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+# RELATIONAL MULTI-INSTANCE LEARNING FOR CONCEPT ANNOTATION FROM MEDICAL TIME SERIES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recent advances in computing technology and sensor design have made it easier to collect longitudinal or time series data from patients, resulting in a gigantic amount of available medical data. Most of the medical time series lack annotations or even when the annotations are available they could be subjective and prone to human errors. Earlier works have developed natural language processing techniques to extract concept annotations and/or clinical narratives from doctor notes. However, these approaches are slow and do not use the accompanying medical time series data. To address this issue, we introduce the problem of concept annotation for the medical time series data, i.e., the task of predicting and localizing medical concepts by using the time series data as input. We propose Relational Multi-Instance Learning (RMIL) - a deep Multi Instance Learning framework based on recurrent neural networks, which uses pooling functions and attention mechanisms for the concept annotation tasks. Empirical results on medical datasets show that our proposed models outperform various multi-instance learning models.
+
+# 1 INTRODUCTION
+
+Clinicians have limited time (e.g., only a few minutes (Howie et al., 1999)) to study and treat each patient. However, they are overloaded with a lot of patient data from multiple sources and in various formats, such as patient medical history and doctor’s notes in free-flowing text, vitals and monitoring data which are captured as time series, and prescriptions and drugs which appear as medical codes including ICD-9 (Organization & Corporation, 1998), LOINC codes (Forrey et al., 1996), etc. This rich information should be summarized and available to clinicians in easily digestible format for faster diagnosis and treatment. Graphical visualizations (Plaisant et al., 1998) are a popular approach to show patient data to doctors. However, recent studies have shown that graphical visualisations are not always helpful for clinicians’ decision-making (Law et al., 2005; Van Der Meulen et al., 2010). Text summaries on the other hand are widely embraced and are usually adopted in practice (Scott et al., 2013). Most existing systems use natural language processing techniques (Afantenos et al., 2005; Giordano et al., 2015) to generate summaries from doctor notes which include test results, discharge reports, observational notes, etc. While these systems are useful, they only use one source of data, i.e., doctor’s notes which might have noisy and erroneous entries, for text summarization. On the other hand, electronic health records have other sources of patient data such as vital signs, monitoring sensors, and lab results in the form of multivariate time-series, which can be more accurate and may contain rich information about patient’s conditions. Few existing patient summarization systems actually extract information directly from these time series for concept prediction and/or summarization. Generating simple text summaries such as trends from time series has been investigated before (Sripada et al., 2003) but is marginally useful since these trends are not mapped to the medical concepts which clinicians can quickly comprehend. Recent works (Pham et al., 2016; Choi et al., 2016a;b; Lipton et al., 2015; Che et al., 2016) have successfully shown that clinical events and outcomes can be predicted using medical codes or clinical time series data. However, directly obtaining medical concept annotations and summaries from the time series data is still an open question.
+
+In this work, we introduce the concept annotation task as the problem of predicting and localizing the medical concepts by modeling the related medical time series data. Figure 1 illustrates a concept annotation example where medical time series data such as heart rate, $\mathrm { p H }$ and blood gas pressure are given, and the goal is to predict the time series of concepts such as intubation, extubation and resuscitate. To solve concept annotation problem, we formulate it as a Multi-Instance Learning (MIL) problem (Dietterich et al., 1997) and propose a deep learning based framework called Relational MultiInstance Learning (RMIL). RMIL uses Recurrent Neural Networks (RNNs) to model multivariate time series data, leverages instance relations via attention mechanisms, and provides concept predictions using pooling functions.
+
+![](images/d138111f217b2a0b650ab49fd7dfba33f87f2e1e94504c7e336ff367fad611f6.jpg)  
+Figure 1: Medical time series to concept annotations.
+
+The main contributions of our work are the following. We present a unified view of the MIL approaches for time series data using RNN models with different pooling functions and attention mechanisms. We show that our RMIL model is capable of learning a good classifier for concept detection (bag label predictions) and concept localization tasks (instance label prediction), even though it is only trained using bag labels. We demonstrate that RMIL obtains promising results on real-world medical datasets and outperforms popular MIL approaches.
+
+The rest of the paper is structured as follows. In the following section, we briefly discuss the related works. Afterwards, we describe MIL framework and describe how RNN can be combined with multi-instance learning framework to obtain our proposed RMIL. In Sections 4 and 5, we present experimental results and conclusions respectively. In the appendix, we demonstrate anomaly detection as another application of our RMIL framework.
+
+# 2 RELATED WORK
+
+Discovering concept annotations from the multivariate time series is a relatively new problem in medical domain with limited prior work. In this section, we will first highlight the related works on annotation tasks and then review related works on multi-instance learning.
+
+Concept Annotation In medical domain, concept annotation is usually addressed in the clinical narrative mining and biomedical text mining literature (Aggarwal & Zhai, 2012; Cohen & Hersh, 2005; Zweigenbaum et al., 2007; Vincze et al., 2008). In other domains such as web-mining and computer vision, the concept annotation is usually analogous to semantic annotation (Kiryakov et al., 2004), image annotation (Jeon et al., 2003), object localization (Lampert et al., 2008) and image captioning (Karpathy et al., 2014).
