedings of the 27th International Conference on Computational Linguistics, pp. 2159–2169, 2018. ",
+        "bbox": [
+            174,
+            406,
+            825,
+            449
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Juntao Yu, Bernd Bohnet, and Massimo Poesio. Named entity recognition as dependency parsing. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pp. 6470–6476, Online, July 2020. Association for Computational Linguistics. doi: 10.18653/v1/ 2020.acl-main.577. URL https://www.aclweb.org/anthology/2020.acl-main. 577. ",
+        "bbox": [
+            174,
+            458,
+            825,
+            527
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Haisong Zhang, Lemao Liu, Haiyun Jiang, Yangming Li, Enbo Zhao, Kun Xu, Linfeng Song, Suncong Zheng, Botong Zhou, Jianchen Zhu, et al. Texsmart: A text understanding system for fine-grained ner and enhanced semantic analysis. arXiv preprint arXiv:2012.15639, 2020. ",
+        "bbox": [
+            174,
+            537,
+            823,
+            580
+        ],
+        "page_idx": 11
+    }
+]

parse/train/EdXhmWvvQV/EdXhmWvvQV.md

@@ -0,0 +1,296 @@
+# CENTER-WISE LOCAL IMAGE MIXTURE FOR CONTRASTIVE REPRESENTATION LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recent advances in unsupervised representation learning have experienced remarkable progress, especially with the achievements of contrastive learning, which regards each image as well its augmentations as a separate class, while does not consider the semantic similarity among images. This paper proposes a new kind of data augmentation, named Center-wise Local Image Mixture, to expand the neighborhood space of an image. CLIM encourages both local similarity and global aggregation while pulling similar images. This is achieved by searching local similar samples of an image, and only selecting images that are closer to the corresponding cluster center, which we denote as center-wise local selection. As a result, similar representations are progressively approaching the clusters, while do not break the local similarity. Furthermore, image mixture is used as a smoothing regularization to avoid overconfidence on the selected samples. Besides, we introduce multi-resolution augmentation, which enables the representation to be scale invariant. Integrating the two augmentations produces better feature representation on several unsupervised benchmarks. Notably, we reach $7 5 . 5 \%$ top-1 accuracy with linear evaluation over ResNet-50, and $5 9 . 3 \%$ top-1 accuracy when fine-tuned with only $1 \%$ labels, as well as consistently outperforming supervised pretraining on several downstream transfer tasks.
+
+# 1 INTRODUCTION
+
+Learning general representations that can be transferable to different downstream tasks is a key challenge in computer vision. This is usually achieved by fully supervised learning paradigm, e.g., making use of ImageNet labels for pretraining over the past several years. Recently, self-supervised learning has attracted more attention due to its free of human labels. In self-supervised learning, the network aims at exploring the intrinsic distributions of images via a series of predefined pretext tasks (Doersch et al., 2015; Gidaris et al., 2018; Noroozi & Favaro, 2016; Pathak et al., 2016). Among them, instance discrimination (Wu et al., 2018) based methods have achieved remarkable progress (Chen et al., 2020a; He et al., 2020; Grill et al., 2020; Caron et al., 2020). The core idea of instance discrimination is to push away different images, and encourage the representation of different transformations (augmentations) of the same image to be similar. Following this paradigm, self-supervised models are able to generate features that are comparable or even better than those produced by supervised pretraining when evaluated on some downstream tasks, e.g.,COCO detection and segmentation (Chen et al., 2020c;b).
+
+In contrastive learning, the positive pairs are simply constrained within different transformations of the same image, e.g., cropping, color distortion, Gaussian blur, rotation, etc.. Recent advances have demonstrated that better data augmentations (Chen et al., 2020a) really help to improve the representation robustness. However, contrasting two images that are de facto similar in semantic space is not applicable for general representations. It is intuitive to pull semantically similar images for better transferability. DeepCluster (Caron et al., 2018) and Local Aggregation (Zhuang et al., 2019) relax the extreme instance discrimination task via discriminating groups of images instead of an individual image. However, due to the lack of labels, it is inevitable that the positive pairs contain noisy samples, which limits the performance.
+
+In this paper, we target at expanding instance discrimination by exploring local similarities among images. Towards this goal, one need to solve two issues: i) how to select similar images as positive pairs of an image, and ii) how to incorporate these positive pairs, which inevitably contain noisy assignments, into contrastive learning. We propose a new kind of data augmentation, named Centerwise Local Image Mixture, to tackle the above two issues in a robust and efficient way. CLIM consists of two core elements, i.e., a center-wise positive sample selection, as well as a data mixing operation. For positive sample selection, the motivation is that a good representation should be endowed with high intra-class similarity, and we find that although MoCo (He et al., 2020) does not explicitly model invariance to similar images, the intra-class similarity becomes higher as the training process goes. Based on this observation, we explicitly enforce semantically similar images towards the center of clusters, and generate representation with higher intra-class similarity, which we find is beneficial for few shot learning. This is achieved by searching nearest neighbors of an image, and only retaining similar samples that are closer to the corresponding cluster center, which we denote as center-wise local sample selection. As a result, an image is pulled towards the center while do not break the local similarity.
+
+Once similar samples are selected, a direct way is to treat these similar samples as multiple positives for contrastive learning. However, since feature representation in high dimensional space is complex, the returned positive samples inevitably contain noisy assignments, which should not be overconfident. Instead, we rely on data mixing as augmented samples, which can be treated as a smoothing regularization in unsupervised learning. In particular, we apply Cutmix (Yun et al., 2019), a widely used data augmentation in supervised learning, where patches are cut and pasted among the positive pairs to generate new samples. Benefit from the center-wise sample selection, the Cutmix augmentation is only constrained within the local neighborhood of an image, and can be treated as an expansion of current neighborhood space. In this way, similar samples are pulled together in a smoother and robust way, which we find is beneficial for general representation.
+
+Furthermore, we propose multi-resolution augmentation, which aims at contrasting the same image (patch) at different resolutions explicitly, to enable the representation to be scale invariant. We argue that although previous operations such as crop and resize introduce multi-resolution implicitly, they do not compare the same patch at different resolutions directly. As comparisons, multi-resolution incorporates scale invariance into contrastive learning, and significantly boosts the performance even based on a strong baseline. The multi-resolution strategy is simple but effective, and can be combined with current data augmentations for further improving performance.
+
+We evaluate the feature representation on several self-supervised learning benchmarks. In particular, on ImageNet linear evaluation protocol, we achieve $7 5 . 5 \%$ top-1 accuracy with a standard ResNet50. In few shot setting, when finetuned with only $1 \%$ labels, we achieve $5 9 . 3 \%$ top-1 accuracy, surpassing previous works by a large margin. We also validate its transferring ability on several downstream tasks, and consistently outperform the fully supervised counterparts.
+
+# 2 RELATED WORK
+
+Unsupervised Representation Learning. Unsupervised learning aims at exploring the intrinsic distribution of data samples via constructing a series of pretext tasks without human labels. These pretext tasks take many forms and vary in utilizing different properties of images. Among them, one family of methods takes advantage of the spatial properties of images, typical pretext tasks include predicting the relative spatial positions of patches (Doersch et al., 2015; Noroozi & Favaro, 2016), or inferring the missing parts of images by inpainting (Pathak et al., 2016), colorization (Zhang et al., 2016), or rotation prediction (Gidaris et al., 2018). Recent progress in self-supervised learning mainly benefits from instance discrimination, which regards each image (and augmentations of itself) as one class for contrastive learning. The motivation behind these works is the InfoMax principle, which aims at maximizing mutual information (Tian et al., 2019; Wu et al., 2018) across different augmentations of the same image (He et al., 2020; Chen et al., 2020a), (Tian et al., 2019).
+
+Data Augmentation. Instance discrimination makes use of several data augmentations, e.g., random cropping, color jittering, horizontal flipping, to define a large view set of vicinities for each image. As has been demonstrated (Chen et al., 2020a; Tian et al., 2020), the effectiveness of instance discrimination methods strongly relies on the type of augmentations. Hoping that the network holds invariance in the local vicinities of each sample. However, current data augmentations are mostly constrained within a single image. An exception is (Shen et al., 2020), where image mixture is used for flattened contrastive predictions. However, such mixture strategy is conducted among all images, which destroys the local similarity when contrasting mixed samples that are semantic dissimilar.
+
+![](images/ffd0a272d956fceeaba42deb11917f1a238c23884dc732393819c4737ac36b63.jpg)  
+Figure 1: An illustration of the proposed CLIM and multi-resolution data augmentations.
+
+Beyond self-supervised learning, mixing samples from different images is widely used to help alleviate overfitting in training deep networks. In particular, Mixup (Zhang et al., 2017) combines two samples linearly on pixel level, where the target of the synthetic image was a linear combination of one-hot labels. Following Mixup, there are a few variants (Verma et al., 2018) as well as a recent effort named Cutmix (Yun et al., 2019), which combined Mixup and Cutout (DeVries & Taylor, 2017) by cutting and pasting patches.
+
+# 3 METHOD
+
+In this section, we start by reviewing contrastive learning for unsupervised representation learning. Then we elaborate our proposed CLIM data augmentation, which targets at pulling similar samples via center-wise similar sample selection, followed by a cutmix data augmentation. We also present multi-resolution augmentation that we observe further improves the performance, as well as detailed analysis with recent methods that share similar targets with our method.
+
+# 3.1 CONTRASTIVE LEARNING
+
+Contrastive learning targets at training an encoder to map positive pairs to similar representations while pushing away the negative samples in the embedding space. Given unlabeled training set $X = \{ x _ { 1 } , x _ { 2 } , . . . , x _ { n } \}$ . Instance-wise contrastive learning aims to learn an encoder $f _ { q }$ that maps the samples $\boldsymbol { X }$ to embedding space $V = \{ v _ { 1 } , v _ { 2 } , . . . , v _ { n } \}$ by optimizing a contrastive loss. Take the Noise Contrastive Estimator (NCE) (Oord et al., 2018) as an example, the contrastive loss is defined as:
+
+$$
+\mathcal { L } _ { n c e } ( x _ { i } , x _ { i } ^ { \prime } ) = - \log \frac { \exp ( f _ { q } ( x _ { i } ) \cdot f _ { k } ( x _ { i } ^ { \prime } ) / \tau ) } { \exp ( f _ { q } ( x _ { i } ) \cdot f _ { k } ( x _ { i } ^ { \prime } ) / \tau ) + \sum _ { j = 1 } ^ { K } \exp ( f _ { q } ( x _ { i } ) \cdot f _ { k } ( x _ { j } ^ { \prime } ) / \tau ) ) } ,
+$$
+
+where $\tau$ is the temperature parameter, and $x _ { i } ^ { \prime }$ and $\boldsymbol { x } _ { j } ^ { \prime }$ denote the positive and negative samples of $x _ { i }$ , respectively. The encoder $f _ { k }$ can be shared (Chen et al., 2020a; Caron et al., 2020) or momentum update of the encoder $f _ { q }$ (He et al., 2020).
+
+# 3.2 CLIM: CENTER-WISE LOCAL IMAGE MIXTURE
+
+In contrastive learning, each sample as well as its augmentations is treated as a separate class, while all other samples are regarded as negative examples and pushed away. In principle, semantically similar samples should have similar feature representation in the embedding space, while current contrastive strategies do not consider the semantic similarities among different samples, and only choose different views of the same sample as positive pairs. To solve this issue, we propose a new kind of data augmentation, termed as Center-wise Local Image Mixture, which pulls samples that are semantically similar in an efficient and robust way. The proposed CLIM augmentation consists of two elements, i.e., center-wise local similar sample selection, and a cutmix data augmentation, which would be described in details in the following.
+
+![](images/95c1d4581531a2c56703083501081c5a75bb3f593b31744aef4ee9ebc8f5df89.jpg)  
+Figure 2: Comparison of three positive sample selection strategies, i.e., k-means, knn, and the proposed center-wise local sample selection.
+
+# 3.2.1 CENTER-WISE LOCAL POSITIVE SAMPLE SELECTION
+
+As noted by (Wang & Isola, 2020), a good representation should satisfy both alignment and uniformity, which encourages similar images to have similar representation in the embedding space, and meanwhile, semantically similar features are well-clustered. Towards this goal, we propose a positive sample selection strategy that considers both local similarity and global aggregation. This is achieved by searching similar samples within a cluster that the anchor sample belongs to, and only retaining samples that are closer to the corresponding cluster center. We denote it as center-wise local selection as these samples are picked out towards the cluster center among the local neighborhood of an image. In this way, similar samples are progressively pulled to the predefined cluster centers, while do not break the local similarity.
+
+Specifically, given a set of unlabeled images $X = \{ x _ { 1 } , x _ { 2 } , . . . , x _ { n } \}$ and the corresponding embedding $V = \{ v _ { 1 } , v _ { 2 } , . . . , v _ { n } \}$ with encoder $f _ { \theta }$ , where $v _ { i } = f _ { \theta } ( x _ { i } )$ . We cluster the representations $V$ using a standard $\mathbf { k }$ -means algorithm, and obtain $m$ centers $\mathbf { C } = \{ c _ { 1 } , c _ { 2 } , . . . , c _ { m } \}$ . Given an anchor $x _ { i }$ with its assigned cluster $c ( x _ { i } ) \in C$ , denote the sample set that belongs to $c ( x _ { i } )$ as $\Omega _ { 1 } = \{ x | c ( x ) = c ( x _ { i } ) \}$ . We search the $k$ nearest neighbors of $x _ { i }$ over the entire space with L2 distance, obtaining sample set $\pmb { \Omega } _ { 2 } = \{ x _ { i 1 } , . . . , x _ { i k } \}$ . The positive samples are selected based on the following rule:
+
+$$
+\Omega _ { p } = \{ x | d ( f _ { \theta } ( x ) , v _ { c ( x _ { i } ) } ) \leq d ( f _ { \theta } ( x _ { i } ) , v _ { c ( x _ { i } ) } ) , x \in \Omega _ { 1 } \cap \Omega _ { 2 } \} ,
+$$
+
+where $d ( \cdot , \cdot )$ denotes the L2 distance of two samples, and $v _ { c ( x _ { i } ) }$ denotes the feature representation of the corresponding cluster center, respectively. In this way, the samples are aggregated towards the predefined clusters, and meanwhile maintaining the local similarity.
+
+Our method combines the advantages of cluster and nearest neighbor methods. An illustration comparing the three methods is shown in Fig. 2. Cluster-based method regards all samples that belong to the same center as positive pairs, which breaks the local similarity among samples especially when the anchor is around the boundary. While nearest neighbor-based method independently pulling samples of an anchor, and does not encourage the well-clustered goal. As a result, the embedding space is not highly concentrated among multiple similar anchors. As comparisons, by center-wise sample selection, similar samples are progressively pulled to the predefined center as well as considering the local similarity. In the experimental section, we would compare the performance of the three methods, and validate the superior performance of our proposed selection strategy.
+
+# 3.2.2 CUTMIX DATA AUGMENTATION
+
+Once we obtain the positive samples of an anchor, one direct way is to treat these samples similar as the augmented ones for contrastive learning. However, similarity computation in high dimensional space inevitably contains noisy samples, which should not be overconfident for contrasting. To solve this issue, we make use of data mixture strategy, which aims at mixing patches from two different images as augmented samples for contrasting. Data mixing is widely used in supervised learning as label smoothing regularization. The highlight is that without image level labels, we are not able to assign new labels to the augmented samples. Instead, we only mixing samples that are similar in representation, and the mixed samples can be treated as an augmented version of the anchor. In this way, these mixed samples, as well as traditional data augmentations, can be pulled together in contrastive learning. Specifically, given a positive pair $( x _ { i } , \tilde { x } _ { i } )$ , we conduct data mixing as follows:
+
+$$
+x _ { m i x } = \mathbf { M } \odot x _ { i } + ( \mathbf { 1 } - \mathbf { M } ) \odot \tilde { x } _ { i } ,
+$$
+
+where $\mathbf { M } \in \{ 0 , 1 \} ^ { W \times H }$ denotes a binary mask indicating the mixed rectangle region of an image, i.e., where to cutout the region in $x _ { i }$ and replaced with a randomly selected patch from ${ \tilde { x } } _ { i }$ , and $W , H$ denotes the wide and height of an image, respectively. 1 is a binary mask filled with ones, and $\odot$ is the element-wise multiplication operation. For mask $\mathbf { M }$ generation, we follow the setting in (Yun et al., 2019). For the mixed sample $x _ { m i x }$ , the positive sample can be either $x _ { i }$ or $\tilde { x } _ { i }$ , and we reformulate the contrastive learning as combing two NCE loss:
+
+$$
+{ \mathcal L } _ { m i x } ( x _ { i } , \tilde { x } _ { i } ) = \lambda \cdot { \mathcal L } _ { n c e } ( x _ { m i x } , x _ { i } ) + ( 1 - \lambda ) \cdot { \mathcal L } _ { n c e } ( x _ { m i x } , \tilde { x } _ { i } ) .
+$$
+
+Where the combination ratio $\lambda$ is sampled from beta distribution $\mathtt { B e t a } ( \alpha , \alpha )$ with parameter $\alpha$ . The proposed data mixing augmentation can be seamlessly incorporated into current contrastive learning. The advantages are twofold: first, mixed samples help to expand the neighborhood space of current anchor sample for better representation; second, minimizing the two terms simultaneously can help to maximize the mutual information between $x _ { i }$ and ${ \tilde { x } } _ { i }$ in a soft manner and perform as smoothing regularization on the prediction for selected positive samples.
+
+# 3.3 MULTI-RESOLUTION DATA AUGMENTATION
+
+Data augmentation plays a key role in current contrastive learning, among them crop augmentation is one of the most effective way (Chen et al., 2020a). In a typical crop augmentation, a sample $x$ with size $H \times W$ is randomly cropped with ratio $\sigma$ , and resized to $K _ { t r a i n } \times K _ { t r a i n }$ as augmented samples, where $K _ { t r a i n } \times K _ { t r a i n }$ denotes the input resolution for model training. Hence the scaling factor w.r.t. sample $x$ can be described as:
+
+$$
+s = \frac { 1 } { \sigma } \cdot \frac { K _ { t r a i n } } { \sqrt { H \times W } } .
+$$
+
+For crop augmentation, the parameter $K _ { t r a i n }$ is fixed, and the crop ratio $\sigma$ is randomly selected among positive pairs. As a result, different crop augmentations usually contain different contents, which can be regarded as modeling occlusion invariance to some extent, where each crop sees one view of an image. In this section, we propose a simple but effective data augmentation strategy, named multi-resolution augmentation, which enables the representation to be scale invariant of an example. The highlight is that it is better for contrasting positive pairs with the same content but different resolutions. Specifically, for each positive we keep the crop ratio $\sigma$ fixed, and adjust $K _ { t r a i n }$ to different resolutions for contrastive loss. An illustration is shown in Fig. 1 .Using multi-resolution, the objective function can be generalized as:
+
+$$
+\mathcal { L } _ { m r } = \sum _ { r , r ^ { \prime } \in \{ r _ { 1 } , \ldots , r _ { n } \} } \mathcal { L } _ { m i x } ( x _ { i } ^ { r } , \tilde { x } _ { i } ^ { r ^ { \prime } } ) ,
+$$
+
+where $\{ r _ { 1 } , . . . , r _ { n } \}$ indicates the resolution set. In this way, the encoder would be encouraged to discriminate the positive samples with different resolutions from a series of negative keys, which will maximize the mutual information between inputs with different resolutions and discard redundant information brought by resolutions.
+
+Relation with Multi-crop Augmentation. There exist recent works that aim at improving crop augmentations, including multi-crop (Caron et al., 2020) and jigsaw-crop (Misra & Maaten, 2020). However, both methods target at reducing crop ratio $\sigma$ in Eq.5 and resolution $K _ { t r a i n }$ simultaneously to bridge different parts of an object, and do not explicitly model scale invariance. As comparisons, our proposed multi-resolution strategy fixes the crop ratio to explicitly model scale invariance. In the experimental section, we would compare these two augmentations to validate the difference.
+
+Table 1: Top-1 accuracies under linear evaluation on ImageNet, using ResNet-50 as encoder   
+
+<table><tr><td>Method Accuracy (%)</td></tr><tr><td>Supervised 76.5</td></tr><tr><td>Colorization (Zhang et al., 2016) 39.6</td></tr><tr><td>Jigsaw (Noroozi &amp; Favaro, 2016) 45.7</td></tr><tr><td>NPID (Wu et al., 2018) 54.0</td></tr><tr><td>LA (Zhuang et al., 2019) 58.8</td></tr><tr><td>MoCo (He et al., 2020) 60.6</td></tr><tr><td>SeLa (YM. et al., 2020) 61.5</td></tr><tr><td>PIRL (Misra &amp; Maaten,2020) 63.6</td></tr><tr><td>CPCv2 (Henaff et al., 2019) 63.8</td></tr><tr><td>PCL (Li et al., 2020) 65.9</td></tr><tr><td>SimCLR (Chen et al., 2020a) 70.0</td></tr><tr><td>MoCo v2 (Chen et al., 2020c) 71.1</td></tr><tr><td>SimCLRv2 (Chen et al., 2020b) 71.7</td></tr><tr><td>InfoMin (Tian et al.,2020) 73.0</td></tr><tr><td>BYOL (Grill et al., 2020) 74.3</td></tr><tr><td>SwAV (Caron et al., 2020) 75.3</td></tr><tr><td>CLIM 75.5</td></tr></table>
+
+Table 2: Semi-supervised learning with few shot ImageNet labels, using ResNet50 as encoder (averaged by 5 trials)   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Top-1/Top-5</td></tr><tr><td>1% labels 10% labels</td><td></td></tr><tr><td>Supervised</td><td>25.4 48.4 56.4 56.4</td></tr><tr><td>PIRL</td><td>30.7 57.2 60.4 83.8</td></tr><tr><td>SimCLR</td><td>48.3 75.5 65.6 87.8</td></tr><tr><td>MoCo v2</td><td>52.4 78.4 65.3 86.6</td></tr><tr><td>BYOL</td><td>53.2 78.4 68.8 89.0</td></tr><tr><td>SwAV</td><td>53.9 78.5 70.2 89.9</td></tr><tr><td>SimCLRv2</td><td>57.9 82.5 68.4 89.2</td></tr><tr><td>CLIM</td><td>59.3 81.6 70.0 89.3</td></tr></table>
+
+Table 3: Transfer learning on VOC object detection (averaged by 5 trials).   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Accuracy (%)</td></tr><tr><td>AP50</td><td>AP75</td></tr><tr><td>Supervised</td><td>81.4</td><td>58.8</td></tr><tr><td>MoCo v2</td><td>82.5</td><td>64.0</td></tr><tr><td>SwAV</td><td>82.6</td><td>:</td></tr><tr><td>CLIM</td><td>82.8</td><td>64.5</td></tr></table>
+
+Relation with Fix-Res. The proposed multi-resolution augmentation is reminiscent of recent work FixRes (Touvron et al., 2019), which also explores resolution issue of better representation, but they are different in both motivation and goal. FixRes is based on the observation that data augmentations induce a significant discrepancy between the size of the objects seen by the classifier at train and test time, and employs different train and test resolutions to fix the train-test resolution discrepancy. The goal is to require less scale invariance for the neural net in FixRes. While our multi-resolution augmentation aims to model the scale invariance explicitly, which is not carefully considered in previous self-supervised learning.
+
+# 4 EXPERIMENTAL RESULTS
+
+In this section, we assess our pretrained feature representation on several unsupervised benchmarks. We evaluate it on ImageNet under linear evaluation and semi-supervised settings. Then we transfer the learned features to different downstream tasks. We also analyze the performance of our representation with detailed ablation studies. For brief expression, except for the ablation study, we denote our method as CLIM, which includes two kinds of data augmentations.
+
+# 4.1 LINEAR EVALUATION ON IMAGENET
+
+The feature representation is trained based on ImageNet 2012 (Russakovsky et al., 2015), using a standard ResNet-50 structure as backbone. We follow the setting in MoCo v2 (Chen et al., 2020c), and the training details are listed in Appendix A. We first evaluate our features by training a linear classifier on top of the frozen representation, following a common protocol in (He et al., 2020; Tian et al., 2019). For linear classifier, the learning rate is initialized as 30 and decayed by 0.1 after 60, 80 epochs, respectively. Table 1 shows the top-1 accuracies with center crop evaluation. Our method achieves an accuracy of $7 5 . 5 \%$ , surpassing MoCo v2 baseline $( 7 1 . 1 \% )$ by $4 . 4 \%$ , and nearly approaching the supervised learning baseline $( 7 6 . 5 \% )$ .
+
+Table 4: Transfer learning on COCO detection and instance segmentation (averaged by 5 trials)   
+
+<table><tr><td rowspan=3 colspan=1>Method</td><td rowspan=1 colspan=2>Mask R-CNN,R50-FPN,Det</td><td rowspan=1 colspan=2>Mask R-CNN,R50-FPN,InsSeg</td></tr><tr><td rowspan=1 colspan=1>1× schedule</td><td rowspan=1 colspan=1>2× schedule</td><td rowspan=1 colspan=1>1× schedule</td><td rowspan=1 colspan=1>2× schedule</td></tr><tr><td rowspan=1 colspan=1>AP66AP0AP</td><td rowspan=1 colspan=1>AP66AP0AP</td><td rowspan=1 colspan=1>APmkAPAP7</td><td rowspan=1 colspan=1>APmkAPAP7</td></tr><tr><td rowspan=1 colspan=1>Supervised</td><td rowspan=1 colspan=1>38.959.642.0</td><td rowspan=1 colspan=1>40.661.344.4</td><td rowspan=1 colspan=1>35.4 56.5 38.1</td><td rowspan=1 colspan=1>36.8 58.1 39.5</td></tr><tr><td rowspan=1 colspan=1>MoCo v2</td><td rowspan=1 colspan=1>39.259.942.7</td><td rowspan=1 colspan=1>41.562.245.3</td><td rowspan=1 colspan=1>35.7 56.8 38.1</td><td rowspan=1 colspan=1>37.5 59.1 40.1</td></tr><tr><td rowspan=1 colspan=1>CLIM</td><td rowspan=1 colspan=1>39.560.043.3</td><td rowspan=1 colspan=1>41.862.345.7</td><td rowspan=1 colspan=1>35.8 57.0 38.6</td><td rowspan=1 colspan=1>37.7 59.4 40.5</td></tr></table>
+
+# 4.2 SEMI-SUPERVISED TRAINING ON IMAGENET
+
+We also evaluate our method by fine-tuning the pretrained model with a small subset of labels, following the semi-supervised settings in (Grill et al., 2020; Kornblith et al., 2019; Chen et al., 2020a; Caron et al., 2020). For fair comparisons, we use the same fixed $1 \%$ and $1 0 \%$ splits of training data as in (Chen et al., 2020a), and fine-tune all layers using SGD optimizer with momentum of 0.9, and learning rate of 0.0001 for backbone, 10 for the newly initialized fc layer. The fine-tune epochs is set as 60, and the learning rate is decayed by 0.1 after every 20 epochs. During training, only random cropping and flipping data augmentations are used for fair comparison. The results are reported in Table 2. CLIM achieves $5 9 . 3 \%$ top-1 accuracy with only $1 \%$ labels, and $7 0 . 0 \%$ with $1 0 \%$ labels. The performance gains are larger with $1 \%$ labels, e.g., $6 . 1 \%$ higher than BYOL, and $5 . 4 \%$ better than SwAV, which demonstrates that the proposed feature representation is mainly suitable for extremely few shot learning. Note that SimCLR v2 makes use of other tricks like more MLP layers for better performance, while our method simply adds one fc layer, and still achieves better performance under both settings.
+
+# 4.3 DOWNSTREAM TASKS
+
+We also evaluate our feature representation on several downstream tasks, including object detection and instance segmentation, to evaluate the transferability of the learned features. For fair comparison, all experiments follow MoCo settings.
+
+PASCAL VOC Object Detection. Following the evaluation protocol in (He et al., 2020), we use Faster R-CNN (Ren et al., 2015) with R50-C4 as backbone. We fine-tune all layers on the trainval set of $\mathrm { \ V O C { 0 7 + 1 2 } }$ for $2 \times$ schedule and evaluate on the test set of VOC2007. We report the performances under the metric of AP50 and AP75. As shown in Table 3, on PASCAL VOC, CLIM achieves $8 2 . 8 \%$ and $6 4 . 5 \%$ mAP under AP50 and AP75 metric, which is 1.4 points and 5.7 points higher than the fully supervised counterparts, and is slightly better than the results of MoCo v2.
+
+COCO Object Detection and Instance Segmentation. We also evaluate the representation learned on a large scale COCO dataset. Following (He et al., 2020), we choose Mask R-CNN with FPN as backbone, and fine-tune all the layers on the train set and evaluate on the val set of COCO2017. In Table 4, we report results under both $1 \times$ and $2 \times$ schedules. We show that CLIM consistently outperforms the supervised pretrained model and MoCo v2. Under 2X schedule, we achieve $4 1 . 8 \%$ and $3 7 . 7 \%$ detection and segmentation accuracies, respectively, which is 1.2 points and 1.1 points better than the supervised couterparts, and also slightly better than the highly optimized MoCo v2.
+
+LVIS Long Tailed Instance Segmentation. Different from VOC and COCO where the number of training samples is comparable, LVIS is a long-tailed dataset, which contains more than 1200 categories, among them some categories only have less than ten instances. The main challenge is to learn accurate few shot models for classes among the tail of the class distribution, for which little data is available. We evaluate our features on this long-tailed dataset to validate how the unsupervised representation boosts the performance. Similarly, we fine-tune the model (Mask R-CNN, R50-FPN) on the train set and evaluate on the val set of Lvis v0.5. Table 5 shows the result under $2 \times$ schedule. CLIM outperforms the supervised pretrained model by a large margin and is slightly better than MoCo v2. We claim that it is mainly to the proposed data mixing data augmentation, which is able to learn generalized representations even with extremely few labeled data.
+
+Table 5: Transfer learning on LVIS long-tailed instance segmentation (averaged by 5 trials)   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Object Det</td><td colspan="3">Instance Seg</td></tr><tr><td>AP66</td><td>AP</td><td>AP</td><td>AP66</td><td>AP</td><td>APm</td></tr><tr><td>Supervised</td><td>24.1</td><td>39.4</td><td>25.0</td><td>24.2</td><td>37.8</td><td>25.1</td></tr><tr><td>MoCo v2</td><td>25.1</td><td>40.4</td><td>26.1</td><td>25.3</td><td>38.4</td><td>27.0</td></tr><tr><td>CLIM</td><td>25.5</td><td>41.2</td><td>26.7</td><td>25.6</td><td>39.5</td><td>27.5</td></tr></table>
+
+Table 6: Impact of different sample selection   
+
+<table><tr><td rowspan="2">Strategy</td><td colspan="2">Accuracy (%)</td></tr><tr><td>no mixing</td><td>+cutmix</td></tr><tr><td>MoCo v2</td><td>67.5</td><td>-</td></tr><tr><td>Random</td><td>62.3</td><td>67.1</td></tr><tr><td>KNN</td><td>68.3</td><td>69.5</td></tr><tr><td>K-means</td><td>68.0</td><td>69.2</td></tr><tr><td>KNN ∩ K-means</td><td>68.5</td><td>69.6</td></tr><tr><td>Center-wise</td><td>69.3</td><td>70.1</td></tr></table>
+
+Table 7: Impact of different multiple resolutions   
+
+<table><tr><td>Method</td><td>Resolution</td><td>Accuracy (%)</td></tr><tr><td>Multi-Crop</td><td>2×224 + 2×96</td><td>69.7</td></tr><tr><td rowspan="4">Multi-Reso</td><td>r,r&#x27;∈ {224,96}</td><td>70.4</td></tr><tr><td>r,r&#x27; ∈ {224,128}</td><td>71.7</td></tr><tr><td>r,r&#x27; ∈ {224,160}</td><td>72.3</td></tr><tr><td>r,r&#x27;∈ {224,224}</td><td>71.4</td></tr></table>
+
+# 4.4 ABLATION STUDY
+
+In this section, we present ablation studies to better understand how each component affects the performance. Detailed comparisons include 1) positive sample selection, 2) cutmix data augmentation, and 3) multi-resolution augmentation. Unless specified, we train the model for 200 epochs over the ImageNet-1000 and report the top-1 classification accuracy under linear evaluation protocol.
+
+Positive Sample Selection. We first analyze the advantages of our proposed center-wise local sample selection strategy. The compared sample selection alternatives include:
+
+• Random selection: Randomly select a sample from all unlabeled data.   
+• KNN selection: Use $\mathbf { k }$ -nearest neighbors to build the correlation map among samples, and randomly select a sample from the Top- $k$ $k = 1 0$ ) nearest neighbors as positive samples.   
+• K-means selection: Use $\mathbf { k }$ -means clustering algorithm to obtain $k$ cluster centers, and randomly select a sample from the corresponding cluster as positive samples.
+
+• KNN ∩ K-means selection: Use K-means clustering algorithm to obtain $k$ cluster centers, and randomly select nearest neighbor within the cluster as positive samples.
+
+The results are shown in the second column of Table 6. In order the inspect the influence of sample selection, we do not conduct cutmix augmentation, and these positive samples are simply pulled via a standard contrastive loss. It can be shown that comparing with the MoCo baseline, both KNN and cluster-based sample selection boost the performance, and notably, simply selecting the union of knn and k-means achieves $\cdot$ accuracy, which is comparable with result that directly using knn. Since for samples not lie around the boundary, it equals to knn, and does not encourage intra-class compactness. As comparison, our proposed center-wise selection strategy outperforms all the above selection methods.
+
+Cutmix Data Augmentation. Data mixing helps to expand the neighborhood space of the target sample, and acts as smoothing regularization for the prediction. As shown in the third column of Table 6, cutmix augmentation consistently improve the performance, comparing with directly pulling similar samples in contrastive loss, and achieve $7 0 . 1 \%$ accuracy with only 200 training epochs. Notably, with randomly selected positive samples, cutmix operation even obtains $6 7 . 1 \%$ accuracy, slightly lower than the MoCo baseline, while significantly better than no mixing with only $6 2 . 3 \%$ accuracy. This can be attributed to the smoothing regularization of cutmix, which is able to alleviate the effect of noisy samples and update model in a more robust way.
+
+Multiple Resolution. Based on CLIM, we further add multi-resolution data augmentation to validate its effectiveness. The results of introducing different resolutions are shown in Table 7. Using multiple resolutions setting with $r , r ^ { \prime } \in \{ 2 2 4 , 1 \bar { 6 } 0 \}$ , our method achieves an accuracy of $7 2 . 3 \%$ with only 200 epochs, which surpasses the baseline of MoCo by $4 . 8 \%$ , and even much better than the results of MoCo with 800 epochs $( 7 1 . 1 \% )$ ).
+
+We also compare our multi-resolution augmentation with multi-crop augmentation proposed in (Caron et al., 2020). $2 \times 2 2 4 + 2 \times 9 6$ denotes using two $2 2 4 \times 2 2 4$ crops with crop-scale $\sigma \sim U ( 0 . 2 , 1 . 0 )$ and two $9 6 \times 9 6$ crops with $\sigma \sim U ( 0 . 0 \bar { 5 } , 0 . 1 4 )$ , referring to (Caron et al., 2020). The main difference is that, the multi-crop strategy targets at capturing relationship between local and global information, while our proposed multiple resolution target at enabling the encoder with scale invariance. We find that multi-crop slightly deteriorates the performance of CLIM $( 7 0 . 1 \%$ versus $6 9 . 7 \%$ ), partially because data mixing behaves like image cropping augmentation, and shares similarity with multi-crop strategy.
+
+# 5 CONCLUSION
+
+In this work, we proposed CLIM data augmentation, to efficiently pull semantically similar samples for better representation in contrastive learning. The main contributions of CLIM consist of two elements, center-wise positive sample selection, which considers both local similarity and global aggregation property. In such way, similar samples are progressively aggregated to a series of predefined clusters, while not breaking the local similarity; and data mixing augmentation, which expands the neighborhood space of an example by mixing two images, and acts as a smoothing regularization for contrastive loss. Furthermore, we present a simple but effective multi-resolution augmentation, which explicitly model scale invariance to further improve the representation. Experiments evaluated on several unsupervised benchmarks demonstrate the effectiveness of our method.
+
+# REFERENCES
+
+Mathilde Caron, Piotr Bojanowski, Armand Joulin, and Matthijs Douze. Deep clustering for unsupervised learning of visual features. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 132–149, 2018.   
+Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. arXiv preprint arXiv:2006.09882, 2020.   
+Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv preprint arXiv:2002.05709, 2020a.   
+Ting Chen, Simon Kornblith, Kevin Swersky, Mohammad Norouzi, and Geoffrey Hinton. Big selfsupervised models are strong semi-supervised learners. arXiv preprint arXiv:2006.10029, 2020b.   
+Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. arXiv preprint arXiv:2003.04297, 2020c.   
+Terrance DeVries and Graham W Taylor. Improved regularization of convolutional neural networks with cutout. arXiv preprint arXiv:1708.04552, 2017.   
+Carl Doersch, Abhinav Gupta, and Alexei A Efros. Unsupervised visual representation learning by context prediction. In Proceedings of the IEEE international conference on computer vision, pp. 1422–1430, 2015.   
+Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. arXiv preprint arXiv:1803.07728, 2018.   
+Jean-Bastien Grill, Florian Strub, Florent Altche, Corentin Tallec, Pierre H Richemond, Elena ´ Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, et al. Bootstrap your own latent: A new approach to self-supervised learning. arXiv preprint arXiv:2006.07733, 2020.   
+Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9729–9738, 2020.
+
+Olivier J Henaff, Aravind Srinivas, Jeffrey De Fauw, Ali Razavi, Carl Doersch, SM Eslami, and ´ Aaron van den Oord. Data-efficient image recognition with contrastive predictive coding. arXiv preprint arXiv:1905.09272, 2019.
+
+Jeff Johnson, Matthijs Douze, and Herve J ´ egou. Billion-scale similarity search with gpus. ´ IEEE Transactions on Big Data, 2019.
+
+Simon Kornblith, Jonathon Shlens, and Quoc V Le. Do better imagenet models transfer better? In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2661–2671, 2019.
+
+Junnan Li, Pan Zhou, Caiming Xiong, Richard Socher, and Steven CH Hoi. Prototypical contrastive learning of unsupervised representations. arXiv preprint arXiv:2005.04966, 2020.
+
+Ishan Misra and Laurens van der Maaten. Self-supervised learning of pretext-invariant representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6707–6717, 2020.
+
+Mehdi Noroozi and Paolo Favaro. Unsupervised learning of visual representations by solving jigsaw puzzles. In European Conference on Computer Vision, pp. 69–84. Springer, 2016.
+
+Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.
+
+Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros. Context encoders: Feature learning by inpainting. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2536–2544, 2016.
+
+Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in neural information processing systems, pp. 91–99, 2015.
+
+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211–252, 2015.
+
+Zhiqiang Shen, Zechun Liu, Zhuang Liu, Marios Savvides, and Trevor Darrell. Rethinking image mixture for unsupervised visual representation learning. arXiv preprint arXiv:2003.05438, 2020.
+
+Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. arXiv preprint arXiv:1906.05849, 2019.
+
+Yonglong Tian, Chen Sun, Ben Poole, Dilip Krishnan, Cordelia Schmid, and Phillip Isola. What makes for good views for contrastive learning. arXiv preprint arXiv:2005.10243, 2020.
+
+Hugo Touvron, Andrea Vedaldi, Matthijs Douze, and Herve J ´ egou. Fixing the train-test resolution ´ discrepancy. In Advances in Neural Information Processing Systems, pp. 8252–8262, 2019.
+
+Vikas Verma, Alex Lamb, Christopher Beckham, Amir Najafi, Ioannis Mitliagkas, Aaron Courville, David Lopez-Paz, and Yoshua Bengio. Manifold mixup: Better representations by interpolating hidden states. arXiv preprint arXiv:1806.05236, 2018.
+
+Tongzhou Wang and Phillip Isola. Understanding contrastive representation learning through alignment and uniformity on the hypersphere. arXiv preprint arXiv:2005.10242, 2020.
+
+Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via nonparametric instance discrimination. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3733–3742, 2018.
+
+Asano YM., Rupprecht C., and Vedaldi A. Self-labelling via simultaneous clustering and representation learning. In International Conference on Learning Representations (ICLR), 2020.
+
+Sangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo. Cutmix: Regularization strategy to train strong classifiers with localizable features. In Proceedings of the IEEE International Conference on Computer Vision, pp. 6023–6032, 2019.
+
+Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017.
+
+Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In European conference on computer vision, pp. 649–666. Springer, 2016.
+
+Chengxu Zhuang, Alex Lin Zhai, and Daniel Yamins. Local aggregation for unsupervised learning of visual embeddings. In Proceedings of the IEEE International Conference on Computer Vision, pp. 6002–6012, 2019.
+
+A IMPLEMENTATION DETAILS
+
+# A.1 IMPLEMENTATION DETAILS FOR CONTRASTIVE PRETRAINING
+
+Architecture and Optimization. We follow the setting in MoCo v2 (Chen et al., 2020c), which relies on two encoders, one for training and the other one for momentum update $m = 0 . 9 9 9$ ) to store negative keys. Following SimCLR (Chen et al., 2020a), we replace the fc head with a 2-layer MLP to project the output of the final pooling layer to 128-d. We use SGD as optimizer, with weight decay setting as 0.0001 and the momentum as 0.9. We use a mini-batch size of 512 on 16 V100 GPUs with a cosine learning rate schedule decayed from 0.06. We train the model for 1200 epochs, as we introducing data mixing augmentation, and usually requires more epochs for better performance as in supervised learning (Yun et al., 2019). 1
+
+Image Augmentations. We combine the proposed augmentations with previous widely used basic augmentation strategies, following the settings in (Chen et al., 2020a; He et al., 2020). The basic augmentations are listed below, as well as the corresponding parameters.
+
+• RandomResizedCrop: A crop of random size (from 0.2 to 1.0) of the original size and a random aspect ratio (from $3 / 4$ to 4/3) of the original aspect ratio is made. • RandomFlip: Randomly horizontally flip the image with a probability of 0.5. • ColorJitter: Randomly change the brightness, contrast and saturation of an image. • RandomGrayscale: Randomly convert RGB image to grayscale with a probability of 0.2. • RandomGaussianBlur: Randomly blur the image with a probability of 0.5. The radius is randomly sampled from 0.1 to 2.0.
+
+# A.2 DETAILS OF POSITIVE SAMPLE SELECTION
+
+We implement k-means and knn by faiss (Johnson et al., 2019). For efficiency, we perform clustering and knn computation every 5 epochs, since each iteration can be finished within minutes, the extra computation cost is marginal comparing with the budget for model training. The number of clusters is set as $1 0 K$ , and we select the top 40 nearest neighbors in knn. In order to balance the contribution of each image, we randomly select 10 positive samples for the following cutmix augmentations. For situations where there remained no more than 10 examples (e.g., the anchor is already around the cluster center), we simply select the most nearest samples among the remained top- $4 0 \mathrm { k n n }$ samples.
+
+# B MORE ABLATION STUDIES
+
+This section gives more detailed analysis w.r.t. some hyperparameters. Unless specified, we train the model for 200 epochs over the ImageNet-1000 and report the top-1 classification accuracy under linear evaluation protocol.
+
+Table 8: Impact of the number of clusters $m$ and $k$ of knn   
+
+<table><tr><td>Number of Clusters (m)</td><td colspan="3">5000</td><td colspan="3">10000</td><td colspan="3">20000</td></tr><tr><td>knn (k）</td><td>20</td><td>40</td><td>60</td><td>20</td><td>40</td><td>60</td><td>20</td><td>40</td><td>60</td></tr><tr><td> Accuracy (%)</td><td>69.1</td><td>69.5</td><td>69.4</td><td>70.0</td><td>70.1</td><td>69.7</td><td>69.6</td><td>69.9</td><td>69.5</td></tr></table>
+
+The number of Clusters $m$ and the $k$ in Knn. Here we inspect the impact of the number of clusters $m$ in $\mathbf { k }$ -means and the $k$ in knn to analyze their effect on the performance. In order to ensure local similarity, we restrict the nearest neighbors within a range from 20 to 60. The results for different clusters and top- $\mathbf { \nabla } \cdot \mathbf { k }$ neighbors are shown in Table 8. We observe that CLIM consistently improves
+
+the performance comparing the baseline Moco $6 7 . 5 \%$ , and is relatively robust to different $m$ and $k$ .   
+Notably, the best performance is achieved when $m = 1 0 0 0 0$ , $k = 4 0$ .
+
+Hyperparameters $\alpha$ in Cutmix. The combination $\lambda$ in cutmix is sampled from the beta distribution $\mathtt { B e t a } ( \alpha , \alpha )$ , where $\alpha$ plays an important role in data mixing augmentation, which controls the strength of interpolation between the anchor and its positive pair. Here we inspect how different $\alpha \in \{ 1 , 1 . 5 , 2 , 2 . 5 \}$ affect the representation. As shown in Table 9. We find that the performance is relatively robust to different $\alpha$ , and the best performance is achieved when $\alpha$ is set as 2.
+
+Table 9: Impact of $\alpha$ in cutmix   
+
+<table><tr><td>a</td><td>1.0</td><td>1.5</td><td>2.0</td><td>2.5</td></tr><tr><td>Accuracy (%)</td><td>69.7</td><td>69.9</td><td>70.1</td><td>69.8</td></tr></table>
+
+Ablation study on mixing strategies. Our method targets at generating new samples that expanding the neighborhood of an anchor. Here we compare performance of using mixup data augmentation, a widely used method in supervised settings. We try different choices of beta distribution for Mixup (Zhang et al., 2017) and choose the best one $\alpha = 0 . 2$ ) for comparison. Table 10 shows that Cutmix performs better than Mixup, partially because mixup destroys the real pixel distribution (destroys the naturality of pixels).
+
+Table 10: Ablation study on the mixing methods   
+
+<table><tr><td>Method</td><td>Accuracy (%)</td></tr><tr><td>Mixup</td><td>69.5</td></tr><tr><td>Cutmix</td><td>70.1</td></tr></table>
+
+Extra ablation experiments for longer training schedule. We compare the improvements brought by different components of our proposed method for longer training schedule (800 epochs). Table 11 shows the top-1 accuracies under linear evaluation protocol. Our method consistently outperforms the MoCo v2 baseline, which demonstrates the effectiveness of our proposed method.
+
+Table 11: Ablation study on the longer training schedule   
+
+<table><tr><td>Method</td><td>Accuracy (%)</td></tr><tr><td>MoCo v2</td><td>71.1</td></tr><tr><td>Center-wise+ cutmix</td><td>73.7</td></tr><tr><td>Center-wise+ cutmix+Multi-reso</td><td>75.2</td></tr></table>
+
+# C MORE EXPERIMENTAL RESULTS
+
+Visualization of Feature Representation. We visualize the feature space to better understand how CLIM augmentation pulls similar samples. Specifically, we randomly choose 10 classes from the validation set and provide the $t$ -sne visualization of feature representation generated by CLIM, supervised training and MoCo v2. As shown in Fig. 3, the same color denotes features with the same label. It can be shown that CLIM takes on higher aggregation property comparing with MoCo, and the fully supervised learned representation reveals the highest aggregation due to it makes use of image labels. Furthermore, we compute the intra-class similarity as the average cosine distance among all intra-class pairwise samples, and report the average similarity across 1000 classes, as shown in Table 12, CLIM achieves an intra-class similarity of 0.65, which is much higher than that in MoCo v2 with similarity of only 0.58. As comparison, we also list the result of supervised learning, with a similarity metric of 0.75.
+
+![](images/55b0297d115f1b5f87a6d8eadfb7c6b2db53a21c9a14da60c5f87e4d83eb84f9.jpg)  
+Figure 3: t-sne visualization of representation learned by MoCo, CLIM and supervised learning.
+
+Table 12: Intra-class similarity for different models   
+
+<table><tr><td>Method</td><td>Intra-class Similarity</td></tr><tr><td>Supervised</td><td>0.75</td></tr><tr><td>MoCo v2</td><td>0.58</td></tr><tr><td>CLIM</td><td>0.65</td></tr></table>
+
+Table 13: Results of different training epochs   
+
+<table><tr><td>Epochs</td><td>Accuracy (%)</td></tr><tr><td>200</td><td>72.3</td></tr><tr><td>800</td><td>75.2</td></tr><tr><td>1200</td><td>75.5</td></tr></table>
+
+Results of Different Training Epochs. In Table 13, we compare CLIM trained with different epochs. Our method achieves an accuracy of $7 2 . 3 \%$ with only 200 epochs, $7 5 . 2 \%$ with 800 epochs, and can be further improved to $7 5 . 5 \%$ when training with 1200 epochs.