+
+Clinical Narratives Mining Automated discovery of temporal relations from clinical narratives (Savova et al., 2009; Zhou et al., 2006; Albright et al., 2013) and doctor notes (Plaisant et al., 1996) to uncover the patterns of disease progression is an important research problem in clinical informatics. Recent efforts such as SemEval competitions (Bethard et al., 2016) have been conducted to study this problem and evaluate/benchmark clinical information extraction systems $\mathrm { X u }$ et al., 2010). These competitions focus on discrete, well-defined tasks which allow for rapid, reliable and repeatable evaluations. However, they only consider identifying and extracting temporal relations from clinical notes and do not use the accompanying medical time series data.
+
+Image Annotation and Captioning Successful object recognition systems have been developed in the past few decades for image annotation, object detection and localization in images and videos. ImageNet (Deng et al., 2009) and PASCAL challenges (Everingham et al., 2010) have greatly accelerated the research in this area. Image captioning and visual-to-text translation, which are more generalized image annotation tasks, have been recently studied in several works (Karpathy et al., 2014; Kojima et al., 2002; Mao et al., 2014; Yu et al., 2016) where the goal is to find a text caption for a given image. Deep learning models such as RNN and sequence-to-sequence models (Sutskever et al., 2014) have achieved excellent results for image annotation/captioning tasks.
+
+Multi-Instance Learning Multi-Instance Learning (MIL), a well known researched topic in machine learning, was first introduced by Dietterich et al. (1997) as a form of weakly supervised learning for drug activity prediction. MIL frameworks have since been applied to many other domains including image and text annotations (Chen & Wang, 2004; Soleimani & Miller, 2017). Andrews et al. (2003) adapted Support Vector Machines (SVM) to the MIL framework and introduced miSVM and MISVM for optimizing instance-level and bag-level classifications respectively. Zhou & Zhang (2007) further extended MIL and proposed MIMLSVM for tackling multi-label problems. Zhou et al. (2009) introduced miGraph and MIGraph to model the structure in each bag. Zhang et al. (2011); Guan et al. (2016) also proposed MIL framework for structure data by leveraging the relational structures at the bag and instance levels. Generative model based MIL frameworks such as Multi-Instance Mixture Model (MIMM)(Foulds & Smyth, 2011), and Dirichlet Process Mixture of Guassians (DPMIL) (Kandemir & Hamprecht, 2014) have also been proposed for binary multi-instance classification. Guan et al. (2016) used an autoregressive hidden Markov model and proposed an MIL framework for activity recognition in time-series data. Garcez & Zaverucha (2012) used recurrent neural networks to combine instance-level preprocessing and bag-level classification in MIL setting. Comprehensive reviews of MIL approaches are provided in Amores (2013); Herrera et al. (2016); Soleimani & Miller (2017). Recently, deep learning models have been successfully applied for MIL framework (Zhu et al., 2017; Wu et al., 2015; Huang et al., 2013; Yan et al., 2016; Kotzias et al., 2014; Kraus et al., 2016) and these approaches are generally termed as deep multi-instance learning models. Most of these works use either convolutional neural networks or deep neural networks in their MIL framework for image annotation, labeling, segmentation or classification tasks. Despite the popularity of deep models for MIL, there are few works which have extended deep MIL models for multivariate time series data. The goal of this paper is to propose and study deep multi-instance learning models for multivariate time series data.
+
+# 3 METHODS
+
+In this section, we will first describe the Multi-instance learning framework, and then present our problem formulation and our proposed relational multi-instance learning models.
+
+# 3.1 MULTI-INSTANCE LEARNING
+
+Multi-instance learning (MIL) is a form of weakly supervised learning where the training data is arranged in sets called bags, and a label is provided for the entire bag. The data points inside a bag are referred to as instances. In the MIL framework, instance labels are not provided during training. The main goal of MIL is to learn a model based on the instances in the bag and the label of the bag - to make bag-level and instance-level predictions. In this work, we only focus on the classification task in MIL, leaving out other learning tasks such as regression. Generally, two broad assumptions can be used to model the relationship between instance label and bag label. In the standard MIL assumption (Dietterich et al., 1997), the bag label is negative if all the instances in the bag have a negative label, and the bag label is positive if at-least one of the instances in the bag has a positive label. Following the notations of Carbonneau et al. (2017), let $X$ denote a bag with $N$ feature vector instances i.e., $\bar { X } = \{ x _ { 1 } , . . . , x _ { N } \}$ . Let each instance $x _ { i }$ in feature space $\mathcal { X }$ be mapped to a class by some process $f : \mathcal { X }  \{ 0 , 1 \}$ , where 0 and 1 correspond to negative and positive labels respectively. The bag classifier, also know as the aggregator function, $g ( X )$ is defined by:
+
+$$
+g ( X ) = { \left\{ \begin{array} { l l } { 1 } & { \quad { \mathrm { i f } } \exists x \in X : f ( X ) = 1 ; } \\ { 0 } & { \quad { \mathrm { o t h e r w i s e } } } \end{array} \right. }
+$$
+
+The standard assumption is quite restrictive for some problem settings, where the positive bags cannot be identified by a single instance. Thus, this assumption can be relaxed to a collective assumption
+
+which says that several positive instances in a bag are necessary to assign a positive label to that bag. In this case, a bag classifier is given by:
+
+$$
+g ( X ) = { \left\{ \begin{array} { l l } { 1 } & { \quad { \mathrm { i f } } \ \sum _ { x \in X } f ( X ) \geq \theta ; } \\ { 0 } & { \quad { \mathrm { o t h e r w i s e } } } \end{array} \right. }
+$$
+
+where $\theta$ is a threshold which indicates the minimum number of instances with positive labels that should be present in a bag to assign a positive label to that bag Weidmann et al. (2003). As discussed in section 2, a plethora of works have adapted machine learning models to the MIL setting to optimize instance-level and/or bag-level predictions.
+
+# 3.2 PROBLEM FORMULATION
+
+We formulate the concept annotation task as the detection and localization of concepts given the medical time series data. Let each patient $i \in \{ 1 , . . , N \}$ be associated with a medical time series (also referred to as feature time series) denoted by $\pmb { X } _ { i } \in \mathbb { R } ^ { T \times D }$ , where $D$ denotes the number of features (such as heart rate, blood pressure) and $T$ denotes the length of time series observations (i.e., amount of time a patient is monitored). Let $\mathcal { C }$ denote the set of all the concepts associated with the $N$ patients, and $\bar { Y _ { i } } \in \left\{ 0 , 1 \right\} ^ { K }$ denote the concepts associated with $i ^ { t h }$ patient where $K = | \mathcal { C } |$ . Let $C _ { i } \in \{ 0 , 1 \} ^ { T \times K }$ denote the concept time series of $X _ { i }$ with $C _ { i } ^ { j k } = 1$ when concept $k$ is present at time-stamp $j$ . In multi-instance learning settings, we treat each time series $X _ { i }$ as one bag, and the observation at each time step $j$ i.e. $\boldsymbol { X } _ { i } ^ { j } \in \mathbb { R } ^ { D }$ as an instance in that bag. We are interested in the following tasks:
+
+• Concept prediction task: for patient $i$ , given $X _ { i }$ , predict $\mathbf { Y } _ { i }$ .   
+• Concept localization task: for patient $i$ , given $X _ { i }$ , predict $C _ { i }$ .
+
+Notice that during training phase, only the input $X _ { i }$ and prediction label $\mathbf { Y } _ { i }$ are available. Though $C _ { i }$ is not known, we usually have some assumptions about the relationship of prediction and localization labels. In this work, $\begin{array} { r } { Y _ { i } ^ { k } = \mathcal { T } \left( \sum _ { 1 \leq j \leq T } C _ { i } ^ { j k } \geq \eta \right) } \end{array}$ , where $\mathcal { T }$ is an indicator function and $\eta$ is a constant which depends on the MIL assumption. For example, in our concept annotation tasks we make the standard assumption, i.e we assume $\eta = 1$ , i.e. the time series label (bag label) for a concept is positive if that concept is present at any one time-stamp (at-least one instance has positive label).
+
+# 3.3 RELATIONAL MULTI-INSTANCE LEARNING FRAMEWORK
+
+Inspired by the recent success of recurrent neural networks in sequence modeling Bahdanau et al. (2014); Sutskever et al. (2014) and classification tasks Krizhevsky et al. (2012); Simonyan & Zisserman (2014), we adapt these models to the MIL framework to model multivariate time series data for concept annotation tasks. We denote all the variables at every time step as an instance and the entire multivariate time series as a bag. Unlike the traditional MIL setting, where the instances within a bag are independent of each other, in our case, the instances have relationships (namely temporal dependencies) among them. To model these dependencies, we propose to combine RNN models such as Long-Short Term Memory Neural Networks (LSTM) and Sequence-to-Sequence models with MIL, and propose our Relational Multi-Instance Learning framework, abbreviated as RMIL. RMIL takes in multivariate time series as input and outputs concept annotations. In RMIL, the outputs of RNN model provide the instance label predictions (i.e. solution for concept localization task) and the aggregation of the instance labels using aggregators such as pooling layer provides the bag label predictions (i.e. solution for concept prediction task). We propose different pooling functions and attention mechanisms which can be easily incorporated into our RMIL to improve the concept annotations.
+
+Pooling Layers for RMIL The bag-level prediction is obtained by using an aggregation gathering on all instance-level predictions. The aggregator function $g ( \cdot ) : \left[ 0 , 1 \right] ^ { T } \mapsto \left[ 0 , 1 \right]$ in RMIL can be modeled using the pooling layers. Without loss of generality we assume that RNN model computes a mapping from the feature time series to the concept time series for each of the concept $k \in \mathcal { C }$ . Let us denote the probability of an instance $j$ belonging to concept $k$ as $p _ { j k }$ . Then, the bag level probability for a concept $k$ is given by $P _ { k } = g ( p _ { 1 k } , p _ { 2 k } , . . . , p _ { T k } )$ . The role of aggregator function $g ( \cdot )$ is to combine the instance probabilities from each class specific feature map $\{ p _ { j k } \}$ into a single bag probability $P _ { k }$ . Several pooling mechanisms shown in Table 1 have been introduced in MIL and deep learning literature which can be used in our RMIL. In Table $1 , r , a , b _ { k }$ , and $r _ { k }$ are parameters which can be fixed or are learned during training, and $\sigma ( \cdot )$ denotes the sigmoid function.
+
+Table 1: Pooling functions for RMIL.   