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+        "text": "CENTER-WISE LOCAL IMAGE MIXTURE FOR CONTRASTIVE REPRESENTATION LEARNING ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
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+        "text": "ABSTRACT ",
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+        "text": "Recent advances in unsupervised representation learning have experienced remarkable progress, especially with the achievements of contrastive learning, which regards each image as well its augmentations as a separate class, while does not consider the semantic similarity among images. This paper proposes a new kind of data augmentation, named Center-wise Local Image Mixture, to expand the neighborhood space of an image. CLIM encourages both local similarity and global aggregation while pulling similar images. This is achieved by searching local similar samples of an image, and only selecting images that are closer to the corresponding cluster center, which we denote as center-wise local selection. As a result, similar representations are progressively approaching the clusters, while do not break the local similarity. Furthermore, image mixture is used as a smoothing regularization to avoid overconfidence on the selected samples. Besides, we introduce multi-resolution augmentation, which enables the representation to be scale invariant. Integrating the two augmentations produces better feature representation on several unsupervised benchmarks. Notably, we reach $7 5 . 5 \\%$ top-1 accuracy with linear evaluation over ResNet-50, and $5 9 . 3 \\%$ top-1 accuracy when fine-tuned with only $1 \\%$ labels, as well as consistently outperforming supervised pretraining on several downstream transfer tasks. ",
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+        "text": "1 INTRODUCTION ",
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+        "text": "Learning general representations that can be transferable to different downstream tasks is a key challenge in computer vision. This is usually achieved by fully supervised learning paradigm, e.g., making use of ImageNet labels for pretraining over the past several years. Recently, self-supervised learning has attracted more attention due to its free of human labels. In self-supervised learning, the network aims at exploring the intrinsic distributions of images via a series of predefined pretext tasks (Doersch et al., 2015; Gidaris et al., 2018; Noroozi & Favaro, 2016; Pathak et al., 2016). Among them, instance discrimination (Wu et al., 2018) based methods have achieved remarkable progress (Chen et al., 2020a; He et al., 2020; Grill et al., 2020; Caron et al., 2020). The core idea of instance discrimination is to push away different images, and encourage the representation of different transformations (augmentations) of the same image to be similar. Following this paradigm, self-supervised models are able to generate features that are comparable or even better than those produced by supervised pretraining when evaluated on some downstream tasks, e.g.,COCO detection and segmentation (Chen et al., 2020c;b). ",
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+        "text": "In contrastive learning, the positive pairs are simply constrained within different transformations of the same image, e.g., cropping, color distortion, Gaussian blur, rotation, etc.. Recent advances have demonstrated that better data augmentations (Chen et al., 2020a) really help to improve the representation robustness. However, contrasting two images that are de facto similar in semantic space is not applicable for general representations. It is intuitive to pull semantically similar images for better transferability. DeepCluster (Caron et al., 2018) and Local Aggregation (Zhuang et al., 2019) relax the extreme instance discrimination task via discriminating groups of images instead of an individual image. However, due to the lack of labels, it is inevitable that the positive pairs contain noisy samples, which limits the performance. ",
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+        "text": "In this paper, we target at expanding instance discrimination by exploring local similarities among images. Towards this goal, one need to solve two issues: i) how to select similar images as positive pairs of an image, and ii) how to incorporate these positive pairs, which inevitably contain noisy assignments, into contrastive learning. We propose a new kind of data augmentation, named Centerwise Local Image Mixture, to tackle the above two issues in a robust and efficient way. CLIM consists of two core elements, i.e., a center-wise positive sample selection, as well as a data mixing operation. For positive sample selection, the motivation is that a good representation should be endowed with high intra-class similarity, and we find that although MoCo (He et al., 2020) does not explicitly model invariance to similar images, the intra-class similarity becomes higher as the training process goes. Based on this observation, we explicitly enforce semantically similar images towards the center of clusters, and generate representation with higher intra-class similarity, which we find is beneficial for few shot learning. This is achieved by searching nearest neighbors of an image, and only retaining similar samples that are closer to the corresponding cluster center, which we denote as center-wise local sample selection. As a result, an image is pulled towards the center while do not break the local similarity. ",
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+        "text": "Once similar samples are selected, a direct way is to treat these similar samples as multiple positives for contrastive learning. However, since feature representation in high dimensional space is complex, the returned positive samples inevitably contain noisy assignments, which should not be overconfident. Instead, we rely on data mixing as augmented samples, which can be treated as a smoothing regularization in unsupervised learning. In particular, we apply Cutmix (Yun et al., 2019), a widely used data augmentation in supervised learning, where patches are cut and pasted among the positive pairs to generate new samples. Benefit from the center-wise sample selection, the Cutmix augmentation is only constrained within the local neighborhood of an image, and can be treated as an expansion of current neighborhood space. In this way, similar samples are pulled together in a smoother and robust way, which we find is beneficial for general representation. ",
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+        "text": "Furthermore, we propose multi-resolution augmentation, which aims at contrasting the same image (patch) at different resolutions explicitly, to enable the representation to be scale invariant. We argue that although previous operations such as crop and resize introduce multi-resolution implicitly, they do not compare the same patch at different resolutions directly. As comparisons, multi-resolution incorporates scale invariance into contrastive learning, and significantly boosts the performance even based on a strong baseline. The multi-resolution strategy is simple but effective, and can be combined with current data augmentations for further improving performance. ",
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+        "text": "We evaluate the feature representation on several self-supervised learning benchmarks. In particular, on ImageNet linear evaluation protocol, we achieve $7 5 . 5 \\%$ top-1 accuracy with a standard ResNet50. In few shot setting, when finetuned with only $1 \\%$ labels, we achieve $5 9 . 3 \\%$ top-1 accuracy, surpassing previous works by a large margin. We also validate its transferring ability on several downstream tasks, and consistently outperform the fully supervised counterparts. ",
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+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Unsupervised Representation Learning. Unsupervised learning aims at exploring the intrinsic distribution of data samples via constructing a series of pretext tasks without human labels. These pretext tasks take many forms and vary in utilizing different properties of images. Among them, one family of methods takes advantage of the spatial properties of images, typical pretext tasks include predicting the relative spatial positions of patches (Doersch et al., 2015; Noroozi & Favaro, 2016), or inferring the missing parts of images by inpainting (Pathak et al., 2016), colorization (Zhang et al., 2016), or rotation prediction (Gidaris et al., 2018). Recent progress in self-supervised learning mainly benefits from instance discrimination, which regards each image (and augmentations of itself) as one class for contrastive learning. The motivation behind these works is the InfoMax principle, which aims at maximizing mutual information (Tian et al., 2019; Wu et al., 2018) across different augmentations of the same image (He et al., 2020; Chen et al., 2020a), (Tian et al., 2019). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Data Augmentation. Instance discrimination makes use of several data augmentations, e.g., random cropping, color jittering, horizontal flipping, to define a large view set of vicinities for each image. As has been demonstrated (Chen et al., 2020a; Tian et al., 2020), the effectiveness of instance discrimination methods strongly relies on the type of augmentations. Hoping that the network holds invariance in the local vicinities of each sample. However, current data augmentations are mostly constrained within a single image. An exception is (Shen et al., 2020), where image mixture is used for flattened contrastive predictions. However, such mixture strategy is conducted among all images, which destroys the local similarity when contrasting mixed samples that are semantic dissimilar. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/ffd0a272d956fceeaba42deb11917f1a238c23884dc732393819c4737ac36b63.jpg",
+        "image_caption": [
+            "Figure 1: An illustration of the proposed CLIM and multi-resolution data augmentations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Beyond self-supervised learning, mixing samples from different images is widely used to help alleviate overfitting in training deep networks. In particular, Mixup (Zhang et al., 2017) combines two samples linearly on pixel level, where the target of the synthetic image was a linear combination of one-hot labels. Following Mixup, there are a few variants (Verma et al., 2018) as well as a recent effort named Cutmix (Yun et al., 2019), which combined Mixup and Cutout (DeVries & Taylor, 2017) by cutting and pasting patches. ",
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+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we start by reviewing contrastive learning for unsupervised representation learning. Then we elaborate our proposed CLIM data augmentation, which targets at pulling similar samples via center-wise similar sample selection, followed by a cutmix data augmentation. We also present multi-resolution augmentation that we observe further improves the performance, as well as detailed analysis with recent methods that share similar targets with our method. ",
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+    {
+        "type": "text",
+        "text": "3.1 CONTRASTIVE LEARNING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Contrastive learning targets at training an encoder to map positive pairs to similar representations while pushing away the negative samples in the embedding space. Given unlabeled training set $X = \\{ x _ { 1 } , x _ { 2 } , . . . , x _ { n } \\}$ . Instance-wise contrastive learning aims to learn an encoder $f _ { q }$ that maps the samples $\\boldsymbol { X }$ to embedding space $V = \\{ v _ { 1 } , v _ { 2 } , . . . , v _ { n } \\}$ by optimizing a contrastive loss. Take the Noise Contrastive Estimator (NCE) (Oord et al., 2018) as an example, the contrastive loss is defined as: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/f174a984ffbd98356c2afd659381ff5ee524af8864854195ef08d07381728ab7.jpg",
+        "text": "$$\n\\mathcal { L } _ { n c e } ( x _ { i } , x _ { i } ^ { \\prime } ) = - \\log \\frac { \\exp ( f _ { q } ( x _ { i } ) \\cdot f _ { k } ( x _ { i } ^ { \\prime } ) / \\tau ) } { \\exp ( f _ { q } ( x _ { i } ) \\cdot f _ { k } ( x _ { i } ^ { \\prime } ) / \\tau ) + \\sum _ { j = 1 } ^ { K } \\exp ( f _ { q } ( x _ { i } ) \\cdot f _ { k } ( x _ { j } ^ { \\prime } ) / \\tau ) ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\tau$ is the temperature parameter, and $x _ { i } ^ { \\prime }$ and $\\boldsymbol { x } _ { j } ^ { \\prime }$ denote the positive and negative samples of $x _ { i }$ , respectively. The encoder $f _ { k }$ can be shared (Chen et al., 2020a; Caron et al., 2020) or momentum update of the encoder $f _ { q }$ (He et al., 2020). ",
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+    {
+        "type": "text",
+        "text": "3.2 CLIM: CENTER-WISE LOCAL IMAGE MIXTURE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In contrastive learning, each sample as well as its augmentations is treated as a separate class, while all other samples are regarded as negative examples and pushed away. In principle, semantically similar samples should have similar feature representation in the embedding space, while current contrastive strategies do not consider the semantic similarities among different samples, and only choose different views of the same sample as positive pairs. To solve this issue, we propose a new kind of data augmentation, termed as Center-wise Local Image Mixture, which pulls samples that are semantically similar in an efficient and robust way. The proposed CLIM augmentation consists of two elements, i.e., center-wise local similar sample selection, and a cutmix data augmentation, which would be described in details in the following. ",
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+    {
+        "type": "image",
+        "img_path": "images/95c1d4581531a2c56703083501081c5a75bb3f593b31744aef4ee9ebc8f5df89.jpg",
+        "image_caption": [
+            "Figure 2: Comparison of three positive sample selection strategies, i.e., k-means, knn, and the proposed center-wise local sample selection. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.2.1 CENTER-WISE LOCAL POSITIVE SAMPLE SELECTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As noted by (Wang & Isola, 2020), a good representation should satisfy both alignment and uniformity, which encourages similar images to have similar representation in the embedding space, and meanwhile, semantically similar features are well-clustered. Towards this goal, we propose a positive sample selection strategy that considers both local similarity and global aggregation. This is achieved by searching similar samples within a cluster that the anchor sample belongs to, and only retaining samples that are closer to the corresponding cluster center. We denote it as center-wise local selection as these samples are picked out towards the cluster center among the local neighborhood of an image. In this way, similar samples are progressively pulled to the predefined cluster centers, while do not break the local similarity. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Specifically, given a set of unlabeled images $X = \\{ x _ { 1 } , x _ { 2 } , . . . , x _ { n } \\}$ and the corresponding embedding $V = \\{ v _ { 1 } , v _ { 2 } , . . . , v _ { n } \\}$ with encoder $f _ { \\theta }$ , where $v _ { i } = f _ { \\theta } ( x _ { i } )$ . We cluster the representations $V$ using a standard $\\mathbf { k }$ -means algorithm, and obtain $m$ centers $\\mathbf { C } = \\{ c _ { 1 } , c _ { 2 } , . . . , c _ { m } \\}$ . Given an anchor $x _ { i }$ with its assigned cluster $c ( x _ { i } ) \\in C$ , denote the sample set that belongs to $c ( x _ { i } )$ as $\\Omega _ { 1 } = \\{ x | c ( x ) = c ( x _ { i } ) \\}$ . We search the $k$ nearest neighbors of $x _ { i }$ over the entire space with L2 distance, obtaining sample set $\\pmb { \\Omega } _ { 2 } = \\{ x _ { i 1 } , . . . , x _ { i k } \\}$ . The positive samples are selected based on the following rule: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/6affe03b1e5fd2949b948f220cf3634d54ab19fd977163f6c69d7e09d7666287.jpg",
+        "text": "$$\n\\Omega _ { p } = \\{ x | d ( f _ { \\theta } ( x ) , v _ { c ( x _ { i } ) } ) \\leq d ( f _ { \\theta } ( x _ { i } ) , v _ { c ( x _ { i } ) } ) , x \\in \\Omega _ { 1 } \\cap \\Omega _ { 2 } \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $d ( \\cdot , \\cdot )$ denotes the L2 distance of two samples, and $v _ { c ( x _ { i } ) }$ denotes the feature representation of the corresponding cluster center, respectively. In this way, the samples are aggregated towards the predefined clusters, and meanwhile maintaining the local similarity. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our method combines the advantages of cluster and nearest neighbor methods. An illustration comparing the three methods is shown in Fig. 2. Cluster-based method regards all samples that belong to the same center as positive pairs, which breaks the local similarity among samples especially when the anchor is around the boundary. While nearest neighbor-based method independently pulling samples of an anchor, and does not encourage the well-clustered goal. As a result, the embedding space is not highly concentrated among multiple similar anchors. As comparisons, by center-wise sample selection, similar samples are progressively pulled to the predefined center as well as considering the local similarity. In the experimental section, we would compare the performance of the three methods, and validate the superior performance of our proposed selection strategy. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2.2 CUTMIX DATA AUGMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Once we obtain the positive samples of an anchor, one direct way is to treat these samples similar as the augmented ones for contrastive learning. However, similarity computation in high dimensional space inevitably contains noisy samples, which should not be overconfident for contrasting. To solve this issue, we make use of data mixture strategy, which aims at mixing patches from two different images as augmented samples for contrasting. Data mixing is widely used in supervised learning as label smoothing regularization. The highlight is that without image level labels, we are not able to assign new labels to the augmented samples. Instead, we only mixing samples that are similar in representation, and the mixed samples can be treated as an augmented version of the anchor. In this way, these mixed samples, as well as traditional data augmentations, can be pulled together in contrastive learning. Specifically, given a positive pair $( x _ { i } , \\tilde { x } _ { i } )$ , we conduct data mixing as follows: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6c26fc314941060999853f8ac0a018e854b4d147e3d0a4dfbc634182399ca5eb.jpg",
+        "text": "$$\nx _ { m i x } = \\mathbf { M } \\odot x _ { i } + ( \\mathbf { 1 } - \\mathbf { M } ) \\odot \\tilde { x } _ { i } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            166,
+            604,
+            184
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { M } \\in \\{ 0 , 1 \\} ^ { W \\times H }$ denotes a binary mask indicating the mixed rectangle region of an image, i.e., where to cutout the region in $x _ { i }$ and replaced with a randomly selected patch from ${ \\tilde { x } } _ { i }$ , and $W , H$ denotes the wide and height of an image, respectively. 1 is a binary mask filled with ones, and $\\odot$ is the element-wise multiplication operation. For mask $\\mathbf { M }$ generation, we follow the setting in (Yun et al., 2019). For the mixed sample $x _ { m i x }$ , the positive sample can be either $x _ { i }$ or $\\tilde { x } _ { i }$ , and we reformulate the contrastive learning as combing two NCE loss: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/94db52c66d7c3eb789897ede435a11c886ed18316c4427428189957e85f985b9.jpg",
+        "text": "$$\n{ \\mathcal L } _ { m i x } ( x _ { i } , \\tilde { x } _ { i } ) = \\lambda \\cdot { \\mathcal L } _ { n c e } ( x _ { m i x } , x _ { i } ) + ( 1 - \\lambda ) \\cdot { \\mathcal L } _ { n c e } ( x _ { m i x } , \\tilde { x } _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Where the combination ratio $\\lambda$ is sampled from beta distribution $\\mathtt { B e t a } ( \\alpha , \\alpha )$ with parameter $\\alpha$ . The proposed data mixing augmentation can be seamlessly incorporated into current contrastive learning. The advantages are twofold: first, mixed samples help to expand the neighborhood space of current anchor sample for better representation; second, minimizing the two terms simultaneously can help to maximize the mutual information between $x _ { i }$ and ${ \\tilde { x } } _ { i }$ in a soft manner and perform as smoothing regularization on the prediction for selected positive samples. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 MULTI-RESOLUTION DATA AUGMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Data augmentation plays a key role in current contrastive learning, among them crop augmentation is one of the most effective way (Chen et al., 2020a). In a typical crop augmentation, a sample $x$ with size $H \\times W$ is randomly cropped with ratio $\\sigma$ , and resized to $K _ { t r a i n } \\times K _ { t r a i n }$ as augmented samples, where $K _ { t r a i n } \\times K _ { t r a i n }$ denotes the input resolution for model training. Hence the scaling factor w.r.t. sample $x$ can be described as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/145d93c9fc68789ba30985c0b4b5c94c692f99c9b7ebb8ac578f254bc4dae07a.jpg",
+        "text": "$$\ns = \\frac { 1 } { \\sigma } \\cdot \\frac { K _ { t r a i n } } { \\sqrt { H \\times W } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For crop augmentation, the parameter $K _ { t r a i n }$ is fixed, and the crop ratio $\\sigma$ is randomly selected among positive pairs. As a result, different crop augmentations usually contain different contents, which can be regarded as modeling occlusion invariance to some extent, where each crop sees one view of an image. In this section, we propose a simple but effective data augmentation strategy, named multi-resolution augmentation, which enables the representation to be scale invariant of an example. The highlight is that it is better for contrasting positive pairs with the same content but different resolutions. Specifically, for each positive we keep the crop ratio $\\sigma$ fixed, and adjust $K _ { t r a i n }$ to different resolutions for contrastive loss. An illustration is shown in Fig. 1 .Using multi-resolution, the objective function can be generalized as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ebe6c860b3329984a35225a4ec167b53200e1ff11068b567b01307689e2a16f.jpg",
+        "text": "$$\n\\mathcal { L } _ { m r } = \\sum _ { r , r ^ { \\prime } \\in \\{ r _ { 1 } , \\ldots , r _ { n } \\} } \\mathcal { L } _ { m i x } ( x _ { i } ^ { r } , \\tilde { x } _ { i } ^ { r ^ { \\prime } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\{ r _ { 1 } , . . . , r _ { n } \\}$ indicates the resolution set. In this way, the encoder would be encouraged to discriminate the positive samples with different resolutions from a series of negative keys, which will maximize the mutual information between inputs with different resolutions and discard redundant information brought by resolutions. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Relation with Multi-crop Augmentation. There exist recent works that aim at improving crop augmentations, including multi-crop (Caron et al., 2020) and jigsaw-crop (Misra & Maaten, 2020). However, both methods target at reducing crop ratio $\\sigma$ in Eq.5 and resolution $K _ { t r a i n }$ simultaneously to bridge different parts of an object, and do not explicitly model scale invariance. As comparisons, our proposed multi-resolution strategy fixes the crop ratio to explicitly model scale invariance. In the experimental section, we would compare these two augmentations to validate the difference. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/9157c868d2a73f30cdc8acb3885ccfb9b8a43d4b0d5e149fb1e81b6e744c8511.jpg",
+        "table_caption": [
+            "Table 1: Top-1 accuracies under linear evaluation on ImageNet, using ResNet-50 as encoder "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method Accuracy (%)</td></tr><tr><td>Supervised 76.5</td></tr><tr><td>Colorization (Zhang et al., 2016) 39.6</td></tr><tr><td>Jigsaw (Noroozi &amp; Favaro, 2016) 45.7</td></tr><tr><td>NPID (Wu et al., 2018) 54.0</td></tr><tr><td>LA (Zhuang et al., 2019) 58.8</td></tr><tr><td>MoCo (He et al., 2020) 60.6</td></tr><tr><td>SeLa (YM. et al., 2020) 61.5</td></tr><tr><td>PIRL (Misra &amp; Maaten,2020) 63.6</td></tr><tr><td>CPCv2 (Henaff et al., 2019) 63.8</td></tr><tr><td>PCL (Li et al., 2020) 65.9</td></tr><tr><td>SimCLR (Chen et al., 2020a) 70.0</td></tr><tr><td>MoCo v2 (Chen et al., 2020c) 71.1</td></tr><tr><td>SimCLRv2 (Chen et al., 2020b) 71.7</td></tr><tr><td>InfoMin (Tian et al.,2020) 73.0</td></tr><tr><td>BYOL (Grill et al., 2020) 74.3</td></tr><tr><td>SwAV (Caron et al., 2020) 75.3</td></tr><tr><td>CLIM 75.5</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/21f6e8c82a1aeee53814d6e11c1bdc3d483163d5f52593a4144cfe0d1de9218a.jpg",
+        "table_caption": [
+            "Table 2: Semi-supervised learning with few shot ImageNet labels, using ResNet50 as encoder (averaged by 5 trials) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Top-1/Top-5</td></tr><tr><td>1% labels 10% labels</td><td></td></tr><tr><td>Supervised</td><td>25.4 48.4 56.4 56.4</td></tr><tr><td>PIRL</td><td>30.7 57.2 60.4 83.8</td></tr><tr><td>SimCLR</td><td>48.3 75.5 65.6 87.8</td></tr><tr><td>MoCo v2</td><td>52.4 78.4 65.3 86.6</td></tr><tr><td>BYOL</td><td>53.2 78.4 68.8 89.0</td></tr><tr><td>SwAV</td><td>53.9 78.5 70.2 89.9</td></tr><tr><td>SimCLRv2</td><td>57.9 82.5 68.4 89.2</td></tr><tr><td>CLIM</td><td>59.3 81.6 70.0 89.3</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/470a26f1863429776b939eb1d50bc8ef012808c70e368b15de1505a8d8a6673e.jpg",
+        "table_caption": [
+            "Table 3: Transfer learning on VOC object detection (averaged by 5 trials). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Accuracy (%)</td></tr><tr><td>AP50</td><td>AP75</td></tr><tr><td>Supervised</td><td>81.4</td><td>58.8</td></tr><tr><td>MoCo v2</td><td>82.5</td><td>64.0</td></tr><tr><td>SwAV</td><td>82.6</td><td>:</td></tr><tr><td>CLIM</td><td>82.8</td><td>64.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Relation with Fix-Res. The proposed multi-resolution augmentation is reminiscent of recent work FixRes (Touvron et al., 2019), which also explores resolution issue of better representation, but they are different in both motivation and goal. FixRes is based on the observation that data augmentations induce a significant discrepancy between the size of the objects seen by the classifier at train and test time, and employs different train and test resolutions to fix the train-test resolution discrepancy. The goal is to require less scale invariance for the neural net in FixRes. While our multi-resolution augmentation aims to model the scale invariance explicitly, which is not carefully considered in previous self-supervised learning. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            664,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we assess our pretrained feature representation on several unsupervised benchmarks. We evaluate it on ImageNet under linear evaluation and semi-supervised settings. Then we transfer the learned features to different downstream tasks. We also analyze the performance of our representation with detailed ablation studies. For brief expression, except for the ablation study, we denote our method as CLIM, which includes two kinds of data augmentations. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 LINEAR EVALUATION ON IMAGENET ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The feature representation is trained based on ImageNet 2012 (Russakovsky et al., 2015), using a standard ResNet-50 structure as backbone. We follow the setting in MoCo v2 (Chen et al., 2020c), and the training details are listed in Appendix A. We first evaluate our features by training a linear classifier on top of the frozen representation, following a common protocol in (He et al., 2020; Tian et al., 2019). For linear classifier, the learning rate is initialized as 30 and decayed by 0.1 after 60, 80 epochs, respectively. Table 1 shows the top-1 accuracies with center crop evaluation. Our method achieves an accuracy of $7 5 . 5 \\%$ , surpassing MoCo v2 baseline $( 7 1 . 1 \\% )$ by $4 . 4 \\%$ , and nearly approaching the supervised learning baseline $( 7 6 . 5 \\% )$ . ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/58bbcbe16b1867be37cadb012d9721ce0e938d1ebf73967976b13981f3c1ede8.jpg",
+        "table_caption": [
+            "Table 4: Transfer learning on COCO detection and instance segmentation (averaged by 5 trials) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=3 colspan=1>Method</td><td rowspan=1 colspan=2>Mask R-CNN,R50-FPN,Det</td><td rowspan=1 colspan=2>Mask R-CNN,R50-FPN,InsSeg</td></tr><tr><td rowspan=1 colspan=1>1× schedule</td><td rowspan=1 colspan=1>2× schedule</td><td rowspan=1 colspan=1>1× schedule</td><td rowspan=1 colspan=1>2× schedule</td></tr><tr><td rowspan=1 colspan=1>AP66AP0AP</td><td rowspan=1 colspan=1>AP66AP0AP</td><td rowspan=1 colspan=1>APmkAPAP7</td><td rowspan=1 colspan=1>APmkAPAP7</td></tr><tr><td rowspan=1 colspan=1>Supervised</td><td rowspan=1 colspan=1>38.959.642.0</td><td rowspan=1 colspan=1>40.661.344.4</td><td rowspan=1 colspan=1>35.4 56.5 38.1</td><td rowspan=1 colspan=1>36.8 58.1 39.5</td></tr><tr><td rowspan=1 colspan=1>MoCo v2</td><td rowspan=1 colspan=1>39.259.942.7</td><td rowspan=1 colspan=1>41.562.245.3</td><td rowspan=1 colspan=1>35.7 56.8 38.1</td><td rowspan=1 colspan=1>37.5 59.1 40.1</td></tr><tr><td rowspan=1 colspan=1>CLIM</td><td rowspan=1 colspan=1>39.560.043.3</td><td rowspan=1 colspan=1>41.862.345.7</td><td rowspan=1 colspan=1>35.8 57.0 38.6</td><td rowspan=1 colspan=1>37.7 59.4 40.5</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 SEMI-SUPERVISED TRAINING ON IMAGENET ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We also evaluate our method by fine-tuning the pretrained model with a small subset of labels, following the semi-supervised settings in (Grill et al., 2020; Kornblith et al., 2019; Chen et al., 2020a; Caron et al., 2020). For fair comparisons, we use the same fixed $1 \\%$ and $1 0 \\%$ splits of training data as in (Chen et al., 2020a), and fine-tune all layers using SGD optimizer with momentum of 0.9, and learning rate of 0.0001 for backbone, 10 for the newly initialized fc layer. The fine-tune epochs is set as 60, and the learning rate is decayed by 0.1 after every 20 epochs. During training, only random cropping and flipping data augmentations are used for fair comparison. The results are reported in Table 2. CLIM achieves $5 9 . 3 \\%$ top-1 accuracy with only $1 \\%$ labels, and $7 0 . 0 \\%$ with $1 0 \\%$ labels. The performance gains are larger with $1 \\%$ labels, e.g., $6 . 1 \\%$ higher than BYOL, and $5 . 4 \\%$ better than SwAV, which demonstrates that the proposed feature representation is mainly suitable for extremely few shot learning. Note that SimCLR v2 makes use of other tricks like more MLP layers for better performance, while our method simply adds one fc layer, and still achieves better performance under both settings. ",
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+    {
+        "type": "text",
+        "text": "4.3 DOWNSTREAM TASKS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We also evaluate our feature representation on several downstream tasks, including object detection and instance segmentation, to evaluate the transferability of the learned features. For fair comparison, all experiments follow MoCo settings. ",
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+    {
+        "type": "text",
+        "text": "PASCAL VOC Object Detection. Following the evaluation protocol in (He et al., 2020), we use Faster R-CNN (Ren et al., 2015) with R50-C4 as backbone. We fine-tune all layers on the trainval set of $\\mathrm { \\ V O C { 0 7 + 1 2 } }$ for $2 \\times$ schedule and evaluate on the test set of VOC2007. We report the performances under the metric of AP50 and AP75. As shown in Table 3, on PASCAL VOC, CLIM achieves $8 2 . 8 \\%$ and $6 4 . 5 \\%$ mAP under AP50 and AP75 metric, which is 1.4 points and 5.7 points higher than the fully supervised counterparts, and is slightly better than the results of MoCo v2. ",
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+    {
+        "type": "text",
+        "text": "COCO Object Detection and Instance Segmentation. We also evaluate the representation learned on a large scale COCO dataset. Following (He et al., 2020), we choose Mask R-CNN with FPN as backbone, and fine-tune all the layers on the train set and evaluate on the val set of COCO2017. In Table 4, we report results under both $1 \\times$ and $2 \\times$ schedules. We show that CLIM consistently outperforms the supervised pretrained model and MoCo v2. Under 2X schedule, we achieve $4 1 . 8 \\%$ and $3 7 . 7 \\%$ detection and segmentation accuracies, respectively, which is 1.2 points and 1.1 points better than the supervised couterparts, and also slightly better than the highly optimized MoCo v2. ",
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+    {
+        "type": "text",
+        "text": "LVIS Long Tailed Instance Segmentation. Different from VOC and COCO where the number of training samples is comparable, LVIS is a long-tailed dataset, which contains more than 1200 categories, among them some categories only have less than ten instances. The main challenge is to learn accurate few shot models for classes among the tail of the class distribution, for which little data is available. We evaluate our features on this long-tailed dataset to validate how the unsupervised representation boosts the performance. Similarly, we fine-tune the model (Mask R-CNN, R50-FPN) on the train set and evaluate on the val set of Lvis v0.5. Table 5 shows the result under $2 \\times$ schedule. CLIM outperforms the supervised pretrained model by a large margin and is slightly better than MoCo v2. We claim that it is mainly to the proposed data mixing data augmentation, which is able to learn generalized representations even with extremely few labeled data. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6158e94de3681a3713fb74f1b42c2f888f74fc645102ee464c981e9c5ead7ca2.jpg",
+        "table_caption": [
+            "Table 5: Transfer learning on LVIS long-tailed instance segmentation (averaged by 5 trials) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">Object Det</td><td colspan=\"3\">Instance Seg</td></tr><tr><td>AP66</td><td>AP</td><td>AP</td><td>AP66</td><td>AP</td><td>APm</td></tr><tr><td>Supervised</td><td>24.1</td><td>39.4</td><td>25.0</td><td>24.2</td><td>37.8</td><td>25.1</td></tr><tr><td>MoCo v2</td><td>25.1</td><td>40.4</td><td>26.1</td><td>25.3</td><td>38.4</td><td>27.0</td></tr><tr><td>CLIM</td><td>25.5</td><td>41.2</td><td>26.7</td><td>25.6</td><td>39.5</td><td>27.5</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/d8fdab28c1242d186af1cfafac89bc4a3127fbe127a1a018014ed29367ac47e2.jpg",
+        "table_caption": [
+            "Table 6: Impact of different sample selection "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Strategy</td><td colspan=\"2\">Accuracy (%)</td></tr><tr><td>no mixing</td><td>+cutmix</td></tr><tr><td>MoCo v2</td><td>67.5</td><td>-</td></tr><tr><td>Random</td><td>62.3</td><td>67.1</td></tr><tr><td>KNN</td><td>68.3</td><td>69.5</td></tr><tr><td>K-means</td><td>68.0</td><td>69.2</td></tr><tr><td>KNN ∩ K-means</td><td>68.5</td><td>69.6</td></tr><tr><td>Center-wise</td><td>69.3</td><td>70.1</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6f75cbd6216896b646e28064eef1c53ecedb9fe480660579062dbdb97ba6a00b.jpg",
+        "table_caption": [
+            "Table 7: Impact of different multiple resolutions "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Resolution</td><td>Accuracy (%)</td></tr><tr><td>Multi-Crop</td><td>2×224 + 2×96</td><td>69.7</td></tr><tr><td rowspan=\"4\">Multi-Reso</td><td>r,r&#x27;∈ {224,96}</td><td>70.4</td></tr><tr><td>r,r&#x27; ∈ {224,128}</td><td>71.7</td></tr><tr><td>r,r&#x27; ∈ {224,160}</td><td>72.3</td></tr><tr><td>r,r&#x27;∈ {224,224}</td><td>71.4</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we present ablation studies to better understand how each component affects the performance. Detailed comparisons include 1) positive sample selection, 2) cutmix data augmentation, and 3) multi-resolution augmentation. Unless specified, we train the model for 200 epochs over the ImageNet-1000 and report the top-1 classification accuracy under linear evaluation protocol. ",
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+        "type": "text",
+        "text": "Positive Sample Selection. We first analyze the advantages of our proposed center-wise local sample selection strategy. The compared sample selection alternatives include: ",
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+        "type": "text",
+        "text": "• Random selection: Randomly select a sample from all unlabeled data.   \n• KNN selection: Use $\\mathbf { k }$ -nearest neighbors to build the correlation map among samples, and randomly select a sample from the Top- $k$ $k = 1 0$ ) nearest neighbors as positive samples.   \n• K-means selection: Use $\\mathbf { k }$ -means clustering algorithm to obtain $k$ cluster centers, and randomly select a sample from the corresponding cluster as positive samples. ",
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+    },
+    {
+        "type": "text",
+        "text": "• KNN ∩ K-means selection: Use K-means clustering algorithm to obtain $k$ cluster centers, and randomly select nearest neighbor within the cluster as positive samples. ",
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+    {
+        "type": "text",
+        "text": "The results are shown in the second column of Table 6. In order the inspect the influence of sample selection, we do not conduct cutmix augmentation, and these positive samples are simply pulled via a standard contrastive loss. It can be shown that comparing with the MoCo baseline, both KNN and cluster-based sample selection boost the performance, and notably, simply selecting the union of knn and k-means achieves $\\cdot$ accuracy, which is comparable with result that directly using knn. Since for samples not lie around the boundary, it equals to knn, and does not encourage intra-class compactness. As comparison, our proposed center-wise selection strategy outperforms all the above selection methods. ",
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+    },
+    {
+        "type": "text",
+        "text": "Cutmix Data Augmentation. Data mixing helps to expand the neighborhood space of the target sample, and acts as smoothing regularization for the prediction. As shown in the third column of Table 6, cutmix augmentation consistently improve the performance, comparing with directly pulling similar samples in contrastive loss, and achieve $7 0 . 1 \\%$ accuracy with only 200 training epochs. Notably, with randomly selected positive samples, cutmix operation even obtains $6 7 . 1 \\%$ accuracy, slightly lower than the MoCo baseline, while significantly better than no mixing with only $6 2 . 3 \\%$ accuracy. This can be attributed to the smoothing regularization of cutmix, which is able to alleviate the effect of noisy samples and update model in a more robust way. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Multiple Resolution. Based on CLIM, we further add multi-resolution data augmentation to validate its effectiveness. The results of introducing different resolutions are shown in Table 7. Using multiple resolutions setting with $r , r ^ { \\prime } \\in \\{ 2 2 4 , 1 \\bar { 6 } 0 \\}$ , our method achieves an accuracy of $7 2 . 3 \\%$ with only 200 epochs, which surpasses the baseline of MoCo by $4 . 8 \\%$ , and even much better than the results of MoCo with 800 epochs $( 7 1 . 1 \\% )$ ). ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We also compare our multi-resolution augmentation with multi-crop augmentation proposed in (Caron et al., 2020). $2 \\times 2 2 4 + 2 \\times 9 6$ denotes using two $2 2 4 \\times 2 2 4$ crops with crop-scale $\\sigma \\sim U ( 0 . 2 , 1 . 0 )$ and two $9 6 \\times 9 6$ crops with $\\sigma \\sim U ( 0 . 0 \\bar { 5 } , 0 . 1 4 )$ , referring to (Caron et al., 2020). The main difference is that, the multi-crop strategy targets at capturing relationship between local and global information, while our proposed multiple resolution target at enabling the encoder with scale invariance. We find that multi-crop slightly deteriorates the performance of CLIM $( 7 0 . 1 \\%$ versus $6 9 . 7 \\%$ ), partially because data mixing behaves like image cropping augmentation, and shares similarity with multi-crop strategy. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we proposed CLIM data augmentation, to efficiently pull semantically similar samples for better representation in contrastive learning. The main contributions of CLIM consist of two elements, center-wise positive sample selection, which considers both local similarity and global aggregation property. In such way, similar samples are progressively aggregated to a series of predefined clusters, while not breaking the local similarity; and data mixing augmentation, which expands the neighborhood space of an example by mixing two images, and acts as a smoothing regularization for contrastive loss. Furthermore, we present a simple but effective multi-resolution augmentation, which explicitly model scale invariance to further improve the representation. Experiments evaluated on several unsupervised benchmarks demonstrate the effectiveness of our method. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            449,
+            285,
+            463
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mathilde Caron, Piotr Bojanowski, Armand Joulin, and Matthijs Douze. Deep clustering for unsupervised learning of visual features. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 132–149, 2018.   \nMathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. arXiv preprint arXiv:2006.09882, 2020.   \nTing Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv preprint arXiv:2002.05709, 2020a.   \nTing Chen, Simon Kornblith, Kevin Swersky, Mohammad Norouzi, and Geoffrey Hinton. Big selfsupervised models are strong semi-supervised learners. arXiv preprint arXiv:2006.10029, 2020b.   \nXinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. arXiv preprint arXiv:2003.04297, 2020c.   \nTerrance DeVries and Graham W Taylor. Improved regularization of convolutional neural networks with cutout. arXiv preprint arXiv:1708.04552, 2017.   \nCarl Doersch, Abhinav Gupta, and Alexei A Efros. Unsupervised visual representation learning by context prediction. In Proceedings of the IEEE international conference on computer vision, pp. 1422–1430, 2015.   \nSpyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. arXiv preprint arXiv:1803.07728, 2018.   \nJean-Bastien Grill, Florian Strub, Florent Altche, Corentin Tallec, Pierre H Richemond, Elena ´ Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, et al. Bootstrap your own latent: A new approach to self-supervised learning. arXiv preprint arXiv:2006.07733, 2020.   \nKaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9729–9738, 2020. ",
+        "bbox": [
+            171,
+            472,
+            826,
+            931
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Olivier J Henaff, Aravind Srinivas, Jeffrey De Fauw, Ali Razavi, Carl Doersch, SM Eslami, and ´ Aaron van den Oord. Data-efficient image recognition with contrastive predictive coding. arXiv preprint arXiv:1905.09272, 2019. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jeff Johnson, Matthijs Douze, and Herve J ´ egou. Billion-scale similarity search with gpus. ´ IEEE Transactions on Big Data, 2019. ",
+        "bbox": [
+            173,
+            154,
+            823,
+            183
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Simon Kornblith, Jonathon Shlens, and Quoc V Le. Do better imagenet models transfer better? In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2661–2671, 2019. ",
+        "bbox": [
+            178,
+            190,
+            823,
+            233
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Junnan Li, Pan Zhou, Caiming Xiong, Richard Socher, and Steven CH Hoi. Prototypical contrastive learning of unsupervised representations. arXiv preprint arXiv:2005.04966, 2020. ",
+        "bbox": [
+            171,
+            242,
+            823,
+            271
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ishan Misra and Laurens van der Maaten. Self-supervised learning of pretext-invariant representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6707–6717, 2020. ",
+        "bbox": [
+            174,
+            279,
+            823,
+            321
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mehdi Noroozi and Paolo Favaro. Unsupervised learning of visual representations by solving jigsaw puzzles. In European Conference on Computer Vision, pp. 69–84. Springer, 2016. ",
+        "bbox": [
+            173,
+            330,
+            823,
+            359
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018. ",
+        "bbox": [
+            171,
+            367,
+            823,
+            396
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros. Context encoders: Feature learning by inpainting. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2536–2544, 2016. ",
+        "bbox": [
+            176,
+            404,
+            823,
+            448
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in neural information processing systems, pp. 91–99, 2015. ",
+        "bbox": [
+            176,
+            454,
+            823,
+            497
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211–252, 2015. ",
+        "bbox": [
+            178,
+            506,
+            821,
+            549
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhiqiang Shen, Zechun Liu, Zhuang Liu, Marios Savvides, and Trevor Darrell. Rethinking image mixture for unsupervised visual representation learning. arXiv preprint arXiv:2003.05438, 2020. ",
+        "bbox": [
+            174,
+            556,
+            820,
+            587
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. arXiv preprint arXiv:1906.05849, 2019. ",
+        "bbox": [
+            173,
+            593,
+            823,
+            622
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yonglong Tian, Chen Sun, Ben Poole, Dilip Krishnan, Cordelia Schmid, and Phillip Isola. What makes for good views for contrastive learning. arXiv preprint arXiv:2005.10243, 2020. ",
+        "bbox": [
+            173,
+            631,
+            823,
+            660
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hugo Touvron, Andrea Vedaldi, Matthijs Douze, and Herve J ´ egou. Fixing the train-test resolution ´ discrepancy. In Advances in Neural Information Processing Systems, pp. 8252–8262, 2019. ",
+        "bbox": [
+            173,
+            667,
+            823,
+            696
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Vikas Verma, Alex Lamb, Christopher Beckham, Amir Najafi, Ioannis Mitliagkas, Aaron Courville, David Lopez-Paz, and Yoshua Bengio. Manifold mixup: Better representations by interpolating hidden states. arXiv preprint arXiv:1806.05236, 2018. ",
+        "bbox": [
+            178,
+            704,
+            823,
+            748
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tongzhou Wang and Phillip Isola. Understanding contrastive representation learning through alignment and uniformity on the hypersphere. arXiv preprint arXiv:2005.10242, 2020. ",
+        "bbox": [
+            174,
+            756,
+            820,
+            786
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via nonparametric instance discrimination. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3733–3742, 2018. ",
+        "bbox": [
+            173,
+            792,
+            823,
+            837
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Asano YM., Rupprecht C., and Vedaldi A. Self-labelling via simultaneous clustering and representation learning. In International Conference on Learning Representations (ICLR), 2020. ",
+        "bbox": [
+            173,
+            843,
+            823,
+            873
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo. Cutmix: Regularization strategy to train strong classifiers with localizable features. In Proceedings of the IEEE International Conference on Computer Vision, pp. 6023–6032, 2019. ",
+        "bbox": [
+            176,
+            881,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In European conference on computer vision, pp. 649–666. Springer, 2016. ",
+        "bbox": [
+            173,
+            140,
+            823,
+            170
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chengxu Zhuang, Alex Lin Zhai, and Daniel Yamins. Local aggregation for unsupervised learning of visual embeddings. In Proceedings of the IEEE International Conference on Computer Vision, pp. 6002–6012, 2019. ",
+        "bbox": [
+            173,
+            179,
+            823,
+            222
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A IMPLEMENTATION DETAILS ",
+        "bbox": [
+            176,
+            102,
+            437,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 IMPLEMENTATION DETAILS FOR CONTRASTIVE PRETRAINING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            642,
+            147
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Architecture and Optimization. We follow the setting in MoCo v2 (Chen et al., 2020c), which relies on two encoders, one for training and the other one for momentum update $m = 0 . 9 9 9$ ) to store negative keys. Following SimCLR (Chen et al., 2020a), we replace the fc head with a 2-layer MLP to project the output of the final pooling layer to 128-d. We use SGD as optimizer, with weight decay setting as 0.0001 and the momentum as 0.9. We use a mini-batch size of 512 on 16 V100 GPUs with a cosine learning rate schedule decayed from 0.06. We train the model for 1200 epochs, as we introducing data mixing augmentation, and usually requires more epochs for better performance as in supervised learning (Yun et al., 2019). 1 ",
+        "bbox": [
+            173,
+            159,
+            825,
+            271
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Image Augmentations. We combine the proposed augmentations with previous widely used basic augmentation strategies, following the settings in (Chen et al., 2020a; He et al., 2020). The basic augmentations are listed below, as well as the corresponding parameters. ",
+        "bbox": [
+            173,
+            277,
+            823,
+            320
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• RandomResizedCrop: A crop of random size (from 0.2 to 1.0) of the original size and a random aspect ratio (from $3 / 4$ to 4/3) of the original aspect ratio is made. • RandomFlip: Randomly horizontally flip the image with a probability of 0.5. • ColorJitter: Randomly change the brightness, contrast and saturation of an image. • RandomGrayscale: Randomly convert RGB image to grayscale with a probability of 0.2. • RandomGaussianBlur: Randomly blur the image with a probability of 0.5. The radius is randomly sampled from 0.1 to 2.0. ",
+        "bbox": [
+            212,
+            332,
+            825,
+            449
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 DETAILS OF POSITIVE SAMPLE SELECTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            465,
+            511,
+            479
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We implement k-means and knn by faiss (Johnson et al., 2019). For efficiency, we perform clustering and knn computation every 5 epochs, since each iteration can be finished within minutes, the extra computation cost is marginal comparing with the budget for model training. The number of clusters is set as $1 0 K$ , and we select the top 40 nearest neighbors in knn. In order to balance the contribution of each image, we randomly select 10 positive samples for the following cutmix augmentations. For situations where there remained no more than 10 examples (e.g., the anchor is already around the cluster center), we simply select the most nearest samples among the remained top- $4 0 \\mathrm { k n n }$ samples. ",
+        "bbox": [
+            173,
+            491,
+            825,
+            603
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B MORE ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            622,
+            433,
+            638
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This section gives more detailed analysis w.r.t. some hyperparameters. Unless specified, we train the model for 200 epochs over the ImageNet-1000 and report the top-1 classification accuracy under linear evaluation protocol. ",
+        "bbox": [
+            178,
+            654,
+            823,
+            696
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/a891f976ad2b72a1059d75c6c961d1abbf079bb75e745224fc40bc1d9952f4ea.jpg",
+        "table_caption": [
+            "Table 8: Impact of the number of clusters $m$ and $k$ of knn "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Number of Clusters (m)</td><td colspan=\"3\">5000</td><td colspan=\"3\">10000</td><td colspan=\"3\">20000</td></tr><tr><td>knn (k）</td><td>20</td><td>40</td><td>60</td><td>20</td><td>40</td><td>60</td><td>20</td><td>40</td><td>60</td></tr><tr><td> Accuracy (%)</td><td>69.1</td><td>69.5</td><td>69.4</td><td>70.0</td><td>70.1</td><td>69.7</td><td>69.6</td><td>69.9</td><td>69.5</td></tr></table>",
+        "bbox": [
+            192,
+            739,
+            807,
+            799
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The number of Clusters $m$ and the $k$ in Knn. Here we inspect the impact of the number of clusters $m$ in $\\mathbf { k }$ -means and the $k$ in knn to analyze their effect on the performance. In order to ensure local similarity, we restrict the nearest neighbors within a range from 20 to 60. The results for different clusters and top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ neighbors are shown in Table 8. We observe that CLIM consistently improves ",
+        "bbox": [
+            173,
+            819,
+            825,
+            875
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "the performance comparing the baseline Moco $6 7 . 5 \\%$ , and is relatively robust to different $m$ and $k$ .   \nNotably, the best performance is achieved when $m = 1 0 0 0 0$ , $k = 4 0$ . ",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameters $\\alpha$ in Cutmix. The combination $\\lambda$ in cutmix is sampled from the beta distribution $\\mathtt { B e t a } ( \\alpha , \\alpha )$ , where $\\alpha$ plays an important role in data mixing augmentation, which controls the strength of interpolation between the anchor and its positive pair. Here we inspect how different $\\alpha \\in \\{ 1 , 1 . 5 , 2 , 2 . 5 \\}$ affect the representation. As shown in Table 9. We find that the performance is relatively robust to different $\\alpha$ , and the best performance is achieved when $\\alpha$ is set as 2. ",
+        "bbox": [
+            173,
+            138,
+            825,
+            209
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/b6cef28b8d7fee69b0ca313fb48006b8c406b05b5d2179a5658aaaec2adec066.jpg",
+        "table_caption": [
+            "Table 9: Impact of $\\alpha$ in cutmix "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>a</td><td>1.0</td><td>1.5</td><td>2.0</td><td>2.5</td></tr><tr><td>Accuracy (%)</td><td>69.7</td><td>69.9</td><td>70.1</td><td>69.8</td></tr></table>",
+        "bbox": [
+            346,
+            253,
+            650,
+            292
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Ablation study on mixing strategies. Our method targets at generating new samples that expanding the neighborhood of an anchor. Here we compare performance of using mixup data augmentation, a widely used method in supervised settings. We try different choices of beta distribution for Mixup (Zhang et al., 2017) and choose the best one $\\alpha = 0 . 2$ ) for comparison. Table 10 shows that Cutmix performs better than Mixup, partially because mixup destroys the real pixel distribution (destroys the naturality of pixels). ",
+        "bbox": [
+            173,
+            314,
+            825,
+            397
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/72626f551e6a809a5de6a4d3e9a5f8c3cfc4458ee59b9bd8f7fa25f6ce4001ef.jpg",
+        "table_caption": [
+            "Table 10: Ablation study on the mixing methods "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Accuracy (%)</td></tr><tr><td>Mixup</td><td>69.5</td></tr><tr><td>Cutmix</td><td>70.1</td></tr></table>",
+        "bbox": [
+            405,
+            441,
+            591,
+            501
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Extra ablation experiments for longer training schedule. We compare the improvements brought by different components of our proposed method for longer training schedule (800 epochs). Table 11 shows the top-1 accuracies under linear evaluation protocol. Our method consistently outperforms the MoCo v2 baseline, which demonstrates the effectiveness of our proposed method. ",
+        "bbox": [
+            173,
+            529,
+            825,
+            584
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/cae2c7ba6dfedcce3d6d80bc279f5a0d62553acb5f4156df834764b6542c3c7b.jpg",
+        "table_caption": [
+            "Table 11: Ablation study on the longer training schedule "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Accuracy (%)</td></tr><tr><td>MoCo v2</td><td>71.1</td></tr><tr><td>Center-wise+ cutmix</td><td>73.7</td></tr><tr><td>Center-wise+ cutmix+Multi-reso</td><td>75.2</td></tr></table>",
+        "bbox": [
+            316,
+            628,
+            679,
+            705
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C MORE EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            738,
+            478,
+            756
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Visualization of Feature Representation. We visualize the feature space to better understand how CLIM augmentation pulls similar samples. Specifically, we randomly choose 10 classes from the validation set and provide the $t$ -sne visualization of feature representation generated by CLIM, supervised training and MoCo v2. As shown in Fig. 3, the same color denotes features with the same label. It can be shown that CLIM takes on higher aggregation property comparing with MoCo, and the fully supervised learned representation reveals the highest aggregation due to it makes use of image labels. Furthermore, we compute the intra-class similarity as the average cosine distance among all intra-class pairwise samples, and report the average similarity across 1000 classes, as shown in Table 12, CLIM achieves an intra-class similarity of 0.65, which is much higher than that in MoCo v2 with similarity of only 0.58. As comparison, we also list the result of supervised learning, with a similarity metric of 0.75. ",
+        "bbox": [
+            173,
+            770,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/55b0297d115f1b5f87a6d8eadfb7c6b2db53a21c9a14da60c5f87e4d83eb84f9.jpg",
+        "image_caption": [
+            "Figure 3: t-sne visualization of representation learned by MoCo, CLIM and supervised learning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            106,
+            812,
+            234
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/c497142066bb34fa3c606c520b32fb532e9dcb8c833edf1ea5ffe13e1b243179.jpg",
+        "table_caption": [
+            "Table 12: Intra-class similarity for different models "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Intra-class Similarity</td></tr><tr><td>Supervised</td><td>0.75</td></tr><tr><td>MoCo v2</td><td>0.58</td></tr><tr><td>CLIM</td><td>0.65</td></tr></table>",
+        "bbox": [
+            222,
+            319,
+            457,
+            395
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/b0cc9c3f895abfc095569f3153ac0fe9eaaffafd8d175d0df1c39bc988619a6f.jpg",
+        "table_caption": [
+            "Table 13: Results of different training epochs "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Epochs</td><td>Accuracy (%)</td></tr><tr><td>200</td><td>72.3</td></tr><tr><td>800</td><td>75.2</td></tr><tr><td>1200</td><td>75.5</td></tr></table>",
+        "bbox": [
+            589,
+            319,
+            751,
+            395
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Results of Different Training Epochs. In Table 13, we compare CLIM trained with different epochs. Our method achieves an accuracy of $7 2 . 3 \\%$ with only 200 epochs, $7 5 . 2 \\%$ with 800 epochs, and can be further improved to $7 5 . 5 \\%$ when training with 1200 epochs. ",
+        "bbox": [
+            174,
+            412,
+            825,
+            455
+        ],
+        "page_idx": 13
+    }
+]

parse/train/HJeTo2VFwH/HJeTo2VFwH.md

@@ -0,0 +1,339 @@
+# A SIGNAL PROPAGATION PERSPECTIVE FOR PRUNING NEURAL NETWORKS AT INITIALIZATION
+
+Namhoon Lee1, Thalaiyasingam Ajanthan2, Stephen Gould2, Philip H. S. Torr1
+
+1University of Oxford 2Australian National University 1{namhoon,phst}@robots.ox.ac.uk 2{thalaiyasingam.ajanthan, stephen.gould}@anu.edu.au
+
+# ABSTRACT
+
+Network pruning is a promising avenue for compressing deep neural networks. A typical approach to pruning starts by training a model and then removing redundant parameters while minimizing the impact on what is learned. Alternatively, a recent approach shows that pruning can be done at initialization prior to training, based on a saliency criterion called connection sensitivity. However, it remains unclear exactly why pruning an untrained, randomly initialized neural network is effective. In this work, by noting connection sensitivity as a form of gradient, we formally characterize initialization conditions to ensure reliable connection sensitivity measurements, which in turn yields effective pruning results. Moreover, we analyze the signal propagation properties of the resulting pruned networks and introduce a simple, data-free method to improve their trainability. Our modifications to the existing pruning at initialization method lead to improved results on all tested network models for image classification tasks. Furthermore, we empirically study the effect of supervision for pruning and demonstrate that our signal propagation perspective, combined with unsupervised pruning, can be useful in various scenarios where pruning is applied to non-standard arbitrarily-designed architectures.
+
+# 1 INTRODUCTION
+
+Deep learning has made great strides in machine learning and been applied to various fields from computer vision and natural language processing, to health care and playing games (LeCun et al., 2015). Despite the immense success, however, it remains challenging to deal with the excessive computational and memory requirements of large neural network models. To this end, lightweight models are often preferred, and network pruning, a technique to reduce parameters in a network, has been widely employed to compress deep neural networks (Han et al., 2016). Nonetheless, designing pruning algorithms has been often purely based on ad-hoc intuition lacking rigorous underpinning, partly because pruning was typically carried out after training the model as a post-processing step or interwoven with the training procedure, without adequate tools to analyze.
+
+Recently, Lee et al. (2019) have shown that pruning can be done on randomly initialized neural networks in a single-shot prior to training (i.e., pruning at initialization). They empirically showed that as long as the initial random weights are drawn from appropriately scaled Gaussians (e.g., Glorot & Bengio (2010)), their pruning criterion called connection sensitivity can be used to prune deep neural networks, often to an extreme level of sparsity while maintaining good accuracy once trained. However, it remains unclear as to why pruning at initialization is effective, how it should be understood theoretically and whether it can be extended further.
+
+In this work, we first look into the effect of initialization on pruning, and find that initial weights have critical impact on connection sensitivity, and therefore, pruning results. Deeper investigation shows that connection sensitivity is determined by an interplay between gradients and weights. Therefore when the initial weights are not chosen appropriately, the propagation of input signals into layers of these random weights can result in saturating error signals (i.e., gradients) under backpropagation, and hence unreliable connection sensitivity, potentially leading to a catastrophic pruning failure.
+
+This result leads us to develop a signal propagation perspective for pruning at initialization, and to provide a formal characterization of how a network needs to be initialized for reliable connection sensitivity measurements and in turn effective pruning. Precisely, we show that a sufficient condition to ensure faithful1 connection sensitivity is layerwise dynamical isometry, which is defined as all singular values of the layerwise Jacobians being concentrated around 1. Our signal propagation perspective is inspired by the recent literature on dynamical isometry and mean field theory (Saxe et al., 2014; Poole et al., 2016; Schoenholz et al., 2017; Pennington et al., 2017), in which the general signal propagation in neural networks is studied. We extend this result to understanding and improving pruning at initialization.
+
+Moreover, we study signal propagation in the pruned sparse networks and its effect on trainability. We find that pruning neural networks can indeed break dynamical isometry, and hence, hinders signal propagation and degrades the training performance of the resulting sparse network. In order to address this issue, we propose a simple, yet effective data-free method to recover the layerwise orthogonality given the sparse topology, which in turn improves the training performance of the compressed network significantly. Our analysis further reveals that in addition to signal propagation, the choice of pruning method and sparsity level can influence trainability in sparse neural networks.
+
+Perfect layerwise dynamical isometry cannot always be ensured in the modern networks that have components such as ReLU nonlinearities (Pennington et al., 2017) and/or batch normalization (Yang et al., 2019). Even in such cases, however, our experiments on various modern architectures (including convolutional and residual neural networks) indicate that connection sensitivity computed based on layerwise dynamical isometry is robust and consistently outperforms pruning based on other initialization schemes. This indicates that the signal propagation perspective is not only important to theoretically understand pruning at initialization, but also it improves the results of pruning for a range of networks of practical interest.
+
+Furthermore, this signal propagation perspective for pruning poses another important question: how informative is the error signal computed on randomly initialized networks, or can we prune neural networks even without supervision? To understand this, we compute connection sensitivity scores with different unsupervised surrogate losses and evaluate the pruning results. Interestingly, our results indicate that we can in fact prune networks in an unsupervised manner to extreme sparsity levels without compromising accuracy, and it often compares competitively to pruning with supervision. Moreover, we test if pruning at initialization can be extended to obtain architectures that yield better performance than standard pre-designed architectures with the same number of parameters. In fact, this process, which we call neural architecture sculpting, compares favorably against hand-designed architectures, taking network pruning one step further towards neural architecture search.
+
+# 2 PRELIMINARIES
+
+Pruning at initialization. The principle behind conventional approaches for network pruning is to find unnecessary parameters, such that by eliminating them the complexity of the model is reduced while minimizing the impact on what is learned (Reed, 1993). Naturally, a typical pruning algorithm starts after convergence to a minimum or training to some degree. This pretraining requirement has been left unattended until Lee et al. (2019) recently showed that pruning can be performed on untrained networks at initiailzation prior to training. They proposed a method called SNIP which relies on a new saliency criterion, namely connection sensitivity, defined as follows:
+
+$$
+s _ { j } ( \mathbf { w } ; \mathcal { D } ) = \frac { | g _ { j } ( \mathbf { w } ; \mathcal { D } ) | } { \sum _ { k = 1 } ^ { m } | g _ { k } ( \mathbf { w } ; \mathcal { D } ) | } , \qquad \mathrm { w h e r e } \quad g _ { j } ( \mathbf { w } ; \mathcal { D } ) = \left. \frac { \partial L ( \mathbf { c } \odot \mathbf { w } ; \mathcal { D } ) } { \partial c _ { j } } \right| _ { \mathbf { c } = 1 } .
+$$
+
+Here, $s _ { j }$ is the saliency of the parameter $j$ , $\mathbf { w _ { \lambda } } \in \mathbb { R } ^ { m }$ is the network parameters, $\mathbf { c } \in \{ 0 , 1 \} ^ { m }$ is the auxiliary indicator variables representing the connectivity of network parameters, $m$ is the total number of parameters in the network, and $\mathcal { D }$ is a given dataset. Also, $g _ { j }$ is the derivative of the loss $L$ with respect to $c _ { j }$ , which turns out to be an infinitesimal approximation of the change in the loss with respect to removing the parameter $j$ . Designed to be computed at initialization, pruning is performed by keeping top- $\kappa$ (where $\kappa$ denotes a desired sparsity level) salient parameters based on the above sensitivity scores.
+
+Dynamical isometry and mean field theory. The success of training deep neural networks is due in large part to the initial weights (Hinton & Salakhutdinov, 2006; Glorot & Bengio, 2010; Pascanu et al., 2013). In essence, the principle behind these random weight initializations is to have the mean squared singular value of a network’s input-output Jacobian close to 1, so that on average, an error vector will preserve its norm under backpropagation; however, this is not sufficient to prevent amplification or attenuation of an error vector on worst case. A stronger condition that having as many singular values as possible near 1 is called dynamical isometry (Saxe et al., 2014). Under this condition, error signals backpropagate isometrically through the network, approximately preserving its norm and all angles between error vectors. Alongside dynamical isometry, mean field theory is used to develop a theoretical understanding of signal propagation in neural networks with random parameters (Poole et al., 2016). Precisely, the mean field approximation states that preactivations of wide, untrained neural networks can be captured as a Gaussian distribution. Recent works revealed a maximum depth through which signals can propagate at initialization, and verified that networks are trainable when signals can travel all the way through them (Schoenholz et al., 2017; Yang & Schoenholz, 2017; Xiao et al., 2018).
+
+# 3 SIGNAL PROPAGATION PERSPECTIVE TO PRUNING RANDOM NETWORKS
+
+Problem setup. Consider a fully-connected, feed-forward neural network with weight matrices $\mathbf { W } ^ { l } \in \mathbb { R } ^ { N \times N }$ , biases $\mathbf { b } ^ { l } \in \mathbb { R } ^ { N }$ , pre-activations $\mathbf { h } ^ { l } \in \mathbb { R } ^ { N }$ , and post-activations $\mathbf { x } ^ { l } \in \mathbb { R } ^ { N }$ , for $l \in$ $\{ 1 \ldots K \}$ up to $K$ layers. Now, the feed-forward dynamics of a network can be written as,
+
+$$
+\mathbf { x } ^ { l } = \phi ( \mathbf { h } ^ { l } ) , \qquad \mathbf { h } ^ { l } = \mathbf { W } ^ { l } \mathbf { x } ^ { l - 1 } + \mathbf { b } ^ { l } ,
+$$
+
+where $\phi : \mathbb { R }  \mathbb { R }$ is an elementwise nonlinearity, and the input is denoted by $\mathbf { x } ^ { 0 }$ . Given the network configuration, the parameters are initialized by sampling from a probability distribution, typically a zero mean Gaussian with scaled variance (LeCun et al., 1998; Glorot & Bengio, 2010).
+
+# 3.1 EFFECT OF INITIALIZATION ON PRUNING
+
+It is observed in Lee et al. (2019) that pruning results tend to improve when initial weights are drawn from a scaled Gaussian, or so-called variance scaling initialization (LeCun et al., 1998; Glorot & Bengio, 2010; He et al., 2015). As we wish to better understand the role of these random initial weights in pruning, we will examine the effect of varying initialization on the pruning results.
+
+In essence, variance scaling schemes introduce normalization factors to adjust the variance $\sigma$ of the weight sampling distribution, which can be summarized as $\begin{array} { r } { \sigma  \frac { \alpha } { \psi _ { l } } \sigma } \end{array}$ , where $\psi _ { l }$ is a layerwise scalar that depends on an architecture specification such as the number of output neurons in the previous layer (e.g., fan-in), and $\alpha$ is a global scalar throughout the network. Notice in case of a network with layers of the same width, the variance can be controlled by a single scalar $\textstyle \gamma = { \frac { \alpha } { \psi } }$ as $\psi _ { l } = \psi$ for all layers $l$ . In particular, we take both linear and tanh multilayer perceptron networks (MLP) of layers $K = 7$ and width $N = 1 0 0$ on MNIST with $\sigma = 1$ as the default, similar to Saxe et al. (2014). We initialize these networks with different $\gamma$ , compute the connection sensitivity, prune it, and then visualize layerwise the resulting sparsity patterns c as well as the corresponding connection sensitivity used for pruning in Figure 1.
+
+It is seen in the sparsity patterns that for the tanh network, unlike the linear case, more parameters tend to be pruned in the later layers than the earlier layers. As a result, this limits the learning capability of the subnetwork critically when a high sparsity level is requested; e.g., for $\bar { \kappa } = 9 0 \%$ , only a few parameters in later layers are retained after pruning. This is explained by the connection sensitivity plot. The sensitivity of parameters in the nonlinear network tends to decrease towards the later layers, and therefore, choosing the top- $\kappa$ parameters globally based on the sensitivity scores results in a subnetwork in which retained parameters are distributed highly non-uniformly and sparsely towards the end of the network. This result implies that the initial weights have a crucial effect on the connection sensitivity, and from there, the pruning results.
+
+![](images/a88651e8398c5c0bf25ce6ab4afa3502637fa3bc5ab864585ebaeabd4504dc3e.jpg)  
+Figure 1: (left) layerwise sparsity patterns $c \in \{ 0 , 1 \} ^ { 1 0 0 \times 1 0 0 }$ obtained as a result of pruning for the sparsity level $\bar { \kappa } = \{ 1 0 , . . , \bar { 9 0 } \} \%$ . Here, black(0)/white(1) pixels refer to pruned/retained parameters; (right) connection sensitivities (CS) measured for the parameters in each layer. All networks are initialized with $\gamma = 1 . 0$ . Unlike the linear case, the sparsity pattern for the tanh network is nonuniform over different layers. When pruning for a high sparsity level (e.g., $\bar { \kappa } = 9 0 \%$ ), this becomes critical and leads to poor learning capability as there are only a few parameters left in later layers. This is explained by the connection sensitivity plot which shows that for the nonlinear network parameters in later layers have saturating, lower connection sensitivities than those in earlier layers.
+
+# 3.2 GRADIENT SIGNAL IN CONNECTION SENSITIVITY
+
+We posit that the unreliability of connection sensitivity observed in Figure 1 is due to poor signal propagation: an initialization that projects the input signal to be strongly amplified or attenuated in the forward pass will saturate the error signal under backpropagation (i.e., gradients), and hence will result in poorly calibrated connection sensitivity scores across layers, which will eventually lead to poor pruning results, potentially with complete disconnection of signal paths (e.g., entire layer).
+
+Precisely, we give the relationship between the connection sensitivity and the gradients as follows. From Equation 1, connection sensitivity is a normalized magnitude of gradients with respect to the connectivity parameters c. Here, we use the vectorized notation where w denotes all learnable parameters and c denotes the corresponding connectivity parameters. From chain rule, we can write:
+
+$$
+\left. \frac { \partial L ( \mathbf { c } \odot \mathbf { w } ; \mathcal { D } ) } { \partial \mathbf { c } } \right| _ { \mathbf { c } = \mathbf { 1 } } = \left. \frac { \partial L ( \mathbf { c } \odot \mathbf { w } ; \mathcal { D } ) } { \partial ( \mathbf { c } \odot \mathbf { w } ) } \right| _ { \mathbf { c } = \mathbf { 1 } } \odot \mathbf { w } = \frac { \partial L ( \mathbf { w } ; \mathcal { D } ) } { \partial \mathbf { w } } \odot \mathbf { w } .
+$$
+
+Therefore, $\partial L / \partial \mathbf { c }$ is the gradients ${ \partial L } / { \partial \mathbf { w } }$ amplified (or attenuated) by the corresponding weights w, i.e., ${ \partial L } / { \partial { \dot { c } } _ { j } } = { \partial L } / { \partial w _ { j } w _ { j } }$ for all $j \in \{ \bar { 1 } \ldots m \}$ . Considering $\bar { \partial } L / \partial c _ { j }$ for a given $j$ , since $w _ { j }$ does not depend on any other layers or signal propagation, the only term that depends on signal propagation in the network is the gradient term $\partial L / \partial w _ { j }$ . Hence, a necessary condition to ensure faithful $\partial L / \partial \mathbf { c }$ (and connection sensitivity) is that the gradients ${ \partial L } / { \partial \mathbf { w } }$ need to be faithful. In the following section, we formalize this from a signal propagation perspective, and characterize an initial condition that ensures reliable connection sensitivity measurement.
+
+# 3.3 LAYERWISE DYNAMICAL ISOMETRY
+
+# 3.3.1 GRADIENTS IN TERMS OF JACOBIANS
+
+From the feed-forward dynamics of a network in Equation 2, the network’s input-output Jacobian corresponding to a given input $\mathbf { x } ^ { 0 }$ can be written, by the chain rule of differentiation, as:
+
+$$
+{ \bf J } ^ { 0 , K } = \frac { \partial { \bf x } ^ { K } } { \partial { \bf x } ^ { 0 } } = \prod _ { l = 1 } ^ { K } { \bf D } ^ { l } { \bf W } ^ { l } ,
+$$
+
+where $\mathbf { D } ^ { l } \in \mathbb { R } ^ { N \times N }$ is a diagonal matrix with entries $\mathbf { D } _ { i j } ^ { l } = \phi ^ { \prime } ( h _ { i } ^ { l } ) \delta _ { i j }$ , with $\phi ^ { \prime }$ denoting the derivative of nonlinearity $\phi$ , and $\delta _ { i j } = \mathbb { I } [ i = j ]$ is the Kronecker delta. Here, we will use $\mathbf { J } ^ { k , l }$ to denote the Jacobian from layer $k$ to layer $l$ . Now, we give the relationship between gradients and Jacobians:
+
+Proposition 1. Let $\epsilon = \partial L / \partial \mathbf { x } ^ { K }$ denote the error signal and $\mathbf { x } ^ { 0 }$ denote the input signal. Then,
+
+1. the gradients satisfy:
+
+$$
+{ \bf g } _ { { \bf w } ^ { l } } ^ { T } = \epsilon { \bf J } ^ { l , K } { \bf D } ^ { l } \otimes { \bf x } ^ { l - 1 } ,
+$$
+
+where $\mathbf { J } ^ { l , K } = \partial \mathbf { x } ^ { K } / \partial \mathbf { x } ^ { l }$ is the Jacobian from layer $l$ to the output and $\otimes$ is the Kronecker product. 2. additionally, for linear networks, i.e., when $\phi$ is the identity:
+
+$$
+{ \bf g } _ { { \bf w } ^ { l } } ^ { T } = \epsilon { \bf J } ^ { l , K } \otimes \left( { \bf J } ^ { 0 , l - 1 } { \bf x } ^ { 0 } + { \bf a } \right) ,
+$$
+
+where $\mathbf { J } ^ { 0 , l - 1 } = \partial \mathbf { x } ^ { l - 1 } / \partial \mathbf { x } ^ { 0 }$ is the Jacobian from the input to layer $l - 1$ and $\mathbf { a } \in \mathbb { R } ^ { N }$ is a constant term that does not depend on $\mathbf { x } ^ { 0 }$ .
+
+Proof. This can be proved by an algebraic manipulation of the chain rule while using the feedforward dynamics in Equation 2. We provide the full derivation in Appendix A. □
+
+Notice that the gradient at layer $l$ constitutes both the backward propagation of the error signal $\epsilon$ up to layer $l$ and the forward propagation of the input signal $\mathbf { x } ^ { 0 }$ up to layer $l - 1$ . Moreover, especially in the linear case, the signal propagation in both directions is governed by the corresponding Jacobians. We believe that this interpretation of gradients is useful as it sheds light on how signal propagation affects the gradients. To this end, we next analyze the conditions on the Jacobians, which would guarantee faithful signal propagation in the network, and consequently, faithful gradients.
+
+# 3.3.2 ENSURING FAITHFUL GRADIENTS
+
+Here, we first consider the layerwise signal propagation which would be useful to derive properties on the initialization to ensure faithful gradients. To this end, let us consider the layerwise Jacobian:
+
+$$
+\mathbf { J } ^ { l - 1 , l } = \frac { \partial \mathbf { x } ^ { l } } { \partial \mathbf { x } ^ { l - 1 } } = \mathbf { D } ^ { l } \mathbf { W } ^ { l } .
+$$
+
+Note that it is sufficient to have layerwise dynamical isometry in order to ensure faithful signal propagation in the network.
+
+Definition 1. (Layerwise dynamical isometry) Let $\begin{array} { r } { \mathbf { J } _ { ~ . ~ . ~ . ~ } ^ { l - 1 , l } = ~ \frac { \partial \mathbf { x } ^ { l } } { \partial \mathbf { x } ^ { l - 1 } } \in ~ \mathbb { R } ^ { N _ { l } \times N _ { l - 1 } } } \end{array}$ be the Jacobian matrix of layer $l$ . The network is said to satisfy layerwise dynamical isometry if the singular values of $\mathbf { J } ^ { l - 1 , l }$ are concentrated near 1 for all layers, i.e., for a given $\epsilon > 0$ , the singular value $\sigma _ { j }$ satisfies $| 1 - \sigma _ { j } | \le \epsilon$ for all $j$ .
+
+This would guarantee that the signal from layer $l$ to $l - 1$ (or vice versa) is propagated without amplification or attenuation in any of its dimension. From Proposition 1 and Equation 7, by induction, it is easy to show that if the layerwise signal propagation is faithful, the error and input signals will faithfully propagate throughout the network, resulting in faithful gradients.
+
+For linear networks, $\mathbf { J } ^ { l - 1 , l } = W ^ { l }$ . Therefore, one can initialize the weight matrix to be orthogonal such that $( \mathbf { W } ^ { l } ) ^ { T } \mathbf { W } ^ { l } = \mathbf { I }$ , where I is the identity matrix of dimension $N$ . In this case, all singular values of $\dot { \mathbf { W } } ^ { l }$ are exactly 1 (i.e., exact dynamical isometry), and such an initialization guarantees faithful gradients. While a linear network is of little practical use, we note that it helps to develop theoretical analysis and provides intuition as to why dynamical isometry is a useful measure.
+
+For nonlinear networks, the diagonal matrix $\mathbf { D } ^ { l }$ needs to be accounted for as it depends on the pre-activations $\mathbf { h } ^ { l }$ at layer $l$ . In this case, it is important to have the pre-activations $\mathbf { h } ^ { l }$ fall into the linear region of the nonlinear function $\phi$ . Precisely, mean-field theory assumes that for large- $N$ limit, the empirical distribution of the pre-activations $\bar { \mathbf { h } } ^ { l }$ converges to a Gaussian with zero mean and variance $q ^ { l }$ , where the variance follows a recursion relation (Poole et al., 2016). Therefore, to achieve layerwise dynamical isometry, the idea becomes to find a fixed point $q ^ { * }$ such that $\mathbf { h } ^ { l } \sim \mathcal { N } ( 0 , q ^ { * } )$ for all $l \in \{ 1 \ldots K \}$ . Such a fixed point makes $\mathbf { D } ^ { l } = \mathbf { D }$ for all layers, and therefore, the preactivations are placed in the linear region of the nonlinearity.2 Then, given the nonlinearity, one can find a rescaling such that $( \mathbf { D } \mathbf { W } ^ { l } ) ^ { T } ( \mathbf { \bar { D } } \mathbf { W } ^ { l } ) = ( \mathbf { W } ^ { l } ) ^ { T } \mathbf { W } ^ { l } / \sigma _ { w } ^ { 2 } = \mathbf { I }$ . The procedure for finding the rescaling $\sigma _ { w } ^ { 2 }$ for various nonlinearities are discussed in Pennington et al. (2017; 2018). Also, this easily extends to convolutional neural networks using the initialization method in Xiao et al. (2018).
+
+Table 1: Jacobian singular values and resulting sparse networks for the 7-layer tanh MLP network considered in section 3.1. SG, CN, and Sparsity refer to Scaled Gaussian, Condition Number (i.e., $s _ { \operatorname* { m a x } } / s _ { \operatorname* { m i n } }$ , where $s _ { \mathrm { m a x } }$ and $s _ { \mathrm { m i n } }$ are the maximum and minimum Jacobian singular values), and a ratio of pruned prameters to the total number of parameters, respectively. SG $( \gamma { = } \mathrm { \breve { 1 0 } ^ { - 2 } } ,$ ) is equivalent to the variance scaling initialization as in LeCun et al. (1998); Glorot & Bengio (2010). The failure cases correspond to unreliable connection sensitivity resulted from poorly conditioned initial Jacobians.   
+
+<table><tr><td></td><td colspan="3">Jacobian singular values</td><td colspan="9">Sparsity in pruned network (across layers)</td></tr><tr><td>Initialization</td><td>Mean</td><td>Std</td><td>CN</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td></td><td>6</td><td>7</td><td>Error</td></tr><tr><td>SG(y=10-4)</td><td>2.46e-07</td><td>9.90e-08</td><td>4.66e+00</td><td>0.97</td><td>0.80</td><td>0.80</td><td>0.80</td><td>0.80</td><td></td><td>0.81</td><td>0.48</td><td>2.66</td></tr><tr><td>SG(y=10-3)</td><td>5.74e-04</td><td>2.45e-04</td><td>8.54e+00</td><td>0.97</td><td>0.80</td><td>0.80</td><td>0.80</td><td></td><td>0.80</td><td>0.81</td><td>0.48</td><td>2.67</td></tr><tr><td>SG(y=10-2)</td><td>4.49e-01</td><td>2.51e-01</td><td>5.14e+01</td><td>0.96</td><td>0.80</td><td>0.80</td><td>0.80</td><td></td><td>0.81</td><td>0.81</td><td>0.49</td><td>2.67</td></tr><tr><td>SG(γ=10-1)</td><td>2.30e+01</td><td>2.56e+01</td><td>2.92e+04</td><td>0.96</td><td>0.81</td><td>0.82</td><td>0.82</td><td></td><td>0.82</td><td>0.80</td><td>0.45</td><td>2.61</td></tr><tr><td>SG(γ=100)</td><td>1.03e+03</td><td>2.61e+03</td><td>3.34e+11</td><td>0.85</td><td>0.88</td><td>0.99</td><td>1.00</td><td></td><td>1.00</td><td>1.00</td><td>0.91</td><td>90.2</td></tr><tr><td>SG(γ=101)</td><td>3.67e+04</td><td>2.64e+05</td><td>inf</td><td>0.84</td><td>0.95</td><td>1.00</td><td>1.00</td><td></td><td>1.00</td><td>1.00</td><td>1.00</td><td>90.2</td></tr></table>
+
+We note that dynamical isometry is in fact a weaker condition than layerwise dynamical isometry. However, in practice, the initialization suggested in the existing works (Pennington et al., 2017; Xiao et al., 2018), i.e., orthogonal initialization for weight matrices in each layer with rescaling based on mean field theory, satisfy layerwise dynamical isometry, even though this term was not mentioned.
+
+Now, recall from Section 3.1 that a network is pruned with a global threshold based on connection sensitivity, and from Section 3.2 that the connection sensitivity is the gradients scaled by the weights. This in turn implies that the connection sensitivity scores across layers are required to be of the same scale. To this end, we require the gradients to be faithful and the weights to be in the same scale for all the layers. Notice, this condition is trivially satisfied when the layerwise dynamical isometry is ensured, as each layer is initialized identically (i.e., orthogonal initialization) and the gradients are guaranteed to be faithful.
+
+Finally, we verify the failure of pruning cases presented in Section 3.1 based on the signal propagation perspective. Specifically, we measure the singular value distribution of the input-output Jacobian $( \mathbf { J } ^ { 0 , K } )$ for the 7-layer tanh MLP network, and the results are reported in Table 1. Note that while connection sensitivity based pruning is robust to moderate changes in the Jacobian singular values, it failed catastrophically when the condition number of the Jacobian is very large $( > 1 \mathrm { e } + 1 1 ) ,$ . In fact, these failure cases correspond to the completely disconnected networks, as a consequence of pruning with unreliable connection sensitivity resulted from poorly conditioned initial Jacobians. As we will show subsequently, these findings extend to modern architectures, and layerwise dynamical isometry yields well-conditioned Jacobians and in turn the best pruning results.
+
+# 4 SIGNAL PROPAGATION IN SPARSE NEURAL NETWORKS
+
+So far, we have shown empirically and theoretically that layerwise dynamical isometry can improve the process of pruning at initialization. One remaining question to address is the following: how well do signals propagate in the pruned sparse networks? In this section, we first examine the effect of sparsity on signal propagation after pruning. We find that indeed pruning can break dynamical isometry, degrading trainability of sparse networks. Then we follow up to present a simple, but effective data-free method to recover approximate dynamical isometry on sparse networks.
+
+Setup. The overall process is summarized as follows: Step 1. Initialize a network with a variance scaling (VS) or layerwise dynamical isometry (LDI) satisfying orthogonal initialization. Step 2. Prune at initialization for a sparsity level $\bar { \kappa }$ based on connection sensitivity (CS); we also test random (Rand) and magnitude (Mag) based pruning for comparison. Step 3. (optional) Enforce approximate dynamical isometry, if specified. Step 4. Train the pruned sparse network using SGD. We measure signal propagation (e.g., Jacobian singular values) on the sparse network right before Step 4, and observe training behavior during Step 4. Different methods are named as {A}-{B}-{C}, where A, B, C stand for initialization scheme, pruning method, (optional) approximate isometry, respectively. We perform this on 7-layer linear and tanh MLP networks as before 3.
+
+![](images/0456d4c99d2986882e19f3354f9afa0e7a4bb0ac07556742ba64a152ca4e555a.jpg)  
+Figure 2: (a) Signal propagation (mean Jacobian singular values) in sparse networks pruned for varying sparsity levels $\bar { \kappa }$ , and (b) training behavior of the sparse network at $\bar { \kappa } ~ = ~ 9 0 \%$ . Signal propagation, pruning scheme, and overparameterization affect trainability of sparse neural networks. We train using SGD with the initial learning rate of 0.1 decayed by $1 / 1 0$ at every 20k iterations. All results are the average over 10 runs. We provide other singular value statistics (max, min, std), accuracy plot, and extended training results for random and magnitude pruning in Appendix C.
+
+Effect of pruning on signal propagation and trainability. Let us first check signal propagation measurements in the pruned networks (see Figure 2a). In general, Jacobian singular values decrease continuously as the sparsity level increases (except for $\{ \cdot \} - \{ \cdot \} - \mathrm { A I }$ which we will explain later), indicating that the more parameters are removed, the less faithful a network is likely to be with regard to propagating signals. Also, notice that the singular values drop more rapidly with random pruning compared to connection sensitivity based pruning methods (i.e., $\{ \cdot \}$ -Rand vs. $\{ \cdot \}$ -CS). This means that pruning using connection sensitivity is more robust to destruction of dynamical isometry and preserve better signal propagation in the sparse network than random pruning. We further note that, albeit marginal, layerwise dynamical isometry allows better signal propagation than variance scaling initialization, with relatively higher mean singular values and much lower standard deviations especially in the low sparsity regime (see Appendix C).
+
+Now, we look into the relation between signal propagation and trainability of the sparse networks. Figure 2b shows training behavior of the pruned networks ( $\bar { \kappa } = 9 0 \%$ ) obtained by different methods. We can see a clear correlation between signal propagation capability of a network and its training performance; i.e., the better a network propagates signals, the faster it converges during training. For instance, compare the trainability of a network before and after pruning. That is, compared to LDI-Dense $\bar { \kappa } = 0$ ), LDI-{CS, Mag, Rand} decrease the loss much slowly; random pruning starts to decrease the loss around 4k iteration, and finally reaches to close to zero loss around $1 0 \mathrm { k }$ iterations (see Appendix C), which is more than an order of magnitude slower than a network pruned by connection sensitivity. Recall that the pruned networks have much smaller singular values.
+
+Enforcing approximate dynamical isometry. The observation above indicates that the better signal propagation is ensured on sparse networks, the better their training performs. This motivates us to think of the following: what if we can repair the broken isometry, before we start training the pruned network, such that we can achieve trainability comparable to that of the dense network? Precisely, we consider the following:
+
+$$
+\operatorname* { m i n } _ { \mathbf { W } ^ { l } } \| ( \mathbf { C } ^ { l } \odot \mathbf { W } ^ { l } ) ^ { T } ( \mathbf { C } ^ { l } \odot \mathbf { W } ^ { l } ) - \mathbf { I } ^ { l } \| _ { F } ,
+$$
+
+where $\mathbf { C } ^ { l } , \mathbf { W } ^ { l } , \mathbf { \Phi } .$ ${ \bf { I } } ^ { l }$ are the sparse mask obtained by pruning, the corresponding weights, the identity matrix at layer $l$ , respectively, and $\| \cdot \| _ { F }$ is the Frobenius norm. We optimize this for all layers identically using gradient descent. Given the sparsity topology $\mathbf { C } ^ { l }$ and initial weights $\mathbf { W } ^ { l }$ , this datafree method attempts to find an optimal $\mathbf { W } ^ { * }$ such that the combination of the sparse topology and the weights to be layerwise orthogonal, potentially to the full rank capacity. This simple method (i.e., $\{ \cdot \} - \{ \cdot \}$ -AI, where AI is named for Approximate Isometry) turns out to be highly effective. The results are provided in Figure 2, and we summarize our key findings below:
+
+• Signal propagation (LDI-{CS, Rand $\}$ vs. LDI-{CS, Rand}-AI). The decreased singular values (by pruning $\bar { \kappa } > 0$ ) bounce up dramatically and become close to the level before pruning. This means that orthogonality enforced by Equation 8 is achieved in the sparse topology of the pruned network (i.e., approximate dynamical isometry), and therefore, signal propagation on the sparse network is likely to behave similarly to the dense network. As expected, the training performance increased significantly (e.g., compare LDI-CS with LDI-CS-AI for trainability). This works more dramatically for random pruning; i.e., even for randomly pruned sparse networks, training speed increases significantly, implying the benefit of ensuring signal propagation.
+
+Table 2: Pruning results for various neural networks on different datasets. All networks are pruned at initialization for the sparsity $\bar { \kappa } = 9 0 \%$ based on connection sensitivity scores as in Lee et al. (2019). We report orthogonality scores (OS) and generalization errors (Error) on CIFAR-10 (VGG16, ResNets) and Tiny-ImageNet (WRN16); all results are the average over 5 runs. The first and second best results are highlighted in each column of errors. The orthogonal initialization with enforced approximate isometry method (i.e., LDI-AI) achieves the best results across all tested architectures.   
+
+<table><tr><td></td><td colspan="2">VGG16</td><td colspan="2">ResNet32</td><td colspan="2">ResNet56</td><td colspan="2">ResNet110</td><td colspan="2">WRN16</td></tr><tr><td>Initialization</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td></tr><tr><td>VS-L</td><td>13.72</td><td>8.16</td><td>4.50</td><td>11.96</td><td>4.64</td><td>10.43</td><td>4.65</td><td>9.13</td><td>11.99</td><td>45.08</td></tr><tr><td>VS-G</td><td>13.60</td><td>8.18</td><td>4.55</td><td>11.89</td><td>4.67</td><td>10.60</td><td>4.67</td><td>9.17</td><td>11.50</td><td>44.56</td></tr><tr><td>VS-H</td><td>15.44</td><td>8.36</td><td>4.41</td><td>12.21</td><td>4.44</td><td>10.63</td><td>4.39</td><td>9.08</td><td>13.49</td><td>46.62</td></tr><tr><td>LDI</td><td>13.33</td><td>8.11</td><td>4.43</td><td>11.55</td><td>4.51</td><td>10.08</td><td>4.57</td><td>8.88</td><td>11.28</td><td>44.20</td></tr><tr><td>LDI-AI</td><td>6.43</td><td>7.99</td><td>2.62</td><td>11.47</td><td>2.79</td><td>9.85</td><td>2.92</td><td>8.78</td><td>6.62</td><td>44.12</td></tr></table>
+
+Table 3: Pruning results for VGG16 and ResNet32 with different activation functions on CIFAR-10. We report generalization errors (avg. over 5 runs), and the first and second best results are highlighted.   
+
+<table><tr><td></td><td colspan="3">VGG16</td><td colspan="3">ResNet32</td></tr><tr><td>Initialization</td><td>tanh</td><td>1-relu</td><td>selu</td><td>tanh</td><td>1-relu</td><td>selu</td></tr><tr><td>VS-L</td><td>9.07</td><td>7.78</td><td>8.70</td><td>13.41</td><td>12.04</td><td>12.26</td></tr><tr><td>VS-G</td><td>9.06</td><td>7.84</td><td>8.82</td><td>13.44</td><td>12.02</td><td>12.32</td></tr><tr><td>VS-H</td><td>9.99</td><td>8.43</td><td>9.09</td><td>13.12</td><td>11.66</td><td>12.21</td></tr><tr><td>LDI</td><td>8.76</td><td>7.53</td><td>8.21</td><td>13.22</td><td>11.58</td><td>11.98</td></tr><tr><td>LDI-AI</td><td>8.72</td><td>7.47</td><td>8.20</td><td>13.14</td><td>11.51</td><td>11.68</td></tr></table>
+
+Table 4: Unsupervised pruning results for $K$ -layer MLP networks on MNIST. All networks are pruned for sparsity $\bar { \kappa } = 9 0 \%$ at orthogonal initialization. We report generalization errors (avg. over 10 runs).   
+
+<table><tr><td>Loss</td><td>Superv.</td><td>K=3</td><td>K=5</td><td>K=7</td></tr><tr><td>GT</td><td>√</td><td>2.46</td><td>2.43</td><td>2.61</td></tr><tr><td>Pred. (raw)</td><td>X</td><td>3.31</td><td>3.38</td><td>3.60</td></tr><tr><td>Pred. (softmax)</td><td>X</td><td>3.11</td><td>3.37</td><td>3.56</td></tr><tr><td>Unif.</td><td>X</td><td>2.77</td><td>2.77</td><td>2.94</td></tr></table>
+
+• Structure (LDI-Rand-AI vs. LDI-CS-AI). Even if the approximate dynamical isometry is enforced identically, the network pruned using connection sensitivity shows better trainability than the randomly pruned network. This potentially means that the sparse topology obtained by different pruning methods also matters, in addition to signal propagation characteristics.
+
+• Overparameterization (LDI-Dense vs. LDI-{CS, Rand}-AI). Even though the singular values are restored to a level close to before pruning with approximate isometry, the non-pruned dense network converges faster than pruned networks. We hypothesize that in addition to signal propagation, overparameterization helps in optimization taking less time to find a minimum.
+
+While being simple and data free (thus fast), our signal propagation perspective not only can be used to improve trainability of sparse neural networks, but also to complement a common explanation for decreased trainability of compressed networks which is often attributed merely to a reduced capacity. Our results also extend to the case of convolutional neural network (see Figure 8 in Appendix C).
+
+# 5 VALIDATION AND EXTENSIONS
+
+In this section, we aim to demonstrate the efficacy of our signal propagation perspective on a wide variety of settings. We first evaluate the idea of employing layerwise dynamical isometry on various modern neural networks. In addition, we further study the role of supervision under the pruning at initialization regime, extending it to unsupervised pruning. Our results show that indeed, pruning can be approached from the signal propagation perspective at varying scale, bringing forth the notion of neural architecture sculpting. The experiment settings used to generate the presented results are detailed in Appendix B. The code can be found here: https://github.com/namhoonlee/spp-public.
+
+# .1 EVALUATION ON VARIOUS NEURAL NETWORKS AND DATASETS
+
+Here, we verify that our signal propagation perspective for pruning neural networks at initialization is indeed valid, by evaluating further on various modern neural networks and datasets. To this end, we provide orthogonality scores (OS) and generalization errors of the sparse networks obtained by different methods and show that layerwise dynamical isometry with enforced approximate isometry results in the best performance; here, we define OS as $\begin{array} { r } { \frac 1 l \sum _ { l } \| ( \mathbf { \dot { C } } ^ { l } \odot \mathbf { W } ^ { l } ) ^ { T } ( \mathbf { C } ^ { l } \odot \mathbf { \dot { W } } ^ { l } ) - \mathbf { I } ^ { l } \| _ { F } } \end{array}$ , which can be used to indicate how close are the weight matrices in each layer of the pruned network to being orthogonal. All results are the average of 5 runs, and we do not optimize anything specific for a particular case (see Appendix B for experiment settings). The results are presented in Table 2.
+
+The best pruning results are achieved when the approximate dynamical isometry is enforced on the pruned sparse network (i.e., LDI-AI), across all tested architectures. Also, the second best results are achieved with the orthogonal initialization that satisfies layerwise dynamical isometry (i.e., LDI). Looking closely, it is evident that there exists a high correlation between the orthogonality scores and the performance of pruned networks; i.e., the network initialized to have the lowest orthogonality scores achieves the best generalization errors after training. Note that the orthogonality scores being close to 0, by definition, states how faithful a network will be with regard to letting signals propagate without being amplified or attenuated. Therefore, the fact that a pruned network with the lowest orthogonality scores tends to yield good generalization errors further validates that our signal propagation perspective is indeed effective for pruning at initialization. Moreover, we test for other nonlinear activation functions (tanh, leaky-relu, selu), and found that the orthogonal initialization consistently outperforms variance scaling methods (see Table 3).
+
+# 5.2 PRUNING WITHOUT SUPERVISION
+
+So far, we have shown that pruning random networks can be approached from a signal propagation perspective by ensuring faithful connection sensitivity. Another factor that constitutes connection sensitivity is the loss term. At a glance, it is not obvious how informative the supervised loss measured on a random network will be for connection sensitivity. In this section, we look into the effect of supervision, by simply replacing the loss computed using ground-truth labels with different unsupervised surrogate losses as follows: replacing the target distribution using ground-truth labels with uniform distribution (Unif.), and using the averaged output prediction of the network (Pred.; softmax/raw). The results for MLP networks are in Table 4. Even though unsupervised pruning results are not as good as the supervised case, the results are still interesting, especially for the uniform case, in that there was no supervision given. We thus experiment further for the uniform case on other networks, and obtain the following results: 8.25, 11.69, 11.01, 8.82 errors $( \% )$ for VGG16, ResNet32, ResNet56, ResNet110, respectively. Surprisingly, the results are often competitive to that of pruning with supervision (i.e., compare to LDI results in Table 2). Notably, previous pruning algorithms assume the existence of supervision a priori. Being the first demonstration, along with the signal propagation perspective, this unsupervised pruning strategy can be useful in scenarios where there are no labels or only weak supervision is available.
+
+To demonstrate further, we also conducted transfer of sparsity experiments such as transferring a pruned network from one task to another (MNIST Fashion-MNIST). Table 5 shows that, while pruning results may degrade if sparsity is transferred, or done without supervision, less impact is caused for unsupervised pruning when transferred to a different task (i.e., 0.52 to 0.14 on MNIST, and 1.11 to $- 0 . 7 8$ on F
+
+Table 5: Transfer of sparsity experiment results for LeNet. We prune for $\bar { \kappa } = 9 7 \%$ at orthogonal initialization, and report gen. errors (average over 10 runs).   
+
+<table><tr><td></td><td colspan="2">Dataset</td><td colspan="2">Error</td><td rowspan="2">Error rand</td></tr><tr><td>Category</td><td>prune</td><td>train&amp;test</td><td>sup.→unsup.</td><td>(△)</td></tr><tr><td>Standard</td><td>MNIST</td><td>MNIST</td><td>2.42 → 2.94</td><td>+0.52</td><td>15.56</td></tr><tr><td>Transfer</td><td>F-MNIST</td><td>MNIST</td><td>2.66→ 2.80</td><td>+0.14</td><td>18.03</td></tr><tr><td>Standard</td><td>F-MNIST</td><td>F-MNIST</td><td>11.90 → 13.01</td><td>+1.11</td><td>24.72</td></tr><tr><td>Transfer</td><td>MNIST</td><td>F-MNIST</td><td>14.17→ 13.39</td><td>-0.78</td><td>24.89</td></tr></table>
+
+MNIST). This indicates that inductive bias exists in data, affecting transfer and unsupervised pruning, and potentially, that “universal” sparse topology might be obtainable if universal data distribution is known (e.g., extremely large dataset in practice). This may help in situations where different tasks from unknown data distribution are to be performed (e.g., continual learning). We also tested two other unsupervised losses, but none performed as well as uniform loss (e.g., Jacobian norms $\| J \| _ { 1 }$ : 5.03, $\| J \| _ { 2 }$ : 3.00 vs. Unif.: 2.94), implying that if pruning is to be unsupervised, the uniform loss would better be used, because other unsupervised losses depend on the input data (thus can suffer from inductive bias). Random pruning degrades significantly at high sparsity for all cases.
+
+We have shown that pruning at initialization, even when no supervision is provided, can be effective based on the signal propagation perspective. This begs the question of whether pruning needs to be limited to pre-shaped architectures or not. In other words, what if pruning is applied to an arbitrarily bulky network and is treated as sculpting an architecture? In order to find out, we conduct the following experiments: we take a popular pre-designed architecture (ResNet20 in He et al. (2016)) as a base network, and consider a range of variants that are originally bigger than the base model, but pruned to have the same number of parameters as the base dense network. Specifically, we consider the following equivalents: (1) the same number of residual blocks, but with larger widths; (2) a reduced number of residual blocks with larger widths; (3) a larger residual block and the same width (see Table 6 in Appendix B for details). The results are presented in Figure 3.
+
+Overall, the sparse equivalents record lower errors than the dense base model. Notice that some models are extremely sparse (e.g., Equivalent 1 pruned for $\bar { \kappa } = 9 8 . 4 \%$ ). While all networks have the same number of parameters, discovered sparse equivalents outperform the dense reference network. This result is well aligned with recent findings in Kalchbrenner et al. (2018): large sparse networks can outperform their small dense counterpart, while enjoying increased computational and memory efficiency via a dedicated implementation for sparsity in practice. Also, it seems that pruning wider networks tends to be more effective in producing a better model than pruning deeper ones (e.g., Equivalent 1 vs. Equivalent 3). We further note that unlike existing prior works, the sparse networks are discovered by sculpting arbitrarily-designed architecture, without pretraining nor supervision.
+
+![](images/68753f2ac064ffc8b1af0b7cbd0c62d195ccf2ffbcbf5ae69f379e5113029a67.jpg)  
+Figure 3: Neural architecture sculpting results on CIFAR-10. We report generalization errors (avg. over 5 runs). All networks have the same number of parameters (269k) and trained identically.
+
+# 6 DISCUSSION AND FUTURE WORK
+
+In this work, we have approached the problem of pruning neural networks at initialization from a signal propagation perspective. Based on observing the effect of varying the initialization, we found that initial weights have a critical impact on connection sensitivity measurements and hence pruning results. This led us to conduct theoretical analysis based on dynamical isometry and a mean field theory, and formally characterize a sufficient condition to ensure faithful signal propagation in a given network. Moreover, our analysis on compressed neural networks revealed that signal propagation characteristics of a sparse network highly correlates with its trainability, and also that pruning can break dynamical isometry ensured on a network at initialization, resulting in degradation of trainability of the compressed network. To address this, we introduced a simple, yet effective data-free method to recover the orthogonality and enhance trainability of the compressed network. Finally, throughout a range of validation and extension experiments, we verified that our signal propagation perspective is effective for understanding, improving, and extending the task of pruning at initialization across various settings. We believe that our results on the increased trainability of compressed networks can take us one step towards finding “winning lottery ticket” (i.e., a set of initial weights that given a sparse topology can quickly reach to a generalization performance that is comparable to the uncompressed network, once trained) suggested in Frankle & Carbin (2019).
+
+We point out, however, that there remains several aspects to consider. While pruning on enforced isometry produces trainable sparse networks, the two-stage orthogonalization process (i.e., prune first and enforce the orthogonality later) can be suboptimal especially at a high sparsity level. Also, network weights change during training, which can affect signal propagation characteristics, and therefore, dynamical isometry may not continue to hold over the course of training. We hypothesize that a potential key to successful neural network compression is to address the complex interplay between optimization and signal propagation, and it might be immensely beneficial if an optimization naturally takes place in the space of isometry. We believe that our signal propagation perspective provides a means to formulate this as an optimization problem by maximizing the trainability of sparse networks while pruning, and we intend to explore this direction as a future work.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the ERC grant ERC-2012-AdG 321162-HELIOS, EPSRC grant Seebibyte EP/M013774/1, EPSRC/MURI grant EP/N019474/1 and the Australian Research Council Centre of Excellence for Robotic Vision (project number CE140100016). We would also like to acknowledge the Royal Academy of Engineering and FiveAI, and thank Richard Hartley, Puneet Dokania and Amartya Sanyal for helpful discussions.
+
+# REFERENCES
+
+Jonathan Frankle and Michael Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks. ICLR, 2019.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. AISTATS, 2010.
+
+Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. ICLR, 2016.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. ICCV, 2015.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CVPR, 2016.
+
+Geoffrey E Hinton and Ruslan R Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 2006.
+
+Nal Kalchbrenner, Erich Elsen, Karen Simonyan, Seb Noury, Norman Casagrande, Edward Lockhart, Florian Stimberg, Aaron van den Oord, Sander Dieleman, and Koray Kavukcuoglu. Efficient neural audio synthesis. ICML, 2018.
+
+Yann LeCun, Léon Bottou, Genevieve B. Orr, and Klaus-Robert Müller. Efficient backprop. Neural Networks: Tricks of the Trade, 1998.
+
+Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 2015.
+
+Namhoon Lee, Thalaiyasingam Ajanthan, and Philip HS Torr. Snip: Single-shot network pruning based on connection sensitivity. ICLR, 2019.
+
+Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio. On the difficulty of training recurrent neural networks. 2013.
+
+Jeffrey Pennington, Samuel Schoenholz, and Surya Ganguli. Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. NeurIPS, 2017.
+
+Jeffrey Pennington, Samuel S Schoenholz, and Surya Ganguli. The emergence of spectral universality in deep networks. AISTATS, 2018.
+
+Ben Poole, Subhaneil Lahiri, Maithra Raghu, Jascha Sohl-Dickstein, and Surya Ganguli. Exponential expressivity in deep neural networks through transient chaos. NeurIPS, 2016.
+
+Russell Reed. Pruning algorithms-a survey. Neural Networks, 1993.
+
+Andrew M Saxe, James L McClelland, and Surya Ganguli. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. ICLR, 2014.
+
+Samuel S Schoenholz, Justin Gilmer, Surya Ganguli, and Jascha Sohl-Dickstein. Deep information propagation. ICLR, 2017.
+
+Wojciech Tarnowski, Piotr Warchoł, Stanisław Jastrz˛ebski, Jacek Tabor, and Maciej A Nowak. Dynamical isometry is achieved in residual networks in a universal way for any activation function. AISTATS, 2019.
+
+Lechao Xiao, Yasaman Bahri, Jascha Sohl-Dickstein, Samuel S Schoenholz, and Jeffrey Pennington. Dynamical isometry and a mean field theory of cnns: How to train 10,000-layer vanilla convolutional neural networks. ICML, 2018.
+
+Ge Yang and Samuel Schoenholz. Mean field residual networks: On the edge of chaos. NeurIPS, 2017.
+
+Greg Yang, Jeffrey Pennington, Vinay Rao, Jascha Sohl-Dickstein, and Samuel S Schoenholz. A mean field theory of batch normalization. ICLR, 2019.
+
+# A GRADIENTS IN TERMS OF JACOBIANS
+
+Proposition 1. Let $\epsilon = \partial L / \partial \mathbf { x } ^ { K }$ denote the error signal and $\mathbf { x } ^ { 0 }$ denote the input signal. Then,
+
+1. the gradients satisfy:
+
+$$
+{ \bf g } _ { { \bf w } ^ { l } } ^ { T } = \epsilon { \bf J } ^ { l , K } { \bf D } ^ { l } \otimes { \bf x } ^ { l - 1 } ,
+$$
+
+where $\mathbf { J } ^ { l , K } = \partial \mathbf { x } ^ { K } / \partial \mathbf { x } ^ { l }$ is the Jacobian from layer $l$ to the output and $\otimes$ is the Kronecker product. 2. additionally, for linear networks, $i . e .$ , when $\phi$ is the identity:
+
+$$
+{ \bf g } _ { { \bf w } ^ { l } } ^ { T } = \epsilon { \bf J } ^ { l , K } \otimes \left( { \bf J } ^ { 0 , l - 1 } { \bf x } ^ { 0 } + { \bf a } \right) ,
+$$
+
+where $\mathbf { J } ^ { 0 , l - 1 } = \partial \mathbf { x } ^ { l - 1 } / \partial \mathbf { x } ^ { 0 }$ is the Jacobian from the input to layer $l - 1$ and $\mathbf { a } \in \mathbb { R } ^ { N }$ is the constant term that does not depend on $\mathbf { x } ^ { 0 }$ .
+
+Proof. The proof is based on a simple algebraic manipulation of the chain rule. The gradient of the loss with respect to the weight matrix $\mathbf { W } ^ { l }$ can be written as:
+
+$$
+{ \bf g } _ { { \bf w } ^ { l } } = \frac { \partial L } { \partial { \bf W } ^ { l } } = \frac { \partial L } { \partial { \bf x } ^ { K } } \frac { \partial { \bf x } ^ { K } } { \partial { \bf x } ^ { l } } \frac { \partial { \bf x } ^ { l } } { \partial { \bf W } ^ { l } } .
+$$
+
+Here, the gradient $\partial \mathbf { y } / \partial \mathbf { x }$ is represented as a matrix of dimension $\mathbf { y } { \mathrm { - s i z e } } \times \mathbf { x }$ -size. For gradients with respect to matrices, their vectorized from is used. Notice,
+
+$$
+\frac { \partial \mathbf { x } ^ { l } } { \partial \mathbf { W } ^ { l } } = \frac { \partial \mathbf { x } ^ { l } } { \partial \mathbf { h } ^ { l } } \frac { \partial \mathbf { h } ^ { l } } { \partial \mathbf { W } ^ { l } } = \mathbf { D } ^ { l } \frac { \partial \mathbf { h } ^ { l } } { \partial \mathbf { W } ^ { l } } .
+$$
+
+Considering the feed-forward dynamics for a particular neuron $i$
+
+$$
+\begin{array} { c } { { h _ { i } ^ { l } = \displaystyle \sum _ { j } W _ { i j } ^ { l } x _ { j } ^ { l - 1 } + b _ { i } ^ { l } , } } \\ { { { } } } \\ { { \displaystyle \frac { \partial h _ { i } ^ { l } } { \partial W _ { i j } ^ { l } } = x _ { j } ^ { l - 1 } . } } \end{array}
+$$
+
+Therefore, using the Kronecker product, we can compactly write:
+
+$$
+\frac { \partial { \mathbf { x } } ^ { l } } { \partial { \mathbf { W } } ^ { l } } = ( { \mathbf { D } } ^ { l } ) ^ { T } \otimes ( { \mathbf { x } } ^ { l - 1 } ) ^ { T } .
+$$
+
+Now, Equation 11 can be written as:
+
+$$
+\begin{array} { r l } & { { \bf g } _ { { \bf w } ^ { l } } = ( \epsilon { \bf J } ^ { l , K } { \bf D } ^ { l } ) ^ { T } \otimes ( { \bf x } ^ { l - 1 } ) ^ { T } , } \\ & { { \bf g } _ { { \bf w } ^ { l } } ^ { T } = \epsilon { \bf J } ^ { l , K } { \bf D } ^ { l } \otimes { \bf x } ^ { l - 1 } . } \end{array}
+$$
+
+Here, $\mathbf { A } ^ { T } \otimes \mathbf { B } ^ { T } = ( \mathbf { A } \otimes \mathbf { B } ) ^ { T }$ is used. Moreover, for linear networks $\mathbf { D } ^ { l } = \mathbf { I }$ and $\mathbf { x } ^ { l } = \mathbf { h } ^ { l }$ for all $l \in \{ 1 \ldots K \}$ . Therefore, $\mathbf { x } ^ { l - 1 }$ can be written as:
+
+$$
+\begin{array} { r l } & { { \displaystyle { \bf x } ^ { l - 1 } = \phi ( { \bf W } ^ { l - 1 } \phi ( { \bf W } ^ { l - 2 } \cdot \dots \phi ( { \bf W } ^ { 1 } { \bf x } ^ { 0 } + { \bf b } ^ { 1 } ) \dots + { \bf b } ^ { l - 2 } ) + { \bf b } ^ { l - 1 } ) } ~ , } \\ & { \quad \quad = { \bf W } ^ { l - 1 } ( { \bf W } ^ { l - 2 } \cdot \dots ( { \bf W } ^ { 1 } { \bf x } ^ { 0 } + { \bf b } ^ { 1 } ) \dots + { \bf b } ^ { l - 2 } ) + { \bf b } ^ { l - 1 } ~ , } \\ & { \quad \quad = \displaystyle \prod _ { k = 1 } ^ { l - 1 } { \bf W } ^ { k } { \bf x } ^ { 0 } + \prod _ { k = 2 } ^ { l - 1 } { \bf W } ^ { k } { \bf b } ^ { 1 } + \dots + { \bf b } ^ { l - 1 } ~ , } \\ & { \quad \quad = { \bf J } ^ { 0 , l - 1 } { \bf x } ^ { 0 } + { \bf a } ~ , } \end{array}
+$$
+
+where $\mathbf { a }$ is the constant term that does not depend on $\mathbf { x } ^ { 0 }$ . Hence, the proof is complete.
+
+# B EXPERIMENT SETTINGS
+
+Pruning at initialization. By default, we perform pruning at initialization based on connection sensitivity scores as in Lee et al. (2019). When computing connection sensitivity, we always use all examples in the training set to prevent stochasticity by a particular mini-batch. Unless stated otherwise, we set the default sparsity level to be $\bar { \kappa } = 9 0 \%$ (i.e., $90 \%$ of the entire parameters in a network is pruned away). For all tested architectures, pruning for such level of sparsity does not lead to a large accuracy drop. Additionally, we perform either random pruning (at initialization) or a magnitude based pruning (at pretrained) for comparison purposes. Random pruning refers to pruning parameters randomly and globally for a given sparsity level. For the magnitude based pruning, we first train a model and simply prune parameters globally in a single-shot based on the magnitude of the pretrained parameters (i.e., keep the large weights while pruning small ones). For initialization methods, we follow either variance scaling initialization schemes (i.e., VS-L, VSG, VS-H, as in LeCun et al. (1998); Glorot & Bengio (2010); He et al. (2015), respectively) or (convolutional) orthogonal initialization schemes (Saxe et al., 2014; Xiao et al., 2018).
+
+Training and evaluation. Throughout experiments, we evaluate pruning results on MNIST, CIFAR-10, and Tiny-ImageNet image classification tasks. For training of the pruned sparse networks, we use SGD with momentum and train up to 80k (for MNIST) or $1 0 0 \mathrm { k }$ (for CIFAR-10 and Tiny-ImageNet) iterations. The initial learning rate is set to be 0.1 and is decayed by $1 / 1 0$ at every 20k (MNIST) or 25k (CIFAR-10 and Tiny-ImageNet). The mini-batch size is set to be 100, 128, 200 for MNIST, CIFAR-10, Tiny-ImageNet, respectively. We do not optimize anything specific for a particular case, and follow the standard training procedure. For all experiments, we use $10 \%$ of training set for the validation set, which corresponds to 5400, 5000, 9000 images for MNIST, CIFAR-10, Tiny-IamgeNet, respectively. We evaluate at every 1k iteration, and record the lowest test error. All results are the average of either 10 (for MNIST) or 5 (for CIFAR-10 and Tiny-ImageNet) runs.
+
+Signal propagation and approximate dynamical isometry. We use the entire training set when computing Jacobian singular values of a network. In order to enforce approximate dynamical isometry when specified, given a pruned sparse network, we optimize for the objective in Equation 8 using gradient descent. The learning rate is set to be 0.1 and we perform $1 0 \mathrm { k }$ gradient update steps (although it usually reaches to convergence far before). This process is data-free and thus fast; e.g., depending on the size of the network and the number of update steps, it can take less than a few seconds on a modern computer.
+
+Neural architecture sculpting. We provide the model details in Table 6.
+
+Table 6: All models (Equivalents 1,2,3) are initially bigger than the base network (ResNet20), by either being wider or deeper, but pruned to have the same number of parameters as the base network (269k). The widening factor $\mathbf { \Psi } ( \mathbf { k } )$ refers to the filter multiplier; e.g., for the basic filter size of 16, the widening factor of ${ \bf k } = 2$ will result in 32 filters. The block size refers to the number of residual blocks in each block layer; all models have three block layers. More/less number of residual blocks means the network is deeper/shallower. The reported generalization errors are averages over 5 runs. We find that the technique of architecture sculpting, pruning randomly initialized neural networks based on our signal propagation perspective even in the absence of ground-truth supervision, can be used to find models of superior performance under the same parameter budget.
+
+<table><tr><td>Model category</td><td>Shape</td><td>Widening (k)</td><td>Block size</td><td>Init.</td><td>GT</td><td>Sparsity</td><td>Error</td></tr><tr><td>Base</td><td>ResNet20 (He et al.,2016)</td><td>1</td><td>3</td><td>VS-H</td><td>√</td><td>0.0</td><td>8.046</td></tr><tr><td>Equivalent 1</td><td>wider</td><td>2</td><td>3</td><td>LDI</td><td>X</td><td>74.8</td><td>7.618</td></tr><tr><td></td><td>wider</td><td>4</td><td>3</td><td>LDI</td><td>X</td><td>93.7</td><td>7.630</td></tr><tr><td></td><td>wider</td><td>6</td><td>3</td><td>LDI</td><td>X</td><td>97.2</td><td>7.708</td></tr><tr><td>Equivalent 2</td><td>wider</td><td>8</td><td>3</td><td>LDI</td><td>X</td><td>98.4</td><td>7.836</td></tr><tr><td></td><td>wider&amp; shallower</td><td>2</td><td>2</td><td>LDI</td><td>X</td><td>60.4</td><td>7.776</td></tr><tr><td></td><td>wider&amp; shallower</td><td>4</td><td>2</td><td>LDI</td><td>X</td><td>90.1</td><td>7.876</td></tr><tr><td></td><td>wider&amp; shallower</td><td>6</td><td>2</td><td>LDI</td><td>X</td><td>95.6</td><td>7.940</td></tr><tr><td>Equivalent 3</td><td>deeper</td><td>1</td><td>5</td><td>LDI</td><td>X</td><td>42.0</td><td>7.912</td></tr></table>
+
+![](images/b5581e9204ada42f22daf0b704642a1ff044bb50d5f1ce5c1db1571bb4df7e29.jpg)  
+Figure 4: Full results for (a) signal propagation (all signular value statistics), and (b) training behavior (including accuracy) for 7-layer linear and tanh MLP networks. We provide results of LDI-Rand, LDI-Rand-AI, VS-CS, LDI-CS, LDI-CS-AI on the linear case for both singular value statistics and training log. We also plot results of LDI-Mag and LDI-Dense on the tanh case for trainability; the training results of non-pruned (LDI-Dense) and magnitude (LDI-Mag) pruning are only reported for the tanh case, because the learning rate had to be lowered for the linear case (otherwise it explodes), which makes the comparison not entirely fair. We provide the singular value statistics for the magnitude pruning in Figure 6 to avoid clutter. Also, extended training logs for random and magnitude based pruning are provided separately in Figure 5 to illustrate the difference in convergence speed.
+
+![](images/d1ed673251181025b8e6f9b38ebb82f8f54fdf9edf9d0f51b8c25f30d0927bbc.jpg)  
+Figure 5: Extended training log (i.e., Loss and Accuracy) for random (Rand) and magnitude (Mag) pruning. The sparse networks obtained by random or magnitude pruning take a much longer time to train than that obtained by pruning based on connection sensitivity. All methods are pruned at the layerwise orthogonal initialization, and trained the same way as before.
+
+![](images/2b750e1f8519d5ee9d681220ed8869910b6d4c70ccc74c576c7b5a7eeae3240c.jpg)  
+Figure 6: Signal propagation measurments (all signular value statistics) for the magnitude based pruning $\left( { \bf M a g } \right)$ on the 7-layer linear and tanh MLP networks. As described in the experiment settings in Appendix B, the magnitude based pruning is performed on a pretrained model. Notice that unlike other cases where pruning is done at initialization (i.e., using either random or connection sensitivity based pruning methods), the singular value distribution changes abruptly when pruned (i.e., note of the sharp change of singular values from 0 to $10 \%$ sparsity). Also, the singular values are not concentrated (note of high standard deviations), which explains rather inferior trainability compared to other methods. We conjecture that naively pruning based on the magnitude of parameters in a single-shot, without pruning gradually or employing some sophisticated tricks such as layerwise thresholding, can lead to a failure of training compressed networks.
+
+![](images/3e97af0711431e9dfb5f7744b19c809dda708b5fe1a341b76b9b70c4b9ffd379.jpg)  
+Figure 7: Signal propagation and training behavior for ReLU and Leaky-ReLU activation functions. They resemble those of the tanh case as in Figure 2, and hence the conclusion holds about the same.
+
+![](images/85d20abf34e6c349a4275751a84c0c829146780e7dc068edeb4a8ec0cefe4ebe.jpg)  
+Figure 8: Training performance (loss and accuracy) by different methods for VGG16 on CIFAR-10. To examine the effect of initialization in isolation on the trainability of sparse neural networks, we remove batch normalization (BN) layers for this experiment, as BN tends to improve training speed as well as generalization performance. As a result, enforcing approximate isometry (LDI-CS-AIF) improves the training speed quite dramatically compared to the pruned network without isometry (LDI-CS). We also find that even compared to the non-pruned dense network (LDI-Dense) which is ensured layerwise dynamical isometry, LDI-CS-AIF trains faster in the early training phase. This result is quite promising and more encouraging than the previous case of MLP (see Figures 2 and 7), as it potentially indicates that an underparameterized network (by connection sensitivity pruning) can even outperform an overparameterized network, at least in the early phase of neural network training. Furthermore, we add results of using the spectral norm in enforcing approximate isometry in Equation 8 (LDI-CS-AIS), and find that it also trains faster than the case of broken isometry (LDI-CS), yet not as much as the case of using the Frobenius norm (LDI-CS-AIF).