+
+<table><tr><td rowspan=1 colspan=1>Pooling Functions</td></tr><tr><td rowspan=1 colspan=1>Noisy-OR poolingPk = g({𝑝jκ})=1-II(1- Pjk)(Zhang et al., 2006)</td></tr><tr><td rowspan=1 colspan=1>Log-sum-exponention (LSE)  Pk =g({𝑝jk}）)=1log(↓∑jexp(rPjκ））(Ramon &amp; De Raedt, 2000)</td></tr><tr><td rowspan=1 colspan=1>Generalized Mean (GM)      Pk =g({𝑝jx}）=（∑jjκ）1/r(Keeler et al., 1991)</td></tr><tr><td rowspan=1 colspan=1>Integrated segmentation       Px=g({pjx}）=（∑-x）Pjk）/(1+∑jr-)Pjkand recognition (ISR)(Keeler et al.,1991)</td></tr><tr><td rowspan=1 colspan=1>Noisy-AND pooling           P=g{pj}）=（（∑/-b）-ab）(Kraus et al., 2016)                                       σ(a(1-bk))-σ(-abk)</td></tr><tr><td rowspan=1 colspan=1>Max pooling                    Pk = g({pjk}) = maxj (Pjk)</td></tr><tr><td rowspan=1 colspan=1>Softmax pooling               Pk =g({𝑝jk})=(∑jPjκ exp(rkpjκ))/(∑,exp(rkPjk))</td></tr></table>
+
+Attention Mechanism for RMIL Instances within each bag have temporal relations between them. We can use attention mechanism to focus on some of the instances and their relations to improve their instance-level predictions. In order to make predictions at time $j$ , the hidden state $\boldsymbol { h } _ { j } \in \bar { \mathbb { R } } ^ { Q }$ of RNN can be used, where $Q$ is the hidden state dimension. However, relevant information may be captured by hidden states at other time steps as well. Thus, we may want to introduce an attention vector or matrix $( a )$ to leverage information of hidden states $\pmb { H } = \left( \hat { \pmb { h } } _ { 1 } , \cdot \cdot \cdot , \pmb { h } _ { T } \right) ^ { \top } \in \mathbb { R } ^ { T \times Q }$ from all time steps. Let us denote the output after attention as $\tilde { H } \in \mathbb { R } ^ { T \times Q }$ . The attention matrix can then be modeled using $\tilde { \cal H }$ in various ways as listed below.
+
+Feature-based Attention One idea is to design the attention matrix based on the feature and its time-stamp. Let us define a feature-based attention matrix as $\pmb { A } = \left( \pmb { a } _ { 1 } , \cdot \cdot \cdot , \pmb { a } _ { T } \right) ^ { \top } \in \mathbb { R } ^ { T \times Q }$ . For each $j = 1 , \cdots , T$ , we have
+
+$$
+\mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { } \mathbf { }
+$$
+
+and $\tilde { H } = A \odot H$ , where $\odot$ and $\oslash$ are element-wise multiplication and division, respectively and $\ b { W } = \left( \ b { w } _ { 1 } , \cdot \cdot \cdot , \ b { w } _ { T } \right) ^ { \top } \in \mathbb { R } ^ { T \times T }$ is the weight matrix which can be learned during training. We call this Attention- $F$ mechanism. We can simplify this attention by averaging the attentions for all hidden dimensions by taking $\begin{array} { r } { a _ { j q }  1 / Q \cdot \sum _ { 1 \leq q ^ { \prime } \leq Q } { a _ { j q ^ { \prime } } } } \end{array}$ . We will denote this as Attention- $F S$ mechanism.
+
+Time-based Attention Attention model (Ma et al., 2017) can be designed to capture the relation between the current time step $j$ and previous time steps $j ^ { \prime } \le j$ , by solely relying on previous hidden states $h _ { j ^ { \prime } }$ . We can define a time-based attention matrix as $\check { A } \in \mathbb { R } ^ { T \times \check { T } }$ . For each $j$ and $j ^ { \prime }$ in $[ 1 , \cdots , T ]$ , we have
+
+$$
+a _ { j , j ^ { \prime } } = \left\{ \begin{array} { l l } { \left( \exp \left( \boldsymbol { w ^ { \intercal } } h _ { j ^ { \prime } } \right) \right) / \left( \displaystyle \sum _ { 1 \leq j ^ { \prime \prime } \leq j } \exp \left( \boldsymbol { w ^ { \intercal } } h _ { j ^ { \prime \prime } } \right) \right) , } & { j ^ { \prime } \leq j ; } \\ { 0 , } & { \mathrm { o t h e r w i l } } \end{array} \right.
+$$
+
+and ${ \tilde { H } } = A \cdot H$ , where ${ \pmb w } \in \mathbb { R } ^ { D }$ is the weight vector to learn. We use Attention- $T$ to represent Time-based attention mechanism.
+
+Interaction-based Attention The time-based attention can be further improved by considering both the previous and current hidden states $h _ { j ^ { \prime } }$ and $h _ { j }$ (Ma et al., 2017). In this case, we have
+
+$$
+\begin{array} { r } { \nu _ { j , j ^ { \prime } } = \left\{ \begin{array} { l l } { \left( \exp \left( v ^ { \top } \operatorname { t a n h } ( W _ { 1 } h _ { j } + W _ { 2 } h _ { j ^ { \prime } } ) \right) \right) / \left( \displaystyle \sum _ { 1 \leq j ^ { \prime \prime } \leq j } \exp \left( v ^ { \top } \operatorname { t a n h } ( W _ { 1 } h _ { j } + W _ { 2 } h _ { j ^ { \prime \prime } } ) \right) \right) , } & { j ^ { \prime } \leq j ; } \\ { 0 , \quad \quad \mathrm { o t h e r w i s e } } \end{array} \right. } \end{array}
+$$
+
+and similarly ${ \tilde { H } } = A \cdot H$ . Here, we need to learn $\pmb { v } \in \mathbb { R } ^ { S } , \pmb { W _ { 1 } } \in \mathbb { R } ^ { S \times Q } , \pmb { W _ { 2 } } \in \mathbb { R } ^ { S \times Q }$ , and we choose $S = Q / 2$ . A simplified version of interaction-based attention can be obtained if we use vector ${ \pmb w } _ { 1 } \in \mathbb { R } ^ { Q } , { \pmb w } _ { 2 } \in \mathbb { R } ^ { Q }$ instead of matrices $W _ { 1 } , W _ { 2 }$ and by setting ${ \pmb v } = { \bf 1 }$ in the above equation. We use Attention- ${ \bf \nabla } \cdot { \cal I }$ and Attention- $. I S$ to represent Interaction-based attention mechanism and simplified version of interaction-based attention mechanism respectively.
+
+The above attention mechanisms usually help both prediction and localization tasks.
+
+# 4 EXPERIMENTS
+
+Here, we demonstrate the performance of our proposed RMIL models on concept annotation tasks i.e. concept prediction and localization tasks, using a real-world health-care dataset and compare its performance to the popular multi-instance learning approaches. In addition, we discuss the impact of using pooling functions and attention mechanism in our RMIL framework.
+
+# 4.1 DATASET DESCRIPTIONS AND EXPERIMENTAL DESIGN
+
+To evaluate our RMIL, we ran experiments on MIMIC-III RESP datasets whose statistics is shown in Table 2.
+
+MIMIC-III RESP Dataset MIMIC-III is a public dataset (Johnson et al., 2016) which has deidentified clinical care data collected at Beth Israel Deaconess Medical Center from 2001 to 2012. It contains over 58,000 hospital admission records of 38,645 adults and 7,875 neonates. For our work, we extracted 21 feature time series from more than 2,000 adult patients who were diagnosed with a respiratory disorder such as Acute Hypoxemic Respiratory Failure (AHRF) (Khemani et al., 2009) at the time of admission. These 21 features are respiratory based features such as peak inspiratory pressure (PIP) and arterial partial pressure of oxygen (PaO2) and were collected during the first 3 days after admission. The feature time series has 4 time stamps and the first time stamp corresponds to the admission time. We denote this dataset as MIMIC-III RESP dataset. In addition, we also generated another feature time series with more time stamps whose results is shown in the appendix.
+
+Table 2: MIMIC-III RESP dataset.   
+
+<table><tr><td>MIMIC-IIIRESP</td></tr><tr><td># of samples (N)</td><td>2014</td></tr><tr><td># of variables (D)</td><td>21</td></tr><tr><td># of time steps</td><td>4</td></tr><tr><td># of concepts</td><td>26</td></tr></table>
+
+Concept Annotations The medical time series data of MIMIC-III dataset does not come with the concept annotations, however the medical concepts are available in the doctor notes of the MIMIC-III database. To obtain the concept annotations, we extract the concept time series from the doctor notes using the NOTEEVENTS table of MIMIC-III database. The total number of doctor notes is 2,083,180, out of which $9 8 . 1 5 \%$ of notes (2,044,634 notes) have no timestamp and $1 . 8 5 \%$ of notes (38,546 notes) have timestamps associated with them. The total number of unique concepts in the doctor notes in the first 3 days data is 6,197. To obtain concept time series for each patient with respiratory disorder such as AHRF, we first identified respiratory-related concepts from the medical literature (Antonelli et al., 2001; Khemani et al., 2009), and obtained their medical codes from the Unified Medical Language System (UMLS) dictionary (Bodenreider, 2004). Then, we mined the patient’s doctor notes from NOTEEVENTS table to extract all the possible medical concepts related to the respiratory system and its disorder. In total, we chose top 26 respiratory concepts to generate concept time series which has the same number of time stamps as feature time series.
+
+Table 3: Concept annotation results on MIMIC-III RESP dataset. Max-pooling function was used in all RMIL models.   
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="2">Prediction</td><td colspan="2">Localization</td></tr><tr><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td rowspan="6">RMIL Models</td><td>S2S</td><td>0.858</td><td>0.756</td><td>0.788</td><td>0.431</td></tr><tr><td>S2S with Attention I</td><td>0.858</td><td>0.758</td><td>0.793</td><td>0.433</td></tr><tr><td>LSTM</td><td>0.860</td><td>0.763</td><td>0.795</td><td>0.444</td></tr><tr><td>LSTM with Attention I</td><td>0.862</td><td>0.766</td><td>0.797</td><td>0.444</td></tr><tr><td>Bi-LSTM</td><td>0.864</td><td>0.769</td><td>0.789</td><td>0.415</td></tr><tr><td>Bi-LSTM with Attention I</td><td>0.864</td><td>0.769</td><td>0.796</td><td>0.420</td></tr><tr><td rowspan="4">MIL Models</td><td>CNN</td><td>0.857</td><td>0.756</td><td>0.785</td><td>0.397</td></tr><tr><td>CNN with Attention I</td><td>0.855</td><td>0.755</td><td>0.785</td><td>0.409</td></tr><tr><td>DPMIL</td><td>0.531</td><td>0.222</td><td>0.516</td><td>0.148</td></tr><tr><td>MISVM</td><td>0.751</td><td>0.613</td><td>0.706</td><td>0.333</td></tr></table>
+
+# 4.2 COMPARISON AND IMPLEMENTATION DETAILS
+
+We compare the performance of our proposed models to the popular MIL models such as MISVM (Andrews et al., 2003), DPMIL (Kandemir & Hamprecht, 2014) and Convolutional Neural Networks (CNN) Kraus et al. (2016), and CNN with attention. We categorize all the evaluated methods into two groups:
+
+1. Multi-Instance Learning models (MIL): We treat MISVM, DPMIL, Convolutional Neural Networks (CNN), and CNN with Attention I as our baseline MIL models.   