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+# DYNAMIC GRAPH REPRESENTATION LEARNING VIA SELF-ATTENTION NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Learning latent representations of nodes in graphs is an important and ubiquitous task with widespread applications such as link prediction, node classification, and graph visualization. Previous methods on graph representation learning mainly focus on static graphs, however, many real-world graphs are dynamic and evolve over time. In this paper, we present Dynamic Self-Attention Network (DySAT), a novel neural architecture that operates on dynamic graphs and learns node representations that capture both structural properties and temporal evolutionary patterns. Specifically, DySAT computes node representations by jointly employing self-attention layers along two dimensions: structural neighborhood and temporal dynamics. We conduct link prediction experiments on two classes of graphs: communication networks and bipartite rating networks. Our experimental results show that DySAT has a significant performance gain over several different stateof-the-art graph embedding baselines.
+
+# 1 INTRODUCTION
+
+Learning latent representations (or embeddings) of nodes in graphs has been recognized as a fundamental learning problem due to its widespread use in various domains such as social media (Perozzi et al., 2014), biology (Grover & Leskovec, 2016), and knowledge bases (Wang et al., 2014). The basic idea is to learn a low-dimensional vector for each node, which encodes the structural properties of a node and its neighborhood (and possibly attributes). Such low-dimensional representations can benefit a plethora of graph analytical tasks such as node classification, link prediction, and graph visualization (Perozzi et al., 2014; Tang et al., 2015; Grover & Leskovec, 2016; Wang et al., 2016).
+
+Previous work on graph representation learning mainly focuses on static graphs, which contain a fixed set of nodes and edges. However, many graphs in real-world applications are intrinsically dynamic, in which graph structures can evolve over time. They are usually represented as a sequence of graph snapshots from different time steps (Leskovec et al., 2007). Examples include academic co-authorship networks where authors may periodically switch their collaboration behaviors and email communication networks whose structures may change dramatically due to sudden events. In such scenarios, modeling temporal evolutionary patterns is important in accurately predicting node properties and future links.
+
+Learning dynamic node representations is challenging, compared to static settings, due to the complex time-varying graph structures: nodes can emerge and leave, links can appear and disappear, and communities can merge and split. This requires the learned embeddings not only to preserve structural proximity of nodes, but also to jointly capture the temporal dependencies over time. Though some recent work attempts to learn node representations in dynamic graphs, they mainly impose a temporal regularizer to enforce smoothness of the node representations from adjacent snapshots (Zhu et al., 2016; Li et al., 2017; Zhou et al., 2018). However, these approaches may fail when nodes exhibit significantly distinct evolutionary behaviors. Trivedi et al. (2017) employ a recurrent neural architecture for temporal reasoning in multi-relational knowledge graphs. However, their temporal node representations are limited to modeling first-order proximity, while ignoring the structure of higher-order graph neighborhoods.
+
+Attention mechanisms have recently achieved great success in many sequential learning tasks such as machine translation (Bahdanau et al., 2015) and reading comprehension (Yu et al., 2018). The key underlying principle is to learn a function that aggregates a variable-sized input, while focusing on the parts most relevant to a certain context. When the attention mechanism uses a single sequence as both the inputs and the context, it is often called self-attention. Though attention mechanisms were initially designed to facilitate Recurrent Neural Networks (RNNs) to capture long-term dependencies, recent work by Vaswani et al. (2017) demonstrates that a fully self-attentional network itself can achieve state-of-the-art performance in machine translation tasks. Velickovic et al. (2018) extend self-attention to graphs by enabling each node to attend over its neighbors, achieving state-of-the-art results for semi-supervised node classification tasks in static graphs.
+
+As dynamic graphs usually include periodical patterns such as recurrent links or communities, attention mechanisms are capable of utilizing information about most relevant historical context, to facilitate future prediction. Inspired by recent work on attention techniques, we present a novel neural architecture named Dynamic Self-Attention Network (DySAT) to learn node representations on dynamic graphs. Specifically, we employ self-attention along two dimensions: structural neighborhoods and temporal dynamics, i.e., DySAT generates a dynamic representation for a node by considering both its neighbors and historical representations, following a self-attentional strategy. Unlike static graph embedding methods that focus entirely on preserving structural proximity, we learn dynamic node representations that reflect the temporal evolution of graph structure over a varying number of historical snapshots. In contrast to temporal smoothness-based methods, DySAT learns attention weights that capture temporal dependencies at a fine-grained node-level granularity.
+
+We evaluate our framework on the dynamic link prediction task using four benchmarks of different sizes including two email communication networks (Klimt & Yang, 2004; Panzarasa et al., 2009) and two bipartite rating networks (Harper & Konstan, 2016). Our evaluation results show that DySAT achieves significant improvements ( $3 . 6 \%$ macro-AUC on average) over several stateof-the-art baselines and maintains a more stable performance over different time steps.
+
+# 2 RELATED WORK
+
+Our framework is related to previous representation learning techniques on static graphs, dynamic graphs, and recent developments in self-attention mechanisms.
+
+Static graph embeddings. Early work on unsupervised graph representation learning exploits the spectral properties of various graph matrix representations, such as Laplacian, etc. to perform dimensionality reduction (Tenenbaum et al., 2000; Belkin & Niyogi, 2001). To improve scalability, some work (Perozzi et al., 2014; Grover & Leskovec, 2016) utilizes Skip-gram methods, inspired by their success in Natural Language Processing (NLP). Recently, several graph neural network architectures based on generalizations of convolutions have achieved tremendous success, among which many methods are designed for supervised or semi-supervised learning tasks (Niepert et al., 2016; Defferrard et al., 2016; Kipf & Welling, 2017; Sankar et al., 2017; Velickovic et al., 2018). Hamilton et al. (2017b) extend graph convolutional methods through trainable neighborhood aggregation functions, to propose a general framework applicable to unsupervised representation learning. However, these methods are not designed to model temporal evolutionary patterns in dynamic graphs.
+
+Dynamic graph embeddings. Most techniques employ temporal smoothness regularization to ensure embedding stability across consecutive time-steps (Zhu et al., 2016). Zhou et al. (2018) additionally use triadic closure (Kossinets & Watts, 2006) as guidance, leading to significant improvements. Neural methods were recently explored in the knowledge graph domain by Trivedi et al. (2017), who employ a recurrent neural architecture for temporal reasoning. However, their model is limited to tracing link evolution, thus limited to capturing first-order proximity. Goyal et al. (2017) learn incremental node embeddings through initialization from the previous time steps, however, this may not guarantee the model to capture long-term graph similarity. A few recent works (Nguyen et al., 2018; Zuo et al., 2018) examine a related setting of temporal graphs with continuous timestamped links for representation learning, which is however orthogonal to the established problem setup of using dynamic graph snapshots. Li et al. (2017) learn node embeddings in dynamic attributed graphs by initially training an offline model, followed by incremental updates over time. However, their key focus is online learning to improve efficiency over re-training static models, while our goal is to improve representation quality by exploiting the temporal evolutionary patterns in graph structure. Unlike previous approaches, our framework captures the most relevant historical contexts through a self-attentional architecture, to learn dynamic node representations.
+
+Self-attention mechanisms. Recent advancements in many NLP tasks have demonstrated the superiority of self-attention in achieving state-of-the-art performance (Vaswani et al., 2017; Lin et al., 2017; Tan et al., 2018; Shen et al., 2018; Shaw et al., 2018). In DySAT, we employ self-attention mechanisms to compute a dynamic node representation by attending over its neighbors and previous historical representations. Our approach of using self-attention over neighbors is closely related to the Graph Attention Network (GAT) (Velickovic et al., 2018), which employs neighborhood attention for semi-supervised node classification in a static graph. As dynamic graphs usually contain periodical patterns, we extend the self-attention mechanisms over the historical representations of a particular node to capture its temporal evolution behaviors.
+
+# 3 PROBLEM DEFINITION
+
+In this work, we address the problem of dynamic graph representation learning. A dynamic graph is defined as a series of observed snapshots, $\mathbb { G } \stackrel { \smile } { = } \{ \bar { \mathcal { G } } ^ { 1 } , \dots , \mathcal { G } ^ { T } \}$ where $T$ is the number of time steps. Each snapshot $\mathcal { G } _ { t } = ( \nu , \mathcal { E } ^ { t } )$ is a weighted undirected graph with a shared node set $\nu$ , a link set $\dot { \mathcal { E } } ^ { t }$ , and weighted adjacency matrix $A ^ { t }$ at time $t$ . Unlike some previous work that assumes links can only be added over time in dynamic works, we also allow to remove links. Dynamic graph representation learning aims to learn latent representations $\boldsymbol { e } _ { v } ^ { t } \in \mathbb { R } ^ { d }$ for each node $v \in \mathcal V$ at time steps $t = 1 , 2 , \dots , T$ , such that $e _ { v } ^ { t }$ preserves both the local graph structures centered at $v$ and its evolutionary behaviors prior to time $t$ .
+
+# 4 DYNAMIC SELF-ATTENTION NETWORK
+
+In this section, we first describe the high-level structure of our model. DySAT consists of two major novel components: structural and temporal self-attention layers, which can be utilized to construct arbitrary graph neural architectures through stacking of layers. Similar to existing studies on attention mechanisms, we employ multi-head attentions to improve model capacity and stability.
+
+DySAT consists of a structural block followed by a temporal block, as illustrated in Figure 1, where each block may contain multiple stacked layers of the corresponding layer type. The structural block extracts features from the local neighborhood through self-attentional aggregation, to compute intermediate node representations for each snapshot. These representations feed as input to the temporal block, which attends over multiple time steps, capturing temporal variations in the graph.
+
+# 4.1 STRUCTURAL SELF-ATTENTION
+
+The input of this layer is a graph snapshot $\mathcal { G } \in \mathbb { G }$ and a set of input node representations $\{ \pmb { x } _ { v } \in \pmb { \mathrm { \Sigma } }$ $\mathbb { R } ^ { D } , \forall v \in \mathcal { V } \}$ where $D$ is the input embedding dimension. The input to the initial layer can be set as 1-hot encoded vectors for each node (or attributes if available). The output is a new set of node representations $\{ z _ { v } \in \mathbb { R } ^ { F } , \forall v \in \mathcal { V } \}$ with $F$ dimensions that capture local structural properties.
+
+Specifically, the structural self-attention layer attends over the immediate neighbors of a node $v$ (in snapshot $\mathcal { G }$ ), by computing attention weights as a function of their input node embeddings. The structural attention layer is a variant of GAT (Velickovic et al., 2018), applied on a single snapshot:
+
+$$
+z _ { v } = \sigma \Big ( \sum _ { u \in \mathcal { N } _ { v } } \alpha _ { u v } W ^ { s } x _ { u } \Big ) , \quad \alpha _ { u v } = \frac { \exp { \Big ( \sigma \Big ( A _ { u v } \cdot a ^ { T } [ W ^ { s } x _ { u } | | W ^ { s } x _ { v } ] \Big ) \Big ) } } { \sum _ { w \in \mathcal { N } _ { v } } \exp { \Big ( \sigma \Big ( A _ { w v } \cdot a ^ { T } [ W ^ { s } x _ { w } | | W ^ { s } x _ { v } ] \Big ) \Big ) } }
+$$
+
+where $\mathcal { N } _ { v } = \{ u \in \mathcal { V } : ( u , v ) \in \mathcal { E } \}$ is the set of immediate neighbors of node $v$ in snapshot $\mathcal { G }$ ; $W ^ { s } \in \mathbb { R } ^ { D \times F }$ is a shared weight transformation applied to each node in the graph; $\pmb { a } \in \mathbb { R } ^ { 2 D }$ is a weight vector parameterizing the attention function implemented as feed-forward layer; $| |$ is the concatenation operation and $\sigma ( \cdot )$ is a non-linear activation function. Note that $A _ { u v }$ is the weight of link $( u , v )$ in the current snapshot $\mathcal { G }$ . The set of learned coefficients $\alpha _ { u v }$ , obtained by a softmax over the neighbors of each node, indicate the importance or contribution of node $u$ to node $v$ at the current snapshot. We use a LeakyRELU non-linearity to compute the attention weights, followed by ELU for the output representations. In our experiments, we employ sparse matrices to implement the masked self-attention over neighbors.
+
+![](images/32baf58276c3dc1eb6603ff6bd610293a41b3f13f9a7410ead38172fcda865c1.jpg)  
+Figure 1: Neural architecture of DySAT: we employ structural attention layers followed by temporal attention layers. Dashed black arrows indicate new links and dashed blue arrows refer to neighborbased structural-attention.
+
+# 4.2 TEMPORAL SELF-ATTENTION
+
+To further capture temporal evolutionary patterns in a dynamic network, we design a temporal self  
+attention layer. The input of this layer is a seqdifferent time steps. Specifically, for each node ence of representations , we define the input as $v$ $v$ $\{ \boldsymbol { x } _ { v } ^ { 1 } , \bar { \boldsymbol { x } } _ { v } ^ { 2 } , \ldots , \boldsymbol { x } _ { v } ^ { T } \} , \boldsymbol { x } _ { v } ^ { t } \in$ where $T$ is the number of time steps and $D ^ { \prime }$ is the dimensionality of the input represen  
+tations. The layer output is a new representation sequence for $v$ at each time step, i.e., $\begin{array} { r l } { z _ { v } } & { { } = } \end{array}$   
+$\{ z _ { v } ^ { 1 } , z _ { v } ^ { 2 } , \ldots , z _ { v } ^ { T } \} , z _ { v } ^ { t } \in \mathbb { R } ^ { F ^ { \prime } }$ with dimensionality $F ^ { \prime }$ . We denote the input and output represen  
+tations of $v$ , packed together across time, by matrices $\boldsymbol { X } _ { v } \in \mathbb { R } ^ { T \times D ^ { \prime } }$ and $\boldsymbol { Z } _ { v } \in \mathbb { R } ^ { T \times F ^ { \prime } }$ respectively.
+
+The key objective of the temporal self-attentional layer is to capture the temporal variations in graph structure over multiple time steps. The input representation of node $v$ at time-step $t$ , $\boldsymbol { x } _ { v } ^ { t }$ , constitutes an encoding of the current local structure around $v$ . We use $\ v x _ { v } ^ { t }$ as the query to attend over its historical representations $( < t )$ , tracing the evolution of the local neighborhood around $v$ . Thus, temporal self-attention facilitates learning of dependencies between various representations of a node across different time steps.
+
+To compute the output representation of node $v$ at $t$ , we use the scaled dot-product form of attention (Vaswani et al., 2017) where the queries, keys, and values are set as the input node representations. The queries, keys, and values are first transformed to a different space by using linear projections matrices $W _ { q } \in \mathbb { R } ^ { D ^ { \prime } \times F ^ { \prime } } , W _ { k } \in \mathbb { R } ^ { D ^ { \prime } \times F ^ { \prime } }$ and $\boldsymbol { W _ { v } } \in \mathbb { R } ^ { D ^ { \prime } \times F ^ { \prime } }$ respectively. Here, we allow each time-step $t$ to attend over all time-steps up to and including $t$ , to prevent leftward information flow and preserve the auto-regressive property. The temporal self-attention is defined as:
+
+$$
+Z _ { v } = \beta _ { v } ( X _ { v } W _ { v } ) , \quad \beta _ { v } ^ { i j } = { \frac { \exp ( e _ { v } ^ { i j } ) } { \displaystyle \sum _ { k = 1 } ^ { T } \exp ( e _ { v } ^ { i k } ) } } , \quad e _ { v } ^ { i j } = \Big ( { \frac { ( ( X _ { v } W _ { q } ) ( X _ { v } W _ { k } ) ^ { T } ) _ { i j } } { \sqrt { F ^ { \prime } } } } + M _ { i j } \Big )
+$$
+
+where $\beta _ { v } \in \mathbb { R } ^ { T \times T }$ is the attention weight matrix obtained by the multiplicative attention function and $\dot { M } \in \mathbb { R } ^ { T \times T }$ is a mask matrix with each entry $M _ { i j } \in \{ - \infty , 0 \}$ . When $M _ { i j } = - \infty$ , the softmax function results in a zero attention weight, i.e., $\beta _ { v } ^ { i j } = 0$ , which switches off the attention from time-step $i$ to $j$ . To encode the temporal order, we define $M$ as:
+
+$$
+M _ { i j } = { \left\{ \begin{array} { l l } { 0 , } & { i \leq j } \\ { - \infty , } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. }
+$$
+
+# 4.3 MULTI-HEAD ATTENTION
+
+We additionally employ multi-head attention (Vaswani et al., 2017) to jointly attend to different subspaces at each input, leading to a leap in model capacity. We use multiple attention heads, followed by concatenation, in both structural and temporal self-attention layers:
+
+Structural multi-head self-attention:
+
+$$
+\begin{array} { r l r l } & { \pmb { h } _ { v } = \mathrm { C o n c a t } ( z _ { v } ^ { 1 } , z _ { v } ^ { 2 } , \ldots , z _ { v } ^ { H } ) } & & { \forall v \in V } \\ & { \pmb { H } _ { v } = \mathrm { C o n c a t } ( \pmb { Z } _ { v } ^ { 1 } , \pmb { Z } _ { v } ^ { 2 } , \ldots , \pmb { Z } _ { v } ^ { H } ) } & & { \forall v \in V } \end{array}
+$$
+
+where $H$ is the number of attention heads, $\boldsymbol { h } _ { v } \in \mathbb { R } ^ { F }$ and $\pmb { H _ { v } } \in \mathbb { R } ^ { T \times F ^ { \prime } }$ are the outputs of structural and temporal multi-head attentions respectively. Note that while structural attention is applied on a single snapshot, temporal attention operates over multiple time-steps.
+
+# 4.4 DYSAT ARCHITECTURE
+
+In this section, we present our neural architecture DySAT for Dynamic Graph Representation Learning, that uses the above defined structural and temporal self-attention layers as fundamental modules. As shown in Figure 1, DySAT has three modules from its top to bottom, (1) structural attention block, (2) temporal attention block, and (3) graph context prediction. The model takes as input a collection of $T$ graph snapshots, and generates outputs latent node representations at each time step.
+
+Structural attention block. This module is composed of multiple stacked structural self-attention layers to extract features from nodes at different distances. We apply each layer independently at different snapshots with shared parameters, as illustrated in Figure 1, to capture local neighborhood structure around a node at each time step. Note that the embeddings input to a layer can potentially vary across different snapshots. We denote the node representations output by the structural attention block, as $\{ h _ { v } ^ { 1 } , h _ { v } ^ { 2 } , \ldots , \hat { h _ { v } ^ { T } } \} , h _ { v } ^ { t } \in \mathbb { R } ^ { f }$ , which feed as input to the temporal attention block.
+
+Temporal attention block. First, we equip the temporal attention module with a sense of ordering through position embeddings (Gehring et al., 2017), $\{ p ^ { 1 } , \ldots , p ^ { T } \} , p ^ { t } \in \mathbb { R } ^ { f }$ , which embed the absolute temporal position of each snapshot. The position embeddings are then combined with the output of the structural attention block to obtain a sequence of input representations: $\{ h _ { v } ^ { 1 } +$ $p ^ { 1 } , h _ { v } ^ { 2 } + p ^ { 2 } , \ldots , h _ { v } ^ { T } + p ^ { T } \}$ for node $v$ across multiple time steps. This block also follows a similar structure with multiple stacked temporal self-attention layers. The outputs of the final layer pass into a position-wise feed-forward layer to give the final node representations $\{ e _ { v } ^ { 1 } , e _ { v } ^ { 2 } , \ldots , e _ { v } ^ { \tilde { T } } \} \stackrel { . } { \forall } v \in V$ .
+
+Graph context prediction. To ensure that the learned representations capture both structural and temporal information, we define an objective function that preserves the local structure around a node, across multiple time steps. We use the dynamic representations of a node $v$ at time step $t$ , $e _ { v } ^ { t }$ to predict the occurrence of nodes appearing the local neighborhood around $v$ at $t$ . In particular, we use a binary cross-entropy loss function at each time step to encourage nodes co-occurring in fixed-length random walks, to have similar representations.
+
+$$
+L _ { v } = \sum _ { t = 1 } ^ { T } \sum _ { u \in N _ { u n a l k } ^ { t } ( v ) } - \log ( \sigma ( < e _ { u } ^ { t } , e _ { v } ^ { t } > ) ) - w _ { n } \cdot \sum _ { u ^ { ' } \in P _ { n } ^ { t } ( v ) } \log ( 1 - \sigma ( < e _ { u ^ { ' } } ^ { t } , e _ { v } ^ { t } > ) )
+$$
+
+where $\sigma$ is the sigmoid function, $< . >$ denotes the inner product operation, $\mathcal { N } _ { w a l k } ^ { t } ( v )$ is the set of nodes that co-occur with $v$ on fixed-length random walks at snapshot $t .$ , $P _ { n } ^ { t }$ is a negative sampling distribution for snapshot $\mathcal { G } ^ { t }$ , and $w _ { n }$ , negative sampling ratio, is a tunable hyper-parameter to balance the positive and negative samples.
+
+# 5 EXPERIMENTS
+
+We evaluate the quality of our learned node representations on the fundamental task of dynamic link prediction. We choose this task since it has been widely used (Trivedi et al., 2017; Goyal et al., 2017; Li et al., 2018) in evaluating the quality of dynamic node representations to predict the temporal evolution in graph structure.
+
+In our experiments, we compare the performance of DySAT against a variety of static and dynamic graph representation learning baselines. Our experimental results on four publicly available benchmarks indicate that DySAT achieves significant performance gains over other methods.
+
+<table><tr><td></td><td colspan="2">Communication Networks</td><td colspan="2"> Rating Networks</td></tr><tr><td>Dataset</td><td>Enron</td><td>UCI</td><td>Yelp</td><td>ML-10M</td></tr><tr><td># Nodes</td><td>143</td><td>1,809</td><td>6,569</td><td>20,537</td></tr><tr><td># Links</td><td>2,347</td><td>16,822</td><td>95,361</td><td>43,760</td></tr><tr><td># Time steps</td><td>10</td><td>13</td><td>12</td><td>13</td></tr></table>
+
+Table 1: Statistics of the datasets used in our experiments
+
+# 5.1 DATASETS
+
+We use four dynamic graph datasets with two communication and bipartite rating networks each.
+
+Communication networks. We consider two publicly available communication network datasets: Enron (Klimt & Yang, 2004) and UCI (Panzarasa et al., 2009). In Enron, the communication links are email interactions between core employees and the links in UCI represent messages sent between users on an online social network platform.
+
+Rating networks. We use two bipartite rating networks from $\mathrm { Y e l p } ^ { \mathrm { 1 } }$ and MovieLens (Harper & Konstan, 2016). In Yelp, the dynamic graph comprises links between two types of nodes, users and businesses, derived from the observed ratings over time. ML-10M consists of a user-tag interaction network where user-tag links connects users with the tags they applied on certain movies.
+
+In each dataset, multiple graph snapshots are created based on the observed interactions in fixedlength time windows. Dataset statistics are shown in Table 1, while Appendix G has further details.
+
+# 5.2 EXPERIMENTAL SETUP
+
+We conduct experiments on the task of link prediction in dynamic graphs, where we learn dynamic node representations on snapshots $\{ \mathcal { G } ^ { 1 } , \ldots , \mathcal { G } ^ { t } \}$ and use $\{ \dot { e } _ { v } ^ { t } , \forall v \in \mathcal { V } \}$ to predict the links at $\mathcal G ^ { t + 1 }$ during evaluation. We compare different models based on their ability to correctly classify each example (node pair) into links and non-links. To further analyze predictive capability, we also evaluate new link prediction, with a focus on new links that appear at each time step, (Appendix B).
+
+We evaluate the performance of different models by training a logistic regression classifier for dynamic link prediction (Zhou et al., 2018). We create evaluation examples from the links in $\mathcal G ^ { t + 1 }$ and an equal number of randomly sampled pairs of unconnected nodes (non-links). A held-out validation set $20 \%$ links) is used to tune the hyper-parameters across all models, which is later discarded. We randomly sample $2 5 \%$ of the examples for training and use the remaining $7 5 \%$ as our test set. We repeat this for 10 randomized runs and report the average performance in our results.
+
+We follow the strategy recommended by Grover & Leskovec (2016) to compute the feature representation for a pair of nodes, using the Hadamard Operator $( e _ { u } ^ { t } \odot e _ { v } ^ { t } )$ , for all methods unless explicitly specified otherwise. The Hadamard operator computes the element-wise product of two vectors and closely mirrors the widely used inner product operation in learning node embeddings. We evaluate the performance of link prediction using Area Under the ROC Curve (AUC) scores (Grover & Leskovec, 2016). We also report the average precision scores in Table 6 of the Appendix.
+
+We implement DySAT in Tensorflow (Abadi et al., 2016) and employ mini-batch gradient descent with Adam optimizer (Kingma & Ba, 2015) for training. For Enron, we use a single layer in both the structural and temporal blocks, with each layer comprising 16 attention heads computing 8 features apiece (for a total of 128 dimensions). In the other datasets, we use two structural self-attentional layers with 16 and 8 heads respectively, each computing 16 features (layer sizes of 256, 128). The model is trained for a maximum of 200 epochs with a batch size of 256 nodes and the best performing model on the validation set, is chosen for evaluation.
+
+# 5.3 BASELINE
+
+We compare the performance of DySAT with several state-of-the-art dynamic graph embedding techniques. In addition, we include several static graph embedding methods in comparison to analyze the gains of using temporal information for dynamic link prediction. To make a fair comparison with static methods, we provide access to the entire history of snapshots by constructing an aggregated graph upto time $t$ , where the weight of each link is defined as the cumulative weight till $t$ agnostic of its occurrence times. We use author-provided implementations for all the baselines and set the final embedding dimension $d = 1 2 8$ .
+
+Table 2: Experiment results on dynamic link prediction (micro and macro averaged AUC with standard deviation). We show GraphSAGE (denoted by G-SAGE) results with the best performing aggregators for each dataset ( $^ *$ represents GCN, $\dagger$ represents LSTM, and $^ \ddag$ represents max-pooling).   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Enron</td><td colspan="2">UCI</td><td colspan="2">Yelp</td><td colspan="2">ML-10M</td></tr><tr><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td></tr><tr><td>node2vec</td><td>83.72 ± 0.7</td><td>83.05 ± 1.2</td><td>79.99 ± 0.4</td><td>80.49±0.6</td><td>67.86 ± 0.2</td><td>65.34± 0.2</td><td>87.74± 0.2</td><td>87.52 ± 0.3</td></tr><tr><td>G-SAGE</td><td>82.48*±0.6</td><td>81.88*±0.5</td><td>79.15*±0.4</td><td>82.89*±0.2</td><td>60.95† ± 0.1</td><td>58.56†± 0.2</td><td>86.19*± 0.3</td><td>89.92*± 0.1</td></tr><tr><td>G-SAGE +GAT</td><td>72.52 ± 0.4</td><td>73.34 ± 0.6</td><td>74.03 ± 0.4</td><td>79.83±0.2</td><td>66.15 ± 0.1</td><td>65.09±0.2</td><td>83.97 ± 0.3</td><td>84.93 ± 0.1</td></tr><tr><td>GCN-AE</td><td>81.55 ± 1.5</td><td>81.71 ± 1.5</td><td>80.53 ± 0.3</td><td>83.50±0.5</td><td>66.71± 0.2</td><td>65.82 ± 0.2</td><td>85.49 ± 0.1</td><td>85.74 ± 0.1</td></tr><tr><td>GAT-AE</td><td>75.71 ± 1.1</td><td>75.97 ± 1.4</td><td>79.98 ± 0.2</td><td>81.86±0.3</td><td>65.92 ± 0.1</td><td>65.37 ± 0.1</td><td>87.01 ± 0.2</td><td>86.75±0.2</td></tr><tr><td>DynamicTriad</td><td>80.26±0.8</td><td>78.98± 0.9</td><td>77.59 ± 0.6</td><td>80.28±0.5</td><td>63.53 ± 0.3</td><td>62.69 ± 0.3</td><td>88.71± 0.2</td><td>88.43 ± 0.1</td></tr><tr><td>Know-Evolve</td><td>61.57 ± 1.1</td><td>62.28 ± 1.5</td><td>71.20 ± 0.5</td><td>80.93±0.2</td><td>56.88± 0.2</td><td>59.68± 0.2</td><td>78.80± 0.5</td><td>83.70±0.2</td></tr><tr><td>DynGEM</td><td>67.83 ± 0.6</td><td>69.72 ± 1.3</td><td>77.49 ± 0.3</td><td>79.82 ± 0.5</td><td>66.02 ± 0.2</td><td>65.94 ± 0.2</td><td>73.69 ± 1.2</td><td>85.96 ± 0.3</td></tr><tr><td>DySAT</td><td>85.71± 0.3</td><td>86.60 ± 0.2</td><td>81.03 ± 0.2</td><td>85.81 ± 0.1</td><td>70.15 ± 0.1</td><td>69.87 ± 0.1</td><td>90.82 ± 0.3</td><td>93.68± 0.1</td></tr></table>
+
+We compare against several state-of-the-art unsupervised static embedding methods: node2vec (Grover & Leskovec, 2016), GraphSAGE (Hamilton et al., 2017b) and graph autoencoders (Hamilton et al., 2017a). We experiment with different aggregators in GraphSAGE, namely, GCN, mean-pooling, max-pooling, and LSTM, to report the performance of the best performing aggregator in each dataset. To provide a fair comparison with GAT (Velickovic et al., 2018), which originally conduct experiments only on node classification, we implement a graph attention layer as an additional aggregator in GraphSAGE, which we denote by GraphSAGE $^ +$ GAT. We also train GCN and GAT as autoencoders for link prediction along the suggested lines of (Zitnik et al., 2018), denoted by GCN-AE and GAT-AE respectively. In the dynamic setting, we evaluate DySAT against the most recent studies on dynamic graph embedding including Know-Evolve (Trivedi et al., 2017), DynamicTriad (Zhou et al., 2018), and DynGEM (Goyal et al., 2017). The details of hyper-parameter tuning for all methods can be found in Appendix F.
+
+# 5.4 EXPERIMENTAL RESULTS
+
+We evaluate the models at each time step $t$ by training separate models up to snapshot $t$ and evaluate at $t + 1$ for each $t = 1 , \dots , T$ . We summarize the micro and macro averaged AUC scores (across all time steps) for all models in Table 2. From the results, we observe that DySAT achieves consistent gains of $3- 4 \%$ macro-AUC, in comparison to the best baseline across all datasets.
+
+Further, we compare the model performance at each time step (Figure 2), to obtain a deep understanding of their temporal behaviors. We fine the performance of DySAT to be relatively more stable than other methods. This contrast is pronounced in the communication networks (Enron and UCI), where we observe drastic drops in performance of static embedding methods at certain time steps.
+
+The runtime per mini-batch of DySAT on ML-10M, using a machine with Nvidia Tesla V100 GPU and 28 CPU cores, is 0.72 seconds. In comparison, a model variant without temporal attention (Appendix A) takes 0.51 seconds, which illustrates the relatively low cost of temporal attention.
+
+# 6 DISCUSSION
+
+Our experimental results provide several interesting observations and insights to the performance of different graph embedding techniques.
+
+First, we observe that GraphSAGE achieves comparable performance to DynamicTriad across different datasets, despite being trained only on static graphs. One possible explanation may be that GraphSAGE uses trainable neighbor-aggregation functions, while DynamicTriad employs Skipgram based methods augmented with temporal smoothness. This leads us to conjecture that the combination of structural and temporal modeling with expressive aggregation functions, such as multi-head attention, is responsible for the consistently superior performance of DySAT on dynamic link prediction. We also observe that node2vec achieves consistent performance agnostic of temporal information, which demonstrates the effectiveness of second-order random walk sampling. This observation points to the direction of applying sampling techniques to further improve DySAT.
+
+![](images/8e7b96d711bd175b7b544f220041db793cb7d0c654976046f77eb0cbc09f2930.jpg)  
+Figure 2: Performance comparison of DySAT with different models across multiple time steps: the solid line represents DySAT; dashed lines represent static graph embedding models; and dotted lines represent dynamic graph embedding models. We truncate the y-axis to avoid visual clutter.
+
+In DySAT, we employ structural attention layers followed by temporal attention layers. We choose this design because graph structures are not stable over time, which makes directly employing structural attention layers after temporal attention layers infeasible. We also consider another alternative design choice that applies self-attention along the two dimensions of neighbors and time together following the strategy similar to (Shen et al., 2018). In practice, this would be computationally expensive due to variable number of neighbors per node across multiple snapshots. We leave exploring other architectural design choices based on structural and temporal self-attentions as future work.
+
+In the current setup, we store the adjacency matrix of each snapshot in memory using sparse matrix, which may pose memory challenges when scaling to large graphs. In the future, we plan to explore DySAT with memory-efficient mini-batch training strategy along the lines of GraphSAGE (Hamilton et al., 2017b). Further, we develop an incremental self-attention network (IncSAT) that is efficient in both computation and memory cost as a direct extension of DySAT. Our initial results are promising as reported in Appendix E, which opens the door to future exploration of self-attentional architectures for incremental (or streaming) graph representation learning. We also evaluate the capability of DySAT on multi-step link prediction or forecasting and observe significant relative improvements of $6 \%$ AUC on average over existing methods, as reported in Appendix C.
+
+# 7 CONCLUSION
+
+In this paper, we introduce a novel self-attentional neural network architecture named DySAT to learn node representations in dynamic graphs. Specifically, DySAT computes dynamic node representations using self-attention over the (1) structural neighborhood and (2) historical node representations, thus effectively captures the temporal evolutionary patterns of graph structures. Our experiment results on various real-world dynamic graph datasets indicate that DySAT achieves significant performance gains over several state-of-the-art static and dynamic graph embedding baselines. Though our experiments are conducted on graphs without node features, DySAT can be easily generalized on feature-rich graphs. Another interesting direction is exploring continuous-time generalization of our framework to incorporate more fine-grained temporal variations.
+
+# REFERENCES
+
+Mart´ın Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, Manjunath Kudlur, Josh Levenberg, Rajat Monga, Sherry Moore, Derek Gordon Murray, Benoit Steiner, Paul A. Tucker, Vijay Vasudevan, Pete Warden, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. Tensorflow: A system for large-scale machine learning. In 12th USENIX Symposium on Operating Systems Design and Implementation, OSDI 2016, Savannah, GA, USA, November 2-4, 2016., pp. 265–283, 2016.
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In International Conference on Learning Representations (ICLR), 2015.
+
+Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in Neural Information Processing Systems 14 [Neural Information Processing Systems: Natural and Synthetic, NIPS 2001, December 3-8, 2001, Vancouver, British Columbia, Canada], pp. 585–591, 2001.
+
+Michael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks ¨ on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems 29: Annual Conference on Neural Information Processing Systems 2016, December 5- 10, 2016, Barcelona, Spain, pp. 3837–3845, 2016.
+
+Jonas Gehring, Michael Auli, David Grangier, Denis Yarats, and Yann N. Dauphin. Convolutional sequence to sequence learning. In Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, pp. 1243–1252, 2017.
+
+Palash Goyal, Nitin Kamra, Xinran He, and Yan Liu. Dyngem: Deep embedding method for dynamic graphs. In IJCAI International Workshop on Representation Learning for Graphs (ReLiG), August 2017.
+
+Aditya Grover and Jure Leskovec. node2vec: Scalable feature learning for networks. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, August 13-17, 2016, pp. 855–864, 2016.
+
+William L Hamilton, Rex Ying, and Jure Leskovec. Representation learning on graphs: Methods and applications. arXiv preprint arXiv:1709.05584, 2017a.
+
+William L. Hamilton, Zhitao Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4-9 December 2017, Long Beach, CA, USA, pp. 1025– 1035, 2017b.
+
+F Maxwell Harper and Joseph A Konstan. The movielens datasets: History and context. ACM Transactions on Interactive Intelligent Systems (TIIS), 5(4):19, 2016.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations (ICLR), 2015.
+
+Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In International Conference for Learning Representations (ICLR), 2017.
+
+Bryan Klimt and Yiming Yang. Introducing the enron corpus. In CEAS 2004 - First Conference on Email and Anti-Spam, July 30-31, 2004, Mountain View, California, USA, 2004.
+
+Gueorgi Kossinets and Duncan J Watts. Empirical analysis of an evolving social network. science, 311(5757):88–90, 2006.
+
+Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. Graph evolution: Densification and shrinking diameters. ACM Transactions on Knowledge Discovery from Data (TKDD), 1(1):2, 2007.
+
+Jundong Li, Harsh Dani, Xia Hu, Jiliang Tang, Yi Chang, and Huan Liu. Attributed network embedding for learning in a dynamic environment. In Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, CIKM 2017, Singapore, November 06 - 10, 2017, pp. 387–396, 2017.
+
+Jundong Li, Kewei Cheng, Liang Wu, and Huan Liu. Streaming link prediction on dynamic attributed networks. In Proceedings of the Eleventh ACM International Conference on Web Search and Data Mining, WSDM 2018, Marina Del Rey, CA, USA, February 5-9, 2018, pp. 369–377, 2018.
+
+Zhouhan Lin, Minwei Feng, Cicero Nogueira dos Santos, Mo Yu, Bing Xiang, Bowen Zhou, and Yoshua Bengio. A structured self-attentive sentence embedding. In International Conference on Learning Representations (ICLR), 2017.
+
+Giang Hoang Nguyen, John Boaz Lee, Ryan A Rossi, Nesreen K Ahmed, Eunyee Koh, and Sungchul Kim. Continuous-time dynamic network embeddings. In 3rd International Workshop on Learning Representations for Big Networks (WWW BigNet), 2018.
+
+Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov. Learning convolutional neural networks for graphs. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pp. 2014–2023, 2016.
+
+Pietro Panzarasa, Tore Opsahl, and Kathleen M. Carley. Patterns and dynamics of users’ behavior and interaction: Network analysis of an online community. JASIST, 60(5):911–932, 2009.
+
+Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: online learning of social representations. In The 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’14, New York, NY, USA - August 24 - 27, 2014, pp. 701–710, 2014.
+
+Aravind Sankar, Xinyang Zhang, and Kevin Chen-Chuan Chang. Motif-based convolutional neural network on graphs. CoRR, abs/1711.05697, 2017.
+
+Peter Shaw, Jakob Uszkoreit, and Ashish Vaswani. Self-attention with relative position representations. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, NAACL-HLT, New Orleans, Louisiana, USA, June 1-6, 2018, Volume 2 (Short Papers), pp. 464–468, 2018.
+
+Tao Shen, Tianyi Zhou, Guodong Long, Jing Jiang, Shirui Pan, and Chengqi Zhang. Disan: Directional self-attention network for rnn/cnn-free language understanding. In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, New Orleans, Louisiana, USA, February 2-7, 2018, 2018.
+
+Nitish Srivastava, Geoffrey E. Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1):1929–1958, 2014.
+
+Zhixing Tan, Mingxuan Wang, Jun Xie, Yidong Chen, and Xiaodong Shi. Deep semantic role labeling with self-attention. In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, New Orleans, Louisiana, USA, February 2-7, 2018.
+
+Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. LINE: large-scale information network embedding. In Proceedings of the 24th International Conference on World Wide Web, WWW 2015, Florence, Italy, May 18-22, 2015, pp. 1067–1077, 2015. doi: 10.1145/ 2736277.2741093.
+
+Joshua B Tenenbaum, Vin De Silva, and John C Langford. A global geometric framework for nonlinear dimensionality reduction. science, 290(5500):2319–2323, 2000.
+
+Rakshit Trivedi, Hanjun Dai, Yichen Wang, and Le Song. Know-evolve: Deep temporal reasoning for dynamic knowledge graphs. In Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, pp. 3462–3471, 2017.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4-9 December 2017, Long Beach, CA, USA, pp. 6000–6010, 2017.
+
+Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. Graph attention networks. In International Conference on Learning Representations (ICLR), 2018.
+
+Daixin Wang, Peng Cui, and Wenwu Zhu. Structural deep network embedding. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, August 13-17, 2016, pp. 1225–1234, 2016. doi: 10.1145/2939672.2939753.
+
+Zhen Wang, Jianwen Zhang, Jianlin Feng, and Zheng Chen. Knowledge graph embedding by translating on hyperplanes. In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, July 27 -31, 2014, Quebec City, Qu ´ ebec, Canada. ´ , pp. 1112–1119, 2014.
+
+Adams Wei Yu, David Dohan, Minh-Thang Luong, Rui Zhao, Kai Chen, Mohammad Norouzi, and Quoc V. Le. Qanet: Combining local convolution with global self-attention for reading comprehension. In International Conference on Learning Representations (ICLR), 2018.
+
+Le-kui Zhou, Yang Yang, Xiang Ren, Fei Wu, and Yueting Zhuang. Dynamic network embedding by modeling triadic closure process. In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, New Orleans, Louisiana, USA, February 2-7, 2018.
+
+Linhong Zhu, Dong Guo, Junming Yin, Greg Ver Steeg, and Aram Galstyan. Scalable temporal latent space inference for link prediction in dynamic social networks. IEEE Trans. Knowl. Data Eng., 28(10):2765–2777, 2016.
+
+Marinka Zitnik, Monica Agrawal, and Jure Leskovec. Modeling polypharmacy side effects with graph convolutional networks. Bioinformatics, 34(13):457466, 2018.
+
+Yuan Zuo, Guannan Liu, Hao Lin, Jia Guo, Xiaoqian Hu, and Junjie Wu. Embedding temporal network via neighborhood formation. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 2857–2866. ACM, 2018.
+
+# A ANALYSIS OF TEMPORAL ATTENTION
+
+In this section, we conduct an in-depth analysis of the proposed temporal attention layer, to demonstrate its utility and examine the distribution of temporal attention weights.
+
+# A.1 EFFECT OF REMOVING TEMPORAL LAYERS
+
+To demonstrate the effectiveness of temporal self-attention layers, we conduct an experimental study that removes the temporal attention block from DySAT to create a simpler architecture. This model is optimized using the same loss function (Eqn. 5) applied on the intermediate representations $\{ h _ { v } ^ { 1 } , h _ { v } ^ { 2 } , \ldots , h _ { v } ^ { T } \}$ for each node $v \in \mathcal V$ . Note that this model is different from static methods since the structural self-attention block is jointly optimized (using Eqn. 5) across all snapshots without any explicit temporal modeling. We use the best configuration of DySAT on each dataset from the original experiments to initialize the new model. The performance comparison is shown in Table 3. We observe that in some datasets, the structural attention block is able to learn some temporal evolution patterns in graph structure, despite the lack of explicit temporal modeling. However, the new model is consistently inferior to DySAT and we observe that the original DySAT has a $3 \%$ average gain in Macro-AUC, which validates our choice of using temporal self-attentional layers.
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Enron</td><td colspan="2">UCI</td><td colspan="2">Yelp</td><td colspan="2">ML-10M</td></tr><tr><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td></tr><tr><td>Original</td><td>85.71±0.3</td><td>86.60± 0.2</td><td>81.03 ± 0.2</td><td>85.81 ± 0.1</td><td>70.15 ± 0.1</td><td>69.87 ± 0.1</td><td>90.82 ± 0.3</td><td>93.68 ± 0.1</td></tr><tr><td>No Temporal</td><td>84.50± 0.3</td><td>85.68± 0.4</td><td>76.61 ± 0.2</td><td>79.97 ± 0.3</td><td>68.34± 0.1</td><td>67.20± 0.3</td><td>89.61± 0.4</td><td>91.10 ± 0.2</td></tr></table>
+
+Table 3: Experimental study on removing temporal attention layers from DySAT (micro and macro averaged AUC with standard deviation)
+
+# A.2 VISUALIZATION OF TEMPORAL ATTENTION WEIGHTS
+
+We conduct a qualitative analysis to obtain deeper insights into the distribution of temporal attention weights learned by DySAT. In this experiment, we examine the temporal attention coefficients learned at each time step $t$ , which indicate the relative importance of each historical snapshot $( < t )$ in predicting the links at $t$ . We choose the Enron dataset to visualize the mean and standard deviation of temporal attention coefficients, over all the nodes. Figure 3 visualizes a heatmap of the learned temporal attention weights on Enron dataset for the first 10 time steps.
+
+From Figure 3, we observe that the mean temporal attention weights are mildly biased towards recent snapshots, while the historical snaphots vary in their importance across different time steps. Further, we find that the standard deviation of attention weights across different nodes is generally high and exhibits more variability. Thus, the temporal attention weights are well distributed across historical snapshots, with significant variance across different nodes in the graph. While this analysis attempts to provide a high-level perspective on the weights learned by temporal attention, an appropriate interpretation of these coefficients (as done by e.g., (Bahdanau et al., 2015)) requires further domain knowlege about the dataset under study, and is left as future work.
+
+![](images/c8a7137a7b861e4a60d8397700bfc604980fb88203ae9be3f8cc73ee2de30447.jpg)  
+Figure 3: Heatmap visualizing mean and standard deviation of temporal attention weights over all nodes in Enron dataset.
+
+# B DYNAMIC NEW LINK PREDICTION
+
+In this section, we additionally report the results of dynamic link prediction evaluated only on the new links at each time step. This provides an in-depth analysis on the capabilities of different methods in predicting relatively unseen links. We follow the same evaluation setup of training a downstream logistic regression classifier for dynamic link prediction. However, a key difference is that the evaluation examples comprise new links at $\mathcal { G } _ { t + 1 }$ (that are not in $\mathcal { G } _ { t }$ ) and an equal number of randomly sampled non-links.
+
+Table 4 summarizes the micro and macro averaged AUC scores for different methods on the four datasets. The absolute performance numbers of all methods are lower than the original evaluation setup of using all links at $\mathcal { G } _ { t + 1 }$ , which is reasonable since accurate prediction of new links at $\mathscr { G } _ { t + 1 }$ is expected to be slightly more challenging in comparison to predicting all the links at $\mathcal { G } _ { t + 1 }$ . From Table 2), we find that DySAT achieves consistent relative gains of $3- 5 \%$ Macro-AUC over the best baselines on dynamic new link prediction as well, thus validating its effectiveness in accurately capturing temporal context for new link prediction.
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Enron</td><td colspan="2">UCI</td><td colspan="2">Yelp</td><td colspan="2">ML-10M</td></tr><tr><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td></tr><tr><td>node2vec</td><td>76.92 ± 1.2</td><td>75.86 ± 0.5</td><td>73.67 ± 0.3</td><td>74.76± 0.8</td><td>67.36± 0.2</td><td>65.17± 0.2</td><td>85.22± 0.2</td><td>84.89 ±0.1</td></tr><tr><td>G-SAGE</td><td>73.92*±0.7</td><td>74.67*±0.6</td><td>76.69*± 0.3</td><td>79.41*± 0.1</td><td>62.25† ± 0.2</td><td>58.81†± 0.3</td><td>85.23+±0.3</td><td>89.14*± 0.2</td></tr><tr><td>G-SAGE + GAT</td><td>67.02 ± 0.8</td><td>68.32 ±0.7</td><td>73.18 ± 0.4</td><td>76.79 ± 0.2</td><td>66.53±0.2</td><td>65.45 ± 0.1</td><td>80.84 ±0.3</td><td>82.53 ± 0.1</td></tr><tr><td>GCN-AE</td><td>74.46 ± 1.1</td><td>74.02 ± 1.6</td><td>74.76 ± 0.1</td><td>76.75 ± 0.6</td><td>66.18 ± 0.2</td><td>65.77 ± 0.3</td><td>82.45 ± 0.3</td><td>82.48 ± 0.2</td></tr><tr><td>GAT-AE</td><td>69.75 ± 2.2</td><td>69.25 ± 1.9</td><td>72.52 ± 0.4</td><td>73.78 ±0.7</td><td>66.07 ± 0.1</td><td>65.91± 0.2</td><td>84.98± 0.2</td><td>84.51 ± 0.3</td></tr><tr><td>DynamicTriad</td><td>69.59 ± 1.2</td><td>68.77 ±1.7</td><td>67.97 ± 0.7</td><td>71.67 ± 0.9</td><td>63.76± 0.2</td><td>62.83 ± 0.3</td><td>84.72 ± 0.2</td><td>84.32 ± 0.2</td></tr><tr><td>Know-Evolve</td><td>59.05± 2.7</td><td>59.63 ± 2.7</td><td>69.10 ± 0.3</td><td>77.48 ± 0.2</td><td>56.95 ± 0.2</td><td>59.72 ± 0.5</td><td>76.83 ± 0.5</td><td>82.23 ±0.2</td></tr><tr><td>DynGEM</td><td>60.73 ± 1.1</td><td>62.85 ± 1.9</td><td>77.49 ± 0.3</td><td>79.82 ± 0.5</td><td>66.42 ± 0.2</td><td>66.84± 0.2</td><td>73.77 ± 0.7</td><td>83.51 ± 0.3</td></tr><tr><td>DySAT</td><td>78.87 ± 0.6</td><td>78.58 ± 0.6</td><td>79.24 ± 0.3</td><td>83.66± 0.2</td><td>69.46 ± 0.1</td><td>69.14 ± 0.1</td><td>89.29 ± 0.2</td><td>92.65 ± 0.1</td></tr></table>
+
+Table 4: Experiment results on dynamic new link prediction (micro and macro averaged AUC with standard deviation). We show GraphSAGE (denoted by G-SAGE) results with the best performing aggregators for each dataset ( $^ *$ represents GCN, $\dagger$ represents LSTM, and $^ \ddag$ represents max-pooling).
+
+# C MULTI-STEP LINK PREDICTION
+
+In this section, we evaluate various dynamic graph representation learning methods on the task of multi-step link prediction or forecasting. Here, each model is trained for a fixed number of time steps, and the latest embeddings are used to predict links at multiple future time steps. In each dataset, we choose the last 6 snapshots to evaluate multi-step link prediction. The model is trained on the previous remaining snapshots, and the latest embeddings are used to forecast links at future time steps. For each future time step $t + \Delta$ ( $1 \le \Delta \le 6$ ), we create examples from the links in $\mathscr { G } _ { t + \Delta }$ and an equal number of randomly sampled pairs of unconnected nodes (non-links). We otherwise use the same evaluation setup of training a downstream logistic regression classifier to evaluate link prediction. In this experiment, we exclude the links formed by nodes that newly appear in the future evaluation snapshots, since most methods cannot be easily support updates for new nodes.
+
+Figure. 4 depicts the variation in model performance of different methods over the 6 evaluation snapshots. As expected, we observe a slight decay in performance over time for all the models. DySATachieves significant performance gains over all other baselines and maintains a highly stable link prediction performance over multiple future time steps. Static embedding methods often exhibit large variations in performance over time steps, while DySAT achieves a stable consistent performance. The historical context captured by the dynamic node embeddings of DySAT, is one of the most likely reasons for its stable multi-step forecasting performance.
+
+![](images/1931632c976d467fb20f0359853ed24f1e28b58e60a0e031cd6f751a9d2cb516.jpg)  
+Figure 4: Performance comparison of DySAT with different models on multi-step link prediction for 6 future time steps on all datasets
+
+# D IMPACT OF UNSEEN NODES ON DYNAMIC LINK PREDICTION
+
+In this section, we analyze the sensitivity of different graph representation learning on link prediction for previously unseen nodes that appear newly at time t.
+
+![](images/8fc9d1bbe6771cef0f507b7fd722c83058405211c39921d11b2977cd27804a86.jpg)  
+Figure 5: Performance comparison of DySAT with different models on link prediction restricted to new nodes at each time step
+
+![](images/c95b3a2557b60b9e354b570437b922ed3e62865a6ce2ec952771ddb6999baaea.jpg)  
+Figure 6: Neural architecture of IncSAT: the components that are excluded from DySAT are wrapped by dashed blue rectangles. The intermediate node representations are directly loaded from saved models trained previously.
+
+A node is considered as a new node at time step $t$ in $\mathcal { G } _ { t }$ if it has not appeared (has no links) in any of the previous $t - 1$ snapshots. In this experiment, the evaluation set at time step $t$ only comprises the subset of links at $\mathscr { G } _ { t + 1 }$ among the new nodes in $\mathcal { G } _ { t }$ and corresponding randomly sampled non-links. Since the number of nodes varies significantly across different time steps, we report the performance of each method along with the number of new nodes at each time step, in Figure 5.
+
+From Figure 5, we observe that DySAT outperforms other baselines in most datasets, demonstrating the ability to characterize new or previously unseen nodes despite their limited history. Although the temporal attention will focus on the latest representation of a new node $v$ due to absence of history, the structural embedding of $v$ recieves backpropagation signals through the temporal attention on neighboring nodes, which indirectly affects the final embedding of $v$ . We hypothesize that this indirect temporal signal is one of the reasons for DySAT to achieve performance improvements over baselines, albeit not designed to explicitly model historical context for previously unseen nodes.
+
+# E INCREMENTAL SELF-ATTENTION NETWORK
+
+In this section, we describe an extension of our dynamic self-attentional architecture to learn incremental node representations. The motivation of incremental learning arises due to the proliferation in sizes of real-world graphs, making it difficult to store multiple snapshots in memory. Thus, the incremental graph representation learning problem imposes the restriction of no access to historical graph snapshots, in contrast to most dynamic graph embedding methods. Specifically, to learn the node embeddings $\{ e _ { v } ^ { t } \in \mathbb R ^ { d } \forall v \in \mathcal { V } \}$ at time $T$ , we require a model to only access to the snapshot $\mathcal { G } ^ { T }$ and a summary of the historical snapshots. For example, DynGEM (Goyal et al., 2017) is an example of an incremental embedding method that uses the embeddings learned at step $t - 1$ , as initialization to learn the embeddings at $t$ .
+
+We propose an extension of our self-attentional architecture named IncSAT to explore solving the incremental graph representation learning problem. To learn node representations at $T$ , we first incrementally train multiple models at $1 \leq t \leq T$ . Unlike the original DySAT where structural self-attention is applied at each snapshot, IncSAT applies the structural block only at the latest graph snapshot $\mathcal { G } ^ { t }$ . We enable incremental learning by storing the intermediate output representations $\{ h _ { v } ^ { \hat { T } } \ \forall v \in \mathcal { V } \}$ of the structural block. As illustrated in Figure 6, these intermediate output representations of historical snapshots $1 \leq t < T$ ) can be directly loaded from previously saved results at $1 \leq t < T$ . Thus, the structural information of the previous historical snapshots are summarized in the stored intermediate representations. The temporal self-attention is only applied to the current snapshot $\mathcal { G } ^ { T }$ over the historical representations of each node to compute the final node embeddings $\{ e _ { v } ^ { \bar { T } } \forall v \in \mathcal { V } \}$ at $T$ , which are trained on random walks sampled from $\mathcal { G } ^ { T }$ .
+
+Table 5: Experimental results of IncSAT in comparison to DySAT (micro and macro averaged AUC with standard deviation)   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Enron</td><td colspan="2">UCI</td><td colspan="2">Yelp</td><td colspan="2">ML-10M</td></tr><tr><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td><td>Micro-AUC</td><td>Macro-AUC</td></tr><tr><td>DySAT</td><td>85.71 ± 0.3</td><td>86.60 ± 0.2</td><td>81.03 ± 0.2</td><td>85.81 ± 0.1</td><td>70.15 ± 0.1</td><td>69.87 ± 0.1</td><td>90.82 ± 0.3</td><td>93.68± 0.1</td></tr><tr><td>DynGEM</td><td>67.83 ± 0.6</td><td>69.72 ± 1.3</td><td>77.49 ± 0.3</td><td>79.82 ± 0.5</td><td>66.02± 0.2</td><td>65.94± 0.2</td><td>73.69 ± 1.2</td><td>85.96± 0.3</td></tr><tr><td>IncSAT</td><td>84.36± 0.2</td><td>85.43 ±0.3</td><td>76.18 ± 0.5</td><td>85.37 ± 0.2</td><td>69.54 ± 0.1</td><td>68.73 ±0.3</td><td>80.13 ± 0.4</td><td>91.14 ± 0.2</td></tr></table>
+
+We evaluate IncSAT using the same experimental setup, with minor modifications in hyperparameters. We use a dropout rate of 0.4 in both the structural and temporal self-attention layers. From our preliminary experiments, we find that a higher dropout rate in the structural block can facilitate avoiding over-fitting the model to the current graph snapshot. In Table 5, we report the performance of IncSAT in comparison to DySAT and DynGEM, which is the only one that can support incremental training from our baseline models. The results show that IncSAT achieves comparable performance to DySAT on most datasets while significantly outperforming DynGEM, albeit with minimal hyper-parameter tuning.
+
+# F DETAILS ON HYPER-PARAMETER SETTINGS AND TUNING
+
+In DySAT, the objective function (Eqn. 5) utilizes positive pairs of nodes co-occurring in fixed-length random walks. We follow the strategy of Deepwalk (Perozzi et al., 2014) to sample walks 10 walks of length 40 per node, each with a context window size of 10. We use 10 negative samples per positive pair, with context distribution $( P _ { n } ^ { t } )$ smoothing over node degrees with a smoothing parameter of 0.75, following (Perozzi et al., 2014; Grover & Leskovec, 2016; Hamilton et al., 2017b). During training, we apply $L _ { 2 }$ regularization with $\lambda = 5 \times 1 0 ^ { - 4 }$ and use dropout rates (Srivastava et al., 2014) of 0.1 and 0.5 in the self-attention layers of the structural and temporal blocks respectively. We use the validation set for tuning the learning rate in the range of $\lbrace 1 0 ^ { - 4 } , 1 0 ^ { - 3 } \rbrace$ and negative sampling ratio $w _ { n }$ in the range $\{ 0 . 0 \bar { 1 } , 0 . 1 , 1 \}$ .
+
+We tune the hyper-parameters of all baselines following their recommended guidelines. For node2vec, we use the default settings as in the paper, with 10 random walks of length 80 per node and context window of 10, trained for a single epoch. We tune the in-out and return hyper-parameters, $p , q$ using grid-search, in the range $\{ 0 . 2 5 , 0 . 5 0 , 1 , 2 , 4 \}$ and report the best results. In case of GraphSAGE, we train a two layer model with respective neighborhood sample sizes 25 and 10, for 10 epochs, as described in the original paper. We evaluate the embeddings at each epoch on the validation set, and choose the best for final evaluation. Note that the results of GraphSAGE reported in Table 2 represent that of best-performing aggregator in each dataset.
+
+For Know-evolve, we tune the two weight-scale hyper-parameters in the range $\{ 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , 0 . 0 1 , 0 . 1 \}$ , learning rates in $\{ 1 0 ^ { - 4 } , \bar { 1 0 } ^ { - 3 } \}$ and choose the best performing model. DynamicTriad (Zhou et al., 2018) was tuned using their two key hyper-parameters determining the effect of smoothness and triadic closure, $\beta _ { 0 }$ and $\beta _ { 1 }$ in the range $\{ 0 . 0 1 , 0 . 1 , 1 , 1 0 \}$ , as advised, while using recommended settings otherwise. We use the $L _ { 1 }$ operator $( | e _ { u } ^ { t } - e _ { u } ^ { t } | )$ instead of Hadamard, as recommended in the paper, which also gives better performance. For DynGEM, we tune the different scaling and regularization hyper-parameters, $\bar { \alpha } ~ \in ~ \{ 1 0 ^ { - 6 } , 1 0 ^ { - 5 } \}$ , $\beta ~ \in ~ \{ 0 . 1 , 1 , 2 , 5 \}$ , $\nu _ { 1 } \in \{ 1 0 ^ { - 6 } , 1 0 ^ { - 4 } \}$ and $\bar { \nu } _ { 2 } \in \{ 1 0 ^ { - 6 } , 1 \bar { 0 } ^ { - 4 } \}$ , while using other default configurations.
+
+# G ADDITIONAL DATASET DETAILS
+
+In this section, we provide some additional, relevant dataset details. Since dynamic graphs often contain continuous timestamps, we split the data into multiple snapshots using suitable time-windows such each snapshot has an equitable yet reasonable number of interactions (communication/ratings). In each snapshot, the weight of a link is determined by the number of interactions between the corresponding pair of users during that time-period. The pre-processed versions of all datasets will be made publicly available, along with the scripts used for processing the raw data.
+
+Table 6: Experiment results on dynamic link prediction (micro and macro average precision with standard deviation). We show GraphSAGE (denoted by G-SAGE) results with the base performing aggregators for each dataset ( $^ *$ represents GCN, $\dagger$ represents LSTM, and $^ \ddag$ represents max-pooling).   
+
+<table><tr><td>Method</td><td colspan="2">Enron</td><td colspan="2">UCI</td><td colspan="2">Yelp</td><td colspan="2">ML-10M</td></tr><tr><td></td><td>Micro-AP</td><td>Macro-AP</td><td>Micro-AP</td><td>Macro-AP</td><td>Micro-AP</td><td>Macro-AP</td><td>Micro-AP</td><td>Macro-AP</td></tr><tr><td>node2vec</td><td>84.26± 0.8</td><td>84.11 ± 1.1</td><td>80.22 ± 0.4</td><td>81.12 ± 0.5</td><td>66.46± 0.2</td><td>63.82 ± 0.2</td><td>88.86±0.2</td><td>88.71± 0.3</td></tr><tr><td>G-SAGE</td><td>83.99*±0.6</td><td>84.02*±0.6</td><td>75.91*± 0.6</td><td>82.36*±0.2</td><td>58.81† ± 0.1</td><td>55.84†± 0.2</td><td>85.45*± 0.3</td><td>90.26*± 0.2</td></tr><tr><td>G-SAGE +GAT</td><td>72.60 ± 0.5</td><td>74.75 ± 0.9</td><td>66.77 ± 0.4</td><td>76.30 ± 0.4</td><td>62.43± 0.1</td><td>61.49 ± 0.3</td><td>81.69± 0.6</td><td>82.35±0.2</td></tr><tr><td>GCN-AE</td><td>81.97 ± 1.5</td><td>83.08 ± 1.4</td><td>80.73 ± 0.4</td><td>84.16 ± 0.6</td><td>65.92 ± 0.2</td><td>65.39 ± 0.2</td><td>86.85± 0.2</td><td>87.43 ± 0.1</td></tr><tr><td>GAT-AE</td><td>76.98 ± 1.1</td><td>78.18 ± 0.8</td><td>80.14 ± 0.4</td><td>83.75 ± 0.3</td><td>65.45 ± 0.2</td><td>65.01 ± 0.2</td><td>88.42 ± 0.2</td><td>88.14 ± 0.2</td></tr><tr><td>DynamicTriad</td><td>82.06 ± 0.9</td><td>81.22 ± 0.9</td><td>76.21 ± 0.8</td><td>80.05 ±0.6</td><td>61.29 ± 0.3</td><td>60.79 ± 0.3</td><td>89.91 ± 0.2</td><td>89.61 ± 0.3</td></tr><tr><td>Know-Evolve</td><td>57.68 ± 1.2</td><td>60.71 ±1.7</td><td>66.99 ± 0.5</td><td>77.49 ± 0.2</td><td>53.98± 0.2</td><td>56.44 ± 0.2</td><td>75.64 ± 0.5</td><td>79.97 ± 0.3</td></tr><tr><td>DynGEM</td><td>70.37 ± 0.5</td><td>72.35 ± 1.0</td><td>78.78 ±0.3</td><td>81.71 ± 0.3</td><td>68.02 ± 0.2</td><td>68.09 ± 0.2</td><td>80.65±0.9</td><td>89.43 ±0.2</td></tr><tr><td>DySAT</td><td>86.82 ± 0.3</td><td>88.25± 0.2</td><td>80.88 ± 0.2</td><td>85.96 ± 0.1</td><td>65.81 ± 0.1</td><td>66.76 ± 0.1</td><td>93.03 ± 0.2</td><td>94.92 ± 0.1</td></tr></table>
+
+Communication Networks. The original un-processed Enron dataset is available at https: //www.cs.cmu.edu/˜./enron/. We use only the email communcations that are between Enron employees, i.e., sent by an Enron employee and have at least one recipient who is an Enron employee. A time-window of 2 months is used to construct 16 snapshots, where the first 5 are used as warm-up (due to sparsity) and the remaining 11 snapshots for evaluation.
+
+The UCI dataset was downloaded from http://networkrepository.com/opsahl_ ucsocial.php. This dataset contains private messages sent between users over a span of six months, on an online social network platform at the University of California, Irvine. The snapshots are created using their communication history with a time-window of 10 days. We discard/merge the terminal snapshots if they do not contain sufficient communications.
+
+Rating Networks. We use the Round 11 version of the Yelp Dataset Challenge https://www. yelp.com/dataset/challenge. To extract a cohesive subset of user-business ratings, we first select all businesses in the state of Arizona (the state with the largest number of ratings) with a selected set of restaurant categories. Further, we filter the data to retain only users and business which have at-least 15 ratings. Finally, we use a time-window of 6 months to extract 12 snapshots in the period of 2009 to 2015.
+
+The ML-10m dynamic user-tag interaction network was downloaded from http:// networkrepository.com/ia-movielens-user2tags-10m.php. This dataset depicts the tagging behavior of MovieLens users, with the tags applied by a user on her rated movies. We use a time-window of 3 months to extract 13 snaphots over the course of 3 years.