+2. Relational Multi-Instance Learning models (RMIL): We evaluate the following deep learning models as part of our RMIL framework: (a) Long Short-Term Memory neural networks (LSTM) (Hochreiter & Schmidhuber, 1997) (b) Bi-directional LSTM (Bi-LSTM) (Graves et al., 2013) (c) Sequence to Sequence models (S2S) (Sutskever et al., 2014) (d) The above three RMIL models with different attention mechanisms (e) The above RMIL models with different pooling functions
+
+For LSTM models, we use two LSTM layers and two dense layers. For S2S models, we use two LSTM layers for both the encoder and the decoder. For Bi-LSTM models, we use two bi-directional LSTM layers. All the models were constructed to have a comparable number of parameters. We train all the Deep learning models with the RMSProp optimization method and we use early stopping to find the best weights on the validation dataset. For baseline MIL models, we follow the suggestions of the corresponding papers to fine-tune the parameters. All the input variables in the training data are normalized to be 0 mean and 1 standard deviation. The inputs to all the models is the same feature time series data. We used Keras (Chollet, 2017) and Python to run the deep models and MISVM models. Matlab code from the original authors was used to obtain DPMIL results. We use the area under ROC (AUROC) and area under precision-recall curve (AUPRC) scores as our evaluation metrics and report the results from 5-fold cross validation for all the evaluated methods.
+
+# 4.3 QUANTITATIVE RESULTS
+
+Table 3 shows the concept annotation results on the MIMIC-III RESP dataset. From this table, we see that RMIL models outperform the non-deep multi-instance learning models by at least $8 \%$ for concept localization task, and by at least $10 \mathrm { - } 1 5 \%$ for concept prediction tasks in terms of AUROC and AUPRC. RMIL performs slightly better than CNN-based models on all the metrics. Among all the RMIL models, we find that LSTM model obtains slightly better overall results compared to the other models for localization task.
+
+To study the impact of pooling and attention, we trained and evaluated LSTM models with different pooling functions and different attention mechanisms, which are described in Section 3. Tables 4 and 5 show the comparison results. From these tables, we observed that (i) all the attention mechanisms except feature-based attention perform similar to each other especially for the prediction task, and (ii) all the pooling functions other than ISR and Noisy-OR obtain similar overall performance. This
+
+Table 4: Results on RMIL LSTM models with different pooling functions and with Attention-I mechanism.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">Prediction</td><td colspan="2">Localization</td></tr><tr><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td>LSTM with ISR Pooling</td><td>0.862</td><td>0.765</td><td>0.733</td><td>0.375</td></tr><tr><td>LSTM with Noisy-AND Pooling</td><td>0.863</td><td>0.767</td><td>0.779</td><td>0.431</td></tr><tr><td>LSTM with Generalized Mean Pooling</td><td>0.863</td><td>0.767</td><td>0.796</td><td>0.450</td></tr><tr><td>LSTM with LSE Pooling</td><td>0.863</td><td>0.767</td><td>0.792</td><td>0.452</td></tr><tr><td>LSTM with Softmax Pooling</td><td>0.863</td><td>0.767</td><td>0.790</td><td>0.450</td></tr><tr><td>LSTM with Noisy-OR Pooling</td><td>0.860</td><td>0.762</td><td>0.703</td><td>0.318</td></tr><tr><td>LSTM with Max Pooling</td><td>0.862</td><td>0.766</td><td>0.797</td><td>0.444</td></tr></table>
+
+![](images/562a122ec127b761f3c67afdea40bd96e5326f2adc2f796ff16eeeca1fecd15e.jpg)
+
+(a) Ground truth annotations of intubation and extubation concepts.
+
+![](images/dc3efad88de512bfd0634a14ef04c3a8e3061f475991bd73a1016598bec4b0e0.jpg)  
+(b) Intubation and extubation concept prediction probabilities identified by attention-based LSTM.   
+Figure 2: Ground truth and predicted concept annotation comparison. In 2(a) white block corresponds to absence of a concept and black block corresponds to presence of a concept. In 2(b), darker gray value indicates higher chance for presence of a concept. X-axis represents time steps, Y-axis corresponds to different patients.
+
+demonstrates that choice of attention does not matter but choice of pooling has some impact in our RMIL framework.
+
+Table 5: Results on RMIL LSTM models with different attention mechanisms and with max pooling.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Prediction</td><td colspan="2">Localization</td></tr><tr><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td>LSTM with Attention-T</td><td>0.864</td><td>0.768</td><td>0.796</td><td>0.445</td></tr><tr><td>LSTMwith Attention-IS</td><td>0.862</td><td>0.767</td><td>0.798</td><td>0.446</td></tr><tr><td>LSTM with Attention-I</td><td>0.862</td><td>0.766</td><td>0.797</td><td>0.444</td></tr><tr><td>LSTMwith Attention-FS</td><td>0.861</td><td>0.765</td><td>0.796</td><td>0.417</td></tr><tr><td>LSTM with Attention-F</td><td>0.861</td><td>0.764</td><td>0.780</td><td>0.408</td></tr><tr><td>LSTM with no attention</td><td>0.860</td><td>0.763</td><td>0.795</td><td>0.444</td></tr></table>
+
+# 4.4 DISCUSSIONS
+
+We can study the interpretability of concept localization by looking at the localization results of our RMIL models, even though the model is trained without the labels for localization. Figure 2 shows the ground truth annotations of two respiratory concepts - intubation and extubation concepts, and the prediction probabilities of these concepts obtained by our RMIL attention-based LSTM models. From figure 2(a) we can make the following observations, (i) intubation usually happens before extubation for the same patient, (ii) intubation and extubation could happen on the same day, and (iii) intubation and extubation occur commonly within the first 24 hours of admission. From the figure 2(b), we see that our RMIL attention based LSTM predicts that the probability of intubation happening on the first day of admission is higher (draker gray means higher probability of concept occurrence) and the probability of extubation happening within first day is lower. This indicates that the model has correctly learnt that intubation should appear before extubation. This also implicitly implies that the RMIL attention-based LSTM models have correctly learnt the instance-level relationships from the medical time series data with only bag-level labels.
+
+# 5 SUMMARY
+
+In this paper, we presented Relational Multi-Instance Learning - a deep multi-instance learning framework using recurrent neural networks for concept annotation from the medical time series data. Empirical results on medical dataset demonstrated that our proposed models outperform the popular state-of-the-art multi-instance learning approaches. Experiments with different pooling and attention mechanisms showed that while attention mechanism does not have a significant impact on model’s performance, certain pooling functions such as ISR and Noisy-OR can negatively impact the instance prediction results.
+
+# REFERENCES
+
+Stergos Afantenos, Vangelis Karkaletsis, and Panagiotis Stamatopoulos. Summarization from medical documents: a survey. Artificial intelligence in medicine, 33(2):157–177, 2005.   
+Charu C Aggarwal and ChengXiang Zhai. Mining text data. Springer Science & Business Media, 2012.   
+Daniel Albright, Arrick Lanfranchi, Anwen Fredriksen, William F Styler IV, Colin Warner, Jena D Hwang, Jinho D Choi, Dmitriy Dligach, Rodney D Nielsen, James Martin, et al. Towards comprehensive syntactic and semantic annotations of the clinical narrative. Journal of the American Medical Informatics Association, 20(5):922–930, 2013.   
+Jaume Amores. Multiple instance classification: Review, taxonomy and comparative study. Artificial Intelligence, 201:81–105, 2013.   
+Stuart Andrews, Ioannis Tsochantaridis, and Thomas Hofmann. Support vector machines for multipleinstance learning. In Advances in neural information processing systems, pp. 577–584, 2003.   
+M Antonelli, G Conti, ML Moro, A Esquinas, G Gonzalez-Diaz, M Confalonieri, P Pelaia, T Principi, C Gregoretti, F Beltrame, et al. Predictors of failure of noninvasive positive pressure ventilation in patients with acute hypoxemic respiratory failure: a multi-center study. Intensive care medicine, 2001.   
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014.   
+Steven Bethard, Guergana Savova, Wei-Te Chen, Leon Derczynski, James Pustejovsky, and Marc Verhagen. Semeval-2016 task 12: Clinical tempeval. Proceedings of SemEval, pp. 1052–1062, 2016.   
+Olivier Bodenreider. The unified medical language system (umls): integrating biomedical terminology. Nucleic acids research, 32(suppl_1):D267–D270, 2004.   
+Alessandro Camerra, Themis Palpanas, Jin Shieh, and Eamonn Keogh. isax 2.0: Indexing and mining one billion time series. In Data Mining (ICDM), 2010 IEEE 10th International Conference on, pp. 58–67. IEEE, 2010.   
+Ricardo JGB Campello, Davoud Moulavi, Arthur Zimek, and Jörg Sander. Hierarchical density estimates for data clustering, visualization, and outlier detection. ACM Transactions on Knowledge Discovery from Data (TKDD), 10(1):5, 2015.   
+Marc-Andre Carbonneau, Veronika Cheplygina, Eric Granger, and Ghyslain Gagnon. Multiple instance learning: A survey of problem characteristics and applications. Pattern Recognition, 2017.   
+Philip K Chan and Matthew V Mahoney. Modeling multiple time series for anomaly detection. In Data Mining, Fifth IEEE International Conference on, pp. 8–pp. IEEE, 2005.   
+Varun Chandola, Arindam Banerjee, and Vipin Kumar. Anomaly detection: A survey. ACM computing surveys (CSUR), 41(3):15, 2009.   
+Varun Chandola, Arindam Banerjee, and Vipin Kumar. Anomaly detection for discrete sequences: A survey. IEEE Transactions on Knowledge and Data Engineering, 24(5):823–839, 2012.   
+Zhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, and Yan Liu. Recurrent neural networks for multivariate time series with missing values. arXiv preprint arXiv:1606.01865, 2016.   
+Yixin Chen and James Z Wang. Image categorization by learning and reasoning with regions. Journal of Machine Learning Research, 5(Aug):913–939, 2004.   
+Edward Choi, Mohammad Taha Bahadori, Elizabeth Searles, Catherine Coffey, Michael Thompson, James Bost, Javier Tejedor-Sojo, and Jimeng Sun. Multi-layer representation learning for medical concepts. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1495–1504. ACM, 2016a.   