parse/train/OPyWRrcjVQw/OPyWRrcjVQw.md

@@ -0,0 +1,410 @@
+# SHAPLEY EXPLAINABILITY ON THE DATA MANIFOLD
+
+Christopher Frye, Damien de Mijolla, Tom Begley, Laurence Cowton, Megan Stanley, Ilya Feige
+
+Faculty, 54 Welbeck Street, London, UK
+
+# ABSTRACT
+
+Explainability in AI is crucial for model development, compliance with regulation, and providing operational nuance to predictions. The Shapley framework for explainability attributes a model’s predictions to its input features in a mathematically principled and model-agnostic way. However, general implementations of Shapley explainability make an untenable assumption: that the model’s features are uncorrelated. In this work, we demonstrate unambiguous drawbacks of this assumption and develop two solutions to Shapley explainability that respect the data manifold. One solution, based on generative modelling, provides flexible access to data imputations; the other directly learns the Shapley value-function, providing performance and stability at the cost of flexibility. While “off-manifold” Shapley values can (i) give rise to incorrect explanations, (ii) hide implicit model dependence on sensitive attributes, and (iii) lead to unintelligible explanations in higher-dimensional data, on-manifold explainability overcomes these problems.
+
+# 1 INTRODUCTION
+
+Explainability in AI is central to the practical impact of AI on society, thus making it critical to get right. While many dichotomies exist within the field — between local and global explanations (Ribeiro et al., 2016), between post hoc and intrinsic interpretability (Rudin, 2019), and between model-agnostic and model-specific methods (Shrikumar et al., 2017) — in this work we focus on local, post-hoc, model-agnostic explainability as it provides insight into individual model predictions, does not limit model expressiveness, and is comparable across model types.
+
+In this context, explainability can be treated as a problem of attribution. Shapley values (Shapley, 1953) provide the unique attribution method satisfying a set of intuitive axioms, e.g. they capture all interactions between features and sum to the model prediction. The Shapley approach to explainability has matured over the last two decades (Lipovetsky & Conklin, 2001; Kononenko et al., 2010; Strumbelj & Kononenko, 2014; Datta et al., 2016; Lundberg & Lee, 2017). ˇ
+
+Implementations of Shapley explainability suffer from a problem common across model-agnostic methods: they involve marginalisation over features, achieved by splicing data points together and evaluating the model on highly unrealistic inputs (e.g. Fig. 1). Such splicing would only be justified if all features were independent; otherwise, spliced data lies off the data manifold.
+
+Outside the Shapley paradigm, emerging explainability methods have begun to address this problem. See e.g. Anders et al. (2020) for a general treatment of the off-manifold problem in gradient-based explainability. See also Chang et al. (2019) and Agarwal et al. (2019) for image-specific explanations that respect the data distribution.
+
+Within Shapley explainability, initial work towards remedying the off-manifold problem has emerged; e.g. Aas et al. (2019) and Sundararajan & Najmi (2019) explore empirical and kernelbased estimation techniques, but these methods do not scale to complex data. A satisfactorily general and performant solution to computing Shapley values on the data manifold has yet to appear and is a focus of this work. Our main contributions are twofold:
+
+• Sec. 3 compares on- and off-manifold explainability, focusing on novel and unambiguous shortcomings of off-manifold Shapley values. In particular, we show that off-manifold explanations are often incorrect, and that they can hide implicit model dependence on sensitive features.
+
+![](images/8c25d6413831a386bb26665106b8693cd0ad7917f7cc1a291daf88623ac13231.jpg)  
+Figure 1: An MNIST digit, a coalition of pixels in a Shapley calculation, and 5 off-manifold splices.
+
+• Sec. 4 develops two methods to compute on-manifold Shapley values on general data sets: (i) a flexible generative-modelling technique to learn the data’s conditional distributions, and (ii) a simple supervised-learning technique that targets the Shapley value-function directly. We demonstrate the effectiveness of these methods on higher-dimensional data with experiments.
+
+# 2 BACKGROUND ON SHAPLEY EXPLAINABILITY
+
+The Shapley value (Shapley, 1953) is a method from cooperative game theory that distributes credit for the total value $v ( N )$ earned by a team $N = \{ 1 , 2 , \dots , n \}$ among its players:
+
+$$
+\phi _ { v } ( i ) = \sum _ { S \subseteq N \setminus \{ i \} } { \frac { | S | ! ( n - | S | - 1 ) ! } { n ! } } \left[ v ( S \cup \{ i \} ) - v ( S ) \right]
+$$
+
+where the value function $v ( S )$ indicates the value that a coalition of players $S$ would earn without their other teammates. The Shapley value $\phi _ { v } ( i )$ represents player $i$ ’s marginal value-added upon joining the team, averaged over all orderings in which the team can be constructed.
+
+In supervised learning, let $f _ { y } ( x )$ be a model’s predicted probability that data point $x$ belongs to class $y$ .1 To apply Shapley attribution to model explainability, one interprets the features $\{ x _ { 1 } , \ldots , x _ { n } \}$ as players in a game and the output $f _ { y } ( x )$ as their earned value. To compute Shapley values, one must define a value function representing the model’s output on a coalition $x s \subseteq \{ x _ { 1 } , \ldots , x _ { n } \}$ .
+
+As the model is undefined on partial input $x _ { S }$ , the standard implementation (Lundberg & Lee, 2017) samples out-of-coalition features, $x _ { \bar { S } } ^ { \prime }$ where ${ \bar { S } } = N \setminus S$ , unconditionally from the data distribution:
+
+$$
+v _ { f _ { y } ( x ) } ^ { ( \mathrm { o f f } ) } ( S ) = \mathbb { E } _ { p ( x ^ { \prime } ) } \big [ f _ { y } ( x _ { S } \sqcup x _ { \bar { S } } ^ { \prime } ) \big ]
+$$
+
+We refer to this value function, and the corresponding Shapley values, as lying off the data manifold since splices $x _ { S } \sqcup x _ { \bar { S } } ^ { \prime }$ generically lie far from the data distribution. Alternatively, conditioning outof-coalition features $\mathit { x } _ { \bar { S } } ^ { \prime }$ on in-coalition features $x _ { S }$ would result in an on-manifold value function:
+
+$$
+v _ { f _ { y } ( x ) } ^ { ( \mathrm { o n } ) } ( S ) = \mathbb { E } _ { p ( x ^ { \prime } | x _ { S } ) } \big [ f _ { y } ( x ^ { \prime } ) \big ]
+$$
+
+The conditional distribution $p ( x ^ { \prime } | x _ { S } )$ is not empirically accessible in practical scenarios with highdimensional data or many-valued (e.g. continuous) features. A performant method to compute onmanifold Shapley values on general data is until-now lacking and a focus of this work.
+
+Shapley values $\phi _ { f _ { y } ( x ) } ( i )$ provide local explainability for the model’s prediction on data point $x$ . To understand the model’s global behaviour, one aggregates the $\phi _ { f _ { y } ( x ) } ( i )$ ’s into global Shapley values:
+
+$$
+\Phi _ { f } ( i ) = \mathbb { E } _ { p ( x , y ) } \bigl [ \phi _ { f _ { y } ( x ) } ( i ) \bigr ]
+$$
+
+where $p ( x , y )$ is the labelled-data distribution. Global Shapley values can be seen as a special case of the global explanation framework introduced by Covert et al. (2020). As a consequence of the axioms (Shapley, 1953) satisfied by the $\phi _ { f _ { y } ( x ) } ( i )$ ’s, global Shapley values satisfy a sum rule:
+
+$$
+\sum _ { i \in N } \Phi _ { f } ( i ) = \mathbb { E } _ { p ( x , y ) } { \big [ } f _ { y } ( x ) { \big ] } - \mathbb { E } _ { p ( x ^ { \prime } ) } \mathbb { E } _ { p ( y ) } { \big [ } f _ { y } ( x ^ { \prime } ) { \big ] }
+$$
+
+One interprets the global Shapley value $\Phi _ { f } ( i )$ as the portion of model accuracy attributable to the $i ^ { \mathrm { { t h } } }$ feature. Indeed, the first term in Eq. (5) is the accuracy one achieves by sampling labels from $f$ ’s predicted probability distribution over classes. The offset term, which relates to class balance, is not attributable to any individual feature.
+
+# 3 EVIDENCE IN FAVOUR OF ON-MANIFOLD EXPLAINABILITY
+
+The key differences between on- and off-manifold Shapley values is a subject of ongoing discussion; see Sundararajan & Najmi (2019) or Chen et al. (2020) for recent overviews. Here we focus on theoretical arguments and experimental evidence yet to appear in the literature, in favour of the onmanifold approach. We begin with mathematically precise differences between on- and off-manifold methods in Sec. 3.1 and present unambiguous drawbacks of off-manifold Shapley values in Sec. 3.2.
+
+# 3.1 ON- VERSUS OFF-MANIFOLD DIFFERENCES MADE PRECISE
+
+Suppose the model’s input features $x _ { 1 } , \ldots , x _ { n }$ are the result of a data-generating process seeded by unobserved latent variables $z _ { 1 } , \ldots , z _ { d }$ . Then there exist functional relationships
+
+$$
+x _ { i } = g _ { i } ( z _ { 1 } , \dots , z _ { d } ; \epsilon _ { i } ) \quad \mathrm { f o r } \quad i = 1 , \dots , n
+$$
+
+where $\epsilon _ { i }$ represents noise in $x _ { i }$ . In the limit of small $\epsilon _ { i }$ ’s, there are $d$ directions in which a data point $( x _ { 1 } , \ldots , x _ { n } )$ can be perturbed while remaining consistent with the data distribution: these correspond to perturbations in $z _ { 1 } , \ldots , z _ { d }$ in Eq. (6). The data thus lives on a $d$ -dimensional manifold in ambient $n$ -dimensional features space, and therefore satisfies $n - d$ constraints on the $x _ { i }$ ’s:
+
+$$
+\Psi _ { k } ( x _ { 1 } , \ldots , x _ { n } ) = 0 \quad { \mathrm { f o r } } \quad k = 1 , \ldots , n - d
+$$
+
+On-manifold Shapley values evaluate the model on inputs that satisfy these constraints, while the off-manifold approach uses spliced data that generically break them. For a more detailed and mathematically precise discussion of the data manifold in this context, see Anders et al. (2020).
+
+# ALGEBRAIC MODEL DEPENDENCE CAN BE MISLEADING
+
+Any model $f _ { y } ( x )$ can be written in many algebraic forms that all evaluate identically on the data manifold. To show this, one can add any of the $n - d$ constraints from Eq. (7) to any of the model’s $n$ input slots. This changes the model’s algebraic form but does not affect the model’s output on the data, since each constraint equals zero on-manifold. The model $f _ { y } ( x )$ thus belongs to an ${ \dot { n } } ( n - d )$ dimensional equivalence class of functions that behave indistinguishably on the data.
+
+On-manifold Shapley values provide the same explanation for any two models that evaluate identically on the data distribution, because on-manifold explanations do not involve evaluation anywhere else. Off-manifold Shapley values provide different explanations for two models in the same equivalence class, as spliced data in the off-manifold value function break the constraints of Eq. (7).
+
+# HIDDEN MODEL DEPENDENCE ON SENSITIVE ATTRIBUTES
+
+This is not an academic concern: it follows that off-manifold explanations are vulnerable to adversarial model perturbations that hide dependence on select input features (Dombrowski et al., 2019; Slack et al., 2020). Dimanov et al. (2020) demonstrated that the off-manifold Shapley value for a sensitive feature like gender could be reduced near zero via this vulnerability.
+
+To see how this can happen, suppose that input feature $x _ { 1 }$ represents gender and formally solve one of the constraints in Eq. (7) for $x _ { 1 }$ . The result, say $x _ { 1 } = \tilde { \Psi } ( x _ { 2 } , \ldots , x _ { n } )$ , can then be used to transform any model $f _ { y } ( x )$ into another
+
+$$
+\tilde { f } _ { y } ( x _ { 2 } , \ldots , x _ { n } ) = f _ { y } \big ( \tilde { \Psi } ( x _ { 2 } , \ldots , x _ { n } ) , x _ { 2 } , \ldots , x _ { n } \big )
+$$
+
+that has no algebraic dependence on gender $x _ { 1 }$ but behaves identically to $f _ { y } ( x )$ on the data manifold. The off-manifold Shapley value for gender in $\tilde { f }$ would vanish, since the off-manifold value function of Eq. (2) depends on $x _ { 1 }$ only through $\tilde { f }$ (i.e. not at all). This result is problematic, since the two models behave equivalently on the data and thus possess the same gender bias.
+
+In contrast, the on-manifold Shapley values for $f$ and $\tilde { f }$ would be identical, as $x _ { 1 }$ dependence enters the on-manifold value function of Eq. (3) through the conditional expectation value. In a sense, on-manifold Shapley values represent the model’s dependence on the information content of each feature, rather than the model’s algebraic dependence.
+
+![](images/a236c98418689fbba54d01e7f134463f98370e8be5bc8bb9612a9ec2bb8ddd69.jpg)  
+Figure 2: (a) Vulnerability of off-manifold explanations to hidden model dependence. (b) Explanations of a fixed model compared to a model that is retrained on each Shapley coalition of features.
+
+We can demonstrate this on UCI Census Income data (Dua & Graff, 2017). We trained a neural network to predict whether an individual’s income exceeds $\$ 50\mathrm { k }$ based on demographic features in the data. Coral bars in Fig. 2(a) display global Shapley values for this “Original model”. (Onmanifold values were computed with the unsupervised method developed in Sec. 4.1.)
+
+We then trained an alternative model by fine-tuning the neural network above on a loss function that penalises model dependence on sex; see App. B for full details of this experiment. This resulted in a “Suppressed model” that makes identical predictions as the original model on $9 8 . 5 \%$ of the data. Teal bars in Fig. 2(a) display global Shapley values for this model. Note that the off-manifold Shapley value for sex is zero despite the similar behaviour exhibited by the original and suppressed models on the data. In contrast, on-manifold Shapley values explain both models similarly.
+
+# ON-MANIFOLD SHAPLEY VALUES IN THE OPTIMAL-MODEL LIMIT
+
+Here we present a result that strengthens the connection between on-manifold Shapley values and the data distribution: in the limit of an optimal model of the data, on-manifold Shapley values converge to an explanation of how the information in the data associates with the labelled outcomes.
+
+To show why this holds, suppose the predicted probability $f _ { y } ( x )$ converges to the true underlying distribution $p ( y | x )$ . In this optimal-model limit (which is approached in the limit of abundant data and high model capacity) the on-manifold value function of Eq. (3) becomes
+
+$$
+v _ { f _ { y } ( x ) } ^ { ( \mathrm { o n } ) } ( S )  \int d x _ { \bar { S } } ^ { \prime } p ( x _ { \bar { S } } ^ { \prime } | x _ { S } ) p ( y | x _ { S } \sqcup x _ { \bar { S } } ^ { \prime } ) = p ( y | x _ { S } )
+$$
+
+which shows that value is attributed to $x _ { i }$ based on $x _ { i }$ ’s predictivity of the label $y$ .
+
+We can demonstrate this on UCI Drug Consumption data (Dua & Graff, 2017). Using the 10 binary features listed in Fig. 2(b) – Mushrooms, Ecstasy, etc. – we trained a random forest $f$ to predict whether individuals had consumed an 11th drug: LSD. As the data contains just 10 binary features, we were able to empirically sample the conditional distributions in the on-manifold value function, Eq. (3). See Fig. 2(b) for the resulting off- and on-manifold global Shapley values.
+
+Next we fit a separate random forest $g _ { S }$ to each coalition $S$ of features, $2 ^ { 1 0 }$ models in total, in the spirit e.g. of Strumbelj et al. (2009). We used the accuracy ˇ $A ( g _ { S } )$ of each model, in the sense of Eq. (5), as the value function for an additional Shapley computation:
+
+$$
+\Phi _ { g } ( i ) = \sum _ { S \subseteq N \backslash i } { \frac { | S | ! ( n - | S | - 1 ) ! } { n ! } } \left[ A ( g _ { S \cup i } ) - A ( g _ { S } ) \right]
+$$
+
+where $\Phi _ { g } ( i )$ is directly the average gain in accuracy that results from adding feature $i$ to the set of inputs. These values are labelled “Model retraining” in Fig. 2(b). Note their agreement with the on-manifold explanation of the fixed random forest $f$ . On-manifold Shapley values thus indicate which features in the data are most predictive of the label.
+
+This consistency check allows us to show in passing that Tree SHAP (Lundberg et al., 2018; 2020) does not provide a method for on-manifold explainability. Observe in Fig. 2(b) that Tree SHAP roughly tracks the off-manifold explanation, albeit larger on the most predictive feature and somewhat smaller on the others. This occurs because trees tend to split on high-predictivity features first, and Tree SHAP privileges early-splitting features in an otherwise off-manifold calculation.
+
+![](images/aea3129eb9b3b9f7f0493d9cc3fea875b37c3b97945d6bd77785e43db084596b.jpg)  
+Figure 3: An individual outlier (a), its off- and on-manifold explanations (b & c), the error rate in explanations (d), and the distribution of model outputs on Shapley coalitions (e & f).
+
+# 3.2 UNAMBIGUOUS SHORTCOMINGS OF OFF-MANIFOLD EXPLAINABILITY
+
+Whereas above we clarified precise differences between on- and off-manifold Shapley values, in this section we focus on unambiguous drawbacks of the off-manifold approach.
+
+# UNCONTROLLED MODEL BEHAVIOUR OFF-MANIFOLD
+
+Sec. 3.1 might lead one to believe that off-manifold Shapley values provide insight into the algebraic dependence of a model. However, the off-manifold approach of evaluating the model on spliced indistribution data does not constitute a controlled study of such dependence. Off-manifold Shapley values serve as a perilously uncontrolled technique, especially in complex nonlinear models such as neural networks. Indeed, it is widely known that deep-learning models are not robust to distributional shift (Nguyen et al., 2015; Goodfellow et al., 2015). Still, off-manifold Shapley values evaluate the model outside its domain of validity, where it is untrained and potentially wildly misbehaved. This garbage-in-garbage-out problem is the clearest reason to avoid the off-manifold approach.
+
+Since this point has been documented in the literature (Hooker & Mentch, 2019), here we simply provide an example: Fig. 1 shows a binary MNIST digit (LeCun & Cortes, 2010), a coalition of pixels, and 5 random splices that would be used to compute an off-manifold explanation.
+
+# OUTLIER DETECTION EXPLAINED INCORRECTLY OFF-MANIFOLD
+
+To demonstrate that off-manifold Shapley values frequently lead to incorrect explanations, here we offer an example on synthetic data where the ground-truth explanation is known. We generated $1 0 ^ { 4 }$ synthetic data points, each consisting of 20 real-valued features, for the purpose of outlier detection. We split the dataset between $9 9 \%$ inliers and $1 \%$ outliers, with the classes generated according to:
+
+$$
+\begin{array} { r l } & { p _ { \mathrm { i n } } ( x _ { 1 } , \dots , x _ { 2 0 } ) = \displaystyle \frac { 1 } { 2 } \sum _ { z = 0 , 1 } \left( \prod _ { i = 1 } ^ { 2 0 }  { \mathcal { N } } [ z , \sigma ^ { 2 } ] ( x _ { i } ) \right) } \\ & { p _ { \mathrm { o u t } } ( x _ { 1 } , \dots , x _ { 2 0 } ) = \displaystyle \frac { 1 } { 2 } \sum _ { z = 0 , 1 } \left( \prod _ { i = 1 } ^ { 5 }  { \mathcal { N } } [ \bar { z } , \sigma ^ { 2 } ] ( x _ { i } ) \right) \left( \prod _ { i = 6 } ^ { 2 0 }  { \mathcal { N } } [ z , \sigma ^ { 2 } ] ( x _ { i } ) \right) } \end{array}
+$$
+
+That is, there is a single binary latent variable $z$ . For inliers, each feature is an independent noisy reading of the latent $z$ . For outliers, the first 5 features are centred instead around its opposite $\bar { z }$ . An example outlier (with $\sigma = 0 . 0 5 )$ ) is shown in Fig. 3(a). We generated one such data set for each $\sigma \in \{ 0 . 0 \bar { 1 } , 0 . 0 3 , \ldots , 0 . 1 5 \}$ in order to study the effect of noise on explanation errors.
+
+![](images/b23c6c1705e84ff4d49fabd59f25bcba831cbcbe89087806136d3730d1f20702.jpg)  
+Figure 4: Negative global Shapley values arise off-manifold, in both (a) synthetic and (b) real data.
+
+We fit an isolation forest (Liu et al., 2008) to perform outlier detection on each synthetic dataset, achieving $100 \%$ accuracy in every case. We computed the off- and on-manifold value functions of Eqs. (2) and (3) for each isolation forest by sampling the probability distributions directly, as these can be inferred from Eqs. (11) and (12). Figs. 3(b) and 3(c) show the resulting local Shapley values for the example outlier from Fig. 3(a).
+
+The ground-truth explanation of why Fig. 3(a) represents an outlier is that its first 5 features break correlations that exist across $9 9 \%$ of the data. The on-manifold explanation of Fig. 3(c) correctly attributes the 5 largest Shapley values to features $x _ { 1 } , \ldots , x _ { 5 }$ . The off-manifold explanation of Fig. 3(b) is unambiguously incorrect: feature $x _ { 7 }$ receives a larger value than $x _ { 2 }$ , $x _ { 4 }$ , and $x _ { 5 }$ . We consider an explanation to be erroneous if $x _ { 1 } , \ldots , x _ { 5 }$ do not receive the 5 largest Shapley values.
+
+To show the frequency of incorrect explanations, Fig. 3(d) displays the off- and on-manifold error rates as a function of noise $\sigma$ in the synthetic data set. Incorrect explanations are commonplace off-manifold: one-quarter are in error in the presence of minimal noise, and two-thirds are incorrect at $\sigma = 0 . 1 5$ . The on-manifold error rate is dramatically lower across this range.
+
+Figs. 3(e) and 3(f) show the root cause of off-manifold errors. These histograms display the distribution of model outputs when evaluated on Shapley coalitions in the off- and on-manifold calculations for $\sigma = 0 . 0 5$ . In particular, Fig. 3(e) shows the model evaluated on “inlier coalitions” which do not include $x _ { 1 } , \ldots , x _ { 5 }$ . Note that model outputs for on-manifold coalitions agree with the model evaluated on the actual data, while off-manifold coalitions follow a very different distribution. In particular, since a positive model output indicates a predicted outlier, Fig. 3(e) shows that the offmanifold calculation itself fabricates outliers through its splicing procedure.
+
+Similarly, Fig. 3(f) shows the model evaluated on “outlier coalitions” which do include $x _ { 1 } , \ldots , x _ { 5 }$ . Note that model outputs are similar for on-manifold coalitions and actual outliers, whereas offmanifold coalitions again differ dramatically. This is a manifestation of uncontrolled model behaviour off the data manifold, and it ultimately leads to erroneous off-manifold explanations.
+
+# BREAKDOWN IN GLOBAL SHAPLEY VALUES OFF-MANIFOLD
+
+To demonstrate that global Shapley values can be misleading off-manifold as well, we generated an additional synthetic data set according to the process in Fig. 4(a). The data has two binary features and a binary label. We fit a decision tree to this data, resulting in a precise match to Fig. 4(a). Note that the features $x _ { 0 }$ and $x _ { 1 }$ are positively correlated, both with each other and with label $y$ . However, with $x _ { 0 }$ fixed, the likelihood of $y = 1$ decreases slightly from $x _ { 1 } = 0$ to $x _ { 1 } = 1$ . One might think of $x _ { 0 }$ as disease severity, $x _ { 1 }$ as treatment intensity, and $y$ as mortality rate.
+
+Fig. 4(b) displays global Shapley values for this model. The global Shapley values are positive onmanifold, consistent with their interpretation as the portion of model accuracy attributable to each feature. Off-manifold, however, a negative value results from placing too much weight on splices, e.g. with $( x _ { 0 } , x _ { 1 } , y ) = ( 0 , 1 , 1 )$ , that occur less frequently in the actual data. The negative value would erroneously indicate that $x _ { 1 }$ is detrimental to the model’s overall performance.
+
+We can demonstrate this on real data using UCI Abalone data (Dua & Graff, 2017). We trained a neural network to classify abalone as younger than or older than the median age based on physical characteristics. Fig. 4(c) displays global Shapley values for this model. (On-manifold values were computed using techniques developed in Sec. 4.1; see App. B for details.)
+
+Observe the drastic difference between the on- and off-manifold explanations in Fig. 4(c). This is due to the tight correlations between features in the data (4 weights and 3 lengths) making the data manifold low-dimensional and important. Notice further the large negative off-manifold global Shapley value, negating its interpretation as the portion of model accuracy attributable to that feature.
+
+# 4 SCALABLE APPROACHES TO ON-MANIFOLD SHAPLEY VALUES
+
+In Sec. 3 we computed on-manifold Shapley values for simple data by estimating $p ( x ^ { \prime } | x _ { S } )$ from the empirical data distribution or, for synthetic data, by knowing this distribution analytically. Here we introduce two performant methods to compute on-manifold Shapley values on general data. Sec. 4.1 develops the theory underlying our methods, and Sec. 4.2 presents additional experimental results.
+
+# 4.1 THEORETICAL DEVELOPMENT OF ON-MANIFOLD METHODS
+
+Here we develop two methods to learn the on-manifold value function: (i) unsupervised learning the conditional distribution $p ( x ^ { \prime } | x _ { S } )$ , and (ii) a supervised technique to learn the value function directly.
+
+# UNSUPERVISED APPROACH
+
+One can use unsupervised learning to learn the conditional distributions $p ( x ^ { \prime } | x _ { S } )$ that appear in the on-manifold value function. Here we take an approach similar to Ivanov et al. (2019) to learn these distributions with variational inference. See Douglas et al. (2017) and Belghazi et al. (2019) for alternative techniques to learning conditional distributions that could be used here instead.
+
+Our specific approach includes two model components. The first is a variational autoencoder (Kingma & Welling, 2014; Rezende et al., 2014), with encoder $q _ { \phi } ( z | x )$ and decoder $p _ { \theta } ( x | z )$ . The second is a masked encoder, $r _ { \psi } ( z | x _ { S } )$ , for which the goal is to map the coalition $x _ { S }$ to a distribution in latent space that agrees with the encoder $q _ { \phi } ( z | x )$ as well as possible. A model of $p ( x ^ { \prime } | x _ { S } )$ is then provided by the composition:
+
+$$
+\hat { p } ( x ^ { \prime } | x _ { S } ) = \int d z p _ { \theta } ( x ^ { \prime } | z ) r _ { \psi } ( z | x _ { S } )
+$$
+
+and a good fit to the data should maximise ${ \hat { p } } ( x ^ { \prime } | x _ { S } )$ . A lower bound to its log-likelihood is given by
+
+$$
+\mathcal { L } _ { 0 } = \mathbb { E } _ { q _ { \phi } ( z | x ^ { \prime } ) } \big [ \log p _ { \theta } ( x ^ { \prime } | z ) \big ] - \mathcal { D } _ { \mathrm { K L } } \big ( q _ { \phi } ( z | x ^ { \prime } ) | | r _ { \psi } ( z | x _ { S } ) \big )
+$$
+
+While $\mathcal { L } _ { 0 }$ could be used on its own as the objective function to learn $\hat { p } ( x ^ { \prime } | x _ { S } )$ , this would leave the variational distribution $q _ { \phi } ( z | x )$ unconstrained, at odds with our goal of learning a smooth-manifold structure in latent space. This concern can be mitigated by ${ \mathcal { L } } _ { \mathrm { r e g } } = - { \mathcal { D } } _ { \mathrm { K L } } \big ( q _ { \phi } ( z | x ) | | p ( z ) \big )$ which regularises $q _ { \phi } ( z | x )$ by penalising differences from a smooth (e.g. unit normal) prior distribution $p ( z )$ . We thus include $\mathcal { L } _ { \mathrm { r e g } }$ as a regularisation term in our unsupervised objective: $\mathcal { L } = \mathcal { L } _ { 0 } + \beta \mathcal { L } _ { \mathrm { r e g } }$ .
+
+# METRIC FOR THE LEARNT VALUE FUNCTION
+
+The unsupervised method presented above leads to a learnt estimate of the conditional distribution, and thus to an estimate of the on-manifold value function: $\hat { v } _ { f _ { y } ( x ) } ( S ) = \mathbb { E } _ { \hat { p } ( x ^ { \prime } \mid x _ { S } ) } [ f _ { y } ( x ^ { \prime } ) ]$ . With the goal of judging the performance of this estimate, consider the following formal quantity:
+
+$$
+\mathrm { m s e } ( x _ { S } , y ) = \mathbb { E } _ { p ( x ^ { \prime } \mid x _ { S } ) } \left. f _ { y } ( x ^ { \prime } ) - \hat { v } _ { f _ { y } ( x ) } ( S ) \right. ^ { 2 }
+$$
+
+This quantity is minimal with respect to $\hat { v } _ { f _ { y } ( x ) } ( S )$ when $\hat { v } _ { f _ { y } ( x ) } ( S ) = \mathbb { E } _ { p ( x ^ { \prime } | x _ { S } ) } [ f _ { y } ( x ^ { \prime } ) ]$ , in agreement with the definition, Eq. (3), of the on-manifold value function. We can then quantitatively judge the performance of the unsupervised model $\hat { p } ( x ^ { \prime } | x _ { S } )$ by computing
+
+$$
+\mathrm { M S E } = \mathbb { E } _ { p ( x ) } \mathbb { E } _ { S \sim \mathrm { S h a p l e y } } \mathbb { E } _ { y \sim \mathrm { U n i f } } \left| f _ { y } ( x ) - \hat { v } _ { f _ { y } ( x ) } ( S ) \right| ^ { 2 }
+$$
+
+![](images/5e9f11f55dcb93ed715e55d944fbd967c3f734a3dbe8390b396317f2230b6480.jpg)  
+Figure 5: Validation of unsupervised and supervised techniques for computing on-manifold Shapley values. Comparison against empirical ground truth, which appeared as “On manifold” in Fig. 2(b).
+
+Table 1: Performance and stability, in terms of MSE, of supervised and unsupervised approaches. Performance is compared with off-manifold splicing and, where accessible, the empirical optimum.   
+
+<table><tr><td>DATA SET</td><td>OFF MANIFOLD</td><td>UNSUPERVISED</td><td>SUPERVISED</td><td>EMPIRICAL</td></tr><tr><td>DRUG</td><td>0.0634</td><td>0.0536 ± 0.0007</td><td>0.0441 ± 0.0002</td><td>0.0436</td></tr><tr><td>ABALONE</td><td>0.0647</td><td>0.0293 ±0.0009</td><td>0.0200 ± 0.0001</td><td>1</td></tr><tr><td>CENSUS</td><td>0.0344</td><td>0.0300 ±0.0006</td><td>0.0250 ±0.0001</td><td>1</td></tr><tr><td>MNIST</td><td>0.0448</td><td>0.0257 ± 0.0005</td><td>0.0121 ± 0.0001</td><td>1</td></tr></table>
+
+Note that this is precisely Eq. (15) averaged over coalitions $S$ drawn from the Shapley sum,2 features $x _ { S } \sim p ( x _ { S } )$ drawn from the data, and labels $y$ drawn uniformly over classes. Moreover, the meansquare-error in Eq. (16) is easy to estimate using the empirical distribution $p ( x )$ and the learnt model $\hat { p ( } x ^ { \prime } | x _ { S } )$ , thus providing an unambiguous metric to judge the outcome of the unsupervised approach.
+
+# SUPERVISED APPROACH
+
+The MSE metric of Eq. (16) supports a supervised approach to learning the on-manifold value function directly: one can define a surrogate model $g _ { y } ( x _ { S } )$ that operates on coalitions of features $x _ { S }$ (e.g. by masking out-of-coalition features) and that is trained to minimise the loss:
+
+$$
+{ \mathcal { L } } = \mathbb { E } _ { p ( x ) } \mathbb { E } _ { S \sim \mathrm { S h a p l e y } } \mathbb { E } _ { y \sim \mathrm { U n i f } } \left| f _ { y } ( x ) - g _ { y } ( x _ { S } ) \right| ^ { 2 }
+$$
+
+As discussed above Eq. (16), this loss is minimised as the surrogate model $g _ { y } ( x _ { S } )$ approaches the on-manifold value function $\mathbb { E } _ { p ( x ^ { \prime } | x _ { S } ) } [ f _ { y } ( x ^ { \prime } ) ]$ of the model-to-be-explained.
+
+# 4.2 ADDITIONAL EXPERIMENTS
+
+Sec. 3 above presented two experiments using the scalable on-manifold methods developed here. In particular, Fig. 2(a) applied the unsupervised method to Census Income data, showing that onmanifold Shapley values detect hidden model dependence on sensitive features, and Fig. 4(c) applied both methods to Abalone data, showing that global Shapley values remain positive and interpretable on-manifold. In this section, we perform additional experiments to study the performance and stability of Sec. 4.1’s methods, as well as their effectiveness on higher-dimensional data.
+
+# PERFORMANCE AND STABILITY
+
+Our implementations of the unsupervised and supervised approaches to on-manifold Shapley values are summarised in Apps. A and B. Both approaches lead to broadly similar results. Fig. 5 compares the two techniques on the Drug Consumption data, where explanations are compared against the ground-truth empirical computation from Fig. 2(b).
+
+The unsupervised approach is flexible but untargeted: $p ( x ^ { \prime } | x _ { S } )$ is data-specific but model-agnostic, accommodating explanations for many models trained on the same data. The supervised approach must be retrained on each model, but it entails direct minimisation of the MSE. The supervised method is thus expected to achieve higher accuracy. We confirmed this on all data sets studied in this paper; see Table 1 for a numerical comparison of the MSEs.
+
+![](images/e1a04e9af51499eb1ad1dc5bdc08dbc77bdec3a5fb1b18f294fa4ce6affc7d3e.jpg)  
+Figure 6: (a) Randomly drawn MNIST digits explained on / off manifold. Red / blue pixels indicate positive / negative Shapley values, and the colour scale in each column is fixed. (b) Shapley summand as a function of coalition size – averaged over coalitions, pixels, and the MNIST test set.
+
+In Table 1, central values indicate the test-set MSE achieved by each method. The table compares the unsupervised and supervised methods against off-manifold splicing, showing significant improvement over this baseline. Note that an MSE of zero is not achievable, because ${ \dot { f } } _ { y } ( x )$ in Eq. (16) or (17) is not fully determined by partial input $x _ { S }$ . For the Drug Consumption data where we can compute $p ( \boldsymbol { x } ^ { \prime } | \boldsymbol { x } _ { S } )$ empirically, the optimal MSE happens to be 0.0436.
+
+Uncertainties in Table 1 represent the standard deviation in test-set MSE upon repeating each method with fixed hyperparameters 10 times. (Uncertainties are absent for the off-manifold and empirical columns, as these do not involve training a separate model.) The table thus indicates that the supervised method offers increased stability as compared to the unsupervised approach.
+
+The supervised method is more efficient as well: while the unsupervised technique estimates the value function by sampling from ${ \hat { p } } ( x ^ { \prime } | x _ { S } )$ , the supervised approach learns the value function directly. The supervised method thus requires far fewer model evaluations to match the standard-error of the unsupervised method: roughly 10 times fewer in our experiments.
+
+# EXAMPLE ON MNIST
+
+To demonstrate on-manifold explainability on higher-dimensional data, we trained a fully connected network on binary MNIST (LeCun & Cortes, 2010) and explained random digits in Fig. 6(a).
+
+Despite having the same sum over pixels – as controlled by the local version of Eq. (5) – and explaining the same model prediction, each on-manifold explanation is more concentrated, with more interpretable structure, than its off-manifold counterpart. The handwritten strokes are clearly visible on-manifold, with key off-stroke regions highlighted as well. Off-manifold explanations generally display lower intensities spread less informatively across the digit-region.
+
+These off-manifold explanations are a result of splices as in Fig. 1. With such unrealistic input, the model’s output is uncontrolled and less informative. In fact, it is only on very large coalitions of pixels, subject to minimal splicing, that the model can make intelligent predictions off-manifold. This is confirmed in Fig. 6(b), which shows the average Shapley summand as a function of coalition size on MNIST. Note that primarily large coalitions underpin off-manifold explanations, whereas far fewer pixels are required on-manifold, consistent with the low-dimensional manifold underlying the data.
+
+# 5 CONCLUSION
+
+In this work, we took a careful study of the off-manifold problem in AI explainability. We presented important distinctions between on- and off-manifold explainability and provided experimental evidence for several novel shortcomings of the off-manifold approach. We then introduced two techniques to compute on-manifold Shapley values on general data: one technique learns to impute features on the data manifold, while the other learns the Shapley value-function directly. In-so-doing, we provided compelling evidence against the use of off-manifold explainability, and demonstrated that on-manifold Shapley values offer a viable approach to AI explainability in real-world contexts.
+
+REFERENCES   
+J. Aas, M. Jullum, and A. Løland. Explaining individual predictions when features are dependent: More accurate approximations to Shapley values, 2019. [arXiv:1903.10464].   
+C. Agarwal, D. Schonfeld, and A. Nguyen. Removing input features via a generative model to explain their attributions to classifier’s decisions, 2019. [arXiv:1910.04256].   
+C. J. Anders, P. Pasliev, A.-K. Dombrowski, K.-R. Muller, and P. Kessel. Fairwashing explanations ¨ with off-manifold detergent. In International Conference on Machine Learning, 2020.   
+M. Belghazi, M. Oquab, and D. Lopez-Paz. Learning about an exponential amount of conditional distributions. In Advances in Neural Information Processing Systems, 2019.   
+C. Chang, E. Creager, A. Goldenberg, and D. Duvenaud. Explaining image classifiers by counterfactual generation. In International Conference on Learning Representations, 2019.   
+H. Chen, J. D. Janizek, S. Lundberg, and S.-I. Lee. True to the model or true to the data?, 2020. [arXiv:2006.16234].   
+I. Covert, S. Lundberg, and S.-I. Lee. Understanding global feature contributions through additive importance measures, 2020. [arXiv:2004.00668].   
+A. Datta, S. Sen, and Y. Zick. Algorithmic transparency via quantitative input influence: Theory and experiments with learning systems. In IEEE Symposium on Security and Privacy, 2016.   
+B. Dimanov, U. Bhatt, M. Jamnik, and A. Weller. You shouldn’t trust me: Learning models which conceal unfairness from multiple explanation methods. In SafeAI @ AAAI, 2020.   
+A.-K. Dombrowski, M. Alber, C. Anders, M. Ackermann, K.-R. Muller, and P. Kessel. Explanations ¨ can be manipulated and geometry is to blame. In Advances in Neural Information Processing Systems, 2019.   
+L. Douglas, I. Zarov, K. Gourgoulias, C. Lucas, C. Hart, A. Baker, M. Sahani, Y. Perov, and S. Johri. A universal marginalizer for amortized inference in generative models. In NIPS Workshop on Approximate Bayesian Inference, 2017.   
+D. Dua and C. Graff. UCI machine learning repository, 2017. [archive.ics.uci.edu/ml].   
+I. J. Goodfellow, J. Shlens, and C. Szegedy. Explaining and harnessing adversarial examples. In International Conference on Learning Representations, 2015.   
+G. Hooker and L. Mentch. Please stop permuting features: An explanation and alternatives, 2019. [arXiv:1905.03151].   
+O. Ivanov, M. Figurnov, and D. Vetrov. Variational autoencoder with arbitrary conditioning. In International Conference on Learning Representations, 2019.   
+D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations, 2015.   
+D. P. Kingma and M. Welling. Auto-encoding variational bayes. In International Conference on Learning Representations, 2014.   
+I. Kononenko et al. An efficient explanation of individual classifications using game theory. Journal of Machine Learning Research, 2010.   
+Y. LeCun and C. Cortes. MNIST database, 2010. [yann.lecun.com/exdb/mnist].   
+S. Lipovetsky and M. Conklin. Analysis of regression in game theory approach. Applied Stochastic Models in Business and Industry, 2001.   
+F. T. Liu, K. M. Ting, and Z.-H. Zhou. Isolation forest. In IEEE International Conference on Data Mining, 2008.
+
+S. M. Lundberg and S.-I. Lee. A unified approach to interpreting model predictions. In Advances in Neural Information Processing Systems, 2017.
+
+S. M. Lundberg, G. G. Erion, and S.-I. Lee. Consistent individualized feature attribution for tree ensembles, 2018. [arXiv:1802.03888].
+
+S. M. Lundberg, G. Erion, H. Chen, A. DeGrave, J. Prutkin, B. Nair, R. Katz, J. Himmelfarb, N. Bansal, and S. Lee. From local explanations to global understanding with explainable AI for trees. Nature Machine Intelligence, 2020.
+
+A. Nguyen, J. Yosinski, and J. Clune. Deep neural networks are easily fooled: High confidence predictions for unrecognizable images. In Conference on Computer Vision and Pattern Recognition, 2015.
+
+D. J. Rezende, S. Mohamed, and D. Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In International Conference on Machine Learning, 2014.
+
+M. T. Ribeiro, S. Singh, and C. Guestrin. Why should I trust you: Explaining the predictions of any classifier. In International Conference on Knowledge Discovery and Data Mining, 2016.
+
+C. Rudin. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature Machine Intelligence, 2019.
+
+L. S. Shapley. A value for $n$ -person games. In Contribution to the theory of games, 1953.
+
+A. Shrikumar, P. Greenside, and A. Kundaje. Learning important features through propagating activation differences. In International Conference on Machine Learning, 2017.   
+D. Slack, S. Hilgard, E. Jia, S. Singh, and H. Lakkaraju. Fooling lime and shap: Adversarial attacks on post hoc explanation methods. In AIES @ AAAI, 2020.   
+E. Strumbelj and I. Kononenko. Explaining prediction models and individual predictions with fea- ˇ ture contributions. Knowledge and information systems, 2014.   
+E. Strumbelj, I. Kononenko, and M. Robnik-Sikonja. Explaining instance classifications with inter- ˇ actions of subsets of feature values. Data Knowl. Eng., 2009.   
+M. Sundararajan and A. Najmi. The many Shapley values for model explanation. In International Conference on Machine Learning, 2019.
+
+# A IMPLEMENTATION DETAILS
+
+For the unsupervised approach, we modelled the encoder $q _ { \phi } ( z | x )$ as a diagonal normal distribution with mean and variance determined by a neural network:
+
+$$
+q _ { \phi } ( z | x ) = \mathcal { N } \big ( \mu _ { \phi } ( x ) , \sigma _ { \phi } ( x ) \big )
+$$
+
+We modelled the decoder $p _ { \theta } ( x | z )$ as a product distribution:
+
+$$
+p _ { \theta } ( x | z ) = \prod _ { i } p _ { \theta } ( x _ { i } | z )
+$$
+
+where the distribution type (e.g. normal, categorical) of each $x _ { i }$ is chosen per-data-set and each distribution’s parameters are determined by a shared neural network. We modelled the masked encoder $r _ { \psi } ( z | x _ { S } )$ as a Gaussian mixture:
+
+$$
+r _ { \psi } ( z | x _ { S } ) = \sum _ { j } w _ { \phi } ^ { ( j ) } ( x ) \mathcal { N } \Big ( \mu _ { \phi } ^ { ( j ) } ( x ) , \sigma _ { \phi } ^ { ( j ) } ( x ) \Big )
+$$
+
+To allow $r _ { \psi } ( z | x _ { S } )$ to accept variable-size coalitions $x _ { S }$ as input, we simply masked out-of-coalition features with a special value $( - 1 )$ that never appears in the data.
+
+The unsupervised method has several hyperparameters: $\beta$ which multiplies the regularisation term, the number of components in Eq. (20), as well the architecture and optimisation of the networks involved. For each experiment in this paper, we tuned hyperparameters to minimise the MSE of Eq. (16) on a held-out validation set; see App. B for numerical details.
+
+For the supervised approach, we modelled $g _ { y } ( x _ { S } )$ using a neural network, again masking out-ofcoalition features (with $^ { - 1 }$ ) to accommodate variable-size coalitions $x _ { S }$ . This method’s hyperparameters, relating to architecture and optimisation, were similarly tuned to minimise the validationset MSE; see App. B for details.
+
+# B DETAILS OF EXPERIMENTS
+
+Here we provide numerical details for the experiments presented in the paper.
+
+# B.1 DRUG CONSUMPTION EXPERIMENT
+
+On the Drug Consumption data from the UCI repository (Dua & Graff, 2017), we used 10 binary features from the data set – Mushrooms, Ecstasy, etc., as displayed in Fig. $5 -$ to predict whether individuals had ever consumed an 11th drug: LSD. The explanations of Fig. 2(b) and Fig. 5 describe a random forest fit with default sklearn parameters and max features $=$ None, which achieves $8 2 . 2 \%$ test-set accuracy amidst a $5 7 : 4 3$ class balance.
+
+In Fig. 2(b), global off-manifold Shapley values were computed using $1 0 ^ { 6 }$ Monte Carlo samples of Eq. (4). For each labelled data point $( x , y )$ sampled from the test set, a single permutation was drawn to estimate Eq. (1) and a single data point $x ^ { \prime }$ was drawn to estimate the off-manifold value function Eq. (2). In all the figures of this paper, bar height represents the mean that resulted from Monte Carlo sampling, and error bars display the standard error of the mean.
+
+Global on-manifold Shapley values in Fig. 2(b) were computed similarly, but in this case using the on-manifold value function of Eq. (3). For each sampled coalition $x _ { S }$ , a random data point $x ^ { \prime }$ was drawn from the test set, with the crucial requirement that $x _ { S } ^ { \prime } = x _ { S }$ . In the text, we refer to this as empirically estimating the conditional distribution $p ( x ^ { \prime } | x _ { S } )$ . Such empirical estimation is only possible because this data set has a small number of all-binary features.
+
+Tree SHAP values in Fig. 2(b) were computed with the SHAP package (Lundberg & Lee, 2017) with model output $=$ margin and feature perturbation $=$ tree path dependent.
+
+The values labelled “Model retraining” in Fig. 2(b) were computed by fitting a separate random forest $g _ { S }$ for each coalition $S$ of features in the data set: $2 ^ { 1 0 }$ models in all. We used these models to compute the sum of Eq. (10), where $A ( g _ { S } )$ represents a variant of model $g _ { S }$ ’s accuracy: it is the accuracy achieved if one predicts labels by drawing stochastically from $g _ { S }$ ’s predicted probability distribution (as opposed to deterministically drawing the maximum-probability class).
+
+Table 2: Optimal hyperparameters found for computing on-manifold Shapley values.   
+
+<table><tr><td>DATA SET</td><td>METHOD</td><td>HIDDEN DIM.</td><td>LEARN.RATE</td><td>LATENT DIM.</td><td>MODES</td><td>β</td></tr><tr><td>DRUG</td><td>SUPERVISED</td><td>512</td><td>10-3</td><td></td><td></td><td></td></tr><tr><td></td><td>UNSUPERVISED</td><td>128</td><td>10-3</td><td>4</td><td>1</td><td>0.5</td></tr><tr><td>ABALONE</td><td>SUPERVISED UNSUPERVISED</td><td>512 256</td><td>10-3 10-3</td><td>2</td><td>1</td><td>0.05</td></tr><tr><td>CENSUS</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>SUPERVISED UNSUPERVISED</td><td>512 128</td><td>10-3 10-3</td><td>8</td><td>1</td><td>1</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MNIST</td><td>SUPERVISED</td><td>512</td><td>10-4</td><td></td><td></td><td></td></tr><tr><td></td><td>UNSUPERVISED</td><td>512</td><td>10-4</td><td>16</td><td>1</td><td>1</td></tr></table>
+
+The global on-manifold Shapley values in Fig. 2(b) appear in Fig. 5 as well, labelled “Empirical”. Fig. 5 also displays on-manifold Shapley values computed using the supervised and unsupervised methods introduced in this paper. As above, these are Monte Carlo estimates of Eq. (4). The supervised method involved training a fully connected network on the MSE loss of Eq. (17). All neural networks in this paper used 2 flat hidden layers, Adam (Kingma & Ba, 2015) for optimisation, and a batch size of 256. We scanned over a grid with
+
+$$
+\begin{array} { c } { { \mathrm { h i d d e n ~ l a y e r ~ s i z e } = \{ 1 2 8 , 2 5 6 , 5 1 2 \} } } \\ { { \mathrm { l e a r n i n g ~ r a t e } = \{ 1 0 ^ { - 3 } , 1 0 ^ { - 4 } \} } } \end{array}
+$$
+
+choosing the point with minimal MSE on a held-out validation set after $1 0 \mathrm { k }$ epochs of training; see Table 2. Each supervised value in Fig. 5 corresponds to $1 0 ^ { 4 }$ Monte Carlo samples.
+
+The unsupervised method involved training a variational autoencoder as described in Sec. 4.1 and App. A. The encoder, decoder, and masked encoder were each modelled using fully connected networks, trained using early stopping with patience 100. We scanned over a grid of hidden layer sizes and learning rates as in Eq. (21) as well as
+
+$$
+\begin{array} { l } { { \mathrm { l a t e n t d i m e n s i o n } = \{ 2 , 4 , 8 , 1 6 \} } } \\ { { \mathrm { l a t e n t m o d e s } = \{ 1 , 2 \} } } \\ { { \mathrm { r e g u l a r i s a t i o n } \beta = \{ 0 . 0 5 , 0 . 1 , 0 . 5 , 1 \} } } \end{array}
+$$
+
+choosing the point with minimal validation-set MSE; see Table 2. Unsupervised values in Fig. 5 correspond to $\mathrm { \dot { 1 } 0 ^ { 6 } }$ Monte Carlo samples.
+
+# B.2 CENSUS INCOME EXPERIMENT
+
+To produce the explanations of Fig. 2(a) we used the Census Income data set from the UCI repository (Dua & Graff, 2017). The data contains $4 9 \mathrm { k }$ individuals from the 1994 US Census, as well as 13 features which we used to predict whether annual income exceeded $\$ 50\mathrm { k }$ . We trained a fully connected network (hidden layer size 50, default sklearn parameters, and early stopping), achieving a test-set accuracy of $8 5 \%$ amidst a $7 6 : 2 4$ class balance.
+
+The Shapley values for this model are labelled “Original model” in Fig. 2(a). These were computed exactly as described in App. B.1, except that the supervised method used $5 \mathrm { k }$ epochs, and the unsupervised method used patience 50. Optimised hyperparameters are given in Table 2. The onmanifold values in Fig. 2(a) were computed using the unsupervised method. While the supervised method does not appear in the figure, it was performed to complete Table 1.
+
+We also fine-tuned the “Original model” to suppress the importance of sex. Motivated by Dimanov et al. (2020) we added a term to the loss that penalises the finite difference in the model output with respect to sex (as this is a discrete feature). The modified loss function thus becomes
+
+$$
+{ \frac { 1 } { N } } \sum _ { i = 1 } ^ { N } { \mathcal { L } } { \big ( } f ( x _ { i } ) , y _ { i } { \big ) } \ + \ \alpha { \Big | } \ f { \big ( } x _ { i } | \operatorname { d o } ( \operatorname { s e x } = 1 ) { \big ) } - f { \big ( } x _ { i } | \operatorname { d o } ( \operatorname { s e x } = 0 ) { \big ) } { \Big | }
+$$
+
+where $\mathcal { L }$ is the cross-entropy loss, $f { \big ( } x _ { i } | \operatorname { d o } ( \operatorname { s e x } = j ) { \big ) }$ denotes $f$ evaluated on the data point $x _ { i }$ with the value for sex replaced with $j$ , and $\alpha$ is a hyperparameter controlling the trade-off between optimising the accuracy and minimising the effect of sex. We fine-tuned the model for an additional 200 epochs with $\alpha = 3$ . The resulting model agrees with the baseline on over $9 8 . 5 \%$ of the data, and has the same test-set accuracy. Shapley values for this model are labelled “Suppressed model” in Fig. 2(a).
+
+# B.3 ABALONE EXPERIMENT
+
+The Abalone data set from the UCI repository (Dua & Graff, 2017) contains 8 features corresponding to physical measurements (see Fig. 4c) which we used to classify abalone as younger than or older than the median age. We trained a neural network to perform this task – with hidden layer size 100, default sklearn parameters, and early stopping – obtaining a test-set accuracy of $78 \%$ .
+
+Shapley values in Fig. 4(c) were computed exactly as described in App. B.1, except that the supervised method involved training for 5k epochs. Optimised hyperparameters are given in Table 2.
+
+# B.4 MNIST EXPERIMENT
+
+For binary MNIST (LeCun & Cortes, 2010), we trained a fully connected network (hidden layer size 512, default sklearn parameters, and early stopping) achieving $98 \%$ test-set accuracy.
+
+The digits in Fig. 6(a) were randomly drawn from the test set. Shapley values in Fig. 6(a) were computed exactly as described in App. B.1, except that the supervised method involved training for 2k epochs, and the on-manifold explanations are based on $1 6 \mathrm { k }$ Monte Carlo samples per pixel. Optimised hyperparameters are given in Table 2. The on-manifold explanations in Fig. 6(a) were computed using the supervised method. While the unsupervised method does not appear in the figure, it was performed to complete Table 1.
+
+The average uncertainty, which is not shown in Fig. 6(a), is roughly 0.002 – stated as a fraction of the maximum Shapley value in each image.

parse/train/OPyWRrcjVQw/OPyWRrcjVQw_content_list.json

@@ -0,0 +1,1989 @@
+[
+    {
+        "type": "text",
+        "text": "SHAPLEY EXPLAINABILITY ON THE DATA MANIFOLD ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            101,
+            803,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Christopher Frye, Damien de Mijolla, Tom Begley, Laurence Cowton, Megan Stanley, Ilya Feige ",
+        "bbox": [
+            183,
+            143,
+            575,
+            176
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Faculty, 54 Welbeck Street, London, UK ",
+        "bbox": [
+            184,
+            194,
+            449,
+            208
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            246,
+            544,
+            261
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Explainability in AI is crucial for model development, compliance with regulation, and providing operational nuance to predictions. The Shapley framework for explainability attributes a model’s predictions to its input features in a mathematically principled and model-agnostic way. However, general implementations of Shapley explainability make an untenable assumption: that the model’s features are uncorrelated. In this work, we demonstrate unambiguous drawbacks of this assumption and develop two solutions to Shapley explainability that respect the data manifold. One solution, based on generative modelling, provides flexible access to data imputations; the other directly learns the Shapley value-function, providing performance and stability at the cost of flexibility. While “off-manifold” Shapley values can (i) give rise to incorrect explanations, (ii) hide implicit model dependence on sensitive attributes, and (iii) lead to unintelligible explanations in higher-dimensional data, on-manifold explainability overcomes these problems. ",
+        "bbox": [
+            233,
+            273,
+            764,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            477,
+            336,
+            492
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Explainability in AI is central to the practical impact of AI on society, thus making it critical to get right. While many dichotomies exist within the field — between local and global explanations (Ribeiro et al., 2016), between post hoc and intrinsic interpretability (Rudin, 2019), and between model-agnostic and model-specific methods (Shrikumar et al., 2017) — in this work we focus on local, post-hoc, model-agnostic explainability as it provides insight into individual model predictions, does not limit model expressiveness, and is comparable across model types. ",
+        "bbox": [
+            174,
+            507,
+            823,
+            590
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this context, explainability can be treated as a problem of attribution. Shapley values (Shapley, 1953) provide the unique attribution method satisfying a set of intuitive axioms, e.g. they capture all interactions between features and sum to the model prediction. The Shapley approach to explainability has matured over the last two decades (Lipovetsky & Conklin, 2001; Kononenko et al., 2010; Strumbelj & Kononenko, 2014; Datta et al., 2016; Lundberg & Lee, 2017). ˇ ",
+        "bbox": [
+            174,
+            598,
+            823,
+            669
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Implementations of Shapley explainability suffer from a problem common across model-agnostic methods: they involve marginalisation over features, achieved by splicing data points together and evaluating the model on highly unrealistic inputs (e.g. Fig. 1). Such splicing would only be justified if all features were independent; otherwise, spliced data lies off the data manifold. ",
+        "bbox": [
+            174,
+            676,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Outside the Shapley paradigm, emerging explainability methods have begun to address this problem. See e.g. Anders et al. (2020) for a general treatment of the off-manifold problem in gradient-based explainability. See also Chang et al. (2019) and Agarwal et al. (2019) for image-specific explanations that respect the data distribution. ",
+        "bbox": [
+            176,
+            739,
+            823,
+            795
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Within Shapley explainability, initial work towards remedying the off-manifold problem has emerged; e.g. Aas et al. (2019) and Sundararajan & Najmi (2019) explore empirical and kernelbased estimation techniques, but these methods do not scale to complex data. A satisfactorily general and performant solution to computing Shapley values on the data manifold has yet to appear and is a focus of this work. Our main contributions are twofold: ",
+        "bbox": [
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+            803,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• Sec. 3 compares on- and off-manifold explainability, focusing on novel and unambiguous shortcomings of off-manifold Shapley values. In particular, we show that off-manifold explanations are often incorrect, and that they can hide implicit model dependence on sensitive features. ",
+        "bbox": [
+            191,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/8c25d6413831a386bb26665106b8693cd0ad7917f7cc1a291daf88623ac13231.jpg",
+        "image_caption": [
+            "Figure 1: An MNIST digit, a coalition of pixels in a Shapley calculation, and 5 off-manifold splices. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            102,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Sec. 4 develops two methods to compute on-manifold Shapley values on general data sets: (i) a flexible generative-modelling technique to learn the data’s conditional distributions, and (ii) a simple supervised-learning technique that targets the Shapley value-function directly. We demonstrate the effectiveness of these methods on higher-dimensional data with experiments. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ON SHAPLEY EXPLAINABILITY ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            295,
+            588,
+            311
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The Shapley value (Shapley, 1953) is a method from cooperative game theory that distributes credit for the total value $v ( N )$ earned by a team $N = \\{ 1 , 2 , \\dots , n \\}$ among its players: ",
+        "bbox": [
+            174,
+            325,
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a28f8c386614a466229ba6a86b9d91ec6ca6dea7ddbab2a64d755e6fd1000d5a.jpg",
+        "text": "$$\n\\phi _ { v } ( i ) = \\sum _ { S \\subseteq N \\setminus \\{ i \\} } { \\frac { | S | ! ( n - | S | - 1 ) ! } { n ! } } \\left[ v ( S \\cup \\{ i \\} ) - v ( S ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            357,
+            681,
+            397
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where the value function $v ( S )$ indicates the value that a coalition of players $S$ would earn without their other teammates. The Shapley value $\\phi _ { v } ( i )$ represents player $i$ ’s marginal value-added upon joining the team, averaged over all orderings in which the team can be constructed. ",
+        "bbox": [
+            174,
+            400,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In supervised learning, let $f _ { y } ( x )$ be a model’s predicted probability that data point $x$ belongs to class $y$ .1 To apply Shapley attribution to model explainability, one interprets the features $\\{ x _ { 1 } , \\ldots , x _ { n } \\}$ as players in a game and the output $f _ { y } ( x )$ as their earned value. To compute Shapley values, one must define a value function representing the model’s output on a coalition $x s \\subseteq \\{ x _ { 1 } , \\ldots , x _ { n } \\}$ . ",
+        "bbox": [
+            173,
+            448,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As the model is undefined on partial input $x _ { S }$ , the standard implementation (Lundberg & Lee, 2017) samples out-of-coalition features, $x _ { \\bar { S } } ^ { \\prime }$ where ${ \\bar { S } } = N \\setminus S$ , unconditionally from the data distribution: ",
+        "bbox": [
+            173,
+            513,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/586a81667275ade5ee525b93dad78c177cb44f9c169b3664bfb1e8a029a4bf85.jpg",
+        "text": "$$\nv _ { f _ { y } ( x ) } ^ { ( \\mathrm { o f f } ) } ( S ) = \\mathbb { E } _ { p ( x ^ { \\prime } ) } \\big [ f _ { y } ( x _ { S } \\sqcup x _ { \\bar { S } } ^ { \\prime } ) \\big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            546,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We refer to this value function, and the corresponding Shapley values, as lying off the data manifold since splices $x _ { S } \\sqcup x _ { \\bar { S } } ^ { \\prime }$ generically lie far from the data distribution. Alternatively, conditioning outof-coalition features $\\mathit { x } _ { \\bar { S } } ^ { \\prime }$ on in-coalition features $x _ { S }$ would result in an on-manifold value function: ",
+        "bbox": [
+            173,
+            570,
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e06d4d4af962be172823a530d97f952a7ca306c512a06847c3f5ff5d82a95bd3.jpg",
+        "text": "$$\nv _ { f _ { y } ( x ) } ^ { ( \\mathrm { o n } ) } ( S ) = \\mathbb { E } _ { p ( x ^ { \\prime } | x _ { S } ) } \\big [ f _ { y } ( x ^ { \\prime } ) \\big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            618,
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+            641
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The conditional distribution $p ( x ^ { \\prime } | x _ { S } )$ is not empirically accessible in practical scenarios with highdimensional data or many-valued (e.g. continuous) features. A performant method to compute onmanifold Shapley values on general data is until-now lacking and a focus of this work. ",
+        "bbox": [
+            174,
+            643,
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+            685
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Shapley values $\\phi _ { f _ { y } ( x ) } ( i )$ provide local explainability for the model’s prediction on data point $x$ . To understand the model’s global behaviour, one aggregates the $\\phi _ { f _ { y } ( x ) } ( i )$ ’s into global Shapley values: ",
+        "bbox": [
+            176,
+            691,
+            823,
+            723
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/29541d57ca070f0174ee6cd1de7d604871a8fb5c0a098abaeb61502e4d9e5d25.jpg",
+        "text": "$$\n\\Phi _ { f } ( i ) = \\mathbb { E } _ { p ( x , y ) } \\bigl [ \\phi _ { f _ { y } ( x ) } ( i ) \\bigr ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            727,
+            593,
+            746
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $p ( x , y )$ is the labelled-data distribution. Global Shapley values can be seen as a special case of the global explanation framework introduced by Covert et al. (2020). As a consequence of the axioms (Shapley, 1953) satisfied by the $\\phi _ { f _ { y } ( x ) } ( i )$ ’s, global Shapley values satisfy a sum rule: ",
+        "bbox": [
+            173,
+            748,
+            826,
+            791
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cf228577b96b2a5e8f3a05337e90c4f127ca27ca33a2e3dfeb6261ae4eedbdf1.jpg",
+        "text": "$$\n\\sum _ { i \\in N } \\Phi _ { f } ( i ) = \\mathbb { E } _ { p ( x , y ) } { \\big [ } f _ { y } ( x ) { \\big ] } - \\mathbb { E } _ { p ( x ^ { \\prime } ) } \\mathbb { E } _ { p ( y ) } { \\big [ } f _ { y } ( x ^ { \\prime } ) { \\big ] }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One interprets the global Shapley value $\\Phi _ { f } ( i )$ as the portion of model accuracy attributable to the $i ^ { \\mathrm { { t h } } }$ feature. Indeed, the first term in Eq. (5) is the accuracy one achieves by sampling labels from $f$ ’s predicted probability distribution over classes. The offset term, which relates to class balance, is not attributable to any individual feature. ",
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+    {
+        "type": "text",
+        "text": "3 EVIDENCE IN FAVOUR OF ON-MANIFOLD EXPLAINABILITY ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "The key differences between on- and off-manifold Shapley values is a subject of ongoing discussion; see Sundararajan & Najmi (2019) or Chen et al. (2020) for recent overviews. Here we focus on theoretical arguments and experimental evidence yet to appear in the literature, in favour of the onmanifold approach. We begin with mathematically precise differences between on- and off-manifold methods in Sec. 3.1 and present unambiguous drawbacks of off-manifold Shapley values in Sec. 3.2. ",
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+    {
+        "type": "text",
+        "text": "3.1 ON- VERSUS OFF-MANIFOLD DIFFERENCES MADE PRECISE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Suppose the model’s input features $x _ { 1 } , \\ldots , x _ { n }$ are the result of a data-generating process seeded by unobserved latent variables $z _ { 1 } , \\ldots , z _ { d }$ . Then there exist functional relationships ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/218ed64f342e3cdfba2fdfd86ae36ab0489920112fbaf3b18d05b6b3c3b49a15.jpg",
+        "text": "$$\nx _ { i } = g _ { i } ( z _ { 1 } , \\dots , z _ { d } ; \\epsilon _ { i } ) \\quad \\mathrm { f o r } \\quad i = 1 , \\dots , n\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\epsilon _ { i }$ represents noise in $x _ { i }$ . In the limit of small $\\epsilon _ { i }$ ’s, there are $d$ directions in which a data point $( x _ { 1 } , \\ldots , x _ { n } )$ can be perturbed while remaining consistent with the data distribution: these correspond to perturbations in $z _ { 1 } , \\ldots , z _ { d }$ in Eq. (6). The data thus lives on a $d$ -dimensional manifold in ambient $n$ -dimensional features space, and therefore satisfies $n - d$ constraints on the $x _ { i }$ ’s: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec9714312bfe7d343cd0ea7df8ee180f13e687e393a6aebf5b456869ffe37297.jpg",
+        "text": "$$\n\\Psi _ { k } ( x _ { 1 } , \\ldots , x _ { n } ) = 0 \\quad { \\mathrm { f o r } } \\quad k = 1 , \\ldots , n - d\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "On-manifold Shapley values evaluate the model on inputs that satisfy these constraints, while the off-manifold approach uses spliced data that generically break them. For a more detailed and mathematically precise discussion of the data manifold in this context, see Anders et al. (2020). ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "ALGEBRAIC MODEL DEPENDENCE CAN BE MISLEADING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Any model $f _ { y } ( x )$ can be written in many algebraic forms that all evaluate identically on the data manifold. To show this, one can add any of the $n - d$ constraints from Eq. (7) to any of the model’s $n$ input slots. This changes the model’s algebraic form but does not affect the model’s output on the data, since each constraint equals zero on-manifold. The model $f _ { y } ( x )$ thus belongs to an ${ \\dot { n } } ( n - d )$ dimensional equivalence class of functions that behave indistinguishably on the data. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "On-manifold Shapley values provide the same explanation for any two models that evaluate identically on the data distribution, because on-manifold explanations do not involve evaluation anywhere else. Off-manifold Shapley values provide different explanations for two models in the same equivalence class, as spliced data in the off-manifold value function break the constraints of Eq. (7). ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "HIDDEN MODEL DEPENDENCE ON SENSITIVE ATTRIBUTES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This is not an academic concern: it follows that off-manifold explanations are vulnerable to adversarial model perturbations that hide dependence on select input features (Dombrowski et al., 2019; Slack et al., 2020). Dimanov et al. (2020) demonstrated that the off-manifold Shapley value for a sensitive feature like gender could be reduced near zero via this vulnerability. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To see how this can happen, suppose that input feature $x _ { 1 }$ represents gender and formally solve one of the constraints in Eq. (7) for $x _ { 1 }$ . The result, say $x _ { 1 } = \\tilde { \\Psi } ( x _ { 2 } , \\ldots , x _ { n } )$ , can then be used to transform any model $f _ { y } ( x )$ into another ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24d0a2a1a88b636690dcc81514279d368c540b1abda3b17e6ddce1c45d27f921.jpg",
+        "text": "$$\n\\tilde { f } _ { y } ( x _ { 2 } , \\ldots , x _ { n } ) = f _ { y } \\big ( \\tilde { \\Psi } ( x _ { 2 } , \\ldots , x _ { n } ) , x _ { 2 } , \\ldots , x _ { n } \\big )\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "that has no algebraic dependence on gender $x _ { 1 }$ but behaves identically to $f _ { y } ( x )$ on the data manifold. The off-manifold Shapley value for gender in $\\tilde { f }$ would vanish, since the off-manifold value function of Eq. (2) depends on $x _ { 1 }$ only through $\\tilde { f }$ (i.e. not at all). This result is problematic, since the two models behave equivalently on the data and thus possess the same gender bias. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In contrast, the on-manifold Shapley values for $f$ and $\\tilde { f }$ would be identical, as $x _ { 1 }$ dependence enters the on-manifold value function of Eq. (3) through the conditional expectation value. In a sense, on-manifold Shapley values represent the model’s dependence on the information content of each feature, rather than the model’s algebraic dependence. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a236c98418689fbba54d01e7f134463f98370e8be5bc8bb9612a9ec2bb8ddd69.jpg",
+        "image_caption": [
+            "Figure 2: (a) Vulnerability of off-manifold explanations to hidden model dependence. (b) Explanations of a fixed model compared to a model that is retrained on each Shapley coalition of features. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We can demonstrate this on UCI Census Income data (Dua & Graff, 2017). We trained a neural network to predict whether an individual’s income exceeds $\\$ 50\\mathrm { k }$ based on demographic features in the data. Coral bars in Fig. 2(a) display global Shapley values for this “Original model”. (Onmanifold values were computed with the unsupervised method developed in Sec. 4.1.) ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We then trained an alternative model by fine-tuning the neural network above on a loss function that penalises model dependence on sex; see App. B for full details of this experiment. This resulted in a “Suppressed model” that makes identical predictions as the original model on $9 8 . 5 \\%$ of the data. Teal bars in Fig. 2(a) display global Shapley values for this model. Note that the off-manifold Shapley value for sex is zero despite the similar behaviour exhibited by the original and suppressed models on the data. In contrast, on-manifold Shapley values explain both models similarly. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "ON-MANIFOLD SHAPLEY VALUES IN THE OPTIMAL-MODEL LIMIT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here we present a result that strengthens the connection between on-manifold Shapley values and the data distribution: in the limit of an optimal model of the data, on-manifold Shapley values converge to an explanation of how the information in the data associates with the labelled outcomes. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To show why this holds, suppose the predicted probability $f _ { y } ( x )$ converges to the true underlying distribution $p ( y | x )$ . In this optimal-model limit (which is approached in the limit of abundant data and high model capacity) the on-manifold value function of Eq. (3) becomes ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6f44a6eedc6112b56db00307ec754064898d7887982eb42a437c5e959dca99b9.jpg",
+        "text": "$$\nv _ { f _ { y } ( x ) } ^ { ( \\mathrm { o n } ) } ( S )  \\int d x _ { \\bar { S } } ^ { \\prime } p ( x _ { \\bar { S } } ^ { \\prime } | x _ { S } ) p ( y | x _ { S } \\sqcup x _ { \\bar { S } } ^ { \\prime } ) = p ( y | x _ { S } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which shows that value is attributed to $x _ { i }$ based on $x _ { i }$ ’s predictivity of the label $y$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We can demonstrate this on UCI Drug Consumption data (Dua & Graff, 2017). Using the 10 binary features listed in Fig. 2(b) – Mushrooms, Ecstasy, etc. – we trained a random forest $f$ to predict whether individuals had consumed an 11th drug: LSD. As the data contains just 10 binary features, we were able to empirically sample the conditional distributions in the on-manifold value function, Eq. (3). See Fig. 2(b) for the resulting off- and on-manifold global Shapley values. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Next we fit a separate random forest $g _ { S }$ to each coalition $S$ of features, $2 ^ { 1 0 }$ models in total, in the spirit e.g. of Strumbelj et al. (2009). We used the accuracy ˇ $A ( g _ { S } )$ of each model, in the sense of Eq. (5), as the value function for an additional Shapley computation: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a5ba2fdfd342a880b8d008810a0e7e6ca36b99c133c43082dce27f1872138c1c.jpg",
+        "text": "$$\n\\Phi _ { g } ( i ) = \\sum _ { S \\subseteq N \\backslash i } { \\frac { | S | ! ( n - | S | - 1 ) ! } { n ! } } \\left[ A ( g _ { S \\cup i } ) - A ( g _ { S } ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\Phi _ { g } ( i )$ is directly the average gain in accuracy that results from adding feature $i$ to the set of inputs. These values are labelled “Model retraining” in Fig. 2(b). Note their agreement with the on-manifold explanation of the fixed random forest $f$ . On-manifold Shapley values thus indicate which features in the data are most predictive of the label. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This consistency check allows us to show in passing that Tree SHAP (Lundberg et al., 2018; 2020) does not provide a method for on-manifold explainability. Observe in Fig. 2(b) that Tree SHAP roughly tracks the off-manifold explanation, albeit larger on the most predictive feature and somewhat smaller on the others. This occurs because trees tend to split on high-predictivity features first, and Tree SHAP privileges early-splitting features in an otherwise off-manifold calculation. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/aea3129eb9b3b9f7f0493d9cc3fea875b37c3b97945d6bd77785e43db084596b.jpg",
+        "image_caption": [
+            "Figure 3: An individual outlier (a), its off- and on-manifold explanations (b & c), the error rate in explanations (d), and the distribution of model outputs on Shapley coalitions (e & f). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 UNAMBIGUOUS SHORTCOMINGS OF OFF-MANIFOLD EXPLAINABILITY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Whereas above we clarified precise differences between on- and off-manifold Shapley values, in this section we focus on unambiguous drawbacks of the off-manifold approach. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "UNCONTROLLED MODEL BEHAVIOUR OFF-MANIFOLD ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Sec. 3.1 might lead one to believe that off-manifold Shapley values provide insight into the algebraic dependence of a model. However, the off-manifold approach of evaluating the model on spliced indistribution data does not constitute a controlled study of such dependence. Off-manifold Shapley values serve as a perilously uncontrolled technique, especially in complex nonlinear models such as neural networks. Indeed, it is widely known that deep-learning models are not robust to distributional shift (Nguyen et al., 2015; Goodfellow et al., 2015). Still, off-manifold Shapley values evaluate the model outside its domain of validity, where it is untrained and potentially wildly misbehaved. This garbage-in-garbage-out problem is the clearest reason to avoid the off-manifold approach. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Since this point has been documented in the literature (Hooker & Mentch, 2019), here we simply provide an example: Fig. 1 shows a binary MNIST digit (LeCun & Cortes, 2010), a coalition of pixels, and 5 random splices that would be used to compute an off-manifold explanation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "OUTLIER DETECTION EXPLAINED INCORRECTLY OFF-MANIFOLD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate that off-manifold Shapley values frequently lead to incorrect explanations, here we offer an example on synthetic data where the ground-truth explanation is known. We generated $1 0 ^ { 4 }$ synthetic data points, each consisting of 20 real-valued features, for the purpose of outlier detection. We split the dataset between $9 9 \\%$ inliers and $1 \\%$ outliers, with the classes generated according to: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d36b067278dc789cedbcf66ed9dea420587f1c0dac9b4892868ce8f508da212c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { p _ { \\mathrm { i n } } ( x _ { 1 } , \\dots , x _ { 2 0 } ) = \\displaystyle \\frac { 1 } { 2 } \\sum _ { z = 0 , 1 } \\left( \\prod _ { i = 1 } ^ { 2 0 }  { \\mathcal { N } } [ z , \\sigma ^ { 2 } ] ( x _ { i } ) \\right) } \\\\ & { p _ { \\mathrm { o u t } } ( x _ { 1 } , \\dots , x _ { 2 0 } ) = \\displaystyle \\frac { 1 } { 2 } \\sum _ { z = 0 , 1 } \\left( \\prod _ { i = 1 } ^ { 5 }  { \\mathcal { N } } [ \\bar { z } , \\sigma ^ { 2 } ] ( x _ { i } ) \\right) \\left( \\prod _ { i = 6 } ^ { 2 0 }  { \\mathcal { N } } [ z , \\sigma ^ { 2 } ] ( x _ { i } ) \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "That is, there is a single binary latent variable $z$ . For inliers, each feature is an independent noisy reading of the latent $z$ . For outliers, the first 5 features are centred instead around its opposite $\\bar { z }$ . An example outlier (with $\\sigma = 0 . 0 5 )$ ) is shown in Fig. 3(a). We generated one such data set for each $\\sigma \\in \\{ 0 . 0 \\bar { 1 } , 0 . 0 3 , \\ldots , 0 . 1 5 \\}$ in order to study the effect of noise on explanation errors. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/b23c6c1705e84ff4d49fabd59f25bcba831cbcbe89087806136d3730d1f20702.jpg",
+        "image_caption": [
+            "Figure 4: Negative global Shapley values arise off-manifold, in both (a) synthetic and (b) real data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We fit an isolation forest (Liu et al., 2008) to perform outlier detection on each synthetic dataset, achieving $100 \\%$ accuracy in every case. We computed the off- and on-manifold value functions of Eqs. (2) and (3) for each isolation forest by sampling the probability distributions directly, as these can be inferred from Eqs. (11) and (12). Figs. 3(b) and 3(c) show the resulting local Shapley values for the example outlier from Fig. 3(a). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The ground-truth explanation of why Fig. 3(a) represents an outlier is that its first 5 features break correlations that exist across $9 9 \\%$ of the data. The on-manifold explanation of Fig. 3(c) correctly attributes the 5 largest Shapley values to features $x _ { 1 } , \\ldots , x _ { 5 }$ . The off-manifold explanation of Fig. 3(b) is unambiguously incorrect: feature $x _ { 7 }$ receives a larger value than $x _ { 2 }$ , $x _ { 4 }$ , and $x _ { 5 }$ . We consider an explanation to be erroneous if $x _ { 1 } , \\ldots , x _ { 5 }$ do not receive the 5 largest Shapley values. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To show the frequency of incorrect explanations, Fig. 3(d) displays the off- and on-manifold error rates as a function of noise $\\sigma$ in the synthetic data set. Incorrect explanations are commonplace off-manifold: one-quarter are in error in the presence of minimal noise, and two-thirds are incorrect at $\\sigma = 0 . 1 5$ . The on-manifold error rate is dramatically lower across this range. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figs. 3(e) and 3(f) show the root cause of off-manifold errors. These histograms display the distribution of model outputs when evaluated on Shapley coalitions in the off- and on-manifold calculations for $\\sigma = 0 . 0 5$ . In particular, Fig. 3(e) shows the model evaluated on “inlier coalitions” which do not include $x _ { 1 } , \\ldots , x _ { 5 }$ . Note that model outputs for on-manifold coalitions agree with the model evaluated on the actual data, while off-manifold coalitions follow a very different distribution. In particular, since a positive model output indicates a predicted outlier, Fig. 3(e) shows that the offmanifold calculation itself fabricates outliers through its splicing procedure. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similarly, Fig. 3(f) shows the model evaluated on “outlier coalitions” which do include $x _ { 1 } , \\ldots , x _ { 5 }$ . Note that model outputs are similar for on-manifold coalitions and actual outliers, whereas offmanifold coalitions again differ dramatically. This is a manifestation of uncontrolled model behaviour off the data manifold, and it ultimately leads to erroneous off-manifold explanations. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "BREAKDOWN IN GLOBAL SHAPLEY VALUES OFF-MANIFOLD ",
+        "text_level": 1,
+        "bbox": [
+            179,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate that global Shapley values can be misleading off-manifold as well, we generated an additional synthetic data set according to the process in Fig. 4(a). The data has two binary features and a binary label. We fit a decision tree to this data, resulting in a precise match to Fig. 4(a). Note that the features $x _ { 0 }$ and $x _ { 1 }$ are positively correlated, both with each other and with label $y$ . However, with $x _ { 0 }$ fixed, the likelihood of $y = 1$ decreases slightly from $x _ { 1 } = 0$ to $x _ { 1 } = 1$ . One might think of $x _ { 0 }$ as disease severity, $x _ { 1 }$ as treatment intensity, and $y$ as mortality rate. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Fig. 4(b) displays global Shapley values for this model. The global Shapley values are positive onmanifold, consistent with their interpretation as the portion of model accuracy attributable to each feature. Off-manifold, however, a negative value results from placing too much weight on splices, e.g. with $( x _ { 0 } , x _ { 1 } , y ) = ( 0 , 1 , 1 )$ , that occur less frequently in the actual data. The negative value would erroneously indicate that $x _ { 1 }$ is detrimental to the model’s overall performance. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We can demonstrate this on real data using UCI Abalone data (Dua & Graff, 2017). We trained a neural network to classify abalone as younger than or older than the median age based on physical characteristics. Fig. 4(c) displays global Shapley values for this model. (On-manifold values were computed using techniques developed in Sec. 4.1; see App. B for details.) ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Observe the drastic difference between the on- and off-manifold explanations in Fig. 4(c). This is due to the tight correlations between features in the data (4 weights and 3 lengths) making the data manifold low-dimensional and important. Notice further the large negative off-manifold global Shapley value, negating its interpretation as the portion of model accuracy attributable to that feature. ",
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+    {
+        "type": "text",
+        "text": "4 SCALABLE APPROACHES TO ON-MANIFOLD SHAPLEY VALUES",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Sec. 3 we computed on-manifold Shapley values for simple data by estimating $p ( x ^ { \\prime } | x _ { S } )$ from the empirical data distribution or, for synthetic data, by knowing this distribution analytically. Here we introduce two performant methods to compute on-manifold Shapley values on general data. Sec. 4.1 develops the theory underlying our methods, and Sec. 4.2 presents additional experimental results. ",
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+    {
+        "type": "text",
+        "text": "4.1 THEORETICAL DEVELOPMENT OF ON-MANIFOLD METHODS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here we develop two methods to learn the on-manifold value function: (i) unsupervised learning the conditional distribution $p ( x ^ { \\prime } | x _ { S } )$ , and (ii) a supervised technique to learn the value function directly. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "UNSUPERVISED APPROACH",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "One can use unsupervised learning to learn the conditional distributions $p ( x ^ { \\prime } | x _ { S } )$ that appear in the on-manifold value function. Here we take an approach similar to Ivanov et al. (2019) to learn these distributions with variational inference. See Douglas et al. (2017) and Belghazi et al. (2019) for alternative techniques to learning conditional distributions that could be used here instead. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our specific approach includes two model components. The first is a variational autoencoder (Kingma & Welling, 2014; Rezende et al., 2014), with encoder $q _ { \\phi } ( z | x )$ and decoder $p _ { \\theta } ( x | z )$ . The second is a masked encoder, $r _ { \\psi } ( z | x _ { S } )$ , for which the goal is to map the coalition $x _ { S }$ to a distribution in latent space that agrees with the encoder $q _ { \\phi } ( z | x )$ as well as possible. A model of $p ( x ^ { \\prime } | x _ { S } )$ is then provided by the composition: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/a1232dc0fbeae6b25c69ee22f429d8163ed2f54d419e428a7c48290d927ba81a.jpg",
+        "text": "$$\n\\hat { p } ( x ^ { \\prime } | x _ { S } ) = \\int d z p _ { \\theta } ( x ^ { \\prime } | z ) r _ { \\psi } ( z | x _ { S } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "and a good fit to the data should maximise ${ \\hat { p } } ( x ^ { \\prime } | x _ { S } )$ . A lower bound to its log-likelihood is given by ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d881628ad6890882ae5680706655d55ff68c1a62f3f99a66a549c4a9f7bbc49b.jpg",
+        "text": "$$\n\\mathcal { L } _ { 0 } = \\mathbb { E } _ { q _ { \\phi } ( z | x ^ { \\prime } ) } \\big [ \\log p _ { \\theta } ( x ^ { \\prime } | z ) \\big ] - \\mathcal { D } _ { \\mathrm { K L } } \\big ( q _ { \\phi } ( z | x ^ { \\prime } ) | | r _ { \\psi } ( z | x _ { S } ) \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            645
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "While $\\mathcal { L } _ { 0 }$ could be used on its own as the objective function to learn $\\hat { p } ( x ^ { \\prime } | x _ { S } )$ , this would leave the variational distribution $q _ { \\phi } ( z | x )$ unconstrained, at odds with our goal of learning a smooth-manifold structure in latent space. This concern can be mitigated by ${ \\mathcal { L } } _ { \\mathrm { r e g } } = - { \\mathcal { D } } _ { \\mathrm { K L } } \\big ( q _ { \\phi } ( z | x ) | | p ( z ) \\big )$ which regularises $q _ { \\phi } ( z | x )$ by penalising differences from a smooth (e.g. unit normal) prior distribution $p ( z )$ . We thus include $\\mathcal { L } _ { \\mathrm { r e g } }$ as a regularisation term in our unsupervised objective: $\\mathcal { L } = \\mathcal { L } _ { 0 } + \\beta \\mathcal { L } _ { \\mathrm { r e g } }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "METRIC FOR THE LEARNT VALUE FUNCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The unsupervised method presented above leads to a learnt estimate of the conditional distribution, and thus to an estimate of the on-manifold value function: $\\hat { v } _ { f _ { y } ( x ) } ( S ) = \\mathbb { E } _ { \\hat { p } ( x ^ { \\prime } \\mid x _ { S } ) } [ f _ { y } ( x ^ { \\prime } ) ]$ . With the goal of judging the performance of this estimate, consider the following formal quantity: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca4e89cf1ae1a40ac18bc92e55df2071e403eb50cfebb538db624cd993722ee6.jpg",
+        "text": "$$\n\\mathrm { m s e } ( x _ { S } , y ) = \\mathbb { E } _ { p ( x ^ { \\prime } \\mid x _ { S } ) } \\left. f _ { y } ( x ^ { \\prime } ) - \\hat { v } _ { f _ { y } ( x ) } ( S ) \\right. ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This quantity is minimal with respect to $\\hat { v } _ { f _ { y } ( x ) } ( S )$ when $\\hat { v } _ { f _ { y } ( x ) } ( S ) = \\mathbb { E } _ { p ( x ^ { \\prime } | x _ { S } ) } [ f _ { y } ( x ^ { \\prime } ) ]$ , in agreement with the definition, Eq. (3), of the on-manifold value function. We can then quantitatively judge the performance of the unsupervised model $\\hat { p } ( x ^ { \\prime } | x _ { S } )$ by computing ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3d3bbfd8ab571cd56c9fad9cf350e9725549614c7a8c6938e0168f80ad07c043.jpg",
+        "text": "$$\n\\mathrm { M S E } = \\mathbb { E } _ { p ( x ) } \\mathbb { E } _ { S \\sim \\mathrm { S h a p l e y } } \\mathbb { E } _ { y \\sim \\mathrm { U n i f } } \\left| f _ { y } ( x ) - \\hat { v } _ { f _ { y } ( x ) } ( S ) \\right| ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/5e9f11f55dcb93ed715e55d944fbd967c3f734a3dbe8390b396317f2230b6480.jpg",
+        "image_caption": [
+            "Figure 5: Validation of unsupervised and supervised techniques for computing on-manifold Shapley values. Comparison against empirical ground truth, which appeared as “On manifold” in Fig. 2(b). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/356054ec137e5393cabcc6f92c2a5f857f4363b48ff570914e9316b4d75ba175.jpg",
+        "table_caption": [
+            "Table 1: Performance and stability, in terms of MSE, of supervised and unsupervised approaches. Performance is compared with off-manifold splicing and, where accessible, the empirical optimum. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>DATA SET</td><td>OFF MANIFOLD</td><td>UNSUPERVISED</td><td>SUPERVISED</td><td>EMPIRICAL</td></tr><tr><td>DRUG</td><td>0.0634</td><td>0.0536 ± 0.0007</td><td>0.0441 ± 0.0002</td><td>0.0436</td></tr><tr><td>ABALONE</td><td>0.0647</td><td>0.0293 ±0.0009</td><td>0.0200 ± 0.0001</td><td>1</td></tr><tr><td>CENSUS</td><td>0.0344</td><td>0.0300 ±0.0006</td><td>0.0250 ±0.0001</td><td>1</td></tr><tr><td>MNIST</td><td>0.0448</td><td>0.0257 ± 0.0005</td><td>0.0121 ± 0.0001</td><td>1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Note that this is precisely Eq. (15) averaged over coalitions $S$ drawn from the Shapley sum,2 features $x _ { S } \\sim p ( x _ { S } )$ drawn from the data, and labels $y$ drawn uniformly over classes. Moreover, the meansquare-error in Eq. (16) is easy to estimate using the empirical distribution $p ( x )$ and the learnt model $\\hat { p ( } x ^ { \\prime } | x _ { S } )$ , thus providing an unambiguous metric to judge the outcome of the unsupervised approach. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "SUPERVISED APPROACH",
+        "text_level": 1,
+        "bbox": [
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+            517,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The MSE metric of Eq. (16) supports a supervised approach to learning the on-manifold value function directly: one can define a surrogate model $g _ { y } ( x _ { S } )$ that operates on coalitions of features $x _ { S }$ (e.g. by masking out-of-coalition features) and that is trained to minimise the loss: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ef9991c73bf3229e615baffaea967a41117a433bfe080a217262f0c60d8daaf.jpg",
+        "text": "$$\n{ \\mathcal { L } } = \\mathbb { E } _ { p ( x ) } \\mathbb { E } _ { S \\sim \\mathrm { S h a p l e y } } \\mathbb { E } _ { y \\sim \\mathrm { U n i f } } \\left| f _ { y } ( x ) - g _ { y } ( x _ { S } ) \\right| ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            585,
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+            608
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As discussed above Eq. (16), this loss is minimised as the surrogate model $g _ { y } ( x _ { S } )$ approaches the on-manifold value function $\\mathbb { E } _ { p ( x ^ { \\prime } | x _ { S } ) } [ f _ { y } ( x ^ { \\prime } ) ]$ of the model-to-be-explained. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 ADDITIONAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Sec. 3 above presented two experiments using the scalable on-manifold methods developed here. In particular, Fig. 2(a) applied the unsupervised method to Census Income data, showing that onmanifold Shapley values detect hidden model dependence on sensitive features, and Fig. 4(c) applied both methods to Abalone data, showing that global Shapley values remain positive and interpretable on-manifold. In this section, we perform additional experiments to study the performance and stability of Sec. 4.1’s methods, as well as their effectiveness on higher-dimensional data. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "PERFORMANCE AND STABILITY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our implementations of the unsupervised and supervised approaches to on-manifold Shapley values are summarised in Apps. A and B. Both approaches lead to broadly similar results. Fig. 5 compares the two techniques on the Drug Consumption data, where explanations are compared against the ground-truth empirical computation from Fig. 2(b). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The unsupervised approach is flexible but untargeted: $p ( x ^ { \\prime } | x _ { S } )$ is data-specific but model-agnostic, accommodating explanations for many models trained on the same data. The supervised approach must be retrained on each model, but it entails direct minimisation of the MSE. The supervised method is thus expected to achieve higher accuracy. We confirmed this on all data sets studied in this paper; see Table 1 for a numerical comparison of the MSEs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e1a04e9af51499eb1ad1dc5bdc08dbc77bdec3a5fb1b18f294fa4ce6affc7d3e.jpg",
+        "image_caption": [
+            "Figure 6: (a) Randomly drawn MNIST digits explained on / off manifold. Red / blue pixels indicate positive / negative Shapley values, and the colour scale in each column is fixed. (b) Shapley summand as a function of coalition size – averaged over coalitions, pixels, and the MNIST test set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Table 1, central values indicate the test-set MSE achieved by each method. The table compares the unsupervised and supervised methods against off-manifold splicing, showing significant improvement over this baseline. Note that an MSE of zero is not achievable, because ${ \\dot { f } } _ { y } ( x )$ in Eq. (16) or (17) is not fully determined by partial input $x _ { S }$ . For the Drug Consumption data where we can compute $p ( \\boldsymbol { x } ^ { \\prime } | \\boldsymbol { x } _ { S } )$ empirically, the optimal MSE happens to be 0.0436. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Uncertainties in Table 1 represent the standard deviation in test-set MSE upon repeating each method with fixed hyperparameters 10 times. (Uncertainties are absent for the off-manifold and empirical columns, as these do not involve training a separate model.) The table thus indicates that the supervised method offers increased stability as compared to the unsupervised approach. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The supervised method is more efficient as well: while the unsupervised technique estimates the value function by sampling from ${ \\hat { p } } ( x ^ { \\prime } | x _ { S } )$ , the supervised approach learns the value function directly. The supervised method thus requires far fewer model evaluations to match the standard-error of the unsupervised method: roughly 10 times fewer in our experiments. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "EXAMPLE ON MNIST ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            555,
+            328,
+            569
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate on-manifold explainability on higher-dimensional data, we trained a fully connected network on binary MNIST (LeCun & Cortes, 2010) and explained random digits in Fig. 6(a). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Despite having the same sum over pixels – as controlled by the local version of Eq. (5) – and explaining the same model prediction, each on-manifold explanation is more concentrated, with more interpretable structure, than its off-manifold counterpart. The handwritten strokes are clearly visible on-manifold, with key off-stroke regions highlighted as well. Off-manifold explanations generally display lower intensities spread less informatively across the digit-region. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "These off-manifold explanations are a result of splices as in Fig. 1. With such unrealistic input, the model’s output is uncontrolled and less informative. In fact, it is only on very large coalitions of pixels, subject to minimal splicing, that the model can make intelligent predictions off-manifold. This is confirmed in Fig. 6(b), which shows the average Shapley summand as a function of coalition size on MNIST. Note that primarily large coalitions underpin off-manifold explanations, whereas far fewer pixels are required on-manifold, consistent with the low-dimensional manifold underlying the data. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we took a careful study of the off-manifold problem in AI explainability. We presented important distinctions between on- and off-manifold explainability and provided experimental evidence for several novel shortcomings of the off-manifold approach. We then introduced two techniques to compute on-manifold Shapley values on general data: one technique learns to impute features on the data manifold, while the other learns the Shapley value-function directly. In-so-doing, we provided compelling evidence against the use of off-manifold explainability, and demonstrated that on-manifold Shapley values offer a viable approach to AI explainability in real-world contexts. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES   \nJ. Aas, M. Jullum, and A. Løland. Explaining individual predictions when features are dependent: More accurate approximations to Shapley values, 2019. [arXiv:1903.10464].   \nC. Agarwal, D. Schonfeld, and A. Nguyen. Removing input features via a generative model to explain their attributions to classifier’s decisions, 2019. [arXiv:1910.04256].   \nC. J. Anders, P. Pasliev, A.-K. Dombrowski, K.-R. Muller, and P. Kessel. Fairwashing explanations ¨ with off-manifold detergent. In International Conference on Machine Learning, 2020.   \nM. Belghazi, M. Oquab, and D. Lopez-Paz. Learning about an exponential amount of conditional distributions. In Advances in Neural Information Processing Systems, 2019.   \nC. Chang, E. Creager, A. Goldenberg, and D. Duvenaud. Explaining image classifiers by counterfactual generation. In International Conference on Learning Representations, 2019.   \nH. Chen, J. D. Janizek, S. Lundberg, and S.-I. Lee. True to the model or true to the data?, 2020. [arXiv:2006.16234].   \nI. Covert, S. Lundberg, and S.-I. Lee. Understanding global feature contributions through additive importance measures, 2020. [arXiv:2004.00668].   \nA. Datta, S. Sen, and Y. Zick. Algorithmic transparency via quantitative input influence: Theory and experiments with learning systems. In IEEE Symposium on Security and Privacy, 2016.   \nB. Dimanov, U. Bhatt, M. Jamnik, and A. Weller. You shouldn’t trust me: Learning models which conceal unfairness from multiple explanation methods. In SafeAI @ AAAI, 2020.   \nA.-K. Dombrowski, M. Alber, C. Anders, M. Ackermann, K.-R. Muller, and P. Kessel. Explanations ¨ can be manipulated and geometry is to blame. In Advances in Neural Information Processing Systems, 2019.   \nL. Douglas, I. Zarov, K. Gourgoulias, C. Lucas, C. Hart, A. Baker, M. Sahani, Y. Perov, and S. Johri. A universal marginalizer for amortized inference in generative models. In NIPS Workshop on Approximate Bayesian Inference, 2017.   \nD. Dua and C. Graff. UCI machine learning repository, 2017. [archive.ics.uci.edu/ml].   \nI. J. Goodfellow, J. Shlens, and C. Szegedy. Explaining and harnessing adversarial examples. In International Conference on Learning Representations, 2015.   \nG. Hooker and L. Mentch. Please stop permuting features: An explanation and alternatives, 2019. [arXiv:1905.03151].   \nO. Ivanov, M. Figurnov, and D. Vetrov. Variational autoencoder with arbitrary conditioning. In International Conference on Learning Representations, 2019.   \nD. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations, 2015.   \nD. P. Kingma and M. Welling. Auto-encoding variational bayes. In International Conference on Learning Representations, 2014.   \nI. Kononenko et al. An efficient explanation of individual classifications using game theory. Journal of Machine Learning Research, 2010.   \nY. LeCun and C. Cortes. MNIST database, 2010. [yann.lecun.com/exdb/mnist].   \nS. Lipovetsky and M. Conklin. Analysis of regression in game theory approach. Applied Stochastic Models in Business and Industry, 2001.   \nF. T. Liu, K. M. Ting, and Z.-H. Zhou. Isolation forest. In IEEE International Conference on Data Mining, 2008. ",
+        "bbox": [
+            171,
+            69,
+            826,
+            925
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "S. M. Lundberg and S.-I. Lee. A unified approach to interpreting model predictions. In Advances in Neural Information Processing Systems, 2017. ",
+        "bbox": [
+            169,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "S. M. Lundberg, G. G. Erion, and S.-I. Lee. Consistent individualized feature attribution for tree ensembles, 2018. [arXiv:1802.03888]. ",
+        "bbox": [
+            168,
+            140,
+            823,
+            170
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "S. M. Lundberg, G. Erion, H. Chen, A. DeGrave, J. Prutkin, B. Nair, R. Katz, J. Himmelfarb, N. Bansal, and S. Lee. From local explanations to global understanding with explainable AI for trees. Nature Machine Intelligence, 2020. ",
+        "bbox": [
+            174,
+            178,
+            823,
+            222
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A. Nguyen, J. Yosinski, and J. Clune. Deep neural networks are easily fooled: High confidence predictions for unrecognizable images. In Conference on Computer Vision and Pattern Recognition, 2015. ",
+        "bbox": [
+            173,
+            231,
+            823,
+            272
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "D. J. Rezende, S. Mohamed, and D. Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In International Conference on Machine Learning, 2014. ",
+        "bbox": [
+            173,
+            281,
+            820,
+            311
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "M. T. Ribeiro, S. Singh, and C. Guestrin. Why should I trust you: Explaining the predictions of any classifier. In International Conference on Knowledge Discovery and Data Mining, 2016. ",
+        "bbox": [
+            173,
+            319,
+            820,
+            349
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C. Rudin. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature Machine Intelligence, 2019. ",
+        "bbox": [
+            176,
+            357,
+            820,
+            387
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "L. S. Shapley. A value for $n$ -person games. In Contribution to the theory of games, 1953. ",
+        "bbox": [
+            169,
+            395,
+            761,
+            411
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A. Shrikumar, P. Greenside, and A. Kundaje. Learning important features through propagating activation differences. In International Conference on Machine Learning, 2017.   \nD. Slack, S. Hilgard, E. Jia, S. Singh, and H. Lakkaraju. Fooling lime and shap: Adversarial attacks on post hoc explanation methods. In AIES @ AAAI, 2020.   \nE. Strumbelj and I. Kononenko. Explaining prediction models and individual predictions with fea- ˇ ture contributions. Knowledge and information systems, 2014.   \nE. Strumbelj, I. Kononenko, and M. Robnik-Sikonja. Explaining instance classifications with inter- ˇ actions of subsets of feature values. Data Knowl. Eng., 2009.   \nM. Sundararajan and A. Najmi. The many Shapley values for model explanation. In International Conference on Machine Learning, 2019. ",
+        "bbox": [
+            171,
+            419,
+            828,
+            603
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            437,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the unsupervised approach, we modelled the encoder $q _ { \\phi } ( z | x )$ as a diagonal normal distribution with mean and variance determined by a neural network: ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/051253ad9fe294653ace34537c90f9959063cc2779ee4df963fa1d59cd208955.jpg",
+        "text": "$$\nq _ { \\phi } ( z | x ) = \\mathcal { N } \\big ( \\mu _ { \\phi } ( x ) , \\sigma _ { \\phi } ( x ) \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            167,
+            596,
+            188
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We modelled the decoder $p _ { \\theta } ( x | z )$ as a product distribution: ",
+        "bbox": [
+            173,
+            194,
+            563,
+            210
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f6c87c78edc9d196354a0131259da4e422d94843268dd51568a816ff76a6f3f.jpg",
+        "text": "$$\np _ { \\theta } ( x | z ) = \\prod _ { i } p _ { \\theta } ( x _ { i } | z )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            217,
+            576,
+            251
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where the distribution type (e.g. normal, categorical) of each $x _ { i }$ is chosen per-data-set and each distribution’s parameters are determined by a shared neural network. We modelled the masked encoder $r _ { \\psi } ( z | x _ { S } )$ as a Gaussian mixture: ",
+        "bbox": [
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+            257,
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+            299
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8abfd7e506db9b73170a5bdee0950139f5ab566ef0c00509f51307778826c02c.jpg",
+        "text": "$$\nr _ { \\psi } ( z | x _ { S } ) = \\sum _ { j } w _ { \\phi } ^ { ( j ) } ( x ) \\mathcal { N } \\Big ( \\mu _ { \\phi } ^ { ( j ) } ( x ) , \\sigma _ { \\phi } ^ { ( j ) } ( x ) \\Big )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            308,
+            651,
+            343
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To allow $r _ { \\psi } ( z | x _ { S } )$ to accept variable-size coalitions $x _ { S }$ as input, we simply masked out-of-coalition features with a special value $( - 1 )$ that never appears in the data. ",
+        "bbox": [
+            171,
+            351,
+            823,
+            380
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The unsupervised method has several hyperparameters: $\\beta$ which multiplies the regularisation term, the number of components in Eq. (20), as well the architecture and optimisation of the networks involved. For each experiment in this paper, we tuned hyperparameters to minimise the MSE of Eq. (16) on a held-out validation set; see App. B for numerical details. ",
+        "bbox": [
+            174,
+            386,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the supervised approach, we modelled $g _ { y } ( x _ { S } )$ using a neural network, again masking out-ofcoalition features (with $^ { - 1 }$ ) to accommodate variable-size coalitions $x _ { S }$ . This method’s hyperparameters, relating to architecture and optimisation, were similarly tuned to minimise the validationset MSE; see App. B for details. ",
+        "bbox": [
+            173,
+            449,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B DETAILS OF EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            526,
+            431,
+            541
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here we provide numerical details for the experiments presented in the paper. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.1 DRUG CONSUMPTION EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            589,
+            464,
+            604
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "On the Drug Consumption data from the UCI repository (Dua & Graff, 2017), we used 10 binary features from the data set – Mushrooms, Ecstasy, etc., as displayed in Fig. $5 -$ to predict whether individuals had ever consumed an 11th drug: LSD. The explanations of Fig. 2(b) and Fig. 5 describe a random forest fit with default sklearn parameters and max features $=$ None, which achieves $8 2 . 2 \\%$ test-set accuracy amidst a $5 7 : 4 3$ class balance. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 2(b), global off-manifold Shapley values were computed using $1 0 ^ { 6 }$ Monte Carlo samples of Eq. (4). For each labelled data point $( x , y )$ sampled from the test set, a single permutation was drawn to estimate Eq. (1) and a single data point $x ^ { \\prime }$ was drawn to estimate the off-manifold value function Eq. (2). In all the figures of this paper, bar height represents the mean that resulted from Monte Carlo sampling, and error bars display the standard error of the mean. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Global on-manifold Shapley values in Fig. 2(b) were computed similarly, but in this case using the on-manifold value function of Eq. (3). For each sampled coalition $x _ { S }$ , a random data point $x ^ { \\prime }$ was drawn from the test set, with the crucial requirement that $x _ { S } ^ { \\prime } = x _ { S }$ . In the text, we refer to this as empirically estimating the conditional distribution $p ( x ^ { \\prime } | x _ { S } )$ . Such empirical estimation is only possible because this data set has a small number of all-binary features. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tree SHAP values in Fig. 2(b) were computed with the SHAP package (Lundberg & Lee, 2017) with model output $=$ margin and feature perturbation $=$ tree path dependent. ",
+        "bbox": [
+            176,
+            845,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The values labelled “Model retraining” in Fig. 2(b) were computed by fitting a separate random forest $g _ { S }$ for each coalition $S$ of features in the data set: $2 ^ { 1 0 }$ models in all. We used these models to compute the sum of Eq. (10), where $A ( g _ { S } )$ represents a variant of model $g _ { S }$ ’s accuracy: it is the accuracy achieved if one predicts labels by drawing stochastically from $g _ { S }$ ’s predicted probability distribution (as opposed to deterministically drawing the maximum-probability class). ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/b55b4850d4f697551121d046f7907c35ab5054efa4f863bd5fa6e94a0e00a013.jpg",
+        "table_caption": [
+            "Table 2: Optimal hyperparameters found for computing on-manifold Shapley values. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>DATA SET</td><td>METHOD</td><td>HIDDEN DIM.</td><td>LEARN.RATE</td><td>LATENT DIM.</td><td>MODES</td><td>β</td></tr><tr><td>DRUG</td><td>SUPERVISED</td><td>512</td><td>10-3</td><td></td><td></td><td></td></tr><tr><td></td><td>UNSUPERVISED</td><td>128</td><td>10-3</td><td>4</td><td>1</td><td>0.5</td></tr><tr><td>ABALONE</td><td>SUPERVISED UNSUPERVISED</td><td>512 256</td><td>10-3 10-3</td><td>2</td><td>1</td><td>0.05</td></tr><tr><td>CENSUS</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>SUPERVISED UNSUPERVISED</td><td>512 128</td><td>10-3 10-3</td><td>8</td><td>1</td><td>1</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MNIST</td><td>SUPERVISED</td><td>512</td><td>10-4</td><td></td><td></td><td></td></tr><tr><td></td><td>UNSUPERVISED</td><td>512</td><td>10-4</td><td>16</td><td>1</td><td>1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The global on-manifold Shapley values in Fig. 2(b) appear in Fig. 5 as well, labelled “Empirical”. Fig. 5 also displays on-manifold Shapley values computed using the supervised and unsupervised methods introduced in this paper. As above, these are Monte Carlo estimates of Eq. (4). The supervised method involved training a fully connected network on the MSE loss of Eq. (17). All neural networks in this paper used 2 flat hidden layers, Adam (Kingma & Ba, 2015) for optimisation, and a batch size of 256. We scanned over a grid with ",
+        "bbox": [
+            173,
+            377,
+            825,
+            462
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f020a701b6e37bc68730368d420be3cd3805b0865e98c2ec3623897cfb8a331.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\mathrm { h i d d e n ~ l a y e r ~ s i z e } = \\{ 1 2 8 , 2 5 6 , 5 1 2 \\} } } \\\\ { { \\mathrm { l e a r n i n g ~ r a t e } = \\{ 1 0 ^ { - 3 } , 1 0 ^ { - 4 } \\} } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            468,
+            622,
+            507
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "choosing the point with minimal MSE on a held-out validation set after $1 0 \\mathrm { k }$ epochs of training; see Table 2. Each supervised value in Fig. 5 corresponds to $1 0 ^ { 4 }$ Monte Carlo samples. ",
+        "bbox": [
+            173,
+            512,
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+            541
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The unsupervised method involved training a variational autoencoder as described in Sec. 4.1 and App. A. The encoder, decoder, and masked encoder were each modelled using fully connected networks, trained using early stopping with patience 100. We scanned over a grid of hidden layer sizes and learning rates as in Eq. (21) as well as ",
+        "bbox": [
+            174,
+            547,
+            825,
+            604
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d5a2aa523da53b41539e023b55ca486d98b118f6d51ed117d5922b650ee3ad1.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\mathrm { l a t e n t d i m e n s i o n } = \\{ 2 , 4 , 8 , 1 6 \\} } } \\\\ { { \\mathrm { l a t e n t m o d e s } = \\{ 1 , 2 \\} } } \\\\ { { \\mathrm { r e g u l a r i s a t i o n } \\beta = \\{ 0 . 0 5 , 0 . 1 , 0 . 5 , 1 \\} } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            611,
+            627,
+            665
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "choosing the point with minimal validation-set MSE; see Table 2. Unsupervised values in Fig. 5 correspond to $\\mathrm { \\dot { 1 } 0 ^ { 6 } }$ Monte Carlo samples. ",
+        "bbox": [
+            173,
+            670,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.2 CENSUS INCOME EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            715,
+            429,
+            729
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To produce the explanations of Fig. 2(a) we used the Census Income data set from the UCI repository (Dua & Graff, 2017). The data contains $4 9 \\mathrm { k }$ individuals from the 1994 US Census, as well as 13 features which we used to predict whether annual income exceeded $\\$ 50\\mathrm { k }$ . We trained a fully connected network (hidden layer size 50, default sklearn parameters, and early stopping), achieving a test-set accuracy of $8 5 \\%$ amidst a $7 6 : 2 4$ class balance. ",
+        "bbox": [
+            174,
+            741,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The Shapley values for this model are labelled “Original model” in Fig. 2(a). These were computed exactly as described in App. B.1, except that the supervised method used $5 \\mathrm { k }$ epochs, and the unsupervised method used patience 50. Optimised hyperparameters are given in Table 2. The onmanifold values in Fig. 2(a) were computed using the unsupervised method. While the supervised method does not appear in the figure, it was performed to complete Table 1. ",
+        "bbox": [
+            174,
+            818,
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+            888
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We also fine-tuned the “Original model” to suppress the importance of sex. Motivated by Dimanov et al. (2020) we added a term to the loss that penalises the finite difference in the model output with respect to sex (as this is a discrete feature). The modified loss function thus becomes ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            727,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/72bb4f06e91c088f2d80551033119f6ed62aeea6e65c12447f8c7091f4bca284.jpg",
+        "text": "$$\n{ \\frac { 1 } { N } } \\sum _ { i = 1 } ^ { N } { \\mathcal { L } } { \\big ( } f ( x _ { i } ) , y _ { i } { \\big ) } \\ + \\ \\alpha { \\Big | } \\ f { \\big ( } x _ { i } | \\operatorname { d o } ( \\operatorname { s e x } = 1 ) { \\big ) } - f { \\big ( } x _ { i } | \\operatorname { d o } ( \\operatorname { s e x } = 0 ) { \\big ) } { \\Big | }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            125,
+            736,
+            169
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L }$ is the cross-entropy loss, $f { \\big ( } x _ { i } | \\operatorname { d o } ( \\operatorname { s e x } = j ) { \\big ) }$ denotes $f$ evaluated on the data point $x _ { i }$ with the value for sex replaced with $j$ , and $\\alpha$ is a hyperparameter controlling the trade-off between optimising the accuracy and minimising the effect of sex. We fine-tuned the model for an additional 200 epochs with $\\alpha = 3$ . The resulting model agrees with the baseline on over $9 8 . 5 \\%$ of the data, and has the same test-set accuracy. Shapley values for this model are labelled “Suppressed model” in Fig. 2(a). ",
+        "bbox": [
+            174,
+            175,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.3 ABALONE EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            277,
+            382,
+            291
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The Abalone data set from the UCI repository (Dua & Graff, 2017) contains 8 features corresponding to physical measurements (see Fig. 4c) which we used to classify abalone as younger than or older than the median age. We trained a neural network to perform this task – with hidden layer size 100, default sklearn parameters, and early stopping – obtaining a test-set accuracy of $78 \\%$ . ",
+        "bbox": [
+            174,
+            303,
+            825,
+            359
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Shapley values in Fig. 4(c) were computed exactly as described in App. B.1, except that the supervised method involved training for 5k epochs. Optimised hyperparameters are given in Table 2. ",
+        "bbox": [
+            173,
+            366,
+            821,
+            395
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.4 MNIST EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            411,
+            367,
+            425
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For binary MNIST (LeCun & Cortes, 2010), we trained a fully connected network (hidden layer size 512, default sklearn parameters, and early stopping) achieving $98 \\%$ test-set accuracy. ",
+        "bbox": [
+            174,
+            436,
+            823,
+            465
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The digits in Fig. 6(a) were randomly drawn from the test set. Shapley values in Fig. 6(a) were computed exactly as described in App. B.1, except that the supervised method involved training for 2k epochs, and the on-manifold explanations are based on $1 6 \\mathrm { k }$ Monte Carlo samples per pixel. Optimised hyperparameters are given in Table 2. The on-manifold explanations in Fig. 6(a) were computed using the supervised method. While the unsupervised method does not appear in the figure, it was performed to complete Table 1. ",
+        "bbox": [
+            174,
+            472,
+            825,
+            556
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The average uncertainty, which is not shown in Fig. 6(a), is roughly 0.002 – stated as a fraction of the maximum Shapley value in each image. ",
+        "bbox": [
+            174,
+            563,
+            823,
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+        ],
+        "page_idx": 13
+    }
+]

parse/train/SkfMWhAqYQ/SkfMWhAqYQ.md

@@ -0,0 +1,208 @@
+# APPROXIMATING CNNS WITH BAG-OF-LOCALFEATURES MODELS WORKS SURPRISINGLY WELL ON IMAGENET
+
+Wieland Brendel and Matthias Bethge
+
+Eberhard Karls University of Tübingen, Germany Werner Reichardt Centre for Integrative Neuroscience, Tübingen, Germany Bernstein Center for Computational Neuroscience, Tübingen, Germany {wieland.brendel, matthias.bethge}@bethgelab.org
+
+# ABSTRACT
+
+Deep Neural Networks (DNNs) excel on many complex perceptual tasks but it has proven notoriously difficult to understand how they reach their decisions. We here introduce a high-performance DNN architecture on ImageNet whose decisions are considerably easier to explain. Our model, a simple variant of the ResNet50 architecture called BagNet, classifies an image based on the occurrences of small local image features without taking into account their spatial ordering. This strategy is closely related to the bag-of-feature (BoF) models popular before the onset of deep learning and reaches a surprisingly high accuracy on ImageNet $( 8 7 . 6 \%$ top-5 for $3 3 \times 3 3$ px features and Alexnet performance for $1 7 \times 1 7 \ : \mathrm { p x }$ features). The constraint on local features makes it straight-forward to analyse how exactly each part of the image influences the classification. Furthermore, the BagNets behave similar to state-of-the art deep neural networks such as VGG-16, ResNet-152 or DenseNet-169 in terms of feature sensitivity, error distribution and interactions between image parts. This suggests that the improvements of DNNs over previous bag-of-feature classifiers in the last few years is mostly achieved by better fine-tuning rather than by qualitatively different decision strategies.
+
+# 1 INTRODUCTION
+
+A big obstacle in understanding the decision-making of DNNs is due to the complex dependencies between input and hidden activations: for one, the effect of any part of the input on a hidden activation depends on the state of many other parts of the input. Likewise, the role of a hidden unit on downstream representations depends on the activity of many other units. This dependency makes it extremely difficult to understand how DNNs reach their decisions.
+
+To circumvent this problem we here formulate a new DNN architecture that is easier to interpret by design. Our architecture is inspired by bag-of-feature (BoF) models which — alongside extensions such as VLAD encoding or Fisher Vectors — have been the most successful approaches to large-scale object recognition before the advent of deep learning (up to $7 5 \%$ top-5 on ImageNet) and classify images based on the counts, but not the spatial relationships, of a set of local image features. This structure makes the decisions of BoF models particularly easy to explain.
+
+To be concise, throughout this manuscript the concept of interpretability refers to the way in which evidence from small image patches is integrated to reach an image-level decision. While basic BoF models perform just a simple and transparent spatial aggregate of the patch-wise evidences, DNNs non-linearly integrate information across the whole image.
+
+In this paper we show that it is possible to combine the performance and flexibility of DNNs with the interpretability of BoF models, and that the resulting model family (called BagNets) is able to reach high accuracy on ImageNet even if limited to fairly small image patches. Given the simplicity of BoF models we imagine many use cases for which it can be desirable to trade a bit of accuracy for better interpretability, just as this is common e.g. for linear function approximation. This includes diagnosing failure cases (e.g. adversarial examples) or non-iid. settings (e.g. domain transfer), benchmarking diagnostic tools (e.g. attribution methods) or serving as interpretable parts of a computer vision pipeline (e.g. with a relational network on top of the local features).
+
+![](images/9a3c1fbd6ebafa77ccbe8786fbc2fcb2a08df40db7c486389e2dc45c699f1852.jpg)  
+Figure 1: Deep bag-of-features models (BagNets). (A) The models extract features from small image patches which are each fed into a linear classifier yielding one logit heatmap per class. These heatmaps are averaged across space and passed through a softmax to get the final class probabilities. (B) Top-5 ImageNet performance over patch size. (C) Correlation with logits of VGG-16.
+
+In addition, we demonstrate similarities between the decision-making behaviour of BagNets and popular DNNs in computer vision. These similarities suggest that current network architectures base their decisions on a large number of relatively weak and local statistical regularities and are not sufficiently encouraged - either through their architecture, training procedure or task specification - to learn more holistic features that can better appreciate causal relationships between different parts of the image.
+
+# 2 NETWORK ARCHITECTURE
+
+We here recount the main elements of a classic bag-of-features model before introducing the simpler DNN-based BagNets in the next paragraph. Bag-of-feature representations can be described by analogy to bag-of-words representations. With bag-of-words, one counts the number of occurrences of words from a vocabulary in a document. This vocabulary contains important words (but not common ones like "and" or "the") and word clusters (i.e. semantically similar words like "gigantic" and "enormous" are subsumed). The counts of each word in the vocabulary are assembled as one long term vector. This is called the bag-of-words document representation because all ordering of the words is lost. Likewise, bag-of-feature representations are based on a vocabulary of visual words which represent clusters of local image features. The term vector for an image is then simply the number of occurrences of each visual word in the vocabulary. This term vector is used as an input to a classifier (e.g. SVM or MLP). Many successful image classification models have been based on this pipeline (Csurka et al., 2004; Jurie & Triggs, 2005; Zhang et al., 2007; Lazebnik et al., 2006), see O’Hara & Draper (2011) for an up-to-date overview.
+
+BoF models are easy to interpret if the classifier on top of the term vector is linear. In this case the influence of a given part of the input on the classifier is independent of the rest of the input.
+
+Based on this insight we construct a linear DNN-based BoF model as follows (see Figure 1): first, we infer a 2048 dimensional feature representation from each image patch of size ${ \bf q } \times { \bf q }$ pixels using multiple stacked ResNet blocks and apply a linear classifier to infer the class evidence for each patch (heatmaps). We average the class evidence across all patches to infer the image-level class evidence (logits). This structure differs from other ResNets (He et al., 2015) only in the replacement of many $3 \times 3$ by $1 \times 1$ convolutions, thereby limiting the receptive field size of the topmost convolutional layer to ${ \bf q } \times { \bf q }$ pixels (see Appendix for details). There is no explicit assignment to visual words. This could be added through a sparse projection into a high-dimensional embedding but we did not see benefits for interpretability. We denote the resulting architecture as BagNet- $q$ and test $q \in [ 9 , 1 7 , 3 3 ]$ .
+
+![](images/620a51b4630b9fed828a2f7913fd5746930e6e971067629fcf60ac5b55588e4e.jpg)  
+Figure 2: Heatmaps showing the class evidence extracted from of each part of the image. The spatial sum over the evidence is the total class evidence.
+
+Note that an important ingredient of our model is the linear classifier on top of the local feature representation. The word linear here refers to the combination of a linear spatial aggregation (a simple average) and a linear classifier on top of the aggregated features. The fact that the classifier and the spatial aggregation are both linear and thus interchangeable allows us to pinpoint exactly how evidence from local image patches is integrated into one image-level decision.
+
+# 3 RELATED LITERATURE
+
+BoF models and DNNs There are some model architectures that fuse elements from DNNs and BoF models. Predominantly, DNNs were used to replace the previously hand-tuned feature extraction stage in BoF models, often using intermediate or higher layer features of pretrained DNNs (Feng et al., 2017; Gong et al., 2014; $\mathrm { N g }$ et al., 2015; Mohedano et al., 2016; Cao et al., 2017; Khan et al., 2016) for tasks such as image retrieval or geographical scene classification. Other work has explored how well insights from DNN training (e.g. data augmentation) transfer to the training of BoF and Improved Fisher Vector models (Chatfield et al., 2014) and how SIFT and CNN feature descriptions perform (Babenko & Lempitsky, 2015). In contrast, our proposed BoF model architecture is simpler and closer to standard DNNs used for object recognition while still maintaining the interpretability of linear BoF models with local features. Furthermore, to our knowledge this is the first work that explores the relationship between the decision-making of BoF and DNN models.
+
+Interpretable DNNs Our work is most closely related to approaches that use DNNs in conjunction with more interpretable elements. Pinheiro & Collobert (2014) adds explicit labelling of single pixels before the aggregation to an image-level label. The label of each pixel, however, is still inferred from the whole image, making the pixel assignments difficult to interpret. Xiao et al. (2015) proposed a multi-step approach combining object-, part- and domain-detectors to reach a classification decision. In this process object-relevant patches of variable sizes are extracted. In contrast, our approach is much simpler, reaches higher accuracy and is easier to interpret. Besides pixel-level attention-based mechanisms there are several attempts to make the evidence accumulation more interpretable. Hinton et al. (2015) introduced soft decision trees that are trained on the predictions of neural networks. While this increases performance of decision trees, the gap to neural networks on data sets like ImageNet is still large. In Li et al. (2017) an autoencoder architecture is combined with a shallow classifier based on prototype representations. Chen et al. (2018) uses a similar approach but is based on a convolutional architecture to extract class-specific prototype patches. The interpretation of the prototype-based classification, however, is difficult because only the L2 norm between the prototypes and the extracted latent representations is considered1. Finally, the class activation maps by Zhou et al. (2015) share similarities to our approach as they also use a CNN with global average pooling and a linear classifier in order to extract class-specific heatmaps. However, their latent representations are extracted from the whole image and it is unclear how the heatmaps in the latent space are related to the pixel space. In our approach the CNN representations are restricted to very small image patches, making it possible to trace exactly how each image patch contributes the final decision.
+
+![](images/a1f30354250e457abb20b7410bcfdbdd7756899e2a22cbd8cf7ebd9f014b758d.jpg)  
+Figure 3: Most informative image patches for BagNets. For each class (row) and each model (column) we plot two subrows: in the top subrow we show patches that caused the highest logit outputs for the given class across all validation images with that label. Patches in the bottom subrow are selected in the same way but from all validation images with a different label (highlighting errors).
+
+Scattering networks Another related work by Oyallon et al. (2017) uses a scattering network with small receptive fields ( $1 4 \times 1 4$ pixels) in conjunction with a two-layer Multilayer Perceptron or a ResNet-10 on top of the scattering network. This approach reduces the overall depth of the model compared to ResNets (with matched classification accuracy) but does not increase interpretability (because of the non-linear classifier on top of the local scattering features).
+
+A set of superficially similar but unrelated approaches are region proposal models (Wei et al., 2016; Tang et al., 2017; 2016; Arandjelovic et al., 2015). Such models typically use the whole image to infer smaller image regions with relevant objects. These regions are then used to extract a spatially aligned subset of features from the highest DNN layer (so information is still integrated far beyond the proposed image patch). Our approach does not rely on region proposals and extracts features only from small local regions.
+
+# 4 RESULTS
+
+In the first two subsections we investigate the classification performance of BagNets for different patch sizes and demonstrate insights we can derive from its interpretable structure. Thereafter we compare the behaviour of BagNets with several widely used high-performance DNNs (e.g. VGG-16, ResNet-50, DenseNet-169) and show evidence that their decision-making shares many similarities.
+
+# 4.1 ACCURACY & RUNTIME OF BAGNETS ON IMAGENET
+
+We train the BagNets directly on ImageNet (see Appendix for details). Surprisingly, patch sizes as small as $1 7 \times 1 7$ pixels suffice to reach AlexNet (Krizhevsky et al., 2012) performance $8 0 . 5 \%$ top-5 performance) while patches sizes $3 3 \times 3 3$ pixels suffice to reach close to $8 7 . 6 \%$ .
+
+We also compare the runtime of BagNet-q $( q = 3 3 , 1 7 , 9 )$ ) in inference mode with images of size $3 \times 2 2 4 \times 2 2 4$ and batch size 64 against a vanilla ResNet-50. Across all receptive field sizes BagNets reach around 155 images/s for BagNets compared to 570 images/s for ResNet-50. The difference in runtime can be attributed to the reduced amount of downsampling in BagNets compared to ResNet-50.
+
+# 4.2 EXPLAINING DECISIONS
+
+For each ${ \bf q } \times { \bf q }$ patch the model infers evidence for each ImageNet classes, thus yielding a highresolution and very precise heatmap that shows which parts of the image contributes most to certain decisions. We display these heatmaps for the predicted class for ten randomly chosen test images in
+
+Figure 2. Clearly, most evidence lies around the shapes of objects (e.g. the crip or the paddles) or certain predictive image features like the glowing borders of the pumpkin. Also, for animals eyes or legs are important. It’s also notable that background features (like the forest in the deer image) are pretty much ignored by the BagNets.
+
+Next we pick a class and run the BagNets across all validation images to find patches with the most class evidence. Some of these patches are taken from images of that class (i.e. they carry "correct" evidence) while other patches are from images of another class (i.e. these patches can lead to misclassifications). In Figure 3 we show the top-7 patches from both correct and incorrect images for several classes (rows) and different BagNets (columns). This visualisation yields many insights: for example, book jackets are identified mainly by the text on the cover, leading to confusion with other text on t-shirts or websites. Similarly, keys of a typewriter are often interpreted as evidence for handheld computers. The tench class, a large fish species, is often identified by fingers on front of a greenish background. Closer inspection revealed that tench images typically feature the fish hold up like a trophy, thus making the hand and fingers holding it a very predictive image feature. Flamingos are detected by their beaks, which makes them easy to confuse with other birds like storks, while grooms are primarily identified by the transition from suit to neck, an image feature present in many other classes.
+
+In Figure 4 we analyse images misclassified by both BagNet-33 and VGG-16. In the first example the ground-truth class "cleaver" was confused with "granny smith" because of the green cucumber at the top of the image. Looking at the three most predictive patches plotted alongside each heatmap, which show the apple-like edges of the green cucumber pieces, this choice looks comprehensible. Similarly, the local patches in the "thimble" image resemble a gas mask if viewed in isolation. The letters in the "miniskirt" image are very salient, thus leading to the "book jacket" prediction while in the last image the green blanket features a glucamole-like texture.
+
+![](images/5d58a0659f38d3a4bc9e22571a4fc3a87e0b3bb515b7a393e16405df944748e2.jpg)  
+Figure 4: Images misclassified by BagNet33 and VGG-16 with heatmaps for true and predicted label and the most predictive image patches. Class probability reported for BagNet-33 (left) and VGG (right).
+
+# 4.3 COMPARING THE DECISION-MAKING OF BAGNETS AND HIGH-PERFORMANCE DNNS
+
+In the next paragraphs we investigate how similar the decision-making of BagNets is to highperformance DNNs like VGG-16, ResNet-50, ResNet-152 and DenseNet-169. There is no single answer or number, partially because we lack a sensible distance metric between networks. One can compare the pearson correlation between logits (for VGG-16, BagNet-9/17/33 reach $0 . 7 0 / 0 . 7 9 /$ 0.88 respectively, see Figure 1C), but this number can only give a first hint as it does not investigate the specific process that led to the decision. However, the decision-making of BagNets does feature certain key characteristics that we can compare to other models.
+
+![](images/bd3f2546d58d9c3934b258b4f4838bf88e0d2c539ff07b9abc27e0d0f6ec6c63.jpg)  
+Figure 5: Examples of original and texturised images. A vanilla VGG-16 still reaches high accuracy on the texturised images while humans suffer greatly from the loss of global shapes in many images.
+
+![](images/d44c10c2bd91c2bbb034230f864a58f702c0e65d36e7928e301e17f9c405899c.jpg)  
+Figure 6: Interaction of spatially separated image parts. (A) Changes in class-evidence when single image patches are masked (centre) versus change when all patches are masked simultaneously (right). For linear BoF models both terms are the same. (B) Masking regions for different patch sizes. (C) Correlation between both terms for different DNNs over different patch sizes. Interactions are greatly depressed for image features larger than $3 0 \times 3 0 \mathrm { p x }$ .
+
+Image Scrambling One core component of the bag-of-feature networks is the neglect of the spatial relationships between image parts. In other words, scrambling the parts across the image while keeping their counts constant does not change the model decision. Is the same true for current computer vision models like VGG or ResNets? Unfortunately, due to the overlapping receptive fields it is generally not straight-forward to scramble an image in a way that leaves the feature histograms invariant. For VGG-16 an algorithm that comes close to this objective is the popular texture synthesis algorithm based on the Gram features of the hidden layer activations (Gatys et al., 2015), Figure 5. For humans, the scrambling severely increases the difficulty of the task while the performance of VGG-16 is little affected $9 0 . 1 \%$ on clean versus $7 9 . 4 \%$ on texturised image).
+
+This suggests that VGG, in stark contrast to humans, does not rely on global shape integration for perceptual discrimination but rather on statistical regularities in the histogram of local image features. It is well known by practioners that the aforementioned texture synthesis algorithm does not work for ResNet- and DenseNet architectures, the reasons of which are not yet fully understood.
+
+Spatially distinct image manipulations do not interact For BoF models with a linear (but not non-linear!) classifier we do not only expect invariance to the spatial arrangement of image parts, but also that the marginal presence or absence of an image part always has the same effect on the evidence accumulation (i.e. is independent of the rest of the image). In other words, for a BoF model an image with five unconnected wheels (and nothing else) would carry more evidence for class "bike" than a regular photo of a bicycle; a linear BoF model simply ignores whether there is also a frame and a saddle. More precisely, let $\ell _ { \mathrm { m o d e l } } ( \mathbf { x } )$ be the class evidence (logit) as a function of the input $\mathbf { x }$ and let $\delta _ { i }$ be spatially separated and non-overlapping input modifications. For a BagNet-q it holds that
+
+$$
+\ell _ { \mathrm { m o d e l } } ( { \bf x } ) - \ell _ { \mathrm { m o d e l } } ( { \bf x } + \sum _ { i } \delta _ { i } ) = \sum _ { i } \left( \ell _ { \mathrm { m o d e l } } ( { \bf x } ) - \ell _ { \mathrm { m o d e l } } ( { \bf x } + \delta _ { i } ) \right) ,
+$$
+
+as long as the modifications are separated by more than $q$ pixels. We use the Pearson correlation between the LHS and RHS of eq. (1) as a measure of non-linear interactions between image parts. In our experiments we partition the image into a grid of non-overlapping square-sized patches with patch size $q$ . We then replace every second patch in every second row (see Figure 6B) with its DC component (the spatial channel average) both in isolation (RHS of eq. (1)) and in combination (LHS of eq. (1)), see Figure 6A. This ensures that the masked patches are spaced by $q$ pixels and that always around $1 / 4$ of the image is masked. Since most objects fill much of the image, we can expect that the masking will remove many class-predictive image features. We measure the Pearson correlation between the LHS and RHS of eq. 1 for different patch sizes $q$ and DNN models (Figure 6C). The results (Figure 6C) show that VGG-16 exhibits few interactions between image parts spaced by more than 30 pixels. The interactions increase for deeper and more performant architectures.
+
+Error distribution In Figure 7 we plot the top-5 accuracy within each ImageNet class of BagNet33 against the accuracy of regular DNNs. For comparison we also plot VGG-11 against VGG-16. The analysis reveals that the error distribution is fairly consistent between models.
+
+Spatial sensitivity To see whether BagNets and DNNs use similar image parts for image classification we follow Zintgraf et al. (2017) and test how the prediction of DNNs is changing when we mask the most predictive image parts. In Figure 8 (top) we compare the decrease in predicted class probability for an increasing number of masked $8 \times 8$ patches. The masking locations are determined by the heatmaps of BagNets which we compare against random maskings as well as several popular attribution techniques (Baehrens et al., 2010; Sundararajan et al., 2017; Kindermans et al., 2018; Shrikumar et al., 2017) (we use the implementations of DeepExplain (Ancona et al., 2017)) which compute heatmaps directly in the tested models. Notice that these attribution methods have an advantage because they compute heatmaps knowing everything about the models (white-box setting). Nonetheless, the heatmaps from BagNets turn out to be more predictive for class-relevant image parts (see also Table 1). In other words, image parts that are relevant to BagNets are similarly relevant for the classification of normal DNNs. VGG-16 is most affected by the masking of local patches while deeper and more performant architectures are more robust to the relatively small masks, which again suggests that deeper architectures take into account larger spatial relationships.
+
+# 5 DISCUSSION & OUTLOOK
+
+In this paper we introduced and analysed a novel interpretable DNN architecture — coined BagNets — that classifies images based on linear bag-of-local-features representations. The results demonstrate that even complex perceptual tasks like ImageNet can be solved just based on small image features and without any notion of spatial relationships. In addition we showed that the key properties of BagNets, in particlar invariance to spatial relationships as well as weak interactions between image features, are also present to varying degrees in many common computer vision models like ResNet-50
+
+![](images/8304e466b46588fd422f2489d0caf623e2d6bb0bd6eb381d4523c9cfb2d10eb1.jpg)  
+Figure 7: Scatter plots of class-conditional top-5 errors for different models.
+
+![](images/7443e54b4a2b4e2eeb8f7e2493e7c77e6881147c23d09efb6bddc0bc8e424fb5.jpg)  
+Figure 8: Similarity of image features used for object classification. (Top) Decrease of leading class probability in VGG-16, ResNet-50, ResNet-152 and DenseNet-169 if increasingly more patches are masked according to the heatmaps of BagNets and several popular attribution methods. The faster the decrease the more closely does the heatmap highlight image parts relevant for the model decisions making. Image parts relevant to the BagNets turn out to be similarly relevant for all models and outperform post-hoc attribution methods. (Bottom) The first four heatmaps show attributions computed on VGG-16, the other three heatmaps show the class evidence of BagNets.
+
+<table><tr><td rowspan="2"></td><td>Sali- ency</td><td>Int. Grad.</td><td>E-LRP</td><td>Deep LIFT</td><td>BN-9</td><td>BN-17</td><td>BN-33</td></tr><tr><td colspan="4">white-box</td><td></td><td>black-box</td><td></td></tr><tr><td>VGG-16</td><td>0.369</td><td>0.250</td><td>0.326</td><td>0.162</td><td>0.158</td><td>0.151</td><td>0.193</td></tr><tr><td>ResNet-50</td><td>0.528</td><td>0.492</td><td>0.545</td><td>0.379</td><td>0.281</td><td>0.263</td><td>0.291</td></tr><tr><td>ResNet-152</td><td>0.602</td><td>0.580</td><td>0.614</td><td>0.479</td><td>0.394</td><td>0.371</td><td>0.393</td></tr><tr><td>DenseNet-169</td><td>0.589</td><td>0.515</td><td>0.571</td><td>0.423</td><td>0.339</td><td>0.326</td><td>0.359</td></tr></table>
+
+Table 1: Average probability of leading class after masking the 100 patches $8 \times 8$ pixels) with the highest attribution according to different heatmaps (columns).
+
+or VGG-16, suggesting that the decision-making of many DNNs trained on ImageNet follows at least in part a similar bag-of-feature strategy. In contrast to the perceived “leap” in performance from bag-of-feature models to deep neural networks, the representations learnt by DNNs may in the end still be similar to the pre-deep learning era.
+
+VGG-16 is particularly close to bag-of-feature models, as demonstrated by the weak interactions (Figure 6) and the sensitivity to the same small image patches as BagNets (Figure 8). Deeper networks, on the other hand, exhibit stronger nonlinear interactions between image parts and are less sensitive to local maskings. This might explain why texturisation (Figure 5) works well in VGG-16 but fails for ResNet- and DenseNet-architectures.
+
+Clearly, ImageNet alone is not sufficient to force DNNs to learn more physical and causal representation of the world — simply because such a representation is not necessary to solve the task (local image features are enough). This might explain why DNNs generalise poorly to distribution shifts: a DNN trained on natural images has learnt to recognize the textures and local image features associated with different objects (like the fur and eyes of a cat or the keys of a typewriter) and will inevitably fail if presented with cartoon-like images as they lack the key local image features upon which it bases its decisions.
+
+One way forward is to define novel tasks that cannot be solved using local statistical regularities. Here the BagNets can serve as a way to evaluate a lower-bound on the task performance as a function of the observable length-scales. Furthermore, BagNets can be an interesting tool in any application in which it is desirable to trade some accuracy for better interpretability. For example, BagNets can make it much easier to spot the relevant spatial locations and image features that are predictive of certain diseases in medical imaging. Likewise, they can serve as diagnostic tools to benchmark feature attribution techniques since ground-truth attributions are directly available. BagNets can also serve as interpretable parts of a larger computer vision pipeline (e.g. in autonomous cars) as they make it easier to understand edge and failure cases. We released the pretrained BagNets (BagNet-9, BagNet17 and BagNet-33) for PyTorch and Keras at https://github.com/wielandbrendel/ bag-of-local-features-models.
+
+Taken together, DNNs might be more powerful than previous hand-tuned bag-of-feature algorithms in discovering weak statistical regularities, but that does not necessarily mean that they learn substantially different representations. We hope that this work will encourage and inspire future work to adapt tasks, architectures and training algorithms to encourage models to learn more causal models of the world.
+
+# ACKNOWLEDGMENTS
+
+This work has been funded, in part, by the German Research Foundation (DFG CRC 1233 on Robust Vision) as well as by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior / Interior Business Center (DoI/IBC) contract number D16PC00003.
+
+# REFERENCES
+
+Marco Ancona, Enea Ceolini, Cengiz Öztireli, and Markus Gross. Towards better understanding of gradient-based attribution methods for deep neural networks, 2017. URL http://arxiv. org/abs/1711.06104.   
+Relja Arandjelovic, Petr Gronát, Akihiko Torii, Tomás Pajdla, and Josef Sivic. Netvlad: CNN architecture for weakly supervised place recognition. CoRR, abs/1511.07247, 2015. URL http: //arxiv.org/abs/1511.07247.   
+Artem Babenko and Victor S. Lempitsky. Aggregating deep convolutional features for image retrieval. CoRR, abs/1510.07493, 2015. URL http://arxiv.org/abs/1510.07493.   
+David Baehrens, Timon Schroeter, Stefan Harmeling, Motoaki Kawanabe, Katja Hansen, and KlausRobert Müller. How to explain individual classification decisions. Journal of Machine Learning Research, 11:1803–1831, 2010.   
+Jiewei Cao, Zi Huang, and Heng Tao Shen. Local deep descriptors in bag-of-words for image retrieval. In Proceedings of the on Thematic Workshops of ACM Multimedia 2017, Thematic Workshops ’17, pp. 52–58, New York, NY, USA, 2017. ACM. ISBN 978-1-4503-5416-5. doi: 10.1145/3126686.3127018.   
+Ken Chatfield, Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Return of the devil in the details: Delving deep into convolutional nets. In Michel François Valstar, Andrew P. French, and Tony P. Pridmore (eds.), BMVC. BMVA Press, 2014.   
+Chaofan Chen, Oscar Li, Alina Barnett, Jonathan Su, and Cynthia Rudin. This looks like that: deep learning for interpretable image recognition. CoRR, abs/1806.10574, 2018. URL http: //arxiv.org/abs/1806.10574.   
+G. Csurka, C. Bray, C. Dance, and L. Fan. Visual categorization with bags of keypoints. Workshop on Statistical Learning in Computer Vision, ECCV, pp. 1–22, 2004.
+
+Jiangfan Feng, Yuanyuan Liu, and Lin Wu. Bag of visual words model with deep spatial features for geographical scene classification. Comp. Int. and Neurosc., 2017:5169675:1–5169675:14, 2017.
+
+Leon A. Gatys, Alexander S. Ecker, and Matthias Bethge. Texture synthesis and the controlled generation of natural stimuli using convolutional neural networks. CoRR, abs/1505.07376, 2015.
+
+Yunchao Gong, Liwei Wang, Ruiqi Guo, and Svetlana Lazebnik. Multi-scale orderless pooling of deep convolutional activation features. In David J. Fleet, Tomás Pajdla, Bernt Schiele, and Tinne Tuytelaars (eds.), ECCV (7), volume 8695 of Lecture Notes in Computer Science, pp. 392–407. Springer, 2014. ISBN 978-3-319-10583-3.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CoRR, abs/1512.03385, 2015.
+
+Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network, 2015.
+
+Frédéric Jurie and Bill Triggs. Creating efficient codebooks for visual recognition. In ICCV, pp. 604–610, 2005.
+
+Fahad Shahbaz Khan, Joost van de Weijer, Rao Muhammad Anwer, Andrew D. Bagdanov, Michael Felsberg, and Jorma Laaksonen. Scale coding bag of deep features for human attribute and action recognition. CoRR, abs/1612.04884, 2016.
+
+Pieter-Jan Kindermans, Kristof T Schütt, Maximilian Alber, Klaus-Robert Müller, Dumitru Erhan, Been Kim, and Sven Dähne. Learning how to explain neural networks: Patternnet and patternattribution. In 6th International Conference on Learning Representations, 2018.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012.
+
+Svetlana Lazebnik, Cordelia Schmid, and Jean Ponce. Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories. 2006.
+
+Oscar Li, Hao Liu, Chaofan Chen, and Cynthia Rudin. Deep learning for case-based reasoning through prototypes: A neural network that explains its predictions. CoRR, abs/1710.04806, 2017. URL http://arxiv.org/abs/1710.04806.
+
+Eva Mohedano, Kevin McGuinness, Noel E. O’Connor, Amaia Salvador, Ferran Marques, and Xavier Giro-i Nieto. Bags of local convolutional features for scalable instance search. In Proceedings of the 2016 ACM on International Conference on Multimedia Retrieval, ICMR ’16, pp. 327–331, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4359-6. doi: 10.1145/2911996.2912061. URL http://doi.acm.org/10.1145/2911996.2912061.
+
+Joe Yue-Hei Ng, Fan Yang, and Larry S. Davis. Exploiting local features from deep networks for image retrieval. In CVPR Workshops, pp. 53–61. IEEE Computer Society, 2015. ISBN 978-1-4673-6759-2.
+
+Stephen O’Hara and Bruce A. Draper. Introduction to the bag of features paradigm for image classification and retrieval. abs/1101.3354, 2011.
+
+Edouard Oyallon, Eugene Belilovsky, and Sergey Zagoruyko. Scaling the scattering transform: Deep hybrid networks. CoRR, abs/1703.08961, 2017. URL http://arxiv.org/abs/1703. 08961.
+
+Pedro H. O. Pinheiro and Ronan Collobert. Weakly supervised semantic segmentation with convolutional networks. CoRR, abs/1411.6228, 2014. URL http://arxiv.org/abs/1411.6228.
+
+Avanti Shrikumar, Peyton Greenside, and Anshul Kundaje. Learning important features through propagating activation differences. CoRR, abs/1704.02685, 2017.
+
+Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. CoRR, abs/1703.01365, 2017.
+
+Peng Tang, Xinggang Wang, Baoguang Shi, Xiang Bai, Wenyu Liu, and Zhuowen Tu. Deep fishernet for object classification. CoRR, abs/1608.00182, 2016. URL http://arxiv.org/abs/ 1608.00182.
+
+Peng Tang, Xinggang Wang, Zilong Huang, Xiang Bai, and Wenyu Liu. Deep patch learning for weakly supervised object classification and discovery. CoRR, abs/1705.02429, 2017. URL http://arxiv.org/abs/1705.02429.
+
+Yunchao Wei, Wei Xia, Min Lin, Junshi Huang, Bingbing Ni, Jian Dong, Yao Zhao, and Shuicheng Yan. Hcp: A flexible cnn framework for multi-label image classification. IEEE Trans. Pattern Anal. Mach. Intell., 38, 2016.
+
+Tianjun Xiao, Yichong Xu, Kuiyuan Yang, Jiaxing Zhang, Yuxin Peng, and Zheng Zhang. The application of two-level attention models in deep convolutional neural network for fine-grained image classification. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2015, Boston, MA, USA, June 7-12, 2015, pp. 842–850, 2015. doi: 10.1109/CVPR.2015.7298685. URL https://doi.org/10.1109/CVPR.2015.7298685.
+
+Jianguo Zhang, Marcin Marszalek, Svetlana Lazebnik, and Cordelia Schmid. Local features and kernels for classification of texture and object categories: A comprehensive study. International Journal of Computer Vision, 73(2):213–238, 2007.
+
+Bolei Zhou, Aditya Khosla, Àgata Lapedriza, Aude Oliva, and Antonio Torralba. Learning deep features for discriminative localization. CoRR, abs/1512.04150, 2015. URL http://arxiv. org/abs/1512.04150.
+
+Luisa M. Zintgraf, Taco S. Cohen, Tameem Adel, and Max Welling. Visualizing deep neural network decisions: Prediction difference analysis. CoRR, abs/1702.04595, 2017.
+
+# A APPENDIX
+
+The architecture of the BagNets is detailed in Figure A.1. Training of the models was performed in PyTorch using the default ImageNet training script of Torchvision (https://github.com/ pytorch/vision, commit 8a4786a) with default parameters. In brief, we used SGD with momentum (0.9), a batchsize of 256 and an initial learning rate of 0.01 which we decreased by a factor of 10 every 30 epochs. Images were resized to 256 pixels (shortest side) after which we extracted a random crop of size $2 2 4 \times 2 2 4$ pixels.
+
+![](images/9c53f7152e6990458a65e87ad5c359ae15ca5d0e0b25c438c1528e24835abb43.jpg)  
+Figure A.1: The BagNet architecture is almost equivalent to the ResNet-50 architectures except for a few changes in the strides and the replacement of most $3 \times 3$ convolutions with $1 \times 1$ convolutions. Each ResNet block has an expansion of size four (that means the number of output feature maps is four times the number of feature maps within the block). The downsampling operation (dashed arrows) is a simple $1 \times 1$ convolution with stride 2.
+
+![](images/8b5381e745dcf3f8da728b8d6bc244f59c6783e3e6b9f0e7370cda8dcaf957c8.jpg)  
+Figure A.2: Feature attributions of VGG generated using different methods (Saliency, Integrated Gradients, $\epsilon$ -LRP and DeepLIFT) and feature attributions of BagNets.
+
+![](images/51bde508f7f813e57373c54b72efeb20bfd22fe19120973df9551265e81d7574.jpg)  
+Figure A.3: Same as Figure 3 but for more classes.
+
+# A.1 EFFECT OF LOGIT THRESHOLDING
+
+We tested how sensitive the classification accuracy of BagNet-33 is with respect to the exact values of the logits for each patch. To this end we thresholded the logits in two ways: first, by setting all values below the threshold to the threshold (and all values above the threshold stay as is). In the second case we binarized the heatmaps by setting all values below the threshold to zero and all values above the threshold to one (this completely removes the amplitude). The results can be found in Figure A.4. Most interestingly, for certain binarization thresholds the top-5 accuracy is within $3 . 4 \%$ of the vanilla BagNet performance. This indicates that the amplitude of the heatmaps is not decisive.
+
+![](images/918e39879d88aee7c986ff16bba4d3635bc3c78143b8517ac0ed0b904d904931.jpg)  
+Figure A.4: Effect of thresholding the logits on the model performance (top-5 accuracy).

parse/train/SkfMWhAqYQ/SkfMWhAqYQ_content_list.json

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+    {
+        "type": "text",
+        "text": "APPROXIMATING CNNS WITH BAG-OF-LOCALFEATURES MODELS WORKS SURPRISINGLY WELL ON IMAGENET ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Wieland Brendel and Matthias Bethge ",
+        "bbox": [
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+            200,
+            624,
+            215
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Eberhard Karls University of Tübingen, Germany Werner Reichardt Centre for Integrative Neuroscience, Tübingen, Germany Bernstein Center for Computational Neuroscience, Tübingen, Germany {wieland.brendel, matthias.bethge}@bethgelab.org ",
+        "bbox": [
+            254,
+            227,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            306,
+            544,
+            321
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep Neural Networks (DNNs) excel on many complex perceptual tasks but it has proven notoriously difficult to understand how they reach their decisions. We here introduce a high-performance DNN architecture on ImageNet whose decisions are considerably easier to explain. Our model, a simple variant of the ResNet50 architecture called BagNet, classifies an image based on the occurrences of small local image features without taking into account their spatial ordering. This strategy is closely related to the bag-of-feature (BoF) models popular before the onset of deep learning and reaches a surprisingly high accuracy on ImageNet $( 8 7 . 6 \\%$ top-5 for $3 3 \\times 3 3$ px features and Alexnet performance for $1 7 \\times 1 7 \\ : \\mathrm { p x }$ features). The constraint on local features makes it straight-forward to analyse how exactly each part of the image influences the classification. Furthermore, the BagNets behave similar to state-of-the art deep neural networks such as VGG-16, ResNet-152 or DenseNet-169 in terms of feature sensitivity, error distribution and interactions between image parts. This suggests that the improvements of DNNs over previous bag-of-feature classifiers in the last few years is mostly achieved by better fine-tuning rather than by qualitatively different decision strategies. ",
+        "bbox": [
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+            337,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            585,
+            336,
+            601
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A big obstacle in understanding the decision-making of DNNs is due to the complex dependencies between input and hidden activations: for one, the effect of any part of the input on a hidden activation depends on the state of many other parts of the input. Likewise, the role of a hidden unit on downstream representations depends on the activity of many other units. This dependency makes it extremely difficult to understand how DNNs reach their decisions. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To circumvent this problem we here formulate a new DNN architecture that is easier to interpret by design. Our architecture is inspired by bag-of-feature (BoF) models which — alongside extensions such as VLAD encoding or Fisher Vectors — have been the most successful approaches to large-scale object recognition before the advent of deep learning (up to $7 5 \\%$ top-5 on ImageNet) and classify images based on the counts, but not the spatial relationships, of a set of local image features. This structure makes the decisions of BoF models particularly easy to explain. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To be concise, throughout this manuscript the concept of interpretability refers to the way in which evidence from small image patches is integrated to reach an image-level decision. While basic BoF models perform just a simple and transparent spatial aggregate of the patch-wise evidences, DNNs non-linearly integrate information across the whole image. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we show that it is possible to combine the performance and flexibility of DNNs with the interpretability of BoF models, and that the resulting model family (called BagNets) is able to reach high accuracy on ImageNet even if limited to fairly small image patches. Given the simplicity of BoF models we imagine many use cases for which it can be desirable to trade a bit of accuracy for better interpretability, just as this is common e.g. for linear function approximation. This includes diagnosing failure cases (e.g. adversarial examples) or non-iid. settings (e.g. domain transfer), benchmarking diagnostic tools (e.g. attribution methods) or serving as interpretable parts of a computer vision pipeline (e.g. with a relational network on top of the local features). ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/9a3c1fbd6ebafa77ccbe8786fbc2fcb2a08df40db7c486389e2dc45c699f1852.jpg",
+        "image_caption": [
+            "Figure 1: Deep bag-of-features models (BagNets). (A) The models extract features from small image patches which are each fed into a linear classifier yielding one logit heatmap per class. These heatmaps are averaged across space and passed through a softmax to get the final class probabilities. (B) Top-5 ImageNet performance over patch size. (C) Correlation with logits of VGG-16. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In addition, we demonstrate similarities between the decision-making behaviour of BagNets and popular DNNs in computer vision. These similarities suggest that current network architectures base their decisions on a large number of relatively weak and local statistical regularities and are not sufficiently encouraged - either through their architecture, training procedure or task specification - to learn more holistic features that can better appreciate causal relationships between different parts of the image. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 NETWORK ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            546,
+            428,
+            563
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We here recount the main elements of a classic bag-of-features model before introducing the simpler DNN-based BagNets in the next paragraph. Bag-of-feature representations can be described by analogy to bag-of-words representations. With bag-of-words, one counts the number of occurrences of words from a vocabulary in a document. This vocabulary contains important words (but not common ones like \"and\" or \"the\") and word clusters (i.e. semantically similar words like \"gigantic\" and \"enormous\" are subsumed). The counts of each word in the vocabulary are assembled as one long term vector. This is called the bag-of-words document representation because all ordering of the words is lost. Likewise, bag-of-feature representations are based on a vocabulary of visual words which represent clusters of local image features. The term vector for an image is then simply the number of occurrences of each visual word in the vocabulary. This term vector is used as an input to a classifier (e.g. SVM or MLP). Many successful image classification models have been based on this pipeline (Csurka et al., 2004; Jurie & Triggs, 2005; Zhang et al., 2007; Lazebnik et al., 2006), see O’Hara & Draper (2011) for an up-to-date overview. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "BoF models are easy to interpret if the classifier on top of the term vector is linear. In this case the influence of a given part of the input on the classifier is independent of the rest of the input. ",
+        "bbox": [
+            174,
+            763,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Based on this insight we construct a linear DNN-based BoF model as follows (see Figure 1): first, we infer a 2048 dimensional feature representation from each image patch of size ${ \\bf q } \\times { \\bf q }$ pixels using multiple stacked ResNet blocks and apply a linear classifier to infer the class evidence for each patch (heatmaps). We average the class evidence across all patches to infer the image-level class evidence (logits). This structure differs from other ResNets (He et al., 2015) only in the replacement of many $3 \\times 3$ by $1 \\times 1$ convolutions, thereby limiting the receptive field size of the topmost convolutional layer to ${ \\bf q } \\times { \\bf q }$ pixels (see Appendix for details). There is no explicit assignment to visual words. This could be added through a sparse projection into a high-dimensional embedding but we did not see benefits for interpretability. We denote the resulting architecture as BagNet- $q$ and test $q \\in [ 9 , 1 7 , 3 3 ]$ . ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/620a51b4630b9fed828a2f7913fd5746930e6e971067629fcf60ac5b55588e4e.jpg",
+        "image_caption": [
+            "Figure 2: Heatmaps showing the class evidence extracted from of each part of the image. The spatial sum over the evidence is the total class evidence. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            104,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that an important ingredient of our model is the linear classifier on top of the local feature representation. The word linear here refers to the combination of a linear spatial aggregation (a simple average) and a linear classifier on top of the aggregated features. The fact that the classifier and the spatial aggregation are both linear and thus interchangeable allows us to pinpoint exactly how evidence from local image patches is integrated into one image-level decision. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 RELATED LITERATURE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "BoF models and DNNs There are some model architectures that fuse elements from DNNs and BoF models. Predominantly, DNNs were used to replace the previously hand-tuned feature extraction stage in BoF models, often using intermediate or higher layer features of pretrained DNNs (Feng et al., 2017; Gong et al., 2014; $\\mathrm { N g }$ et al., 2015; Mohedano et al., 2016; Cao et al., 2017; Khan et al., 2016) for tasks such as image retrieval or geographical scene classification. Other work has explored how well insights from DNN training (e.g. data augmentation) transfer to the training of BoF and Improved Fisher Vector models (Chatfield et al., 2014) and how SIFT and CNN feature descriptions perform (Babenko & Lempitsky, 2015). In contrast, our proposed BoF model architecture is simpler and closer to standard DNNs used for object recognition while still maintaining the interpretability of linear BoF models with local features. Furthermore, to our knowledge this is the first work that explores the relationship between the decision-making of BoF and DNN models. ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Interpretable DNNs Our work is most closely related to approaches that use DNNs in conjunction with more interpretable elements. Pinheiro & Collobert (2014) adds explicit labelling of single pixels before the aggregation to an image-level label. The label of each pixel, however, is still inferred from the whole image, making the pixel assignments difficult to interpret. Xiao et al. (2015) proposed a multi-step approach combining object-, part- and domain-detectors to reach a classification decision. In this process object-relevant patches of variable sizes are extracted. In contrast, our approach is much simpler, reaches higher accuracy and is easier to interpret. Besides pixel-level attention-based mechanisms there are several attempts to make the evidence accumulation more interpretable. Hinton et al. (2015) introduced soft decision trees that are trained on the predictions of neural networks. While this increases performance of decision trees, the gap to neural networks on data sets like ImageNet is still large. In Li et al. (2017) an autoencoder architecture is combined with a shallow classifier based on prototype representations. Chen et al. (2018) uses a similar approach but is based on a convolutional architecture to extract class-specific prototype patches. The interpretation of the prototype-based classification, however, is difficult because only the L2 norm between the prototypes and the extracted latent representations is considered1. Finally, the class activation maps by Zhou et al. (2015) share similarities to our approach as they also use a CNN with global average pooling and a linear classifier in order to extract class-specific heatmaps. However, their latent representations are extracted from the whole image and it is unclear how the heatmaps in the latent space are related to the pixel space. In our approach the CNN representations are restricted to very small image patches, making it possible to trace exactly how each image patch contributes the final decision. ",
+        "bbox": [
+            173,
+            648,
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+            885
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a1f30354250e457abb20b7410bcfdbdd7756899e2a22cbd8cf7ebd9f014b758d.jpg",
+        "image_caption": [
+            "Figure 3: Most informative image patches for BagNets. For each class (row) and each model (column) we plot two subrows: in the top subrow we show patches that caused the highest logit outputs for the given class across all validation images with that label. Patches in the bottom subrow are selected in the same way but from all validation images with a different label (highlighting errors). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Scattering networks Another related work by Oyallon et al. (2017) uses a scattering network with small receptive fields ( $1 4 \\times 1 4$ pixels) in conjunction with a two-layer Multilayer Perceptron or a ResNet-10 on top of the scattering network. This approach reduces the overall depth of the model compared to ResNets (with matched classification accuracy) but does not increase interpretability (because of the non-linear classifier on top of the local scattering features). ",
+        "bbox": [
+            174,
+            443,
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+            513
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A set of superficially similar but unrelated approaches are region proposal models (Wei et al., 2016; Tang et al., 2017; 2016; Arandjelovic et al., 2015). Such models typically use the whole image to infer smaller image regions with relevant objects. These regions are then used to extract a spatially aligned subset of features from the highest DNN layer (so information is still integrated far beyond the proposed image patch). Our approach does not rely on region proposals and extracts features only from small local regions. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            625,
+            282,
+            641
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the first two subsections we investigate the classification performance of BagNets for different patch sizes and demonstrate insights we can derive from its interpretable structure. Thereafter we compare the behaviour of BagNets with several widely used high-performance DNNs (e.g. VGG-16, ResNet-50, DenseNet-169) and show evidence that their decision-making shares many similarities. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 ACCURACY & RUNTIME OF BAGNETS ON IMAGENET ",
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+        "text": "We train the BagNets directly on ImageNet (see Appendix for details). Surprisingly, patch sizes as small as $1 7 \\times 1 7$ pixels suffice to reach AlexNet (Krizhevsky et al., 2012) performance $8 0 . 5 \\%$ top-5 performance) while patches sizes $3 3 \\times 3 3$ pixels suffice to reach close to $8 7 . 6 \\%$ . ",
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+        "text": "We also compare the runtime of BagNet-q $( q = 3 3 , 1 7 , 9 )$ ) in inference mode with images of size $3 \\times 2 2 4 \\times 2 2 4$ and batch size 64 against a vanilla ResNet-50. Across all receptive field sizes BagNets reach around 155 images/s for BagNets compared to 570 images/s for ResNet-50. The difference in runtime can be attributed to the reduced amount of downsampling in BagNets compared to ResNet-50. ",
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+        "text": "4.2 EXPLAINING DECISIONS ",
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+        "text": "For each ${ \\bf q } \\times { \\bf q }$ patch the model infers evidence for each ImageNet classes, thus yielding a highresolution and very precise heatmap that shows which parts of the image contributes most to certain decisions. We display these heatmaps for the predicted class for ten randomly chosen test images in ",
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+        "text": "Figure 2. Clearly, most evidence lies around the shapes of objects (e.g. the crip or the paddles) or certain predictive image features like the glowing borders of the pumpkin. Also, for animals eyes or legs are important. It’s also notable that background features (like the forest in the deer image) are pretty much ignored by the BagNets. ",
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+        "text": "Next we pick a class and run the BagNets across all validation images to find patches with the most class evidence. Some of these patches are taken from images of that class (i.e. they carry \"correct\" evidence) while other patches are from images of another class (i.e. these patches can lead to misclassifications). In Figure 3 we show the top-7 patches from both correct and incorrect images for several classes (rows) and different BagNets (columns). This visualisation yields many insights: for example, book jackets are identified mainly by the text on the cover, leading to confusion with other text on t-shirts or websites. Similarly, keys of a typewriter are often interpreted as evidence for handheld computers. The tench class, a large fish species, is often identified by fingers on front of a greenish background. Closer inspection revealed that tench images typically feature the fish hold up like a trophy, thus making the hand and fingers holding it a very predictive image feature. Flamingos are detected by their beaks, which makes them easy to confuse with other birds like storks, while grooms are primarily identified by the transition from suit to neck, an image feature present in many other classes. ",
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+        "text": "In Figure 4 we analyse images misclassified by both BagNet-33 and VGG-16. In the first example the ground-truth class \"cleaver\" was confused with \"granny smith\" because of the green cucumber at the top of the image. Looking at the three most predictive patches plotted alongside each heatmap, which show the apple-like edges of the green cucumber pieces, this choice looks comprehensible. Similarly, the local patches in the \"thimble\" image resemble a gas mask if viewed in isolation. The letters in the \"miniskirt\" image are very salient, thus leading to the \"book jacket\" prediction while in the last image the green blanket features a glucamole-like texture. ",
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+        "image_caption": [
+            "Figure 4: Images misclassified by BagNet33 and VGG-16 with heatmaps for true and predicted label and the most predictive image patches. Class probability reported for BagNet-33 (left) and VGG (right). "
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+        "text": "4.3 COMPARING THE DECISION-MAKING OF BAGNETS AND HIGH-PERFORMANCE DNNS ",
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+        "text": "In the next paragraphs we investigate how similar the decision-making of BagNets is to highperformance DNNs like VGG-16, ResNet-50, ResNet-152 and DenseNet-169. There is no single answer or number, partially because we lack a sensible distance metric between networks. One can compare the pearson correlation between logits (for VGG-16, BagNet-9/17/33 reach $0 . 7 0 / 0 . 7 9 /$ 0.88 respectively, see Figure 1C), but this number can only give a first hint as it does not investigate the specific process that led to the decision. However, the decision-making of BagNets does feature certain key characteristics that we can compare to other models. ",
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+        "image_caption": [
+            "Figure 5: Examples of original and texturised images. A vanilla VGG-16 still reaches high accuracy on the texturised images while humans suffer greatly from the loss of global shapes in many images. "
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+        "image_caption": [
+            "Figure 6: Interaction of spatially separated image parts. (A) Changes in class-evidence when single image patches are masked (centre) versus change when all patches are masked simultaneously (right). For linear BoF models both terms are the same. (B) Masking regions for different patch sizes. (C) Correlation between both terms for different DNNs over different patch sizes. Interactions are greatly depressed for image features larger than $3 0 \\times 3 0 \\mathrm { p x }$ . "
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+        "type": "text",
+        "text": "Image Scrambling One core component of the bag-of-feature networks is the neglect of the spatial relationships between image parts. In other words, scrambling the parts across the image while keeping their counts constant does not change the model decision. Is the same true for current computer vision models like VGG or ResNets? Unfortunately, due to the overlapping receptive fields it is generally not straight-forward to scramble an image in a way that leaves the feature histograms invariant. For VGG-16 an algorithm that comes close to this objective is the popular texture synthesis algorithm based on the Gram features of the hidden layer activations (Gatys et al., 2015), Figure 5. For humans, the scrambling severely increases the difficulty of the task while the performance of VGG-16 is little affected $9 0 . 1 \\%$ on clean versus $7 9 . 4 \\%$ on texturised image). ",
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+        "type": "text",
+        "text": "This suggests that VGG, in stark contrast to humans, does not rely on global shape integration for perceptual discrimination but rather on statistical regularities in the histogram of local image features. It is well known by practioners that the aforementioned texture synthesis algorithm does not work for ResNet- and DenseNet architectures, the reasons of which are not yet fully understood. ",
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+        "text": "Spatially distinct image manipulations do not interact For BoF models with a linear (but not non-linear!) classifier we do not only expect invariance to the spatial arrangement of image parts, but also that the marginal presence or absence of an image part always has the same effect on the evidence accumulation (i.e. is independent of the rest of the image). In other words, for a BoF model an image with five unconnected wheels (and nothing else) would carry more evidence for class \"bike\" than a regular photo of a bicycle; a linear BoF model simply ignores whether there is also a frame and a saddle. More precisely, let $\\ell _ { \\mathrm { m o d e l } } ( \\mathbf { x } )$ be the class evidence (logit) as a function of the input $\\mathbf { x }$ and let $\\delta _ { i }$ be spatially separated and non-overlapping input modifications. For a BagNet-q it holds that ",
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+    {
+        "type": "equation",
+        "img_path": "images/6f0c28ae44525033f2e483192db2da5c8d3da803c9baab41e02ff098dbcbcd7e.jpg",
+        "text": "$$\n\\ell _ { \\mathrm { m o d e l } } ( { \\bf x } ) - \\ell _ { \\mathrm { m o d e l } } ( { \\bf x } + \\sum _ { i } \\delta _ { i } ) = \\sum _ { i } \\left( \\ell _ { \\mathrm { m o d e l } } ( { \\bf x } ) - \\ell _ { \\mathrm { m o d e l } } ( { \\bf x } + \\delta _ { i } ) \\right) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "as long as the modifications are separated by more than $q$ pixels. We use the Pearson correlation between the LHS and RHS of eq. (1) as a measure of non-linear interactions between image parts. In our experiments we partition the image into a grid of non-overlapping square-sized patches with patch size $q$ . We then replace every second patch in every second row (see Figure 6B) with its DC component (the spatial channel average) both in isolation (RHS of eq. (1)) and in combination (LHS of eq. (1)), see Figure 6A. This ensures that the masked patches are spaced by $q$ pixels and that always around $1 / 4$ of the image is masked. Since most objects fill much of the image, we can expect that the masking will remove many class-predictive image features. We measure the Pearson correlation between the LHS and RHS of eq. 1 for different patch sizes $q$ and DNN models (Figure 6C). The results (Figure 6C) show that VGG-16 exhibits few interactions between image parts spaced by more than 30 pixels. The interactions increase for deeper and more performant architectures. ",
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+        "text": "Error distribution In Figure 7 we plot the top-5 accuracy within each ImageNet class of BagNet33 against the accuracy of regular DNNs. For comparison we also plot VGG-11 against VGG-16. The analysis reveals that the error distribution is fairly consistent between models. ",
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+        "type": "text",
+        "text": "Spatial sensitivity To see whether BagNets and DNNs use similar image parts for image classification we follow Zintgraf et al. (2017) and test how the prediction of DNNs is changing when we mask the most predictive image parts. In Figure 8 (top) we compare the decrease in predicted class probability for an increasing number of masked $8 \\times 8$ patches. The masking locations are determined by the heatmaps of BagNets which we compare against random maskings as well as several popular attribution techniques (Baehrens et al., 2010; Sundararajan et al., 2017; Kindermans et al., 2018; Shrikumar et al., 2017) (we use the implementations of DeepExplain (Ancona et al., 2017)) which compute heatmaps directly in the tested models. Notice that these attribution methods have an advantage because they compute heatmaps knowing everything about the models (white-box setting). Nonetheless, the heatmaps from BagNets turn out to be more predictive for class-relevant image parts (see also Table 1). In other words, image parts that are relevant to BagNets are similarly relevant for the classification of normal DNNs. VGG-16 is most affected by the masking of local patches while deeper and more performant architectures are more robust to the relatively small masks, which again suggests that deeper architectures take into account larger spatial relationships. ",
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+        "type": "text",
+        "text": "5 DISCUSSION & OUTLOOK ",
+        "text_level": 1,
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+        "text": "In this paper we introduced and analysed a novel interpretable DNN architecture — coined BagNets — that classifies images based on linear bag-of-local-features representations. The results demonstrate that even complex perceptual tasks like ImageNet can be solved just based on small image features and without any notion of spatial relationships. In addition we showed that the key properties of BagNets, in particlar invariance to spatial relationships as well as weak interactions between image features, are also present to varying degrees in many common computer vision models like ResNet-50 ",
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+        "img_path": "images/8304e466b46588fd422f2489d0caf623e2d6bb0bd6eb381d4523c9cfb2d10eb1.jpg",
+        "image_caption": [
+            "Figure 7: Scatter plots of class-conditional top-5 errors for different models. "
+        ],
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+        "img_path": "images/7443e54b4a2b4e2eeb8f7e2493e7c77e6881147c23d09efb6bddc0bc8e424fb5.jpg",
+        "image_caption": [
+            "Figure 8: Similarity of image features used for object classification. (Top) Decrease of leading class probability in VGG-16, ResNet-50, ResNet-152 and DenseNet-169 if increasingly more patches are masked according to the heatmaps of BagNets and several popular attribution methods. The faster the decrease the more closely does the heatmap highlight image parts relevant for the model decisions making. Image parts relevant to the BagNets turn out to be similarly relevant for all models and outperform post-hoc attribution methods. (Bottom) The first four heatmaps show attributions computed on VGG-16, the other three heatmaps show the class evidence of BagNets. "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/fcc638f67104ee1ec77e8fae3261f61864bfe2dedd5d558f8e49a47f8ff335d1.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td>Sali- ency</td><td>Int. Grad.</td><td>E-LRP</td><td>Deep LIFT</td><td>BN-9</td><td>BN-17</td><td>BN-33</td></tr><tr><td colspan=\"4\">white-box</td><td></td><td>black-box</td><td></td></tr><tr><td>VGG-16</td><td>0.369</td><td>0.250</td><td>0.326</td><td>0.162</td><td>0.158</td><td>0.151</td><td>0.193</td></tr><tr><td>ResNet-50</td><td>0.528</td><td>0.492</td><td>0.545</td><td>0.379</td><td>0.281</td><td>0.263</td><td>0.291</td></tr><tr><td>ResNet-152</td><td>0.602</td><td>0.580</td><td>0.614</td><td>0.479</td><td>0.394</td><td>0.371</td><td>0.393</td></tr><tr><td>DenseNet-169</td><td>0.589</td><td>0.515</td><td>0.571</td><td>0.423</td><td>0.339</td><td>0.326</td><td>0.359</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Average probability of leading class after masking the 100 patches $8 \\times 8$ pixels) with the highest attribution according to different heatmaps (columns). ",
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+    {
+        "type": "text",
+        "text": "or VGG-16, suggesting that the decision-making of many DNNs trained on ImageNet follows at least in part a similar bag-of-feature strategy. In contrast to the perceived “leap” in performance from bag-of-feature models to deep neural networks, the representations learnt by DNNs may in the end still be similar to the pre-deep learning era. ",
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+    {
+        "type": "text",
+        "text": "VGG-16 is particularly close to bag-of-feature models, as demonstrated by the weak interactions (Figure 6) and the sensitivity to the same small image patches as BagNets (Figure 8). Deeper networks, on the other hand, exhibit stronger nonlinear interactions between image parts and are less sensitive to local maskings. This might explain why texturisation (Figure 5) works well in VGG-16 but fails for ResNet- and DenseNet-architectures. ",
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+        "type": "text",
+        "text": "Clearly, ImageNet alone is not sufficient to force DNNs to learn more physical and causal representation of the world — simply because such a representation is not necessary to solve the task (local image features are enough). This might explain why DNNs generalise poorly to distribution shifts: a DNN trained on natural images has learnt to recognize the textures and local image features associated with different objects (like the fur and eyes of a cat or the keys of a typewriter) and will inevitably fail if presented with cartoon-like images as they lack the key local image features upon which it bases its decisions. ",
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+    {
+        "type": "text",
+        "text": "One way forward is to define novel tasks that cannot be solved using local statistical regularities. Here the BagNets can serve as a way to evaluate a lower-bound on the task performance as a function of the observable length-scales. Furthermore, BagNets can be an interesting tool in any application in which it is desirable to trade some accuracy for better interpretability. For example, BagNets can make it much easier to spot the relevant spatial locations and image features that are predictive of certain diseases in medical imaging. Likewise, they can serve as diagnostic tools to benchmark feature attribution techniques since ground-truth attributions are directly available. BagNets can also serve as interpretable parts of a larger computer vision pipeline (e.g. in autonomous cars) as they make it easier to understand edge and failure cases. We released the pretrained BagNets (BagNet-9, BagNet17 and BagNet-33) for PyTorch and Keras at https://github.com/wielandbrendel/ bag-of-local-features-models. ",
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+    {
+        "type": "text",
+        "text": "Taken together, DNNs might be more powerful than previous hand-tuned bag-of-feature algorithms in discovering weak statistical regularities, but that does not necessarily mean that they learn substantially different representations. We hope that this work will encourage and inspire future work to adapt tasks, architectures and training algorithms to encourage models to learn more causal models of the world. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "This work has been funded, in part, by the German Research Foundation (DFG CRC 1233 on Robust Vision) as well as by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior / Interior Business Center (DoI/IBC) contract number D16PC00003. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            489,
+            285,
+            503
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Marco Ancona, Enea Ceolini, Cengiz Öztireli, and Markus Gross. Towards better understanding of gradient-based attribution methods for deep neural networks, 2017. URL http://arxiv. org/abs/1711.06104.   \nRelja Arandjelovic, Petr Gronát, Akihiko Torii, Tomás Pajdla, and Josef Sivic. Netvlad: CNN architecture for weakly supervised place recognition. CoRR, abs/1511.07247, 2015. URL http: //arxiv.org/abs/1511.07247.   \nArtem Babenko and Victor S. Lempitsky. Aggregating deep convolutional features for image retrieval. CoRR, abs/1510.07493, 2015. URL http://arxiv.org/abs/1510.07493.   \nDavid Baehrens, Timon Schroeter, Stefan Harmeling, Motoaki Kawanabe, Katja Hansen, and KlausRobert Müller. How to explain individual classification decisions. Journal of Machine Learning Research, 11:1803–1831, 2010.   \nJiewei Cao, Zi Huang, and Heng Tao Shen. Local deep descriptors in bag-of-words for image retrieval. In Proceedings of the on Thematic Workshops of ACM Multimedia 2017, Thematic Workshops ’17, pp. 52–58, New York, NY, USA, 2017. ACM. ISBN 978-1-4503-5416-5. doi: 10.1145/3126686.3127018.   \nKen Chatfield, Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Return of the devil in the details: Delving deep into convolutional nets. In Michel François Valstar, Andrew P. French, and Tony P. Pridmore (eds.), BMVC. BMVA Press, 2014.   \nChaofan Chen, Oscar Li, Alina Barnett, Jonathan Su, and Cynthia Rudin. This looks like that: deep learning for interpretable image recognition. CoRR, abs/1806.10574, 2018. URL http: //arxiv.org/abs/1806.10574.   \nG. Csurka, C. Bray, C. Dance, and L. Fan. Visual categorization with bags of keypoints. Workshop on Statistical Learning in Computer Vision, ECCV, pp. 1–22, 2004. ",
+        "bbox": [
+            171,
+            513,
+            828,
+            929
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jiangfan Feng, Yuanyuan Liu, and Lin Wu. Bag of visual words model with deep spatial features for geographical scene classification. Comp. Int. and Neurosc., 2017:5169675:1–5169675:14, 2017. ",
+        "bbox": [
+            171,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Leon A. Gatys, Alexander S. Ecker, and Matthias Bethge. Texture synthesis and the controlled generation of natural stimuli using convolutional neural networks. CoRR, abs/1505.07376, 2015. ",
+        "bbox": [
+            171,
+            141,
+            823,
+            171
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yunchao Gong, Liwei Wang, Ruiqi Guo, and Svetlana Lazebnik. Multi-scale orderless pooling of deep convolutional activation features. In David J. Fleet, Tomás Pajdla, Bernt Schiele, and Tinne Tuytelaars (eds.), ECCV (7), volume 8695 of Lecture Notes in Computer Science, pp. 392–407. Springer, 2014. ISBN 978-3-319-10583-3. ",
+        "bbox": [
+            174,
+            179,
+            826,
+            237
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CoRR, abs/1512.03385, 2015. ",
+        "bbox": [
+            173,
+            246,
+            821,
+            275
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network, 2015. ",
+        "bbox": [
+            173,
+            284,
+            821,
+            300
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Frédéric Jurie and Bill Triggs. Creating efficient codebooks for visual recognition. In ICCV, pp. 604–610, 2005. ",
+        "bbox": [
+            173,
+            309,
+            825,
+            338
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fahad Shahbaz Khan, Joost van de Weijer, Rao Muhammad Anwer, Andrew D. Bagdanov, Michael Felsberg, and Jorma Laaksonen. Scale coding bag of deep features for human attribute and action recognition. CoRR, abs/1612.04884, 2016. ",
+        "bbox": [
+            174,
+            347,
+            826,
+            390
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Pieter-Jan Kindermans, Kristof T Schütt, Maximilian Alber, Klaus-Robert Müller, Dumitru Erhan, Been Kim, and Sven Dähne. Learning how to explain neural networks: Patternnet and patternattribution. In 6th International Conference on Learning Representations, 2018. ",
+        "bbox": [
+            173,
+            400,
+            826,
+            443
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012. ",
+        "bbox": [
+            176,
+            452,
+            825,
+            494
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Svetlana Lazebnik, Cordelia Schmid, and Jean Ponce. Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories. 2006. ",
+        "bbox": [
+            171,
+            505,
+            825,
+            534
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Oscar Li, Hao Liu, Chaofan Chen, and Cynthia Rudin. Deep learning for case-based reasoning through prototypes: A neural network that explains its predictions. CoRR, abs/1710.04806, 2017. URL http://arxiv.org/abs/1710.04806. ",
+        "bbox": [
+            176,
+            542,
+            825,
+            585
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Eva Mohedano, Kevin McGuinness, Noel E. O’Connor, Amaia Salvador, Ferran Marques, and Xavier Giro-i Nieto. Bags of local convolutional features for scalable instance search. In Proceedings of the 2016 ACM on International Conference on Multimedia Retrieval, ICMR ’16, pp. 327–331, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4359-6. doi: 10.1145/2911996.2912061. URL http://doi.acm.org/10.1145/2911996.2912061. ",
+        "bbox": [
+            173,
+            594,
+            826,
+            666
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Joe Yue-Hei Ng, Fan Yang, and Larry S. Davis. Exploiting local features from deep networks for image retrieval. In CVPR Workshops, pp. 53–61. IEEE Computer Society, 2015. ISBN 978-1-4673-6759-2. ",
+        "bbox": [
+            173,
+            675,
+            826,
+            718
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Stephen O’Hara and Bruce A. Draper. Introduction to the bag of features paradigm for image classification and retrieval. abs/1101.3354, 2011. ",
+        "bbox": [
+            169,
+            727,
+            825,
+            757
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Edouard Oyallon, Eugene Belilovsky, and Sergey Zagoruyko. Scaling the scattering transform: Deep hybrid networks. CoRR, abs/1703.08961, 2017. URL http://arxiv.org/abs/1703. 08961. ",
+        "bbox": [
+            176,
+            765,
+            826,
+            809
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Pedro H. O. Pinheiro and Ronan Collobert. Weakly supervised semantic segmentation with convolutional networks. CoRR, abs/1411.6228, 2014. URL http://arxiv.org/abs/1411.6228. ",
+        "bbox": [
+            169,
+            818,
+            826,
+            847
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Avanti Shrikumar, Peyton Greenside, and Anshul Kundaje. Learning important features through propagating activation differences. CoRR, abs/1704.02685, 2017. ",
+        "bbox": [
+            173,
+            856,
+            823,
+            886
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. CoRR, abs/1703.01365, 2017. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Peng Tang, Xinggang Wang, Baoguang Shi, Xiang Bai, Wenyu Liu, and Zhuowen Tu. Deep fishernet for object classification. CoRR, abs/1608.00182, 2016. URL http://arxiv.org/abs/ 1608.00182. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Peng Tang, Xinggang Wang, Zilong Huang, Xiang Bai, and Wenyu Liu. Deep patch learning for weakly supervised object classification and discovery. CoRR, abs/1705.02429, 2017. URL http://arxiv.org/abs/1705.02429. ",
+        "bbox": [
+            176,
+            155,
+            823,
+            196
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yunchao Wei, Wei Xia, Min Lin, Junshi Huang, Bingbing Ni, Jian Dong, Yao Zhao, and Shuicheng Yan. Hcp: A flexible cnn framework for multi-label image classification. IEEE Trans. Pattern Anal. Mach. Intell., 38, 2016. ",
+        "bbox": [
+            178,
+            207,
+            823,
+            250
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Tianjun Xiao, Yichong Xu, Kuiyuan Yang, Jiaxing Zhang, Yuxin Peng, and Zheng Zhang. The application of two-level attention models in deep convolutional neural network for fine-grained image classification. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2015, Boston, MA, USA, June 7-12, 2015, pp. 842–850, 2015. doi: 10.1109/CVPR.2015.7298685. URL https://doi.org/10.1109/CVPR.2015.7298685. ",
+        "bbox": [
+            173,
+            257,
+            828,
+            328
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jianguo Zhang, Marcin Marszalek, Svetlana Lazebnik, and Cordelia Schmid. Local features and kernels for classification of texture and object categories: A comprehensive study. International Journal of Computer Vision, 73(2):213–238, 2007. ",
+        "bbox": [
+            173,
+            337,
+            823,
+            381
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Bolei Zhou, Aditya Khosla, Àgata Lapedriza, Aude Oliva, and Antonio Torralba. Learning deep features for discriminative localization. CoRR, abs/1512.04150, 2015. URL http://arxiv. org/abs/1512.04150. ",
+        "bbox": [
+            171,
+            388,
+            825,
+            433
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Luisa M. Zintgraf, Taco S. Cohen, Tameem Adel, and Max Welling. Visualizing deep neural network decisions: Prediction difference analysis. CoRR, abs/1702.04595, 2017. ",
+        "bbox": [
+            171,
+            443,
+            825,
+            470
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The architecture of the BagNets is detailed in Figure A.1. Training of the models was performed in PyTorch using the default ImageNet training script of Torchvision (https://github.com/ pytorch/vision, commit 8a4786a) with default parameters. In brief, we used SGD with momentum (0.9), a batchsize of 256 and an initial learning rate of 0.01 which we decreased by a factor of 10 every 30 epochs. Images were resized to 256 pixels (shortest side) after which we extracted a random crop of size $2 2 4 \\times 2 2 4$ pixels. ",
+        "bbox": [
+            173,
+            136,
+            826,
+            219
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/9c53f7152e6990458a65e87ad5c359ae15ca5d0e0b25c438c1528e24835abb43.jpg",
+        "image_caption": [
+            "Figure A.1: The BagNet architecture is almost equivalent to the ResNet-50 architectures except for a few changes in the strides and the replacement of most $3 \\times 3$ convolutions with $1 \\times 1$ convolutions. Each ResNet block has an expansion of size four (that means the number of output feature maps is four times the number of feature maps within the block). The downsampling operation (dashed arrows) is a simple $1 \\times 1$ convolution with stride 2. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            328,
+            234,
+            676,
+            816
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/8b5381e745dcf3f8da728b8d6bc244f59c6783e3e6b9f0e7370cda8dcaf957c8.jpg",
+        "image_caption": [
+            "Figure A.2: Feature attributions of VGG generated using different methods (Saliency, Integrated Gradients, $\\epsilon$ -LRP and DeepLIFT) and feature attributions of BagNets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            90,
+            815,
+            890
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/51bde508f7f813e57373c54b72efeb20bfd22fe19120973df9551265e81d7574.jpg",
+        "image_caption": [
+            "Figure A.3: Same as Figure 3 but for more classes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            70,
+            821,
+            905
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.1 EFFECT OF LOGIT THRESHOLDING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            104,
+            455,
+            117
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We tested how sensitive the classification accuracy of BagNet-33 is with respect to the exact values of the logits for each patch. To this end we thresholded the logits in two ways: first, by setting all values below the threshold to the threshold (and all values above the threshold stay as is). In the second case we binarized the heatmaps by setting all values below the threshold to zero and all values above the threshold to one (this completely removes the amplitude). The results can be found in Figure A.4. Most interestingly, for certain binarization thresholds the top-5 accuracy is within $3 . 4 \\%$ of the vanilla BagNet performance. This indicates that the amplitude of the heatmaps is not decisive. ",
+        "bbox": [
+            173,
+            125,
+            825,
+            223
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/918e39879d88aee7c986ff16bba4d3635bc3c78143b8517ac0ed0b904d904931.jpg",
+        "image_caption": [
+            "Figure A.4: Effect of thresholding the logits on the model performance (top-5 accuracy). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            236,
+            818,
+            467
+        ],
+        "page_idx": 14
+    }
+]

parse/train/SyAbZb-0Z/SyAbZb-0Z.md

@@ -0,0 +1,218 @@
+# TRANSFER LEARNING TO LEARN WITH MULTITASK NEURAL MODEL SEARCH
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Deep learning models require extensive architecture design exploration and hyperparameter optimization to perform well on a given task. The exploration of the model design space is often made by a human expert, and optimized using a combination of grid search and search heuristics over a large space of possible choices. Neural Architecture Search (NAS) is a Reinforcement Learning approach that has been proposed to automate architecture design. NAS has been successfully applied to generate Neural Networks that rival the best human-designed architectures. However, NAS requires sampling, constructing, and training hundreds to thousands of models to achieve well-performing architectures. This procedure needs to be executed from scratch for each new task. The application of NAS to a wide set of tasks currently lacks a way to transfer generalizable knowledge across tasks.
+
+In this paper, we present the Multitask Neural Model Search (MNMS) controller. Our goal is to learn a generalizable framework that can condition model construction on successful model searches for previously seen tasks, thus significantly speeding up the search for new tasks. We demonstrate that MNMS can conduct an automated architecture search for multiple tasks simultaneously while still learning well-performing, specialized models for each task. We then show that pre-trained MNMS controllers can transfer learning to new tasks. By leveraging knowledge from previous searches, we find that pre-trained MNMS models start from a better location in the search space and reduce search time on unseen tasks, while still discovering models that outperform published human-designed models.
+
+# 1 INTRODUCTION
+
+Designing deep learning models that work well for a task requires an extensive process of iterative architecture engineering and tuning. These design decisions are largely made by human experts guided by a combination of intuition, grid search, and search heuristics.
+
+Meta-learning aims to automate model design by using machine learning to discover good architecture and hyperparameter choices. Recent advances in meta-learning using Reinforcement Learning (RL) have made promising strides towards accelerating or even eliminating the manual parameter search. For example, Neural Architecture Search (NAS) has successfully discovered novel network architectures that rival or surpass the best human-designed architectures on challenging benchmark image recognition tasks (Zoph & Le, 2017; Zoph et al., 2017). However, naively applying reinforcement learning to each new task for automated model construction requires sampling, constructing, and training hundreds to thousands of networks to relearn how to generate models from scratch. Human experts, on the other hand, can design and tune networks based on knowledge about underlying dependencies in the search space and experience with prior tasks. We therefore aim to automatically learn and leverage the same information.
+
+In this paper, we present Multitask Neural Model Search (MNMS), an automated model construction framework that finds the best performing models in the search space for multiple tasks simultaneously. We then show that a MNMS framework that has been pre-trained on previous tasks can construct the best performing model for entirely new tasks in significantly less time.
+
+# 2 RELATED WORK
+
+The Neural Architecture Search (NAS) method was introduced in (Zoph & Le, 2017), where it was applied to construct Convolutional Neural Networks (CNNs) for the CIFAR-10 task and Recurrent Neural Networks (RNNs) for the Penn Treebank tasks. Later work by the same authors attempted to address the computational cost of using Neural Architecture Search for more challenging tasks (Zoph et al., 2017). To engineer a convolutional architecture for ImageNet classification, this paper demonstrated that it was possible to train the NAS controller on the simpler, proxy CIFAR-10 task and then transfer the architecture to ImageNet classification by stacking it. However, this work did not attempt to transfer learn the NAS controller itself across multiple tasks, relying instead on the human expert intuition that additional network depth was necessary for the more challenging classification task. Additionally, the final generated architectures required additional tuning, to choose hyperparameters such as the learning rate, before evaluation on the test set.
+
+The complexity of model engineering in machine learning is widely recognized. Optimization methods have been proposed, ranging from random search over the space of possible architectures (Bergstra & Bengio, 2012) to parameter modeling (Bergstra et al., 2013). Recent publications apply RL to automate architecture generation. These include MetaQNN, a Q-learning algorithm that sequentially chooses CNN layers (Baker et al., 2016). MetaQNN uses an aggressive exploration to reduce search time, though it can cause the resulting architectures to underperform. Cai et al. (2017) also propose an RL agent that transforms existing architectures incrementally to avoid generating entire networks from scratch. More recently, an emerging body of modern neuro-evolution research has adapted genetic algorithms as an alternate optimization method for these complex searches (Conti et al., 2017).
+
+Our work also draws on prior research in transfer learning and simultaneous multitask training. Transfer learning has been shown to achieve excellent results as an initialization method for deep networks, including for models trained using RL (Yosinski et al., 2014; Sharif Razavian et al., 2014; Zhan & Taylor, 2015). Simultaneous multitask training can also facilitate learning between tasks with a common structure, though effectively retaining knowledge across tasks is still an active area of research Kirkpatrick et al. (2017); Teh et al. (2017).
+
+# 3 METHODS
+
+# 3.1 NEURAL ARCHITECTURE SEARCH OVERVIEW
+
+![](images/13eb34e98f4eee0a727b484955f7bbe7b852ca7a655b6318a00acd02b07cad6c.jpg)  
+Figure 1: The base Neural Architecture Search framework.
+
+Neural Architecture Search uses an RNN to generate model designs that maximize expected performance on a given task (Figure 1) (Zoph & Le, 2017). Specifically, an RNN controller iteratively samples architectures as a sequence of actions. Every action is a discretized design choice, such as CNN filter heights, widths, and strides. Child networks are then constructed with these architectures and trained to convergence. The performance metric of the child network is used as a reward to update the controller through a policy gradient algorithm. The controller learns a distribution over the architecture search space that is updated to increase the probability of the best performing architectures, allowing it to sample better architectures over time. The original Neural Architecture Search framework sampled neural network models over a search space of strictly architectural parameters, by generating descriptions of each layer in the network at a time. This framework was used to successfully specify a convolutional neural network architecture for image classification, and a recurrent network cell for language tasks (Zoph & Le, 2017). Later work has shown that this framework can be extended to automatically search over other model design parameters and domains, such as update rules for network optimizers (Bello et al., 2017).
+
+# 3.2 SIMULTANEOUS MULTITASK TRAINING IN NEURAL ARCHITECTURE SEARCH
+
+In this section, we describe the Multitask Neural Model Search (MNMS) controller, which allows simultaneous model search over multiple different tasks. Many deep learning models require the same common design decisions, such as choice of network depth, learning rate, and number of training iterations; using a generally defined search space of widely applicable architecture and hyperparameter choices, the controller can therefore engineer a wide range of models applicable to many common machine learning tasks. Multitask training over this space can then allow the controller to learn more broadly applicable relationships between search space actions, by leveraging shared behavior across tasks.
+
+We implement a controller capable of simultaneous multitask training through three key modifications:
+
+# 1. Learned task representation and task conditioning.
+
+The MNMS controller can be trained synchronously on a set of N tasks. The controller learns to build differentiated architectures for each task. This is achieved by sampling a task uniformly at the beginning of each controller training iteration. The task is then mapped to a unique embedding vector. The tasks embeddings are randomly initialized, and are trained jointly with the controller. The task embedding is then used to condition the model construction on the task. This is achieved by concatenating the task embedding to every input that is fed to the controller RNN.
+
+Specifically, in single-task NAS, the controller RNN generates an output at each timestep that determines the distribution over the current set of actions. An action is sampled according to this action distribution and then embedded. The action embedding is then passed back into the RNN as input to the next timestep.
+
+Here, in multi-task training for MNMS, for multi-task training, the task embedding is now concatenated with the action embedding to form the RNN input, allowing it to condition each action on a specific task (Figure 2).
+
+# 2. Off-policy training using multitask replay.
+
+Previous works train the NAS controller using the REINFORCE policy gradient algorithm (Zoph & Le, 2017), and more recently, the PPO algorithm (Zoph et al., 2017; Bello et al., 2017).
+
+Here, we train using off-policy PPO, an actor-critic algorithm in which an actor controller generates sampled models and a critic controller trains on a replay bank of the sampled models and rewards (Schulman et al., 2017). Preliminary experiments with on-policy training found that the controller shows a reduced ability to learn a differentiated model for each task. Specifically, the controller is prone to premature convergence to a single model design that works generally well for all tasks but is not the best model for some of the tasks.
+
+On-policy sampling is biased toward more recent predictions of the optimal parameter distribution. Our hypothesis is that in multitask training, on-policy sampling can prematurely reduce exploration of better parameters for each individual task, while off-policy training allows the actor controller to continue to explore separate parameter choices for each task, and better learn a differentiated distribution over the parameter search space to maximize expected performance for each.
+
+# 3. Per-task baseline and reward distributions normalization.
+
+Each task can define a different performance metric to be used as reward. The rewards affect the amplitude of the updates on the controller, so we need to make sure that the distributions of each task rewards are aligned to have same mean and similar variance.
+
+![](images/a9ed4b597f20ad0feae68f43ad9c748b3099e5814ec74d1b7c18c60a272f77d0.jpg)  
+Figure 2: Overview of the multitask controller RNN. (1) A task embedding table is maintained and updated with controller gradients to learn differentiated task embeddings over time. (2) At each iteration of the multitask training, a task is randomly sampled. The task embedding is passed into the controller RNN along with the sampled action embedding at each RNN timestep. The full sequence of output actions defines the child architecture trained on the chosen task.
+
+The mean of each tasks reward distribution is aligned to 0 by scaling the gradients with the advantage instead of the reward. The advantage, $A ( \boldsymbol { a } , t )$ , of a given model, $a$ , applied to a task, $t$ , is defined as the difference between the reward, $R ( \boldsymbol { a } , t )$ , and the expected reward for the given task, $b ( t )$ :
+
+$$
+A ( a , t ) = R ( a , t ) - b ( t )
+$$
+
+$b ( t )$ is often referred to as the baseline. This is a standard RL technique that is usually applied with the aim of increasing training stability. During multitask training, the baseline is conditioned on the sampled task. We keep track of a separate baseline for each task, computed as an exponential moving average of the rewards recorded for each task.
+
+The range of each tasks reward distribution is normalized by dividing the advantage by the baseline:
+
+$$
+A ^ { \prime } ( a , t ) = { \frac { R ( a , t ) - b ( t ) } { b ( t ) } }
+$$
+
+We refer to $A ^ { \prime }$ as the normalized advantage. Notice that the division by the baseline does not compromise the convergence criteria, as it can be seen as using a distinct adaptive learning rate for each task.
+
+Using the normalized advantage to scale the gradients instead of the raw reward allows MNMS to use any performance metric as a reward even when training on multiple tasks.
+
+# 3.3 TRANSFER LEARNING FOR AUTOMATED MODEL SEARCH
+
+Using the multitask framework, we can transfer learn pretrained controllers by simply reusing the weights of the pretrained controller and adding a randomly initialized task embedding for each new task. The controller weights and the new task embedding are then updated with standard policy gradient steps.
+
+In our experiments, we also restart the experience replay bank used by the off-policy critic, so that only rewards obtained on the new task are sampled. However, future work could retain and continue to sample from previously seen tasks in order to better retain controller memory of the former tasks.
+
+# 4 EXPERIMENTS AND RESULTS
+
+We apply MNMS to the NLP setting, demonstrating that the framework can be trained simultaneously to design models for two separate text classification tasks. We then transfer learn the MNMS model to two new text classification tasks, and demonstrate that the pre-trained framework achieves significant speedups in model search.
+
+Additional details about the experimental procedures and results follow.
+
+# 4.1 EXPERIMENT SETUP
+
+Tasks
+
+For multitask training, we trained the MNMS framework simultaneously on two text classification tasks:
+
+1. Binary sentiment classification on the Stanford Sentiment Treebank (SST) dataset (Socher et al., 2013). 2. Binary Spanish language identification on a dataset consisting of each of the 5,000 highest frequency Wikipedia tokens in English, Spanish, German, and Japanese. The example label is a binary label denoting whether the token is Spanish or not Spanish.
+
+These tasks were chosen specifically for their differences in task complexity, language, and potential for overfitting. This would require a controller capable of true multitask model search to differentiate between the tasks when choosing optimal model parameters for each task.
+
+For transfer learning, we then trained the pre-trained MNMS framework on two new text classification tasks:
+
+1. Binary sentiment classification on the IMDB Large Movie Review dataset, which consists of 50,000 English movie reviews (Maas et al., 2011).   
+2. Binary sentiment classification on the CorpusCine dataset, which consists of 3,878 Spanish movie reviews from the MuchoCine website (Cruz et al., 2008).
+
+These tasks were chosen so that an effectively transfer learned framework could conceivably leverage knowledge from previous searches. As a baseline to compare search convergence rates, we also trained MNMS models from scratch on the transfer learning tasks.
+
+# Search Space
+
+For all four tasks, we define a single general search space consisting of 7 common model parameters, with 2-6 discrete parameter choices specified for each (Table 1). A naive grid search over all parameters would therefore need to try 15,360 parameter combinations to search over all possible models. These parameters represent general architectural and training design choices applicable to any text classification task.
+
+Child networks are then constructed as feed-forward neural networks using the sampled parameters. Specifically, for a sampled parameter sequence consisting of word embedding $W$ , word embedding trainability $T$ , number of neural network layers $N _ { l a y e r s }$ , number of nodes per layer $N _ { n o d e s }$ , learning rate $L$ , number of training iterations $I$ , and $L 2$ regularization weight $w$ , we construct a feedforward network with $N _ { l a y e r s }$ RELU-activated layers and $N _ { n o d e s }$ per layer. For each task, the network receives tokens embedded using $W$ , where we continue to gradient update the entire word embedding table if $T$ is true. The child model is trained for $I$ iterations using learning rate $L$ and $L 2$ regularization weight $w$ . All child models end with a final fully-connected softmax layer, and are trained using the Proximal Adagrad optimizer on batches of 100 training examples at each iteration.
+
+# Training Details
+
+The actor and critic controller RNNs used in off-policy PPO training are 2-layer LSTMs with hidden layer size 50. At each RNN timestep, both action and task embeddings have size 25, resulting in an RNN input of size 50 after concatenation. Both controller and embedding weights are initialized uniformly at random between -0.08 and 0.08.
+
+Table 1: The search space, consisting of six commonly-tuned architectural and training parameters for NLP tasks.
+
+# PARAMETER
+
+# PARAMETER CHOICES
+
+Word embedding tables Word embedding trainability Number of neural network layers Number of nodes per hidden layer Learning Rate Number of training iterations L2 Regularization weight
+
+{Spanish, German, Japanese, English-small, English-big, English-wiki}   
+{True, False}   
+{1, 2, 3, 5, 10}   
+{5, 10, 50, 100}   
+{0.001, 0.01, 0.05, 0.1}   
+{5,000, 10,000, 15,000, 20,000}   
+{0, 0.0001, 0.001, 0.01}
+
+Table 2: Details of the Word embedding tables.   
+
+<table><tr><td>LANG./ID</td><td>DIMENSIONS</td><td>VOCAB SIZE</td><td>TRAINING</td><td>TOKENS</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Spanish</td><td>128</td><td>995k</td><td>Cont. BOW</td><td>50B</td></tr><tr><td>German</td><td>128</td><td>998k</td><td>Cont. BOW</td><td>30B</td></tr><tr><td> Japanese</td><td>128</td><td>993k</td><td>Cont. BOW</td><td>6B</td></tr><tr><td>English-small</td><td>50</td><td>982k</td><td>Cont. BOW</td><td>7B</td></tr><tr><td>English-big</td><td>128</td><td>999k</td><td>Cont. BOW</td><td>200B</td></tr><tr><td>English-wiki</td><td>250</td><td>1M</td><td>Skipgram</td><td>4B</td></tr></table>
+
+When training, the controller that receives gradient updates is trained on batches of size 20 with learning rate $5 \cdot 1 0 ^ { - 4 }$ , and updated for 25 gradient steps before the weights between the two controllers are averaged with Polyak Average weight 0.9. The reward used for updating the controller is the cubed accuracy on a validation set.
+
+# 4.2 SIMULTANEOUS MULTITASK TRAINING RESULTS
+
+![](images/fa108ff709369b04be9319b2eab9b662395e2f9e68ac9d697d5c44bb6e0e6d2b.jpg)  
+Figure 3: Smoothed sampled model accuracy curves for multitask NMS when training simultaneously on the Spanish language identification and SST tasks, and the best validation accuracy achieved for each task. Curves shown use Savitzky-Golay filtering with $\mathrm { { n } = } 1 0 1$ for clarity. The reference benchmark accuracy by Socher (2013) was achieved on the SST task.
+
+We train $\mathrm { n } { = } 3$ MNMS models simultaneously on the SST and Spanish language identification tasks. Accuracies achieved by the sampled child models over time are shown in Figure 3, as well as the validation accuracy achieved by the best sampled models for each task. In each model, the accuracy of sampled models improves over time on a per-task basis, even while the tasks clearly have different baseline accuracies.
+
+Additionally, we find that the best discovered model design outperforms the hand-tuned state-ofthe-art model within the subset of models that use a similar BOW approach (Socher et al., 2013) The best performance on the task is obtained by more complex architectures that are not within the scope of our search space (Le & Mikolov, 2014).
+
+We also find that the MNMS framework can differentiate between the tasks to choose optimal parameters for each. In Figure 4, we show that MNMS learns differentiated distributions over the parameter search space for the separate tasks. For example, MNMS learns to choose a word embedding pre-trained on Spanish documents for the Spanish language identification task, while choosing word embeddings pre-trained on an English dataset for the Stanford Sentiment Treebank task.
+
+Finally, we find that MNMS learns that for the trivial language identification task, there is no significant difference between continuing to train the word embedding vectors or simply using the fixed, pre-trained word embeddings. For the SST task, which contains longer and more complex examples, the model learns that it must continue training the word embeddings to achieve better performance. Similarly, the search converges to favor higher hidden layer dimensions and more training iterations for the more complex SST task.
+
+![](images/f99aea98f94aa59357de777b3a838497615067c7179f80474c1fe0ebf4516506.jpg)  
+Figure 4: Heatmap showing the learned per-task distributions over the parameter search space from a representative MNMS model.
+
+Figure 5 compares the smoothed validation accuracy curves of baseline MNMS models trained from scratch on the IMDB and Corpus Cine tasks with MNMS models pre-trained on SST and Spanish language identification. We observe that transfer learning allows MNMS to start from a better initial location in the parameter search space, train more consistently and stably, and converge much more quickly to finding good parameters for the tasks. Additionally, we find that the best learned models discovered by MNMS perform essentially identically regardless of whether the search is started from scratch or transfer learned from a pre-trained model, demonstrating that the search is not so biased towards pre-training that it converges prematurely to local optima. When compared against other hand-tuned, state-of-the-art benchmarks also using averaged word vector inputs, we find that MNMS discovers models that outperform documented benchmarks on both tasks (Maas et al., 2011; Calvo, 2017).
+
+![](images/a8f8fefb0d104e76ff768a287697de8cc3396360e78c02c99923f1fab09e6d48.jpg)  
+Figure 5: Smoothed sampled model accuracy curves for $\mathrm { n } { = } 3$ MNMS models trained on IMDB and Corpus Cine, comparing models trained from scratch without transfer learning, and models transfer learned after pre-training. Curves smoothed using Savitzky-Golay filtering $\scriptstyle ( \mathrm { n = 1 0 1 }$ ) for clarity.
+
+We also find that MNMS learns task embeddings that encode expected relationships between the tasks (Figure 6). For example, we see a strong learned correlation between the IMDB and SST task embeddings, and separately between the Spanish language identification and Corpus Cine task embeddings. While correlation offers one metric to compare these embeddings with intuitive relationships between the tasks, however, future work could attempt to learn more disentangled and interpretable representations. Notably, for example, the Corpus Cine task embeddings are not as strongly correlated with the SST task embeddings, even though both are sentiment analysis tasks. Future work could explore which dimensions within the task embeddings actually differ, and attempt to draw human-interpretable insights that could improve future model designs on similar tasks.
+
+![](images/92b92bdbd1d14fac4d997ea04c0388232d3e4ff8dc6c5c8049f25f7279eeaf2c.jpg)  
+Figure 6: Heatmap showing correlations between learned task embeddings in a pre-trained MNMS controller transfer learned on (a) the Corpus Cine and (b) the IMDB sentiment classification tasks.
+
+# 5 DISCUSSION
+
+Summary. Machine learning model design choices do not exist in a vacuum. Human experts design good models by leveraging significant prior knowledge about the intuitive relationships between these model parameters, and the performance obtained by different model designs on similar tasks. Automated model design algorithms, too, can and should learn from the models they have discovered for prior tasks. This paper demonstrates that Multitask Neural Model Search can discover good, differentiated model designs for multiple tasks simultaneously, while learning task embeddings that encode meaningful relationships between tasks. We then show that multitask training provides a good baseline for transfer learning to future tasks, allowing the MNMS framework to start from a better location in the search space and converge more quickly to high-performing designs.
+
+Limitations and future work. While the current work demonstrates that the MNMS framework can be used for multitask training and transferable architecture searches, much work remains to determine the scalability of this approach. The results of this study offer several particularly promising avenues for future research. First, studying the effects of additional simultaneous tasks on framework performance is an obvious next step in multitask training. The current framework trains the learned task embeddings by passing them directly into the controller RNN along with the sampled action embeddings. We anticipate that a more complex pre-processing structure, such as a simple encoder-decoder, could better transform these task embeddings to be used by the controller. Additionally, we currently leverage the distributed training structure described by Zoph and Le, which trains multiple sampled child architectures in parallel and asynchronously updates a shared controller parameter server (Zoph & Le, 2017). However, as we continue to scale the MNMS framework for additional simultaneous tasks, future work remains to optimize a parallel training structure and schedule specifically for efficient multitask training.
+
+Experimenting with broader richer hyperparameter search spaces also offers an exciting line of future work. For our current tasks, we defined a search space that encompassed a range of general design choices, including both real-valued parameters (such as learning rates and regularization weights) and higher-level parameters (such as the choice of word embedding table). However, we are actively adapting the controller to sample continuous real-valued parameters, rather than discrete choices from a set of predefined values, which would give the framework much greater flexibility in specifying models. Additionally, we plan to continue expanding the range of modular, higher-level parameter choices in the search space. Allowing the controller to compose these building blocks, rather than more granular design choices, can allow the framework to construct more complex architectures in much less time.
+
+Finally, much work remains to explore cases when transfer learning is and is not effective within RL-based architecture search frameworks such as MNMS. We are particularly interested in studying how transfer learning can be used to design architectures for tasks that were previously considered too resource intensive for standard NAS. For example, Zoph et al. (2017) adapted NAS for the ImageNet classification task by directly modifying the architecture designed for a simpler image classification task. However, pretraining the architecture search framework itself on more computationally feasible tasks, rather than transferring the discovered architectures, would be a significant step towards tackling these difficult search domains.
+
+# REFERENCES
+
+Bowen Baker, Otkrist Gupta, Nikhil Naik, and Ramesh Raskar. Designing neural network architectures using reinforcement learning. arXiv preprint arXiv:1611.02167, 2016.
+
+Irwan Bello, Barret Zoph, Vijay Vasudevan, and Quoc V Le. Neural optimizer search with reinforcement learning. arXiv preprint arXiv:1709.07417, 2017.
+
+James Bergstra and Yoshua Bengio. Random search for hyper-parameter optimization. Journal of Machine Learning Research, 13(Feb):281–305, 2012.
+
+James Bergstra, Daniel Yamins, and David Cox. Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures. In International Conference on Machine Learning, pp. 115–123, 2013.
+
+Han Cai, Tianyao Chen, Weinan Zhang, Yong Yu, and Jun Wang. Reinforcement learning for architecture search by network transformation. arXiv preprint arXiv:1707.04873, 2017.
+
+David Vilares Calvo. Compositional language processing for multilingual sentiment analysis. PhD thesis, Universidade da Coruna, 2017. ˜
+
+Edoardo Conti, Vashisht Madhavan, Felipe Petroski Such, Joel Lehman, Kenneth O Stanley, and Jeff Clune. Improving exploration in evolution strategies for deep reinforcement learning via a population of novelty-seeking agents. arXiv preprint arXiv:1712.06560, 2017.
+
+Fermın L Cruz, Jose A Troyano, Fernando Enriquez, and Javier Ortega. Clasificacion de documentos ´ basada en la opinion: experimentos con un corpus de crıticas de cine en espanol. ´ Procesamiento del lenguaje natural, 41:73–80, 2008.
+
+James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, et al. Overcoming catastrophic forgetting in neural networks. Proceedings of the National Academy of Sciences, pp. 201611835, 2017.
+
+Quoc V. Le and Tomas Mikolov. Distributed representations of sentences and documents. CoRR, abs/1405.4053, 2014. URL http://arxiv.org/abs/1405.4053.
+
+Andrew L Maas, Raymond E Daly, Peter T Pham, Dan Huang, Andrew $\mathrm { ~ Y ~ N ~ g ~ } _ { }$ , and Christopher Potts. Learning word vectors for sentiment analysis. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies-Volume 1, pp. 142–150. Association for Computational Linguistics, 2011.
+
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.
+
+Ali Sharif Razavian, Hossein Azizpour, Josephine Sullivan, and Stefan Carlsson. Cnn features offthe-shelf: an astounding baseline for recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition workshops, pp. 806–813, 2014.
+
+Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher D Manning, Andrew $\mathrm { N g }$ , and Christopher Potts. Recursive deep models for semantic compositionality over a sentiment treebank. In Proceedings of the 2013 conference on empirical methods in natural language processing, pp. 1631–1642, 2013.
+
+Yee Whye Teh, Victor Bapst, Wojciech Marian Czarnecki, John Quan, James Kirkpatrick, Raia Hadsell, Nicolas Heess, and Razvan Pascanu. Distral: Robust multitask reinforcement learning. arXiv preprint arXiv:1707.04175, 2017.
+
+Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? In Advances in neural information processing systems, pp. 3320–3328, 2014.
+
+Yusen Zhan and Matthew E Taylor. Online transfer learning in reinforcement learning domains. arXiv preprint arXiv:1507.00436, 2015.
+
+Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. 2017. URL https://arxiv.org/abs/1611.01578.
+
+Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V. Le. Learning transferable architectures for scalable image recognition. CoRR, abs/1707.07012, 2017. URL http://arxiv.org/ abs/1707.07012.

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+        "text": "TRANSFER LEARNING TO LEARN WITH MULTITASK NEURAL MODEL SEARCH ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
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+        "text": "ABSTRACT ",
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+    },
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+        "type": "text",
+        "text": "Deep learning models require extensive architecture design exploration and hyperparameter optimization to perform well on a given task. The exploration of the model design space is often made by a human expert, and optimized using a combination of grid search and search heuristics over a large space of possible choices. Neural Architecture Search (NAS) is a Reinforcement Learning approach that has been proposed to automate architecture design. NAS has been successfully applied to generate Neural Networks that rival the best human-designed architectures. However, NAS requires sampling, constructing, and training hundreds to thousands of models to achieve well-performing architectures. This procedure needs to be executed from scratch for each new task. The application of NAS to a wide set of tasks currently lacks a way to transfer generalizable knowledge across tasks. ",
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+    {
+        "type": "text",
+        "text": "In this paper, we present the Multitask Neural Model Search (MNMS) controller. Our goal is to learn a generalizable framework that can condition model construction on successful model searches for previously seen tasks, thus significantly speeding up the search for new tasks. We demonstrate that MNMS can conduct an automated architecture search for multiple tasks simultaneously while still learning well-performing, specialized models for each task. We then show that pre-trained MNMS controllers can transfer learning to new tasks. By leveraging knowledge from previous searches, we find that pre-trained MNMS models start from a better location in the search space and reduce search time on unseen tasks, while still discovering models that outperform published human-designed models. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Designing deep learning models that work well for a task requires an extensive process of iterative architecture engineering and tuning. These design decisions are largely made by human experts guided by a combination of intuition, grid search, and search heuristics. ",
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+        "text": "Meta-learning aims to automate model design by using machine learning to discover good architecture and hyperparameter choices. Recent advances in meta-learning using Reinforcement Learning (RL) have made promising strides towards accelerating or even eliminating the manual parameter search. For example, Neural Architecture Search (NAS) has successfully discovered novel network architectures that rival or surpass the best human-designed architectures on challenging benchmark image recognition tasks (Zoph & Le, 2017; Zoph et al., 2017). However, naively applying reinforcement learning to each new task for automated model construction requires sampling, constructing, and training hundreds to thousands of networks to relearn how to generate models from scratch. Human experts, on the other hand, can design and tune networks based on knowledge about underlying dependencies in the search space and experience with prior tasks. We therefore aim to automatically learn and leverage the same information. ",
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+        "text": "In this paper, we present Multitask Neural Model Search (MNMS), an automated model construction framework that finds the best performing models in the search space for multiple tasks simultaneously. We then show that a MNMS framework that has been pre-trained on previous tasks can construct the best performing model for entirely new tasks in significantly less time. ",
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+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    },
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+        "text": "The Neural Architecture Search (NAS) method was introduced in (Zoph & Le, 2017), where it was applied to construct Convolutional Neural Networks (CNNs) for the CIFAR-10 task and Recurrent Neural Networks (RNNs) for the Penn Treebank tasks. Later work by the same authors attempted to address the computational cost of using Neural Architecture Search for more challenging tasks (Zoph et al., 2017). To engineer a convolutional architecture for ImageNet classification, this paper demonstrated that it was possible to train the NAS controller on the simpler, proxy CIFAR-10 task and then transfer the architecture to ImageNet classification by stacking it. However, this work did not attempt to transfer learn the NAS controller itself across multiple tasks, relying instead on the human expert intuition that additional network depth was necessary for the more challenging classification task. Additionally, the final generated architectures required additional tuning, to choose hyperparameters such as the learning rate, before evaluation on the test set. ",
+        "bbox": [
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+    {
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+        "text": "The complexity of model engineering in machine learning is widely recognized. Optimization methods have been proposed, ranging from random search over the space of possible architectures (Bergstra & Bengio, 2012) to parameter modeling (Bergstra et al., 2013). Recent publications apply RL to automate architecture generation. These include MetaQNN, a Q-learning algorithm that sequentially chooses CNN layers (Baker et al., 2016). MetaQNN uses an aggressive exploration to reduce search time, though it can cause the resulting architectures to underperform. Cai et al. (2017) also propose an RL agent that transforms existing architectures incrementally to avoid generating entire networks from scratch. More recently, an emerging body of modern neuro-evolution research has adapted genetic algorithms as an alternate optimization method for these complex searches (Conti et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "Our work also draws on prior research in transfer learning and simultaneous multitask training. Transfer learning has been shown to achieve excellent results as an initialization method for deep networks, including for models trained using RL (Yosinski et al., 2014; Sharif Razavian et al., 2014; Zhan & Taylor, 2015). Simultaneous multitask training can also facilitate learning between tasks with a common structure, though effectively retaining knowledge across tasks is still an active area of research Kirkpatrick et al. (2017); Teh et al. (2017). ",
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+        "text": "3 METHODS ",
+        "text_level": 1,
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+    {
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+        "text": "3.1 NEURAL ARCHITECTURE SEARCH OVERVIEW ",
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+        "image_caption": [
+            "Figure 1: The base Neural Architecture Search framework. "
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+        "text": "Neural Architecture Search uses an RNN to generate model designs that maximize expected performance on a given task (Figure 1) (Zoph & Le, 2017). Specifically, an RNN controller iteratively samples architectures as a sequence of actions. Every action is a discretized design choice, such as CNN filter heights, widths, and strides. Child networks are then constructed with these architectures and trained to convergence. The performance metric of the child network is used as a reward to update the controller through a policy gradient algorithm. The controller learns a distribution over the architecture search space that is updated to increase the probability of the best performing architectures, allowing it to sample better architectures over time. The original Neural Architecture Search framework sampled neural network models over a search space of strictly architectural parameters, by generating descriptions of each layer in the network at a time. This framework was used to successfully specify a convolutional neural network architecture for image classification, and a recurrent network cell for language tasks (Zoph & Le, 2017). Later work has shown that this framework can be extended to automatically search over other model design parameters and domains, such as update rules for network optimizers (Bello et al., 2017). ",
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+        "text": "3.2 SIMULTANEOUS MULTITASK TRAINING IN NEURAL ARCHITECTURE SEARCH ",
+        "text_level": 1,
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+    {
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+        "text": "In this section, we describe the Multitask Neural Model Search (MNMS) controller, which allows simultaneous model search over multiple different tasks. Many deep learning models require the same common design decisions, such as choice of network depth, learning rate, and number of training iterations; using a generally defined search space of widely applicable architecture and hyperparameter choices, the controller can therefore engineer a wide range of models applicable to many common machine learning tasks. Multitask training over this space can then allow the controller to learn more broadly applicable relationships between search space actions, by leveraging shared behavior across tasks. ",
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+        "text": "We implement a controller capable of simultaneous multitask training through three key modifications: ",
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+        "text": "1. Learned task representation and task conditioning. ",
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+        "text": "The MNMS controller can be trained synchronously on a set of N tasks. The controller learns to build differentiated architectures for each task. This is achieved by sampling a task uniformly at the beginning of each controller training iteration. The task is then mapped to a unique embedding vector. The tasks embeddings are randomly initialized, and are trained jointly with the controller. The task embedding is then used to condition the model construction on the task. This is achieved by concatenating the task embedding to every input that is fed to the controller RNN. ",
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+        "text": "Specifically, in single-task NAS, the controller RNN generates an output at each timestep that determines the distribution over the current set of actions. An action is sampled according to this action distribution and then embedded. The action embedding is then passed back into the RNN as input to the next timestep. ",
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+        "text": "Here, in multi-task training for MNMS, for multi-task training, the task embedding is now concatenated with the action embedding to form the RNN input, allowing it to condition each action on a specific task (Figure 2). ",
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+    {
+        "type": "text",
+        "text": "Previous works train the NAS controller using the REINFORCE policy gradient algorithm (Zoph & Le, 2017), and more recently, the PPO algorithm (Zoph et al., 2017; Bello et al., 2017). ",
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+        "text": "Here, we train using off-policy PPO, an actor-critic algorithm in which an actor controller generates sampled models and a critic controller trains on a replay bank of the sampled models and rewards (Schulman et al., 2017). Preliminary experiments with on-policy training found that the controller shows a reduced ability to learn a differentiated model for each task. Specifically, the controller is prone to premature convergence to a single model design that works generally well for all tasks but is not the best model for some of the tasks. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "On-policy sampling is biased toward more recent predictions of the optimal parameter distribution. Our hypothesis is that in multitask training, on-policy sampling can prematurely reduce exploration of better parameters for each individual task, while off-policy training allows the actor controller to continue to explore separate parameter choices for each task, and better learn a differentiated distribution over the parameter search space to maximize expected performance for each. ",
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+    {
+        "type": "text",
+        "text": "3. Per-task baseline and reward distributions normalization. ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Each task can define a different performance metric to be used as reward. The rewards affect the amplitude of the updates on the controller, so we need to make sure that the distributions of each task rewards are aligned to have same mean and similar variance. ",
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+    {
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+        "img_path": "images/a9ed4b597f20ad0feae68f43ad9c748b3099e5814ec74d1b7c18c60a272f77d0.jpg",
+        "image_caption": [
+            "Figure 2: Overview of the multitask controller RNN. (1) A task embedding table is maintained and updated with controller gradients to learn differentiated task embeddings over time. (2) At each iteration of the multitask training, a task is randomly sampled. The task embedding is passed into the controller RNN along with the sampled action embedding at each RNN timestep. The full sequence of output actions defines the child architecture trained on the chosen task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The mean of each tasks reward distribution is aligned to 0 by scaling the gradients with the advantage instead of the reward. The advantage, $A ( \\boldsymbol { a } , t )$ , of a given model, $a$ , applied to a task, $t$ , is defined as the difference between the reward, $R ( \\boldsymbol { a } , t )$ , and the expected reward for the given task, $b ( t )$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b259e8aafebb38f78c8a0174510982216ad1a13ddba4d1e4a8c110f180c47993.jpg",
+        "text": "$$\nA ( a , t ) = R ( a , t ) - b ( t )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "$b ( t )$ is often referred to as the baseline. This is a standard RL technique that is usually applied with the aim of increasing training stability. During multitask training, the baseline is conditioned on the sampled task. We keep track of a separate baseline for each task, computed as an exponential moving average of the rewards recorded for each task. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The range of each tasks reward distribution is normalized by dividing the advantage by the baseline: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e4eb38220721ce24526103ca317999418e1115572b99a048db4df7064515832.jpg",
+        "text": "$$\nA ^ { \\prime } ( a , t ) = { \\frac { R ( a , t ) - b ( t ) } { b ( t ) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We refer to $A ^ { \\prime }$ as the normalized advantage. Notice that the division by the baseline does not compromise the convergence criteria, as it can be seen as using a distinct adaptive learning rate for each task. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Using the normalized advantage to scale the gradients instead of the raw reward allows MNMS to use any performance metric as a reward even when training on multiple tasks. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 TRANSFER LEARNING FOR AUTOMATED MODEL SEARCH",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Using the multitask framework, we can transfer learn pretrained controllers by simply reusing the weights of the pretrained controller and adding a randomly initialized task embedding for each new task. The controller weights and the new task embedding are then updated with standard policy gradient steps. ",
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+    {
+        "type": "text",
+        "text": "In our experiments, we also restart the experience replay bank used by the off-policy critic, so that only rewards obtained on the new task are sampled. However, future work could retain and continue to sample from previously seen tasks in order to better retain controller memory of the former tasks. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS AND RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We apply MNMS to the NLP setting, demonstrating that the framework can be trained simultaneously to design models for two separate text classification tasks. We then transfer learn the MNMS model to two new text classification tasks, and demonstrate that the pre-trained framework achieves significant speedups in model search. ",
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+    {
+        "type": "text",
+        "text": "Additional details about the experimental procedures and results follow. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENT SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Tasks ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For multitask training, we trained the MNMS framework simultaneously on two text classification tasks: ",
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+    },
+    {
+        "type": "text",
+        "text": "1. Binary sentiment classification on the Stanford Sentiment Treebank (SST) dataset (Socher et al., 2013). 2. Binary Spanish language identification on a dataset consisting of each of the 5,000 highest frequency Wikipedia tokens in English, Spanish, German, and Japanese. The example label is a binary label denoting whether the token is Spanish or not Spanish. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "These tasks were chosen specifically for their differences in task complexity, language, and potential for overfitting. This would require a controller capable of true multitask model search to differentiate between the tasks when choosing optimal model parameters for each task. ",
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+    {
+        "type": "text",
+        "text": "For transfer learning, we then trained the pre-trained MNMS framework on two new text classification tasks: ",
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+    {
+        "type": "text",
+        "text": "1. Binary sentiment classification on the IMDB Large Movie Review dataset, which consists of 50,000 English movie reviews (Maas et al., 2011).   \n2. Binary sentiment classification on the CorpusCine dataset, which consists of 3,878 Spanish movie reviews from the MuchoCine website (Cruz et al., 2008). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "These tasks were chosen so that an effectively transfer learned framework could conceivably leverage knowledge from previous searches. As a baseline to compare search convergence rates, we also trained MNMS models from scratch on the transfer learning tasks. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Search Space ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For all four tasks, we define a single general search space consisting of 7 common model parameters, with 2-6 discrete parameter choices specified for each (Table 1). A naive grid search over all parameters would therefore need to try 15,360 parameter combinations to search over all possible models. These parameters represent general architectural and training design choices applicable to any text classification task. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Child networks are then constructed as feed-forward neural networks using the sampled parameters. Specifically, for a sampled parameter sequence consisting of word embedding $W$ , word embedding trainability $T$ , number of neural network layers $N _ { l a y e r s }$ , number of nodes per layer $N _ { n o d e s }$ , learning rate $L$ , number of training iterations $I$ , and $L 2$ regularization weight $w$ , we construct a feedforward network with $N _ { l a y e r s }$ RELU-activated layers and $N _ { n o d e s }$ per layer. For each task, the network receives tokens embedded using $W$ , where we continue to gradient update the entire word embedding table if $T$ is true. The child model is trained for $I$ iterations using learning rate $L$ and $L 2$ regularization weight $w$ . All child models end with a final fully-connected softmax layer, and are trained using the Proximal Adagrad optimizer on batches of 100 training examples at each iteration. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Training Details ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The actor and critic controller RNNs used in off-policy PPO training are 2-layer LSTMs with hidden layer size 50. At each RNN timestep, both action and task embeddings have size 25, resulting in an RNN input of size 50 after concatenation. Both controller and embedding weights are initialized uniformly at random between -0.08 and 0.08. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 1: The search space, consisting of six commonly-tuned architectural and training parameters for NLP tasks. ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "PARAMETER ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "PARAMETER CHOICES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Word embedding tables Word embedding trainability Number of neural network layers Number of nodes per hidden layer Learning Rate Number of training iterations L2 Regularization weight ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "{Spanish, German, Japanese, English-small, English-big, English-wiki}   \n{True, False}   \n{1, 2, 3, 5, 10}   \n{5, 10, 50, 100}   \n{0.001, 0.01, 0.05, 0.1}   \n{5,000, 10,000, 15,000, 20,000}   \n{0, 0.0001, 0.001, 0.01} ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/fe5b2428438ae06e5c645f4aee38cfc3a9366b7f4337c08c46475969df210def.jpg",
+        "table_caption": [
+            "Table 2: Details of the Word embedding tables. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>LANG./ID</td><td>DIMENSIONS</td><td>VOCAB SIZE</td><td>TRAINING</td><td>TOKENS</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Spanish</td><td>128</td><td>995k</td><td>Cont. BOW</td><td>50B</td></tr><tr><td>German</td><td>128</td><td>998k</td><td>Cont. BOW</td><td>30B</td></tr><tr><td> Japanese</td><td>128</td><td>993k</td><td>Cont. BOW</td><td>6B</td></tr><tr><td>English-small</td><td>50</td><td>982k</td><td>Cont. BOW</td><td>7B</td></tr><tr><td>English-big</td><td>128</td><td>999k</td><td>Cont. BOW</td><td>200B</td></tr><tr><td>English-wiki</td><td>250</td><td>1M</td><td>Skipgram</td><td>4B</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "When training, the controller that receives gradient updates is trained on batches of size 20 with learning rate $5 \\cdot 1 0 ^ { - 4 }$ , and updated for 25 gradient steps before the weights between the two controllers are averaged with Polyak Average weight 0.9. The reward used for updating the controller is the cubed accuracy on a validation set. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 SIMULTANEOUS MULTITASK TRAINING RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/fa108ff709369b04be9319b2eab9b662395e2f9e68ac9d697d5c44bb6e0e6d2b.jpg",
+        "image_caption": [
+            "Figure 3: Smoothed sampled model accuracy curves for multitask NMS when training simultaneously on the Spanish language identification and SST tasks, and the best validation accuracy achieved for each task. Curves shown use Savitzky-Golay filtering with $\\mathrm { { n } = } 1 0 1$ for clarity. The reference benchmark accuracy by Socher (2013) was achieved on the SST task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We train $\\mathrm { n } { = } 3$ MNMS models simultaneously on the SST and Spanish language identification tasks. Accuracies achieved by the sampled child models over time are shown in Figure 3, as well as the validation accuracy achieved by the best sampled models for each task. In each model, the accuracy of sampled models improves over time on a per-task basis, even while the tasks clearly have different baseline accuracies. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Additionally, we find that the best discovered model design outperforms the hand-tuned state-ofthe-art model within the subset of models that use a similar BOW approach (Socher et al., 2013) The best performance on the task is obtained by more complex architectures that are not within the scope of our search space (Le & Mikolov, 2014). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We also find that the MNMS framework can differentiate between the tasks to choose optimal parameters for each. In Figure 4, we show that MNMS learns differentiated distributions over the parameter search space for the separate tasks. For example, MNMS learns to choose a word embedding pre-trained on Spanish documents for the Spanish language identification task, while choosing word embeddings pre-trained on an English dataset for the Stanford Sentiment Treebank task. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we find that MNMS learns that for the trivial language identification task, there is no significant difference between continuing to train the word embedding vectors or simply using the fixed, pre-trained word embeddings. For the SST task, which contains longer and more complex examples, the model learns that it must continue training the word embeddings to achieve better performance. Similarly, the search converges to favor higher hidden layer dimensions and more training iterations for the more complex SST task. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/f99aea98f94aa59357de777b3a838497615067c7179f80474c1fe0ebf4516506.jpg",
+        "image_caption": [
+            "Figure 4: Heatmap showing the learned per-task distributions over the parameter search space from a representative MNMS model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 compares the smoothed validation accuracy curves of baseline MNMS models trained from scratch on the IMDB and Corpus Cine tasks with MNMS models pre-trained on SST and Spanish language identification. We observe that transfer learning allows MNMS to start from a better initial location in the parameter search space, train more consistently and stably, and converge much more quickly to finding good parameters for the tasks. Additionally, we find that the best learned models discovered by MNMS perform essentially identically regardless of whether the search is started from scratch or transfer learned from a pre-trained model, demonstrating that the search is not so biased towards pre-training that it converges prematurely to local optima. When compared against other hand-tuned, state-of-the-art benchmarks also using averaged word vector inputs, we find that MNMS discovers models that outperform documented benchmarks on both tasks (Maas et al., 2011; Calvo, 2017). ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/a8f8fefb0d104e76ff768a287697de8cc3396360e78c02c99923f1fab09e6d48.jpg",
+        "image_caption": [
+            "Figure 5: Smoothed sampled model accuracy curves for $\\mathrm { n } { = } 3$ MNMS models trained on IMDB and Corpus Cine, comparing models trained from scratch without transfer learning, and models transfer learned after pre-training. Curves smoothed using Savitzky-Golay filtering $\\scriptstyle ( \\mathrm { n = 1 0 1 }$ ) for clarity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also find that MNMS learns task embeddings that encode expected relationships between the tasks (Figure 6). For example, we see a strong learned correlation between the IMDB and SST task embeddings, and separately between the Spanish language identification and Corpus Cine task embeddings. While correlation offers one metric to compare these embeddings with intuitive relationships between the tasks, however, future work could attempt to learn more disentangled and interpretable representations. Notably, for example, the Corpus Cine task embeddings are not as strongly correlated with the SST task embeddings, even though both are sentiment analysis tasks. Future work could explore which dimensions within the task embeddings actually differ, and attempt to draw human-interpretable insights that could improve future model designs on similar tasks. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/92b92bdbd1d14fac4d997ea04c0388232d3e4ff8dc6c5c8049f25f7279eeaf2c.jpg",
+        "image_caption": [
+            "Figure 6: Heatmap showing correlations between learned task embeddings in a pre-trained MNMS controller transfer learned on (a) the Corpus Cine and (b) the IMDB sentiment classification tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Summary. Machine learning model design choices do not exist in a vacuum. Human experts design good models by leveraging significant prior knowledge about the intuitive relationships between these model parameters, and the performance obtained by different model designs on similar tasks. Automated model design algorithms, too, can and should learn from the models they have discovered for prior tasks. This paper demonstrates that Multitask Neural Model Search can discover good, differentiated model designs for multiple tasks simultaneously, while learning task embeddings that encode meaningful relationships between tasks. We then show that multitask training provides a good baseline for transfer learning to future tasks, allowing the MNMS framework to start from a better location in the search space and converge more quickly to high-performing designs. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Limitations and future work. While the current work demonstrates that the MNMS framework can be used for multitask training and transferable architecture searches, much work remains to determine the scalability of this approach. The results of this study offer several particularly promising avenues for future research. First, studying the effects of additional simultaneous tasks on framework performance is an obvious next step in multitask training. The current framework trains the learned task embeddings by passing them directly into the controller RNN along with the sampled action embeddings. We anticipate that a more complex pre-processing structure, such as a simple encoder-decoder, could better transform these task embeddings to be used by the controller. Additionally, we currently leverage the distributed training structure described by Zoph and Le, which trains multiple sampled child architectures in parallel and asynchronously updates a shared controller parameter server (Zoph & Le, 2017). However, as we continue to scale the MNMS framework for additional simultaneous tasks, future work remains to optimize a parallel training structure and schedule specifically for efficient multitask training. ",
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+    {
+        "type": "text",
+        "text": "Experimenting with broader richer hyperparameter search spaces also offers an exciting line of future work. For our current tasks, we defined a search space that encompassed a range of general design choices, including both real-valued parameters (such as learning rates and regularization weights) and higher-level parameters (such as the choice of word embedding table). However, we are actively adapting the controller to sample continuous real-valued parameters, rather than discrete choices from a set of predefined values, which would give the framework much greater flexibility in specifying models. Additionally, we plan to continue expanding the range of modular, higher-level parameter choices in the search space. Allowing the controller to compose these building blocks, rather than more granular design choices, can allow the framework to construct more complex architectures in much less time. ",
+        "bbox": [
+            174,
+            454,
+            825,
+            593
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finally, much work remains to explore cases when transfer learning is and is not effective within RL-based architecture search frameworks such as MNMS. We are particularly interested in studying how transfer learning can be used to design architectures for tasks that were previously considered too resource intensive for standard NAS. For example, Zoph et al. (2017) adapted NAS for the ImageNet classification task by directly modifying the architecture designed for a simpler image classification task. However, pretraining the architecture search framework itself on more computationally feasible tasks, rather than transferring the discovered architectures, would be a significant step towards tackling these difficult search domains. ",
+        "bbox": [
+            174,
+            599,
+            825,
+            710
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            734,
+            285,
+            750
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Bowen Baker, Otkrist Gupta, Nikhil Naik, and Ramesh Raskar. Designing neural network architectures using reinforcement learning. arXiv preprint arXiv:1611.02167, 2016. ",
+        "bbox": [
+            173,
+            758,
+            821,
+            786
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Irwan Bello, Barret Zoph, Vijay Vasudevan, and Quoc V Le. Neural optimizer search with reinforcement learning. arXiv preprint arXiv:1709.07417, 2017. ",
+        "bbox": [
+            173,
+            799,
+            821,
+            828
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "James Bergstra and Yoshua Bengio. Random search for hyper-parameter optimization. Journal of Machine Learning Research, 13(Feb):281–305, 2012. ",
+        "bbox": [
+            169,
+            840,
+            826,
+            869
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "James Bergstra, Daniel Yamins, and David Cox. Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures. In International Conference on Machine Learning, pp. 115–123, 2013. ",
+        "bbox": [
+            176,
+            881,
+            823,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Han Cai, Tianyao Chen, Weinan Zhang, Yong Yu, and Jun Wang. Reinforcement learning for architecture search by network transformation. arXiv preprint arXiv:1707.04873, 2017. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "David Vilares Calvo. Compositional language processing for multilingual sentiment analysis. PhD thesis, Universidade da Coruna, 2017. ˜ ",
+        "bbox": [
+            173,
+            141,
+            823,
+            170
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Edoardo Conti, Vashisht Madhavan, Felipe Petroski Such, Joel Lehman, Kenneth O Stanley, and Jeff Clune. Improving exploration in evolution strategies for deep reinforcement learning via a population of novelty-seeking agents. arXiv preprint arXiv:1712.06560, 2017. ",
+        "bbox": [
+            176,
+            178,
+            823,
+            222
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fermın L Cruz, Jose A Troyano, Fernando Enriquez, and Javier Ortega. Clasificacion de documentos ´ basada en la opinion: experimentos con un corpus de crıticas de cine en espanol. ´ Procesamiento del lenguaje natural, 41:73–80, 2008. ",
+        "bbox": [
+            174,
+            231,
+            823,
+            272
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, et al. Overcoming catastrophic forgetting in neural networks. Proceedings of the National Academy of Sciences, pp. 201611835, 2017. ",
+        "bbox": [
+            173,
+            281,
+            826,
+            338
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Quoc V. Le and Tomas Mikolov. Distributed representations of sentences and documents. CoRR, abs/1405.4053, 2014. URL http://arxiv.org/abs/1405.4053. ",
+        "bbox": [
+            173,
+            347,
+            823,
+            377
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Andrew L Maas, Raymond E Daly, Peter T Pham, Dan Huang, Andrew $\\mathrm { ~ Y ~ N ~ g ~ } _ { }$ , and Christopher Potts. Learning word vectors for sentiment analysis. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies-Volume 1, pp. 142–150. Association for Computational Linguistics, 2011. ",
+        "bbox": [
+            173,
+            385,
+            825,
+            441
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017. ",
+        "bbox": [
+            173,
+            450,
+            823,
+            479
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ali Sharif Razavian, Hossein Azizpour, Josephine Sullivan, and Stefan Carlsson. Cnn features offthe-shelf: an astounding baseline for recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition workshops, pp. 806–813, 2014. ",
+        "bbox": [
+            174,
+            488,
+            823,
+            531
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher D Manning, Andrew $\\mathrm { N g }$ , and Christopher Potts. Recursive deep models for semantic compositionality over a sentiment treebank. In Proceedings of the 2013 conference on empirical methods in natural language processing, pp. 1631–1642, 2013. ",
+        "bbox": [
+            173,
+            539,
+            825,
+            597
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yee Whye Teh, Victor Bapst, Wojciech Marian Czarnecki, John Quan, James Kirkpatrick, Raia Hadsell, Nicolas Heess, and Razvan Pascanu. Distral: Robust multitask reinforcement learning. arXiv preprint arXiv:1707.04175, 2017. ",
+        "bbox": [
+            174,
+            604,
+            823,
+            648
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? In Advances in neural information processing systems, pp. 3320–3328, 2014. ",
+        "bbox": [
+            171,
+            656,
+            821,
+            686
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yusen Zhan and Matthew E Taylor. Online transfer learning in reinforcement learning domains. arXiv preprint arXiv:1507.00436, 2015. ",
+        "bbox": [
+            173,
+            694,
+            820,
+            723
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. 2017. URL https://arxiv.org/abs/1611.01578. ",
+        "bbox": [
+            174,
+            732,
+            820,
+            761
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V. Le. Learning transferable architectures for scalable image recognition. CoRR, abs/1707.07012, 2017. URL http://arxiv.org/ abs/1707.07012. ",
+        "bbox": [
+            174,
+            770,
+            826,
+            813
+        ],
+        "page_idx": 9
+    }
+]

parse/train/mSAKhLYLSsl/mSAKhLYLSsl.md

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+# DATASET CONDENSATION WITH GRADIENT MATCHING
+
+Bo Zhao, Konda Reddy Mopuri, Hakan Bilen School of Informatics, The University of Edinburgh {bo.zhao, kmopuri, hbilen}@ed.ac.uk
+
+# ABSTRACT
+
+As the state-of-the-art machine learning methods in many fields rely on larger datasets, storing datasets and training models on them become significantly more expensive. This paper proposes a training set synthesis technique for data-efficient learning, called Dataset Condensation, that learns to condense large dataset into a small set of informative synthetic samples for training deep neural networks from scratch. We formulate this goal as a gradient matching problem between the gradients of deep neural network weights that are trained on the original and our synthetic data. We rigorously evaluate its performance in several computer vision benchmarks and demonstrate that it significantly outperforms the state-of-the-art methods1. Finally we explore the use of our method in continual learning and neural architecture search and report promising gains when limited memory and computations are available.
+
+# 1 INTRODUCTION
+
+Large-scale datasets, comprising millions of samples, are becoming the norm to obtain state-ofthe-art machine learning models in multiple fields including computer vision, natural language processing and speech recognition. At such scales, even storing and preprocessing the data becomes burdensome, and training machine learning models on them demands for specialized equipment and infrastructure. An effective way to deal with large data is data selection – identifying the most representative training samples – that aims at improving data efficiency of machine learning techniques. While classical data selection methods, also known as coreset construction (Agarwal et al., 2004; Har-Peled & Mazumdar, 2004; Feldman et al., 2013), focus on clustering problems, recent work can be found in continual learning (Rebuffi et al., 2017; Toneva et al., 2019; Castro et al., 2018; Aljundi et al., 2019) and active learning (Sener & Savarese, 2018) where there is typically a fixed budget in storing and labeling training samples respectively. These methods commonly first define a criterion for representativeness (e.g. in terms of compactness (Rebuffi et al., 2017; Castro et al., 2018), diversity (Sener & Savarese, 2018; Aljundi et al., 2019), forgetfulness (Toneva et al., 2019)), then select the representative samples based on the criterion, finally use the selected small set to train their model for a downstream task.
+
+Unfortunately, these methods have two shortcomings: they typically rely on i) heuristics (e.g. picking cluster centers) that does not guarantee any optimal solution for the downstream task (e.g. image classification), ii) presence of representative samples, which is neither guaranteed. A recent method, Dataset Distillation (DD) (Wang et al., 2018) goes beyond these limitations by learning a small set of informative images from large training data. In particular, the authors model the network parameters as a function of the synthetic training data and learn them by minimizing the training loss over the original training data w.r.t. synthetic data. Unlike in the coreset methods, the synthesized data are directly optimized for the downstream task and thus the success of the method does not rely on the presence of representative samples.
+
+Inspired from DD (Wang et al., 2018), we focus on learning to synthesize informative samples that are optimized to train neural networks for downstream tasks and not limited to individual samples in original dataset. Like DD, our goal is to obtain the highest generalization performance with a model trained on a small set of synthetic images, ideally comparable performance to that of a model trained on the original images (see Figure 1(a)). In particular, we investigate the following questions. Is it possible to i) compress a large image classification dataset into a small synthetic set, ii) train an image classification model on the synthetic set that can be further used to classify real images, iii) learn a single set of synthetic images that can be used to train different neural network architectures? To this end, we propose a Dataset Condensation method to learn a small set of “condensed” synthetic samples such that a deep neural network trained on them obtains not only similar performance but also a close solution to a network trained on the large training data in the network parameter space. We formulate this goal as a minimization problem between two sets of gradients of the network parameters that are computed for a training loss over a large fixed training set and a learnable condensed set (see Figure 1(b)). We show that our method enables effective learning of synthetic images and neural networks trained on them, outperforms (Wang et al., 2018) and coreset methods with a wide margin in multiple computer vision benchmarks. In addition, learning a compact set of synthetic samples also benefits other learning problems when there is a fixed budget on training images. We show that our method outperforms popular data selection methods by providing more informative training samples in continual learning. Finally, we explore a promising use case of our method in neural architecture search, and show that – once our condensed images are learned – they can be used to train numerous network architectures extremely efficiently.
+
+![](images/9db99752d4ff8fbc0baa4b6f5061edebe089d16a3887336fa26da423efb30d55.jpg)  
+Figure 1: Dataset Condensation (left) aims to generate a small set of synthetic images that can match the performance of a network trained on a large image dataset. Our method (right) realizes this goal by learning a synthetic set such that a deep network trained on it and the large set produces similar gradients w.r.t. its weights. The synthetic data can later be used to train a network from scratch in a small fraction of the original computational load. CE denotes Cross-Entropy.
+
+Our method is related to knowledge distillation (KD) techniques (Hinton et al., 2015; Bucilua et al., ˇ 2006; Ba & Caruana, 2014; Romero et al., 2014) that transfer the knowledge in an ensemble of models to a single one. Unlike KD, we distill knowledge of a large training set into a small synthetic set. Our method is also related to Generative Adversarial Networks (Goodfellow et al., 2014a; Mirza & Osindero, 2014; Radford et al., 2015) and Variational AutoEncoders (Kingma & Welling, 2013) that synthesize high-fidelity samples by capturing the data distribution. In contrast, our goal is to generate informative samples for training deep neural networks rather than to produce “real-looking” samples. Finally our method is related to the methods that produce image patches by projecting the feature activations back to the input pixel space (Zeiler & Fergus, 2014), reconstruct the input image by matching the feature activations (Mahendran & Vedaldi, 2015), recover private training images for given training gradients (Zhu et al., 2019; Zhao et al., 2020), synthesize features from semantic embeddings for zero-shot learning (Sariyildiz & Cinbis, 2019). Our goal is however to synthesize a set of condensed training images not to recover the original or missing training images.
+
+In the remainder of this paper, we first review the problem of dataset condensation and introduce our method in section 2, present and analyze our results in several image recognition benchmarks in section 3.1, showcase applications in continual learning and network architecture search in section 3.2, and conclude the paper with remarks for future directions in section 4.
+
+# 2 METHOD
+
+Suppose we are given a large dataset consisting of $| \tau |$ pairs of a training image and its class label $\mathcal { T } = \{ ( \boldsymbol { x } _ { i } , y _ { i } ) \} | _ { i = 1 } ^ { | \mathcal { T } | }$ where $\pmb { x } \in \mathcal { X } \subset \mathbb { R } ^ { d }$ , $y \in \{ 0 , \ldots , C - 1 \}$ , $\mathcal { X }$ is a ${ \mathrm { d } }$ -dimensional input space and $C$ is the number of classes. We wish to learn a differentiable function $\phi$ (i.e. deep neural network)
+
+# 2.1 DATASET CONDENSATION
+
+with parameters $\pmb \theta$ that correctly predicts labels of previously unseen images, i.e. $y = \phi _ { \pmb { \theta } } ( \pmb { x } )$ . One can learn the parameters of this function by minimizing an empirical loss term over the training set:
+
+$$
+\pmb { \theta } ^ { \mathcal { T } } = \arg \operatorname* { m i n } _ { \pmb { \theta } } \mathcal { L } ^ { \mathcal { T } } ( \pmb { \theta } )
+$$
+
+where $\begin{array} { r } { \mathcal L ^ { \mathcal T } ( \pmb { \theta } ) = \frac { 1 } { | \mathcal T | } \sum _ { ( \pmb { x } , \pmb { y } ) \in \mathcal T } \ell ( \phi _ { \pmb { \theta } } ( \pmb { x } ) , \pmb { y } ) , \ell ( \cdot , \cdot ) } \end{array}$ is a task specific loss (i.e. cross-entropy) and $\pmb { \theta } ^ { \mathcal { T } }$ is the minimizer of $\mathcal { L } ^ { \mathcal { T } }$ . The generalization performance of the obtained model $\phi _ { \pmb { \theta } ^ { \tau } }$ can be written as $\mathbb { E } _ { { \pmb x } \sim P _ { \mathcal { D } } } [ \ell ( \phi _ { \pmb \theta ^ { \top } } ( { \pmb x } ) , y ) ]$ where $P _ { \mathcal { D } }$ is the data distribution. Our goal is to generate a small set of condensed synthetic samples with their labels, $\boldsymbol { S } = \{ ( s _ { i } , y _ { i } ) \} | _ { i = 1 } ^ { | S | }$ where $\boldsymbol { s } \in \mathbb { R } ^ { d }$ and $y \in \mathcal { V }$ , $| S | \ll | T |$ . Similar to eq. (1), once the condensed set is learned, one can train $\phi$ on them as follows
+
+$$
+\pmb { \theta } ^ { S } = \operatorname * { a r g m i n } _ { \pmb { \theta } } \mathcal { L } ^ { S } ( \pmb { \theta } )
+$$
+
+where $\begin{array} { r } { \begin{array} { r } { \mathcal { L } ^ { S } ( \pmb { \theta } ) = \frac { 1 } { | \mathcal { S } | } \sum _ { ( s , y ) \in \mathcal { S } } \ell \big ( \phi _ { \pmb { \theta } } \big ( \pmb { s } \big ) , y \big ) } \end{array} } \end{array}$ and $\theta ^ { S }$ is the minimizer of $\mathcal { L } ^ { s }$ . As the synthetic set $s$ is significantly smaller (2-3 orders of magnitude), we expect the optimization in eq. (2) to be significantly faster than that in eq. (1). We also wish the generalization performance of $\phi _ { \pmb { \theta } ^ { s } }$ to be close to $\phi _ { \pmb { \theta } ^ { \tau } }$ , i.e. $\begin{array} { r } { \mathbb { E } _ { { \pmb x } \sim P _ { \mathcal { D } } } [ \ell ( \phi _ { \pmb \theta ^ { \top } } ( \bar { \pmb x } ) , y ) ] \simeq \mathbb { E } _ { { \pmb x } \sim P _ { \mathcal { D } } } [ \ell ( \phi _ { \pmb \theta ^ { s } } ( \bar { \pmb x } ) , y ) ] } \end{array}$ over the real data distribution $P _ { \mathcal { D } }$ .
+
+Discussion. The goal of obtaining comparable generalization performance by training on the condensed data can be formulated in different ways. One approach, which is proposed in (Wang et al., 2018) and extended in (Sucholutsky & Schonlau, 2019; Bohdal et al., 2020; Such et al., 2020), is to pose the parameters $\theta ^ { S }$ as a function of the synthetic data $s$ :
+
+$$
+{ \mathcal { S } } ^ { * } = \underset { S } { \arg \operatorname* { m i n } } { \mathcal { L } } ^ { { \mathcal { T } } } ( \theta ^ { S } ( S ) ) \qquad \mathrm { s u b j e c t ~ t o } \qquad \theta ^ { S } ( S ) = \underset { \theta } { \arg \operatorname* { m i n } } { \mathcal { L } } ^ { S } ( \theta ) .
+$$
+
+The method aims to find the optimum set of synthetic images ${ \boldsymbol { S } } ^ { * }$ such that the model $\phi _ { \pmb { \theta } ^ { s } }$ trained on them minimizes the training loss over the original data. Optimizing eq. (3) involves a nested loop optimization and solving the inner loop for $\check { \theta ^ { S } } ( S )$ at each iteration to recover the gradients for $s$ which requires a computationally expensive procedure – unrolling the recursive computation graph for $s$ over multiple optimization steps for $\pmb \theta$ (see (Samuel & Tappen, 2009; Domke, 2012)). Hence, it does not scale to large models and/or accurate inner-loop optimizers with many steps. Next we propose an alternative formulation for dataset condensation.
+
+# 2.2 DATASET CONDENSATION WITH PARAMETER MATCHING
+
+Here we aim to learn $s$ such that the model $\phi _ { \pmb { \theta } ^ { s } }$ trained on them achieves not only comparable generalization performance to $\phi _ { \pmb { \theta } ^ { \top } }$ but also converges to a similar solution in the parameter space (i.e. $\mathbf { \boldsymbol { \theta } } ^ { s } \approx \mathbf { \boldsymbol { \theta } } ^ { \mathcal { T } }$ ). Let $\phi _ { \pmb { \theta } }$ be a locally smooth function2, similar weights $( \pmb { \theta } ^ { S } \approx \pmb { \theta } ^ { \dagger } )$ ) imply similar mappings in a local neighborhood and thus generalization performance, i.e. $\mathbb { E } _ { { \pmb x } \sim P _ { \mathcal { D } } } [ \ell ( \phi _ { \pmb \theta ^ { \top } } ( { \pmb x } ) , y ) ] \simeq$ $\mathbb { E } _ { { \pmb x } \sim P _ { D } } [ \ell ( \phi _ { \pmb \theta } s ( { \pmb x } ) , y ) ]$ . Now we can formulate this goal as
+
+$$
+\operatorname* { m i n } _ { \boldsymbol { \mathcal { S } } } D \big ( \boldsymbol { \theta } ^ { \boldsymbol { \mathcal { S } } } , \boldsymbol { \theta } ^ { \mathcal { T } } \big ) \quad \mathrm { s u b j e c t ~ t o } \quad \boldsymbol { \theta } ^ { \boldsymbol { \mathcal { S } } } ( \boldsymbol { \mathcal { S } } ) = \underset { \boldsymbol { \theta } } { \arg \operatorname* { m i n } } \mathcal { L } ^ { \boldsymbol { S } } ( \boldsymbol { \theta } )
+$$
+
+where $\begin{array} { r } { \pmb { \theta } ^ { \mathcal { T } } = \arg \operatorname* { m i n } _ { \pmb { \theta } } \mathcal { L } ^ { \mathcal { T } } ( \pmb { \theta } ) } \end{array}$ and $D ( \cdot , \cdot )$ is a distance function. In a deep neural network, $\pmb { \theta } ^ { \mathcal { T } }$ typically depends on its initial values $\pmb { \theta } _ { 0 }$ . However, the optimization in eq. (4) aims to obtain an optimum set of synthetic images only for one model $\phi _ { \pmb { \theta } ^ { \tau } }$ with the initialization $\pmb { \theta } _ { 0 }$ , while our actual goal is to generate samples that can work with a distribution of random initializations $P _ { \pmb { \theta } _ { 0 } }$ . Thus we modify eq. (4) as follows:
+
+$$
+\operatorname* { m i n } _ { \mathcal { S } } \mathrm { E } _ { \theta _ { 0 } \sim P _ { \theta _ { 0 } } } [ D ( \theta ^ { S } ( \theta _ { 0 } ) , \theta ^ { T } ( \theta _ { 0 } ) ) ] \quad \mathrm { s u b j e c t ~ t o } \quad \theta ^ { S } ( \mathcal { S } ) = \operatorname * { a r g m i n } _ { \theta } \mathcal { L } ^ { S } ( \theta ( \theta _ { 0 } ) )
+$$
+
+where $\begin{array} { r } { \pmb { \theta } ^ { \mathcal { T } } = \arg \operatorname* { m i n } _ { \pmb { \theta } } \mathcal { L } ^ { \mathcal { T } } ( \pmb { \theta } ( \pmb { \theta } _ { 0 } ) ) } \end{array}$ . For brevity, we use only $\pmb { \theta } ^ { S }$ and $\pmb { \theta } ^ { \mathcal { T } }$ to indicate $\theta ^ { S } ( \theta _ { 0 } )$ and $\pmb { \theta } ^ { \mathcal { T } } ( \pmb { \theta } _ { 0 } )$ respectively in the next sections. The standard approach to solving eq. (5) employs implicit differentiation (see (Domke, 2012) for details), which involves solving an inner loop optimization for $\theta ^ { S }$ . As the inner loop optimization $\begin{array} { r } { \pmb { \theta } ^ { S } ( S ) = \arg \operatorname* { m i n } _ { \pmb { \theta } } \mathcal { L } ^ { S } ( \pmb { \theta } ) } \end{array}$ can be computationally expensive in case of large-scale models, one can adopt the back-optimization approach in (Domke, 2012) which re-defines $\bar { \pmb { \theta } } ^ { S }$ as the output of an incomplete optimization:
+
+$$
+{ \pmb \theta } ^ { S } ( S ) = \mathrm { o p t } - \mathrm { a 1 q } _ { \pmb { \theta } } ( \mathcal { L } ^ { S } ( \pmb { \theta } ) , \varsigma )
+$$
+
+where $\tt { o p t - a l g }$ is a specific optimization procedure with a fixed number of steps (ς).
+
+In practice, $\pmb { \theta } ^ { \mathcal { T } }$ for different initializations can be trained first in an offline stage and then used as the target parameter vector in eq. (5). However, there are two potential issues by learning to regress $\pmb { \theta } ^ { \mathcal { T } }$ as the target vector. First the distance between $\pmb { \theta } ^ { \mathcal { T } }$ and intermediate values of $\pmb { \theta } ^ { S }$ can be too big in the parameter space with multiple local minima traps along the path and thus it can be too challenging to reach. Second $\tt { o p t - a l g }$ involves a limited number of optimization steps as a tradeoff between speed and accuracy which may not be sufficient to take enough steps for reaching the optimal solution. These problems are similar to those of (Wang et al., 2018), as they both involve parameterizing $\theta ^ { S }$ with $s$ and $\pmb { \theta } _ { 0 }$ .
+
+# 2.3 DATASET CONDENSATION WITH CURRICULUM GRADIENT MATCHING
+
+Here we propose a curriculum based approach to address the above mentioned challenges. The key idea is that we wish $\theta ^ { S }$ to be close to not only the final $\pmb { \theta } ^ { \mathcal { T } }$ but also to follow a similar path to $\pmb { \theta } ^ { \check { T } }$ throughout the optimization. While this can restrict the optimization dynamics for $\pmb { \theta }$ , we argue that it also enables a more guided optimization and effective use of the incomplete optimizer. We can now decompose eq. (5) into multiple subproblems:
+
+$$
+\begin{array} { c } { \displaystyle \operatorname* { m i n } _ { \mathcal { S } } \mathrm { E } _ { \theta _ { 0 } \sim P _ { \theta _ { 0 } } } [ \sum _ { t = 0 } ^ { T - 1 } D ( \theta _ { t } ^ { \mathcal { S } } , \theta _ { t } ^ { \mathcal { T } } ) ] \quad \mathrm { s u b j e c t ~ t o } } \\ { \theta _ { t + 1 } ^ { \mathcal { S } } ( \mathcal { S } ) = \mathrm { o p t - } \mathsf { a l g } _ { \theta } ( \mathcal { L } ^ { \mathcal { S } } ( \theta _ { t } ^ { \mathcal { S } } ) , \varsigma ^ { \mathcal { S } } ) \quad \mathrm { a n d } \quad \theta _ { t + 1 } ^ { \mathcal { T } } = \mathrm { o p t - } \mathsf { a l g } _ { \theta } ( \mathcal { L } ^ { \mathcal { T } } ( \theta _ { t } ^ { \mathcal { T } } ) , \varsigma ^ { \mathcal { T } } ) } \end{array}
+$$
+
+where $T$ is the number of iterations, $\varsigma ^ { \mathcal { S } }$ and $\varsigma ^ { \mathcal { T } }$ are the numbers of optimization steps for $\pmb { \theta } ^ { S }$ and $\pmb { \theta } ^ { \mathcal { T } }$ respectively. In words, we wish to generate a set of condensed samples $s$ such that the network parameters trained on them $( \pmb { \theta } _ { t } ^ { S } )$ are similar to the ones trained on the original training set $( \pmb { \theta } _ { t } ^ { \mathcal { T } } )$ at each iteration $t$ . In our preliminary experiments, we observe that $\theta _ { t + 1 } ^ { S }$ , which is parameterized with $s$ , can successfully track $\theta _ { t + 1 } ^ { \mathcal { T } }$ by updating $s$ and minimizing $D ( \theta _ { t } ^ { S } , \theta _ { t } ^ { T } )$ close to zero.
+
+In the case of one step gradient descent optimization for opt-alg, the update rule is:
+
+$$
+\pmb { \theta } _ { t + 1 } ^ { S }  \pmb { \theta } _ { t } ^ { S } - \eta _ { \theta } \nabla _ { \pmb { \theta } } \mathcal { L } ^ { S } ( \pmb { \theta } _ { t } ^ { S } ) \quad \mathrm { a n d } \quad \pmb { \theta } _ { t + 1 } ^ { T }  \pmb { \theta } _ { t } ^ { T } - \eta _ { \theta } \nabla _ { \pmb { \theta } } \mathcal { L } ^ { T } ( \pmb { \theta } _ { t } ^ { T } ) ,
+$$
+
+where $\eta _ { \theta }$ is the learning rate. Based on our observation $( D ( \theta _ { t _ { - } } ^ { S } , \theta _ { t } ^ { T } ) \approx 0 )$ ), we simplify the formulation in eq. (7) by replacing $\theta _ { t } ^ { \mathcal { T } }$ with $\pmb { \theta } _ { t } ^ { S }$ and use $\pmb { \theta }$ to denote $\pmb { \theta } ^ { S }$ in the rest of the paper:
+
+$$
+\operatorname* { m i n } _ { \boldsymbol { S } } \mathrm { E } _ { \boldsymbol { \theta } _ { 0 } \sim P _ { \boldsymbol { \theta } _ { 0 } } } [ \sum _ { t = 0 } ^ { T - 1 } \boldsymbol { D } ( \nabla _ { \boldsymbol { \theta } } \mathcal { L } ^ { \boldsymbol { S } } ( \boldsymbol { \theta } _ { t } ) , \nabla _ { \boldsymbol { \theta } } \mathcal { L } ^ { T } ( \boldsymbol { \theta } _ { t } ) ) ] .
+$$
+
+We now have a single deep network with parameters $\pmb \theta$ trained on the synthetic set $s$ which is optimized such that the distance between the gradients for the loss over the training samples $\mathcal { L } ^ { \mathcal { T } }$ w.r.t. $\pmb \theta$ and the gradients for the loss over the condensed samples $\mathcal { L } ^ { s }$ w.r.t. $\pmb \theta$ is minimized. In words, our goal reduces to matching the gradients for the real and synthetic training loss w.r.t. $\pmb \theta$ via updating the condensed samples. This approximation has the key advantage over (Wang et al., 2018) and eq. (5) that it does not require the expensive unrolling of the recursive computation graph over the previous parameters $\big \{ \pmb { \theta } _ { 0 } , \dots , \pmb { \theta } _ { t - 1 } \big \}$ . The important consequence is that the optimization is significantly faster, memory efficient and thus scales up to the state-of-the-art deep neural networks (e.g. ResNet (He et al., 2016)).
+
+Discussion. The synthetic data contains not only samples but also their labels $( s , y )$ that can be jointly learned by optimizing eq. (9) in theory. However, their joint optimization is challenging, as the content of the samples depend on their label and vice-versa. Thus in our experiments we learn to synthesize images for fixed labels, e.g. one synthetic image per class.
+
+Algorithm. We depict the optimization details in Alg. 1. At the outer level, it contains a loop over random weight initializations, as we want to obtain condensed images that can later be used to train previously unseen models. Once $\pmb \theta$ is randomly initialized, we use $\phi _ { \pmb { \theta } }$ to first compute the loss over both the training samples $( \mathcal { L } ^ { \mathcal { T } } )$ , synthetic samples $( \mathcal { L } ^ { s } )$ and their gradients w.r.t. $\pmb \theta$ , then optimize the synthetic samples $s$ to match these gradients $\nabla _ { \pmb { \theta } } \mathcal { L } ^ { S }$ to $\nabla _ { \pmb { \theta } } \mathcal { L } ^ { T }$ by applying $\varsigma _ { S }$ gradient descent steps with learning rate $\eta _ { S }$ . We use the stochastic gradient descent optimization for both $\mathsf { o p t } - \mathsf { a l g } _ { \theta }$ and $\mathtt { o p t - a l g } _ { S }$ . Next we train $\pmb \theta$ on the updated synthetic images by minimizing the loss $\mathcal { L } ^ { s }$ with learning rate $\eta _ { \theta }$ for $\varsigma _ { \pmb { \theta } }$ steps. Note that we sample each real and synthetic batch pair from $\tau$ and $s$ containing samples from a single class and the synthetic data for each class are separately (or parallelly) updated at each iteration $\mathbf { \rho } ( t )$ for the following reasons: i) this reduces memory use at train time, ii) imitating the mean gradients w.r.t. the data from single class is easier compared to those of multiple classes. This does not bring any extra computational cost.
+
+# Algorithm 1: Dataset condensation with gradient matching
+
+Input: Training set $\tau$   
+1 Required: Randomly initialized set of synthetic samples $s$ for $C$ classes, probability distribution over randomly initialized weights $P _ { \pmb { \theta } _ { 0 } }$ , deep neural network $\phi _ { \pmb { \theta } }$ , number of outer-loop steps $K$ , number of inner-loop steps $T$ , number of steps for updating weights $\varsigma _ { \pmb { \theta } }$ and synthetic samples $\varsigma s$ in each inner-loop step respectively, learning rates for updating weights $\eta _ { \theta }$ and synthetic samples $\eta _ { S }$ .   
+2 for $k = 0 , \cdots , K - 1$ do   
+3 Initialize $\theta _ { 0 } \sim P _ { \theta _ { 0 } }$   
+4 for $t = 0 , \cdots , T - 1 \boldsymbol { \mathfrak { c } }$ o   
+5 for $c = 0 , \cdots , C - 1$ do   
+6 Sample a minibatch pair $B _ { c } ^ { \tau } \sim \tau$ and $B _ { c } ^ { s } \sim \mathcal { S } \qquad \triangleright B _ { c } ^ { \tau }$ and $B _ { c } ^ { S }$ are of the same class $c$ .   
+7 Compute $\begin{array} { r } { \mathcal { L } _ { c } ^ { \mathcal { T } } = \frac { 1 } { | B _ { c } ^ { \mathcal { T } } | } \sum _ { ( \pmb { x } , y ) \in B _ { c } ^ { \mathcal { T } } } \ell ( \phi _ { \pmb { \theta } _ { t } } ( \pmb { x } ) , y ) } \end{array}$ and $\begin{array} { r } { \mathcal { L } _ { c } ^ { S } = \frac { 1 } { | B _ { c } ^ { S } | } \sum _ { ( s , y ) \in B _ { c } ^ { S } } \ell ( \phi _ { \pmb { \theta } _ { t } } ( s ) , y ) } \end{array}$   
+8 Update $\boldsymbol { \mathcal { S } } _ { c } \gets \mathrm { o p t - a l g } _ { \boldsymbol { s } } ( D ( \nabla _ { \boldsymbol { \theta } } \mathcal { L } _ { c } ^ { S } ( \boldsymbol { \theta } _ { t } ) , \nabla _ { \boldsymbol { \theta } } \mathcal { L } _ { c } ^ { T } ( \boldsymbol { \theta } _ { t } ) ) , \varsigma _ { \boldsymbol { S } } , \eta _ { S } )$   
+9 Update $\pmb { \theta } _ { t + 1 } \gets \mathsf { o p t } - \mathsf { a l g } _ { \pmb { \theta } } \big ( \mathcal { L } ^ { S } ( \pmb { \theta } _ { t } ) , \varsigma \pmb { \theta } , \eta _ { \pmb { \theta } } \big )$ . Use the whole $s$
+
+Output: S
+
+Gradient matching loss. The matching loss $D ( \cdot , \cdot )$ in eq. (9) measures the distance between the gradients for $\mathcal { L } ^ { s }$ and $\mathcal { L } ^ { \mathcal { T } }$ w.r.t. $\pmb \theta$ . When $\phi _ { \pmb { \theta } }$ is a multi-layered neural network, the gradients correspond to a set of learnable 2D $( \mathsf { o u t } \times \mathrm { i n } )$ ) and 4D $\mathsf { o u t } \times \mathrm { i n } \times \mathrm { h } \times \mathrm { w } )$ weights for each fully connected (FC) and convolutional layer resp where out, in, h, w are number of output and input channels, kernel height and width resp. The matching loss can be decomposed into a sum of layerwise losses as $\begin{array} { r } { D ( \nabla _ { \theta } \bar { \mathcal { L } } ^ { \mathcal { S } } , \nabla _ { \theta } \mathcal { L } ^ { \mathcal { T } } ) = \sum _ { l = 1 } ^ { L } d \big ( \nabla _ { \theta ^ { ( l ) } } \mathcal { L } ^ { \mathcal { S } } , \nabla _ { \theta ^ { ( l ) } } \mathcal { L } ^ { \mathcal { T } } \big ) } \end{array}$ where $l$ is the layer index, $L$ is the number of layers with weights and
+
+$$
+d ( \mathbf { A } , \mathbf { B } ) = \sum _ { i = 1 } ^ { \mathrm { o u t } } \left( 1 - { \frac { \mathbf { A _ { i } } \cdot \mathbf { B _ { i } } . } { \| \mathbf { A _ { i } } . \| \| \mathbf { B _ { i } } . \| } } \right)
+$$
+
+where $\mathbf { A } _ { i }$ · and $\mathbf { B } _ { i }$ · are flattened vectors of gradients corresponding to each output node $i$ , which is in dimensional for FC weights and $\mathtt { i n } \times \mathtt { h } \times \mathtt { w }$ dimensional for convolutional weights. In contrast to (Lopez-Paz et al., 2017; Aljundi et al., 2019; Zhu et al., 2019) that ignore the layer-wise structure by flattening tensors over all layers to one vector and then computing the distance between two vectors, we group them for each output node. We found that this is a better distance for gradient matching (see the supplementary) and enables using a single learning rate across all layers.
+
+# 3 EXPERIMENTS
+
+# 3.1 DATASET CONDENSATION
+
+First we evaluate classification performance with the condensed images on four standard benchmark datasets: digit recognition on MNIST (LeCun et al., 1998), SVHN (Netzer et al., 2011) and object classification on FashionMNIST (Xiao et al., 2017), CIFAR10 (Krizhevsky et al., 2009). We test our method using six standard deep network architectures: MLP, ConvNet (Gidaris & Komodakis, 2018), LeNet (LeCun et al., 1998), AlexNet (Krizhevsky et al., 2012), VGG-11 (Simonyan & Zisserman, 2014) and ResNet-18 (He et al., 2016). MLP is a multilayer perceptron with two nonlinear hidden layers, each has 128 units. ConvNet is a commonly used modular architecture in few-shot learning (Snell et al., 2017; Vinyals et al., 2016; Gidaris & Komodakis, 2018) with $D$ duplicate blocks, and each block has a convolutional layer with $W$ $\left( 3 \times 3 \right)$ filters, a normalization layer $N$ , an activation layer $A$ and a pooling layer $P$ , denoted as $[ W , N , A , P ] \times D$ . The default ConvNet (unless specified otherwise) includes 3 blocks, each with 128 filters, followed by InstanceNorm (Ulyanov et al., 2016), ReLU and AvgPooling modules. The final block is followed by a linear classifier. We use Kaiming initialization (He et al., 2015) for network weights. The synthetic images can be initialized from Gaussian noise or randomly selected real training images. More details about the datasets, networks and hyper-parameters can be found in the supplementary.
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Img/Cls</td><td rowspan="2">Ratio %</td><td colspan="4">Coreset Selection</td><td rowspan="2">Ours</td><td rowspan="2">Whole Dataset</td></tr><tr><td>Random</td><td>Herding</td><td>K-Center</td><td>Forgetting</td></tr><tr><td rowspan="3">MNIST</td><td>1</td><td>0.017</td><td>64.9±3.5</td><td>89.2±1.6</td><td>89.3±1.5</td><td>35.5±5.6</td><td>91.7±0.5</td><td rowspan="3">99.6±0.0</td></tr><tr><td>10</td><td>0.17</td><td>95.1±0.9</td><td>93.7±0.3</td><td>84.4±1.7</td><td>68.1±3.3</td><td>97.4±0.2</td></tr><tr><td>50</td><td>0.83</td><td>97.9±0.2</td><td>94.9±0.2</td><td>97.4±0.3</td><td>88.2±1.2</td><td>98.8±0.2</td></tr><tr><td rowspan="3">FashionMNIST</td><td>1</td><td>0.017</td><td>51.4±3.8</td><td>67.0±1.9</td><td>66.9±1.8</td><td>42.0±5.5</td><td>70.5±0.6</td><td rowspan="3">93.5±0.1</td></tr><tr><td>10</td><td>0.17</td><td>73.8±0.7</td><td>71.1±0.7</td><td>54.7±1.5</td><td>53.9±2.0</td><td>82.3±0.4</td></tr><tr><td>50</td><td>0.83</td><td>82.5±0.7</td><td>71.9±0.8</td><td>68.3±0.8</td><td>55.0±1.1</td><td>83.6±0.4</td></tr><tr><td rowspan="3">SVHN</td><td>1</td><td>0.014</td><td>14.6±1.6</td><td>20.9±1.3</td><td>21.0±1.5</td><td>12.1±1.7</td><td>31.2±1.4</td><td rowspan="3">95.4±0.1</td></tr><tr><td>10</td><td>0.14</td><td>35.1±4.1</td><td>50.5±3.3</td><td>14.0±1.3</td><td>16.8±1.2</td><td>76.1±0.6</td></tr><tr><td>50</td><td>0.7</td><td>70.9±0.9</td><td>72.6±0.8</td><td>20.1±1.4</td><td>27.2±1.5</td><td>82.3±0.3</td></tr><tr><td rowspan="3">CIFAR10</td><td>1</td><td>0.02</td><td>14.4±2.0</td><td>21.5±1.2</td><td>21.5±1.3</td><td>13.5±1.2</td><td>28.3±0.5</td><td rowspan="3">84.8±0.1</td></tr><tr><td>10</td><td>0.2</td><td>26.0±1.2</td><td>31.6±0.7</td><td>14.7±0.9</td><td>23.3±1.0</td><td>44.9±0.5</td></tr><tr><td>50</td><td>1</td><td>43.4±1.0</td><td>40.4±0.6</td><td>27.0±1.4</td><td>23.3±1.1</td><td>53.9±0.5</td></tr></table>
+
+Table 1: The performance comparison to coreset methods. This table shows the testing accuracies $( \% )$ of different methods on four datasets. ConvNet is used for training and testing. $\mathrm { I m g / C l s } .$ : image(s) per class, Ratio $( \% )$ : the ratio of condensed images to whole training set.
+
+The pipeline for dataset condensation has two stages: learning the condensed images (denoted as C) and training classifiers from scratch on them (denoted as T). Note that the model architectures used in two stages might be different. For the coreset baselines, the coreset is selected in the first stage. We investigate three settings: 1, 10 and 50 image/class learning, which means that the condensed set or coreset contains 1, 10 and 50 images per class respectively. Each method is run for 5 times, and 5 synthetic sets are generated in the first stage; each generated synthetic set is used to train 20 randomly initialized models in the second stage and evaluated on the test set, which amounts to evaluating 100 models in the second stage. In all experiments, we report the mean and standard deviation of these 100 testing results.
+
+Baselines. We compare our method to four coreset baselines (Random, Herding, K-Center and Forgetting) and also to DD (Wang et al., 2018). In Random, the training samples are randomly selected as the coreset. Herding baseline, which selects closest samples to the cluster center, is based on (Welling, 2009) and used in (Rebuffi et al., 2017; Castro et al., 2018; Wu et al., 2019; Belouadah & Popescu, 2020). K-Center (Wolf, 2011; Sener & Savarese, 2018) picks multiple center points such that the largest distance between a data point and its nearest center is minimized. For Herding and K-Center, we use models trained on the whole dataset to extract features, compute $l _ { 2 }$ distance to centers. Forgetting method (Toneva et al., 2019) selects the training samples which are easy to forget during training. We do not compare to GSS-Greedy (Aljundi et al., 2019), because it is also a similarity based greedy algorithm like K-Center, but GSS-Greedy trains an online learning model to measure the similarity of samples, which is different from general image classification problem. More detailed comparisons can be found in the supplementary.
+
+Comparison to coreset methods. We first compare our method to the coreset baselines on MNIST, FashionMNIST, SVHN and CIFAR10 in Table 1 using the default ConvNet in classification accuracy. Whole dataset indicates training on the whole original set which serves as an approximate upper-bound performance. First we observe that our method outperforms all the baselines significantly and achieves a comparable result $( 9 8 . 8 \% )$ in case of 50 images per class to the upper bound $( 9 9 . 6 \% )$ in MNIST which uses two orders of magnitude more training images per class (6000). We also obtain promising results in FashionMNIST, however, the gap between our method and upper bound is bigger in SVHN and CIFAR10 which contain more diverse images with varying foregrounds and backgrounds. We also observe that, (i) the random selection baseline is competitive to other coreset methods in 10 and 50 images per class and (ii) herding method is on average the best coreset technique. We visualize the condensed images produced by our method under 1 image/class setting in Figure 2. Interestingly they are interpretable and look like “prototypes” of each class.
+
+![](images/33d2777d8c901f7c795ee20141f00794aa28593aa6adaf863c4cfc99d648c3ff.jpg)  
+Figure 2: Visualization of condensed 1 image/class with ConvNet for MNIST, FashionMNIST, SVHN and CIFAR10.
+
+Table 2: Cross-architecture performance in testing accuracy $( \% )$ for condensed 1 image/class in MNIST.   
+
+<table><tr><td>C\T</td><td>MLP</td><td>ConvNet</td><td>LeNet</td><td>AlexNet</td><td>VGG</td><td>ResNet</td></tr><tr><td>MLP</td><td>70.5±1.2</td><td>63.9±6.5</td><td>77.3±5.8</td><td>70.9±11.6</td><td>53.2±7.0</td><td>80.9±3.6</td></tr><tr><td>ConvNet</td><td>69.6±1.6</td><td>91.7±0.5</td><td>85.3±1.8</td><td>85.1±3.0</td><td>83.4±1.8</td><td>90.0±0.8</td></tr><tr><td>LeNet</td><td>71.0±1.6</td><td>90.3±1.2</td><td>85.0±1.7</td><td>84.7±2.4</td><td>80.3±2.7</td><td>89.0±0.8</td></tr><tr><td>AlexNet</td><td>72.1±1.7</td><td>87.5±1.6</td><td>84.0±2.8</td><td>82.7±2.9</td><td>81.2±3.0</td><td>88.9±1.1</td></tr><tr><td>VGG</td><td></td><td>70.3±1.6 90.1±0.7</td><td>83.9±2.7</td><td>83.4±3.7</td><td></td><td>81.7±2.6 89.1±0.9</td></tr><tr><td>ResNet</td><td>73.6±1.2</td><td>91.6±0.5</td><td>86.4±1.5</td><td>85.4±1.9</td><td>83.4±2.4</td><td>89.4±0.9</td></tr></table>
+
+Table 3: Comparison to DD (Wang et al., 2018) in terms of testing accuracy $( \% )$ .   
+
+<table><tr><td>Dataset</td><td>Img/Cls</td><td>DD</td><td>Ours</td><td>| Whole Dataset</td></tr><tr><td>MNIST</td><td>1 10</td><td>79.5±8.1 -</td><td>85.0±1.6 93.9±0.6</td><td>99.5±0.0</td></tr><tr><td>CIFAR10</td><td>1 10</td><td>-</td><td>24.2±0.9 36.8±1.2 39.1±1.2</td><td>83.1±0.2</td></tr></table>
+
+Table 4: Neural Architecture Search. Methods are compared in performance, ranking correlation, time and memory cost.
+
+<table><tr><td></td><td></td><td>Random Herding</td><td></td><td>Ours Early-stopping</td><td>|Whole Dataset</td></tr><tr><td>Performance (%)</td><td>76.2</td><td>76.2</td><td>84.5</td><td>84.5</td><td>85.9</td></tr><tr><td>Correlation</td><td>-0.21</td><td>-0.20</td><td>0.79</td><td>0.42</td><td>1.00</td></tr><tr><td>Time cost (min)</td><td>18.8</td><td>18.8</td><td>18.8</td><td>18.8</td><td>8604.3</td></tr><tr><td>Storage (imgs)</td><td>10²</td><td>102</td><td>10²</td><td>104</td><td>5×104</td></tr></table>
+
+Comparison to DD (Wang et al., 2018). Unlike the setting in Table 1, DD (Wang et al., 2018) reports results only for 10 images per class on MNIST and CIFAR10 over LeNet and AlexCifarNet (a customized AlexNet). We strictly follow the experimental setting in (Wang et al., 2018), use the same architectures and report our and their original results in Table 3 for a fair comparison. Our method achieves significantly better performance than DD on both benchmarks; obtains $5 \%$ higher accuracy with only 1 synthetic sample per class than DD with 10 samples per class. In addition, our method obtains consistent results over multiple runs with a standard deviation of only $0 . 6 \%$ on MNIST, while DD’s performance significantly vary over different runs $( 8 . 1 \% )$ . Finally our method trains 2 times faster than DD and requires $5 \dot { 0 } \%$ less memory on CIFAR10 experiments. More detailed runtime and qualitative comparison can be found in the supplementary.
+
+Cross-architecture generalization. Another key advantage of our method is that the condensed images learned using one architecture can be used to train another unseen one. Here we learn 1 condensed image per class for MNIST over a diverse set of networks including MLP, ConvNet (Gidaris & Komodakis, 2018), LeNet (LeCun et al., 1998), AlexNet (Krizhevsky et al., 2012), VGG-11 (Simonyan & Zisserman, 2014) and ResNet-18 (He et al., 2016) (see Table 2). Once the condensed sets are synthesized, we train every network on all the sets separately from scratch and evaluate their cross architecture performance in terms of classification accuracy on the MNIST test set. Table 2 shows that the condensed images, especially the ones that are trained with convolutional networks, perform well and are thus architecture generic. MLP generated images do not work well for training convolutional architectures which is possibly due to the mismatch between translation invariance properties of MLP and convolutional networks. Interestingly, MLP achieves better performance with convolutional network generated images than the MLP generated ones. The best results are obtained in most cases with ResNet generated images and ConvNet or ResNet as classifiers which is inline with their performances when trained on the original dataset.
+
+Number of condensed images. We also study the test performance of a ConvNet trained on them for MNIST, FashionMNIST, SVHN and CIFAR10 for various number of condensed images per class in Figure 3 in absolute and relative terms – normalized by its upper-bound. Increasing the number of condensed images improves the accuracies in all benchmarks and further closes the gap with the upper-bound performance especially in MNIST and FashionMNIST, while the gap remains larger in SVHN and CIFAR10. In addition, our method outperforms the coreset method - Herding by a large margin in all cases.
+
+Activation, normalization $\pmb { \& }$ pooling. We also study the effect of various activation (sigmoid, ReLU (Nair & Hinton, 2010; Zeiler et al., 2013), leaky ReLU (Maas et al., 2013)), pooling (max, average) and normalization functions (batch (Ioffe & Szegedy, 2015), group (Wu & He, 2018), layer (Ba et al., 2016), instance norm (Ulyanov et al., 2016)) and have the following observations: i) leaky ReLU over ReLU and average pooling over max pooling enable learning better condensed images, as they allow for denser gradient flow; ii) instance normalization obtains better classification performance than its alternatives when used in the networks that are trained on a small set of condensed images. We refer to the supplementary for detailed results and discussion.
+
+![](images/65d0fba993df8bfd499c3ebdab629152da49d451b0467a0f02bd565002a165a1.jpg)  
+Figure 3: Absolute and relative testing accuracies for varying the number of condensed images/class for MNIST, FashionMNIST, SVHN and CIFAR10. The relative accuracy means the ratio compared to its upperbound, i.e. training with the whole dataset.
+
+![](images/a5e8ba0d35c1d6ab80dd013dc8aa19a6b906d27c9e5cdb7cbf097a128c1d52ca.jpg)  
+Figure 4: Continual learning performance in accuracy $( \% )$ . Herding denotes the original E2E (Castro et al., 2018). T1, T2, T3 are three learning stages. The performance at each stage is the mean testing accuracy on all learned tasks.
+
+# 3.2 APPLICATIONS
+
+Continual Learning First we apply our method to a continual-learning scenario (Rebuffi et al., 2017; Castro et al., 2018) where new tasks are learned incrementally and the goal is to preserve the performance on the old tasks while learning the new ones. We build our model on E2E method in (Castro et al., 2018) that uses a limited budget rehearsal memory (we consider 10 images/class here) to keep representative samples from the old tasks and knowledge distillation (KD) to regularize the network’s output w.r.t. to previous predictions. We replace its sample selection mechanism (herding) with ours such that a set of condensed images are generated and stored in the memory, keep the rest of the model same and evaluate this model on the task-incremental learning problem on the digit recognition datasets, SVHN (Netzer et al., 2011), MNIST (LeCun et al., 1998) and USPS (Hull, 1994) in the same order. MNIST and USPS images are reshaped to $3 2 \times 3 2$ RGB images.
+
+We compare our method to E2E (Castro et al., 2018), depicted as herding in Figure 4, with and without KD regularization. The experiment contains 3 incremental training stages $( \mathrm { S V H N { \to } M N I S T { \to } U S P S } )$ and testing accuracies are computed by averaging over the test sets of the previous and current tasks after each stage. The desired outcome is to obtain high mean classification accuracy at T3. The results indicate that the condensed images are more data-efficient than the ones sampled by herding and thus our method outperforms E2E in both settings, while by a larger margin $2 . 3 \%$ at T3) when KD is not employed.
+
+Neural Architecture Search. Here we explore the use of our method in a simple neural architecture search (NAS) experiment on CIFAR10 which typically requires expensive training of numerous architectures multiple times on the whole training set and picking the best performing ones on a validation set. Our goal is to verify that our condensed images can be used to efficiently train multiple networks to identify the best network. To this end, we construct the search space of 720 ConvNets as described in Section 3.1 by varying hyper-parameters $W , N , A , P ,$ $D$ over an uniform grid (see supplementary for more details), train them for 100 epochs on three small proxy datasets (10 images/class) that are obtained with Random sampling, Herding and our method. Note that we train the condensed images for once only with the default ConvNet architecture and use them to train all kinds of architectures. We also compare to early-stopping (Li & Talwalkar, 2020) in which the model is trained on whole training set but with the same number of training iterations as the one required for the small proxy datasets, in other words, for the same amount of computations.
+
+Table 4 depicts i) the average test performance of the best selected model over 5 runs when trained on the whole dataset, ii) Spearman’s rank correlation coefficient between the validation accuracies obtained by training the selected top 10 models on the proxy dataset and whole dataset, iii) time for training 720 architectures on a NVIDIA GTX1080-Ti GPU, and iv) memory print of the training images. Our method achieves the highest testing performance $( 8 4 . 5 \% )$ and performance correlation (0.79), meanwhile significantly decreases the the searching time $( \mathrm { f r o m ~ } 8 6 0 4 . 3 $ to 18.8 minutes) and storage space (from $5 \times \mathrm { { 1 0 ^ { 4 } } }$ to $1 \times 1 0 ^ { 2 }$ images) compared to whole-dataset training. The competitive early-stopping baseline achieves on par performance for the best performing model with ours, however, the rank correlation (0.42) of top 10 models is significantly lower than ours (0.79) which indicates unreliable correlation of performances between early-stopping and whole-dataset training. Furthermore, early-stopping needs 100 times as many training images as ours needs. Note that the training time for synthetic images is around 50 minutes (for $K = 5 0 0$ ) which is one time off and negligible cost when training thousands even millions of candidate architectures in NAS.
+
+# 4 CONCLUSION
+
+In this paper, we propose a dataset condensation method that learns to synthesize a small set of informative images. We show that these images are significantly more data-efficient than the same number of original images and the ones produced by the previous method, and they are not architecture dependent, can be used to train different deep networks. Once trained, they can be used to lower the memory print of datasets and efficiently train numerous networks which are crucial in continual learning and neural architecture search respectively. For future work, we plan to explore the use of condensed images in more diverse and thus challenging datasets like ImageNet (Deng et al., 2009) that contain higher resolution images with larger variations in appearance and pose of objects, background.
+
+Acknowledgment. This work is funded by China Scholarship Council 201806010331 and the EPSRC programme grant Visual AI EP/T028572/1. We thank Iain Murray and Oisin Mac Aodha for their valuable feedback.
+
+# REFERENCES
+
+Pankaj K Agarwal, Sariel Har-Peled, and Kasturi R Varadarajan. Approximating extent measures of points. Journal of the ACM (JACM), 51(4):606–635, 2004.
+
+Rahaf Aljundi, Min Lin, Baptiste Goujaud, and Yoshua Bengio. Gradient based sample selection for online continual learning. In Advances in Neural Information Processing Systems, pp. 11816– 11825, 2019.
+
+Jimmy Ba and Rich Caruana. Do deep nets really need to be deep? In Advances in neural information processing systems, pp. 2654–2662, 2014.
+
+Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016.
+
+Eden Belouadah and Adrian Popescu. Scail: Classifier weights scaling for class incremental learning. In The IEEE Winter Conference on Applications of Computer Vision, 2020.
+
+Ondrej Bohdal, Yongxin Yang, and Timothy Hospedales. Flexible dataset distillation: Learn labels instead of images. Neural Information Processing Systems Workshop, 2020.
+
+Cristian Bucilua, Rich Caruana, and Alexandru Niculescu-Mizil. Model compression. In ˇ Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 535–541, 2006.
+
+Francisco M Castro, Manuel J Mar´ın-Jimenez, Nicol ´ as Guil, Cordelia Schmid, and Karteek Alahari.´ End-to-end incremental learning. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 233–248, 2018.
+
+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.
+
+Justin Domke. Generic methods for optimization-based modeling. In Artificial Intelligence and Statistics, pp. 318–326, 2012.
+
+Dan Feldman, Melanie Schmidt, and Christian Sohler. Turning big data into tiny data: Constant-size coresets for k-means, pca and projective clustering. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pp. 1434–1453. SIAM, 2013.
+
+Spyros Gidaris and Nikos Komodakis. Dynamic few-shot visual learning without forgetting. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4367– 4375, 2018.
+
+Jack Goetz and Ambuj Tewari. Federated learning via synthetic data. arXiv preprint arXiv:2008.04489, 2020.
+
+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, 2014a.
+
+Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. arXiv preprint arXiv:1412.6572, 2014b.
+
+Sariel Har-Peled and Soham Mazumdar. On coresets for $\mathbf { k }$ -means and $\mathbf { k }$ -median clustering. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, 2004.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In Proceedings of the IEEE international conference on computer vision, pp. 1026–1034, 2015.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.
+
+Jonathan J. Hull. A database for handwritten text recognition research. IEEE Transactions on pattern analysis and machine intelligence, 16(5):550–554, 1994.
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. ArXiv, abs/1502.03167, 2015.
+
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.
+
+Pang Wei Koh and Percy Liang. Understanding black-box predictions via influence functions. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1885– 1894. JMLR. org, 2017.
+
+Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012.
+
+Yann LeCun, Leon Bottou, Yoshua Bengio, Patrick Haffner, et al. Gradient-based learning applied ´ to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Guang Li, Ren Togo, Takahiro Ogawa, and Miki Haseyama. Soft-label anonymous gastric x-ray image distillation. In 2020 IEEE International Conference on Image Processing (ICIP), pp. 305– 309. IEEE, 2020.
+
+Liam Li and Ameet Talwalkar. Random search and reproducibility for neural architecture search. In Uncertainty in Artificial Intelligence, pp. 367–377. PMLR, 2020.
+
+Raphael Gontijo Lopes, Stefano Fenu, and Thad Starner. Data-free knowledge distillation for deep neural networks. In LLD Workshop at Neural Information Processing Systems (NIPS ), 2017.
+
+David Lopez-Paz et al. Gradient episodic memory for continual learning. In Advances in Neural Information Processing Systems, pp. 6467–6476, 2017.
+
+Andrew L Maas, Awni Y Hannun, and Andrew Y Ng. Rectifier nonlinearities improve neural network acoustic models. In International conference on machine learning (ICML), 2013.
+
+Aravindh Mahendran and Andrea Vedaldi. Understanding deep image representations by inverting them. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 5188–5196, 2015.
+
+Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. arXiv preprint arXiv:1411.1784, 2014.
+
+Vinod Nair and Geoffrey E Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th international conference on machine learning (ICML-10), pp. 807–814, 2010.
+
+Gaurav Kumar Nayak, Konda Reddy Mopuri, Vaisakh Shaj, Venkatesh Babu Radhakrishnan, and Anirban Chakraborty. Zero-shot knowledge distillation in deep networks. In Proceedings of the 36th International Conference on Machine Learning, 2019.
+
+Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning. 2011.
+
+Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015.
+
+Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, Georg Sperl, and Christoph H Lampert. icarl: Incremental classifier and representation learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2001–2010, 2017.
+
+Salah Rifai, Yoshua Bengio, Yann Dauphin, and Pascal Vincent. A generative process for sampling contractive auto-encoders. arXiv preprint arXiv:1206.6434, 2012.
+
+Adriana Romero, Nicolas Ballas, Samira Ebrahimi Kahou, Antoine Chassang, Carlo Gatta, and Yoshua Bengio. Fitnets: Hints for thin deep nets. arXiv preprint arXiv:1412.6550, 2014.
+
+Kegan GG Samuel and Marshall F Tappen. Learning optimized map estimates in continuouslyvalued mrf models. In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 477–484. IEEE, 2009.
+
+Mert Bulent Sariyildiz and Ramazan Gokberk Cinbis. Gradient matching generative networks for zero-shot learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2168–2178, 2019.
+
+Ozan Sener and Silvio Savarese. Active learning for convolutional neural networks: A core-set approach. ICLR, 2018.
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.
+
+Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Advances in neural information processing systems, pp. 4077–4087, 2017.
+
+Felipe Petroski Such, Aditya Rawal, Joel Lehman, Kenneth O Stanley, and Jeff Clune. Generative teaching networks: Accelerating neural architecture search by learning to generate synthetic training data. International Conference on Machine Learning, 2020.
+
+Ilia Sucholutsky and Matthias Schonlau. Soft-label dataset distillation and text dataset distillation. arXiv preprint arXiv:1910.02551, 2019.
+
+Mariya Toneva, Alessandro Sordoni, Remi Tachet des Combes, Adam Trischler, Yoshua Bengio, and Geoffrey J Gordon. An empirical study of example forgetting during deep neural network learning. ICLR, 2019.
+
+Dmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky. Instance normalization: The missing ingredient for fast stylization. arXiv preprint arXiv:1607.08022, 2016.
+
+Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In Advances in neural information processing systems, 2016.
+
+Tongzhou Wang, Jun-Yan Zhu, Antonio Torralba, and Alexei A Efros. Dataset distillation. arXiv preprint arXiv:1811.10959, 2018.
+
+Max Welling. Herding dynamical weights to learn. In Proceedings of the 26th Annual International Conference on Machine Learning, pp. 1121–1128. ACM, 2009.
+
+G W Wolf. Facility location: concepts, models, algorithms and case studies. 2011.
+
+Yue Wu, Yinpeng Chen, Lijuan Wang, Yuancheng Ye, Zicheng Liu, Yandong Guo, and Yun Fu. Large scale incremental learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 374–382, 2019.
+
+Yuxin Wu and Kaiming He. Group normalization. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 3–19, 2018.
+
+Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017.
+
+Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pp. 818–833. Springer, 2014.
+
+Matthew D. Zeiler, Marc’Aurelio Ranzato, Rajat Monga, Mark Z. Mao, Kyeongcheol Yang, Quoc V. Le, Patrick Nguyen, Andrew W. Senior, Vincent Vanhoucke, Jeffrey Dean, and Geoffrey E. Hinton. On rectified linear units for speech processing. 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 3517–3521, 2013.
+
+Bo Zhao, Konda Reddy Mopuri, and Hakan Bilen. idlg: Improved deep leakage from gradients. arXiv preprint arXiv:2001.02610, 2020.
+
+Yanlin Zhou, George Pu, Xiyao Ma, Xiaolin Li, and Dapeng Wu. Distilled one-shot federated learning. arXiv preprint arXiv:2009.07999, 2020.
+
+Ligeng Zhu, Zhijian Liu, and Song Han. Deep leakage from gradients. In Advances in Neural Information Processing Systems, pp. 14747–14756, 2019.
+
+# A IMPLEMENTATION DETAILS
+
+In this part, we explain the implementation details for the dataset condensation, continual learning and neural architecture search experiments.
+
+Dataset condensation. The presented experiments involve tuning of six hyperparameters – the number of outer-loop $K$ and inner-loop steps $T$ , learning rates $\eta _ { S }$ and number of optimization steps $\varsigma _ { S }$ for the condensed samples, learning rates $\eta _ { \theta }$ and number of optimization steps $\varsigma _ { \pmb { \theta } }$ for the model weights. In all experiments, we set $K = 1 0 0 0$ , $\eta _ { S } = 0 . 1$ , $\eta _ { \pmb { \theta } } = 0 . 0 1$ , $\varsigma s = 1$ and employ Stochastic Gradient Descent (SGD) as the optimizer. The only exception is that we set $\eta _ { S }$ to 0.01 for synthesizing data with MLP in cross-architecture experiments (Table 2), as MLP requires a slightly different treatment. Note that while $K$ is the maximum number of outer-loop steps, the optimization can early-stop automatically if it converges before $K$ steps. For the remaining hyperparameters, we use different sets for 1, 10 and 50 image(s)/class learning. We set $T = 1$ , $\varsigma _ { \pmb { \theta } } = 1$ for 1 image/class, $T = 1 0$ , $\varsigma _ { \pmb { \theta } } = 5 0$ for 10 images/class, $T = 5 0$ , $\varsigma _ { \pmb { \theta } } = 1 0$ for 50 images/class learning. Note that when $T = 1$ , it is not required to update the model parameters (Step 9 in Algorithm 1), as this model is not further used. For those experiments where more than 10 images/class are synthesized, we set $T$ to be the same number as the synthetic images per class and ${ \varsigma } _ { \pmb { \theta } } = 5 0 0 / T$ , e.g. $T = 2 0$ , $\zeta _ { \pmb { \theta } } = 2 5$ for 20 images/class learning. The ablation study on hyper-parameters are given in Appendix B which shows that our method is not sensitive to varying hyper-parameters.
+
+We do separate-class mini-batch sampling for Step 6 in Algorithm 1. Specifically, we sample a mini-batch pair $B _ { c } ^ { \mathcal { T } }$ and $B _ { c } ^ { S }$ that contain real and synthetic images from the same class $c$ at each inner iteration. Then, the matching loss for each class is computed with the sampled mini-batch pair and used to update corresponding synthetic images $ { \boldsymbol { S } } _ { c }$ by back-propogation (Step 7 and 8). This is repeated separately (or parallelly given enough computational resources) for every class. Training as such is not slower than using mixed-class batches. Although our method still works well when we randomly sample the real and synthetic mini-batches with mixed labels, we found that separateclass strategy is faster to train as matching gradients w.r.t. data from single class is easier compared to those of multiple classes. In experiments, we randomly sample 256 real images of a class as a mini-batch to calculate the mean gradient and match it with the mean gradient that is averaged over all synthetic samples with the same class label. The performance is not sensitive to the size of real-image mini-batch if it is greater than 64.
+
+In all experiments, we use the standard train/test splits of the datasets – the train/test statistics are shown in Table T5. We apply data augmentation (crop, scale and rotate) only for experiments (coreset methods and ours) on MNIST. The only exception is that we also use data augmentation when compared to DD (Wang et al., 2018) on CIFAR10 with AlexCifarNet, and data augmentation is also used in (Wang et al., 2018). For initialization of condensed images, we tried both Gaussian noise and randomly selected real training images, and obtained overall comparable performances in different settings and datasets. Then, we used Gaussian noise for initialization in experiments.
+
+<table><tr><td></td><td></td><td></td><td>USPS MNIST FashionMNIST SVHN</td><td></td><td>CIFAR10</td><td>CIFAR100</td></tr><tr><td>Train</td><td>7,291</td><td>60.000</td><td>60.000</td><td>73,257</td><td>50,000</td><td>50,000</td></tr><tr><td>Test</td><td>2.007</td><td>10.000</td><td>10.000</td><td>26.032</td><td>10,000</td><td>10,000</td></tr></table>
+
+Table T5: Train/test statistics for USPS, MNIST, FashionMNIST, SVHN, CIFAR10 and CIFAR100 datasets.
+
+In the first stage – while training the condensed images –, we use Batch Normalization in the VGG and ResNet networks. For reliable estimation of the running mean and variance, we sample many real training data to estimate the running mean and variance and then freeze them ahead of Step 7. In the second stage – while training a deep network on the condensed set –, we replace Batch Normalization layers with Instance Normalization in VGG and ResNet, due to the fact that the batch statistics are not reliable when training networks with few condensed images. Another minor modification that we apply to the standard network ResNet architecture in the first stage is replacing the strided convolutions where $s t r i d e = 2$ with convolutional layers where $s t r i d e = 1$ coupled with an average pooling layer. We observe that this change enables more detailed (per pixel) gradients w.r.t. the condensed images and leads to better condensed images.
+
+Continual learning. In this experiment, we focus on a task-incremental learning on SVHN, MNIST and USPS with the given order. The three tasks share the same label space, however have significantly different image statistics. The images of the three datasets are reshaped to $3 2 \times 3 2$ RGB size for standardization. We use the standard splits for training sets and randomly sample 2,000 test images for each datasets to obtain a balanced evaluation over three datasets. Thus each model is tested on a growing test set with 2,000, 4,000 and 6,000 images at the three stages respectively. We use the default ConvNet in this experiment and set the weight of distillation loss to 1.0 and the temperature to 2. We run 5,000 and 500 iterations for training and balanced finetuning as in (Castro et al., 2018) with the learning rates 0.01 and 0.001 respectively. We run 5 experiments and report the mean and standard variance in Figure 4.
+
+![](images/02e8afc717626841e0a346152ad95409b0561438c4ccba9a56a6d8769dfe5d82.jpg)  
+Figure F5: The performance correlation between the training on proxy dataset and whole-dataset. For each proxy dataset, the best 10 models are selected based on validation set performance. In the figure, each point represents an architecture.
+
+<table><tr><td>C\T</td><td>Sigmoid</td><td>ReLu</td><td>LeakyReLu</td></tr><tr><td>Sigmoid</td><td>86.7±0.7</td><td>91.2±0.6</td><td>91.2±0.6</td></tr><tr><td>ReLu</td><td>86.1±0.9</td><td>91.7±0.5</td><td>91.7±0.5</td></tr><tr><td>LeakyReLu</td><td>86.3±0.9</td><td>91.7±0.5</td><td>91.7±0.4</td></tr></table>
+
+Table T6: Cross-activation experiments in accuracy $( \% )$ for 1 condensed image/class in MNIST.
+
+<table><tr><td>C\T</td><td>None</td><td>MaxPooling</td><td>AvgPooling</td></tr><tr><td>None</td><td>78.7±3.0</td><td>80.8±3.5</td><td>88.3±1.0</td></tr><tr><td>MaxPooling</td><td>81.2±2.8</td><td>89.5±1.1</td><td>91.1±0.6</td></tr><tr><td>Avgpooing</td><td>81.8±2.9</td><td>90.2±0.8</td><td>91.7±0.5</td></tr></table>
+
+Table T7: Cross-pooling experiments in accuracy $( \% )$ for 1 condensed image/class in MNIST.
+
+Neural Architecture Search. To construct the searching space of 720 ConvNets, we vary hyperparameters $W ~ \in ~ \{ 3 2 , 6 4 , 1 2 8 , 2 5 6 \}$ , $D \in \{ 1 , 2 , 3 , 4 \}$ , $N \in$ {None, BatchNorm, LayerNorm, InstanceNorm, $\mathrm { G r o u p N o r m } \}$ , $A \in$ {Sigmoid, ReLu, LeakyReLu}, $P \in$ {None, MaxPooling, AvgPooling}. We randomly sample 5,000 images from the 50,000 training images in CIFAR10 as the validation set. Every candidate ConvNet is trained with the proxy dataset, and then evaluated on the validation set. These candidate ConvNets are ranked by the validation performance. 10 architectures with top validation accuracies are selected to calculate Spearman’s rank correlation coefficient, because the best model that we want will come from the top 10 architectures. We train each ConvNet for 5 times to get averaged validation and testing accuracies.
+
+We visualize the performance correlation for different proxy datasets in Figure F5. Obviously, the condensed proxy dataset produced by our method achieves the highest performance correlation (0.79) which significantly higher than early-stopping (0.42). It means our method can produce more reliable results for NAS.
+
+# B FURTHER ANALYSIS
+
+Next we provide additional results on ablative studies over various deep network layers including activation, pooling and normalization functions and also over depth and width of deep network architecture. We also study the selection of hyper-parameters and the gradient distance metric. An additional qualitative analysis on the learned condensed images is also given.
+
+Ablation study on activation functions. Here we study the use of three activation functions – Sigmoid, ReLU, LeakyReLu (negative slope is set to 0.01) – in two stages, when training condensed images (denoted as C) and when training a ConvNet from scratch on the learned condensed images (denoted as T). The experiments are conducted in MNIST dataset for 1 condensed image/class setting. Table T6 shows that all three activation functions are good for the first stage while generating good condensed images, however, Sigmoid performs poor in the second stage while learning a classifier on the condensed images – its testing accuracies are lower than ReLu and LeakyReLu by around $5 \%$ . This suggests that ReLU can provide sufficiently informative gradients for learning condensed images, though the gradient of ReLU w.r.t. to its input is typically sparse.
+
+Table T8: Cross-normalization experiments in accuracy $( \% )$ for 1 condensed image/class in MNIST.   
+
+<table><tr><td>C\T</td><td>None</td><td>BatchNorm</td><td>LayerNorm</td><td>InstanceNorm</td><td>GroupNorm</td></tr><tr><td>None</td><td>79.0±2.2</td><td>80.8±2.0</td><td>85.8±1.7</td><td>90.7±0.7</td><td>85.9±1.7</td></tr><tr><td>BatchNorm</td><td>78.6±2.1</td><td>80.7±1.8</td><td>85.7±1.6</td><td>90.9±0.6</td><td>85.9±1.5</td></tr><tr><td>LayerNorm</td><td>81.2±1.8</td><td>78.6±3.0</td><td>87.4±1.3</td><td>90.7±0.7</td><td>87.3±1.4</td></tr><tr><td>InstanceNorm</td><td>72.9±7.1</td><td>56.7±6.5</td><td>82.7±5.3</td><td>91.7±0.5</td><td>84.3±4.2</td></tr><tr><td>GroupNorm</td><td>79.5±2.1</td><td>81.8±2.3</td><td>87.3±1.2</td><td>91.6±0.5</td><td>87.2±1.2</td></tr></table>
+
+<table><tr><td>C\T</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>1</td><td></td><td></td><td>61.3±3.5 78.2±3.0 77.1±4.0 76.4±3.5</td><td></td></tr><tr><td>2</td><td>78.3±2.3</td><td>89.0±0.8</td><td>91.0±0.6</td><td>89.4±0.8</td></tr><tr><td>3</td><td>81.6±1.5</td><td>89.8±0.8</td><td>91.7±0.5</td><td>90.4±0.6</td></tr><tr><td>4</td><td>82.5±1.3</td><td>89.9±0.8</td><td>91.9±0.5</td><td>90.6±0.4</td></tr></table>
+
+Table T9: Cross-depth performance in accuracy $( \% )$ for 1 condensed image/class in MNIST.
+
+<table><tr><td>C\T</td><td>32</td><td>64</td><td>128</td><td>256</td></tr><tr><td>32</td><td></td><td></td><td>90.6±0.8 91.4±0.591.5±0.5 91.3±0.6</td><td></td></tr><tr><td>64</td><td>91.0±0.8</td><td>91.6±0.6</td><td>91.8±0.5</td><td>91.4±0.6</td></tr><tr><td>128</td><td>90.8±0.7</td><td>91.5±0.6</td><td>91.7±0.5</td><td>91.2±0.7</td></tr><tr><td>256</td><td>91.0±0.7</td><td>91.6±0.6</td><td>91.7±0.5</td><td>91.4±0.5</td></tr></table>
+
+Table T10: Cross-width performance in accuracy $( \% )$ for 1 condensed image/class in MNIST.
+
+Ablation study on pooling functions. Next we investigate the performance of two pooling functions – average pooling and max pooling – also no pooling for 1 image/class dataset condensation with ConvNet in MNIST in terms of classification accuracy. Table T7 shows that max and average pooling both perform significantly better than no pooling (None) when they are used in the second stage. When the condensed samples are trained and tested on models with average pooling, the best testing accuracy $( 9 1 . 7 \pm 0 . 5 \hat { \% } )$ is obtained, possibly, because average pooling provides more informative and smooth gradients for the whole image rather than only for its discriminative parts.
+
+Ablation study on normalization functions. Next we study the performance of four normalization options – No normalization, Batch (Ioffe & Szegedy, 2015), Layer (Ba et al., 2016), Instance (Ulyanov et al., 2016) and Group Normalization (Wu & He, 2018) (number of groups is set to be four) – for 1 image/class dataset condensation with ConvNet architecture in MNIST classification accuracy. Table T8 shows that the normalization layer has little influence for learning the condensed set, while the choice of normalization layer is important for training networks on the condensed set. LayerNorm and GroupNorm have similar performance, and InstanceNorm is the best choice for training a model on condensed images. BatchNorm obtains lower performance which is similar to None (no normalization), as it is known to perform poorly when training models on few condensed samples as also observed in (Wu & He, 2018). Note that Batch Normalization does not allow for a stable training in the first stage (C); thus we replace its running mean and variance for each batch with those of randomly sampled real training images.
+
+Ablation study on network depth and width. Here we study the effect of network depth and width for 1 image/class dataset condensation with ConvNet architecture in MNIST in terms of classification accuracy. To this end we conduct multiple experiments by varying the depth and width of the networks that are used to train condensed synthetic images and that are trained to classify testing data in ConvNet architecture and report the results in Table T9 and Table T10. In Table T9, we observe that deeper ConvNets with more blocks generate better condensed images that results in better classification performance when a network is trained on them, while ConvNet with 3 blocks performs best as classifier. Interestingly, Table T10 shows that the best results are obtained with the classifier that has 128 filters at each block, while network width (number of filters at each block) in generation has little overall impact on the final classification performance.
+
+Ablation study on hyper-parameters. Our performance is not sensitive to hyper-parameter selection. The testing accuracy for various $K$ and $T$ , when learning 10 images/class condensed sets, is depicted in Figure F6. The results show that the optimum $K$ and $T$ are around similar values across all datasets. Thus we simply set $K$ to 1000 and $T$ to 10 for all datasets. Similarly, for the remaining ones including learning rate, weight decay, we use a single set of hyperparameters that are observed to work well for all datasets and architectures in our preliminary experiments.
+
+Ablation study on gradient distance metric. To prove the effectiveness and robustness of the proposed distance metric for gradients (or weights), we compare to the traditional ones (LopezPaz et al., 2017; Aljundi et al., 2019; Zhu et al., 2019) which vectorize and concatenate the whole gradient, $\mathbf { G } ^ { \mathcal { T } } , \mathbf { G } ^ { s } \mathbf { \Sigma } ^ { } \in \mathbb { R } ^ { D }$ , and compute the squared Euclidean distance $\| \mathbf { G } ^ { T } - \mathbf { G } ^ { S } \| ^ { 2 }$ and the Cosine distance $1 - \cos \left( \mathbf { G } ^ { \mathcal { T } } , \mathbf { G } ^ { \mathcal { S } } \right)$ , where $D$ is the number of all network parameters. We do 1 image/class learning experiment on MNIST with different architectures. For simplicity, the synthetic images are learned and tested on the same architecture in this experiment. Table T11 shows that the proposed gradient distance metric remarkably outperforms others on complex architectures (e.g. LeNet, AlexNet, VGG and ResNet) and achieves the best performances in most settings, which means it is more effective and robust than the traditional ones. Note that we set $\eta _ { S } ~ = ~ 0 . 1$ for MLP-Euclidean and MLP-Cosine because it works better than $\eta _ { S } = 0 . 0 1$ .
+
+![](images/7893e0b3cbf2b7c064bc7cdcc10b6009ef19a6ed2f8d9a0bd820a2622dbbac27.jpg)  
+Figure F6: Ablation study on the hyper-parameters $K$ and $T$ when learning 10 images/class condensed sets.   
+Table T11: Ablation study on different gradient distance metrics. Obviously, the proposed distance metric is more effective and robust. Euclidean: squared Euclidean distance, Cosine: Cosine distance.
+
+<table><tr><td></td><td>MLP</td><td>ConvNet</td><td>LeNet</td><td>AlexNet</td><td>VGG</td><td>ResNet</td></tr><tr><td>Euclidean</td><td>69.3±0.9</td><td>92.7±0.3</td><td>65.0±5.1</td><td>66.2±5.6</td><td>57.1±7.0</td><td>68.0±5.2</td></tr><tr><td>Cosine</td><td>45.2±3.6</td><td>69.2±2.7</td><td>61.1±8.2</td><td>58.3±4.1</td><td>55.0±5.0</td><td>68.8±7.8</td></tr><tr><td>Ours</td><td>70.5±1.2</td><td>91.7±0.5</td><td>85.0±1.7</td><td>82.7±2.9</td><td>81.7±2.6</td><td>89.4±0.9</td></tr></table>
+
+Further qualitative analysis We first depict the condensed images that are learned on MNIST, FashionMNIST, SVHN and CIFAR10 datasets in one experiment using the default ConvNet in 10 images/class setting in Figure F7. It is interesting that the 10 images/class results in Figure F7 are diverse which cover the main variations, while the condensed images for 1 image/class setting (see Figure 2) look like the “prototype” of each class. For example, in Figure F7 (a), the ten images of “four” indicate ten different styles. The ten “bag” images in Figure F7 (b) are significantly different from each other, similarly “wallet” (1st row), “shopping bag” (3rd row), “handbag” (8th row) and “schoolbag” (10th row). Figure F7 (c) also shows the diverse house numbers with different shapes, colors and shadows. Besides, different poses of a “horse” have been learned in Figure F7 (d).
+
+# C COMPARISON TO MORE BASELINES
+
+Optimal random selection. One interesting and strong baseline is Optimal Random Selection (ORS) that we implement random selection experiments for 1,000 times and pick the best ones. Table T12 presents the performance comparison to the selected Top 1000 (all), Top 100 and Top 10 coresets. These optimal coresets are selected by ranking their performance. Obviously, the condensed set generated by our method surpasses the selected Top 10 of 1000 coresets with a large margin on all four datasets.
+
+Generative model. We also compare to the popular generative model, namely, Conditional Generative Adversarial Networks (cGAN) (Mirza & Osindero, 2014). The generator has two blocks which consists of the Up-sampling (scale facto $^ { - 2 }$ ), Convolution (stride $^ { \cdot = 1 }$ ), BatchNorm and LeakyReLu layers. The discriminator has three blocks which consists of Convolution (strid ${ \boldsymbol { \ O } } { \boldsymbol { \ O } } { \boldsymbol { \ O } } { \boldsymbol { \ O } } = 2 { \mathrm { \Omega } }$ ), BatchNorm and LeakyReLu layers. In additional to the random noise, we also input the class label as the condition. We generate 1 and 10 images per class for each dataset with random noise. Table T12 shows that the images produced by cGAN have similar performances to those randomly selected coresets (i.e. Top 1000). It is reasonable, because the aim of cGAN is to generate real-look images. In contrast, our method aims to generate images that can train deep neural networks efficiently.
+
+Analysis of coreset performances We find that K-Center (Wolf, 2011; Sener & Savarese, 2018) and Forgetting (Toneva et al., 2019) don’t work as well as other general coreset methods, namely
+
+![](images/2c52f7feccd4dbddd6d099ea4ea73d2e1415bb094c3c2552192d191cdb422458.jpg)  
+Figure F7: The synthetic images for MNIST, FashionMNIST, SVHN and CIFAR10 produced by our method with ConvNet under 10 images/class setting.
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Img/Cls</td><td rowspan="2">Ratio %</td><td colspan="3">Optimal Random Selection</td><td rowspan="2">cGAN</td><td rowspan="2">Ours</td><td rowspan="2">Whole Dataset</td></tr><tr><td>Top 1000</td><td>Top100</td><td>Top10</td></tr><tr><td rowspan="2">MNIST</td><td>1</td><td>0.017</td><td>64.3±6.1</td><td>74.4±1.8</td><td>78.2±1.7</td><td>64.0±3.2</td><td>91.7±0.5</td><td rowspan="2">99.6±0.0</td></tr><tr><td>10</td><td>0.17</td><td>94.8±0.7</td><td>96.0±0.2</td><td>96.4±0.1</td><td>94.9±0.6</td><td>97.4±0.2</td></tr><tr><td rowspan="2">FashionMNIST</td><td>1</td><td>0.017</td><td>51.3±5.4</td><td>59.6±1.3</td><td>62.4±0.9</td><td>51.1±0.8</td><td>70.5±0.6</td><td rowspan="2">93.5±0.1</td></tr><tr><td>10</td><td>0.17</td><td>73.8±1.6</td><td>76.4±0.6</td><td>77.6±0.2</td><td>73.9±0.7</td><td>82.3±0.4</td></tr><tr><td rowspan="2">SVHN</td><td>1</td><td>0.014</td><td>14.3±2.1</td><td>18.1±0.9</td><td>19.9±0.2</td><td>16.1±0.9</td><td>31.2±1.4</td><td rowspan="2">95.4±0.1</td></tr><tr><td>10</td><td>0.14</td><td>34.6±3.2</td><td>40.3±1.3</td><td>42.9±0.9</td><td>33.9±1.1</td><td>76.1±0.6</td></tr><tr><td rowspan="2">CIFAR10</td><td>1 10</td><td>0.02</td><td>15.0±2.0</td><td>18.5±0.8</td><td>20.1±0.5</td><td>16.3±1.4</td><td>28.3±0.5</td><td rowspan="2">84.8±0.1</td></tr><tr><td></td><td>0.2</td><td>27.1±1.6</td><td>29.8±0.7</td><td>31.4±0.2</td><td>27.9±1.1</td><td>44.9±0.5</td></tr></table>
+
+Table T12: The performance comparison to optimal random selection (ORS) and conditional generative adversarial networks (cGAN) baselines. This table shows the testing accuracies $( \% )$ of different methods on four datasets. ConvNet is used for training and testing. Img/Cls: image(s) per class, Ratio $( \% )$ : the ratio of condensed images to whole training set. Top 1000, Top 100 and Top 10 means the selected 1000, 100 and 10 optimal coresets by ranking their performances.
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Img/Cls</td><td rowspan="2">Ratio%</td><td colspan="4">Core-set Selection</td><td rowspan="2">LD†</td><td rowspan="2">Ours</td><td rowspan="2">Whole Dataset</td></tr><tr><td>Random</td><td>Herding</td><td>K-Center</td><td>Forgetting</td></tr><tr><td rowspan="2">CIFAR100</td><td>1</td><td>0.2</td><td>4.2±0.3</td><td>8.4±0.3</td><td>8.3±0.3</td><td>3.5±0.3</td><td>11.5±0.4</td><td>12.8±0.3</td><td rowspan="2">56.2±0.3</td></tr><tr><td>10</td><td>2</td><td>14.6±0.5</td><td>17.3±0.3</td><td>7.1±0.3</td><td>9.8±0.2</td><td>-</td><td>25.2±0.3</td></tr></table>
+
+Table T13: The performance comparison on CIFAR100. This table shows the testing accuracies $( \% )$ of different methods. ConvNet is used for training and testing except that $\mathrm { L D ^ { \dag } }$ uses AlexNet. $\mathrm { I m g / C l s }$ : image(s) per class, Ratio $( \% )$ : the ratio of condensed images to whole training set.
+
+<table><tr><td>Method</td><td>MLP</td><td>ConvNet</td><td>LeNet</td><td>AlexNet</td><td>VGG</td><td>ResNet</td></tr><tr><td>DD</td><td>72.7±2.8</td><td>77.6±2.9</td><td>79.5±8.1</td><td>51.3±19.9</td><td>11.4±2.6</td><td>63.6±12.7</td></tr><tr><td>Ours</td><td>83.0±2.5</td><td>92.9±0.5</td><td>93.9±0.6</td><td>90.6±1.9</td><td>92.9±0.5</td><td>94.5±0.4</td></tr></table>
+
+Table T14: Generalization ability comparison to DD. The 10 condensed images per class are trained with LeNet, and tested on various architectures. It shows that condensed images generated by our method have better generalization ability.
+
+Random and Herding (Rebuffi et al., 2017), in this experimental setting. After analyzing the algorithms and coresets, we find two main reasons. 1) K-Center and Forgetting are not designed for training deep networks from scratch, instead they are for active learning and continual learning respectively. 2) The two algorithms both tend to select “hard” samples which are often outliers when only a small number of images are selected. These outliers confuse the training, which results in worse performance. Specifically, the first sample per class in K-Center coreset is initialized by selecting the one closest to each class center. The later ones selected by the greedy criterion that pursues maximum coverage are often outliers which confuse the training.
+
+Performance on CIFAR100. We supplement the performance comparison on CIFAR100 dataset which includes 10 times as many classes as other benchmarks. More classes while fewer images per class makes CIFAR100 significantly more challenging than other datasets. We use the same set of hyper-parameters for CIFAR100 as other datasets. Table T13 depicts the performances of coreset selection methods, Label Distillation (LD) Bohdal et al. (2020) and ours. Our method achieves $12 . 8 \%$ and $2 5 . 2 \%$ testing accuracies on CIFAR100 when learning 1 and 10 images per class, which are the best compared with others.
+
+# D FURTHER COMPARISON TO DD (WANG ET AL., 2018)
+
+Next we compare our method to DD (Wang et al., 2018) first quantitatively in terms of crossarchitecture generalization, then qualitatively in terms of synthetic image quality, and finally in terms of computational load for training synthetic images. Note that we use the original source code to obtain the results for DD that is provided by the authors of DD in the experiments.
+
+Generalization ability comparison. Here we compare the generalization ability across different deep network architectures to DD. To this end, we use the synthesized 10 images/class data learned with LeNet on MNIST to train MLP, ConvNet, LeNet, AlexNet, VGG11 and ResNet18 and report the results in Table T14. We see that that the condensed set produced by our method achieves good classification performances with all architectures, while the synthetic set produced by DD perform poorly when used to trained some architectures, e.g. AlexNet, VGG and ResNet. Note that DD generates learning rates to be used in every training step in addition to the synthetic data. This is in contrast to our method which does not learn learning rates for specific training steps. Although the tied learning rates improve the performance of DD while training and testing on the same architecture, they will hinder the generalization to unseen architectures.
+
+<table><tr><td>Method</td><td>Dataset</td><td>Architecture</td><td>Memory (MB)</td><td>Time (min)</td><td>Test Acc.</td></tr><tr><td>DD</td><td>MNIST</td><td>LeNet</td><td>785</td><td>160</td><td>79.5±8.1</td></tr><tr><td>Ours</td><td>MNIST</td><td>LeNet</td><td>653</td><td>46</td><td>93.9±0.6</td></tr><tr><td>DD</td><td></td><td>CIFAR10 AlexCifarNet</td><td>3211</td><td>214</td><td>36.8±1.2</td></tr><tr><td>Ours</td><td>CIFAR10</td><td>）AlexCifarNet</td><td>1445</td><td>105</td><td>39.1±1.2</td></tr></table>
+
+Table T15: Time and memory use for training DD and our method in 10 images/class setting.
+
+![](images/d246b3b1d03476776a7e0546cf648d915af6d884352adf476cb53c4554af990f.jpg)  
+Figure F8: Qualitative comparison between the condensed images produced by DD and ours under $1 0 \ \mathrm { i m }$ - ages/class setting. LeNet and AlexCifarNet are utilized for MNIST and CIFAR10 respectively.
+
+Qualitative comparison. We also provide a qualitative comparison to to DD in terms of image quality in Figure F8. Note that both of the synthetic sets are trained with LeNet on MNIST and AlexCifarNet on CIFAR10. Our method produces more interpretable and realistic images than DD, although it is not our goal. The MNIST images produced by DD are noisy, and the CIFAR10 images produced by DD do not show any clear structure of the corresponding class. In contrast, the MNIST and CIFAR10 images produced by our method are both visually meaningful and diverse.
+
+Training memory and time. One advantage of our method is that we decouple the model weights from its previous states in training, while DD requires to maintain the recursive computation graph which is not scalable to large models and inner-loop optimizers with many steps. Hence, our method requires less training time and memory cost. We compare the training time and memory cost required by DD and our method with one NVIDIA GTX1080-Ti GPU. Table T15 shows that our method requires significantly less memory and training time than DD and provides an approximation reduction of $1 7 \%$ and $5 5 \%$ in memory and $7 1 \%$ and $5 1 \%$ in train time to learn MNIST and CIFAR10 datasets respectively. Furthermore, our training time and memory cost can be significantly decreased by using smaller hyper-parameters, e.g. $K$ , $T$ and the batch size of sampled real images, with a slight performance decline (refer to Figure F6).
+
+# E EXTENDED RELATED WORK
+
+Variations of Dataset Distillation. There exists recent work that extends Dataset Distillation (Wang et al., 2018). For example, (Sucholutsky & Schonlau, 2019; Bohdal et al., 2020) aim to improve DD by learning soft labels with/without synthetic images. (Such et al., 2020) utilizes a generator to synthesize images instead of directly updating image pixels. However, the reported quantitative and qualitative improvements over DD are minor compared to our improvements. In addition, none of these methods have thoroughly verified the cross-architecture generalization ability of the synthetic images.
+
+Zero-shot Knowledge Distillation. Recent zero-shot KD methods (Lopes et al., 2017; Nayak et al., 2019) aim to perform KD from a trained model in the absence of training data by generating synthetic data as the intermediate production to further use. Unlike them, our method does not require pretrained teacher models to provide the knowledge, i.e. to obtain the features and labels.
+
+Data Privacy & Federated Learning. Synthetic dataset is also a promising solution to protecting data privacy and enabling safe federated learning. There exists some work that uses synthetic dataset to protect the privacy of medical dataset (Li et al., 2020) and reduce the communication rounds in federated learning (Zhou et al., 2020). Although transmitting model weights or gradients (Zhu et al., 2019; Zhao et al., 2020) may increase the transmission security, the huge parameters of modern deep neural networks are prohibitive to transmit frequently. In contrast, transmitting small-scale synthetic dataset between clients and server is low-cost (Goetz & Tewari, 2020).