+Youngduck Choi, Chill Yi-I Chiu, and David Sontag. Learning low-dimensional representations of medical concepts. AMIA Summits on Translational Science Proceedings, 2016:41, 2016b.   
+François Chollet. Keras (2015). URL http://keras. io, 2017.   
+Aaron M Cohen and William R Hersh. A survey of current work in biomedical text mining. Briefings in bioinformatics, 6(1):57–71, 2005.   
+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.   
+Thomas G Dietterich, Richard H Lathrop, and Tomás Lozano-Pérez. Solving the multiple instance problem with axis-parallel rectangles. Artificial intelligence, 89(1):31–71, 1997.   
+Mark Everingham, Luc Van Gool, Christopher KI Williams, John Winn, and Andrew Zisserman. The pascal visual object classes (voc) challenge. International journal of computer vision, 88(2): 303–338, 2010.   
+Arden W Forrey, Clement J Mcdonald, Georges DeMoor, Stanley M Huff, Dennis Leavelle, Diane Leland, Tom Fiers, Linda Charles, Brian Griffin, Frank Stalling, et al. Logical observation identifier names and codes (loinc) database: a public use set of codes and names for electronic reporting of clinical laboratory test results. Clinical Chemistry, 42(1):81–90, 1996.   
+James Foulds and Padhraic Smyth. Multi-instance mixture models and semi-supervised learning. In Proceedings of the 2011 SIAM International Conference on Data Mining, pp. 606–617. SIAM, 2011.   
+AS d’Avila Garcez and Gerson Zaverucha. Multi-instance learning using recurrent neural networks. In Neural Networks (IJCNN), The 2012 International Joint Conference on, pp. 1–6. IEEE, 2012.   
+Daniela Giordano, Isaak Kavasidis, and Concetto Spampinato. Automatic summary creation by applying natural language processing on unstructured medical records. In International Conference on Computer Analysis of Images and Patterns, pp. 383–393. Springer, 2015.   
+Alex Graves, Abdel-rahman Mohamed, and Geoffrey Hinton. Speech recognition with deep recurrent neural networks. In Acoustics, speech and signal processing (icassp), 2013 ieee international conference on, pp. 6645–6649. IEEE, 2013.   
+Xinze Guan, Raviv Raich, and Weng-Keen Wong. Efficient multi-instance learning for activity recognition from time series data using an auto-regressive hidden markov model. In International Conference on Machine Learning, pp. 2330–2339, 2016.   
+Fabio Guigou, Pierre Collet, and Pierre Parrend. The artificial immune ecosystem: a bio-inspired metaalgorithm for boosting time series anomaly detection with expert input. In European Conference on the Applications of Evolutionary Computation, pp. 573–588. Springer, 2017.   
+Manish Gupta, Jing Gao, Charu C Aggarwal, and Jiawei Han. Outlier detection for temporal data: A survey. IEEE Transactions on Knowledge and Data Engineering, 26(9):2250–2267, 2014.   
+Francisco Herrera, Sebastián Ventura, Rafael Bello, Chris Cornelis, Amelia Zafra, Dánel SánchezTarragó, and Sarah Vluymans. Introduction. In Multiple Instance Learning, pp. 1–16. Springer, 2016.   
+Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8): 1735–1780, 1997.   
+Victoria Hodge and Jim Austin. A survey of outlier detection methodologies. Artificial intelligence review, 22(2):85–126, 2004.   
+John GR Howie, David J Heaney, Margaret Maxwell, Jeremy J Walker, George K Freeman, and Harbinder Rai. Quality at general practice consultations: cross sectional survey. Bmj, 319(7212): 738–743, 1999.   
+Yan Huang, Wei Wang, Liang Wang, and Tieniu Tan. Multi-task deep neural network for multi-label learning. In Image Processing (ICIP), 2013 20th IEEE International Conference on, pp. 2897–2900. IEEE, 2013.   
+Jiwoon Jeon, Victor Lavrenko, and Raghavan Manmatha. Automatic image annotation and retrieval using cross-media relevance models. In Proceedings of the 26th annual international ACM SIGIR conference on Research and development in informaion retrieval, pp. 119–126. ACM, 2003.   
+AEW Johnson, TJ Pollard, L Shen, L Lehman, M Feng, M Ghassemi, B Moody, P Szolovits, LA Celi, and RG Mark. Mimic-iii, a freely accessible critical care database. Scientific Data, 2016.   
+Michael Jones, Daniel Nikovski, Makoto Imamura, and Takahisa Hirata. Exemplar learning for extremely efficient anomaly detection in real-valued time series. Data Mining and Knowledge Discovery, 30(6):1427–1454, 2016.   
+Melih Kandemir and Fred A Hamprecht. Instance label prediction by dirichlet process multiple instance learning. In UAI, pp. 380–389, 2014.   
+Andrej Karpathy, George Toderici, Sanketh Shetty, Thomas Leung, Rahul Sukthankar, and Li Fei-Fei. Large-scale video classification with convolutional neural networks. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pp. 1725–1732, 2014.   
+James D Keeler, David E Rumelhart, and Wee Kheng Leow. Integrated segmentation and recognition of hand-printed numerals. In Advances in neural information processing systems, pp. 557–563, 1991.   
+Robinder G Khemani, David Conti, Todd A Alonzo, Robert D Bart, and Christopher JL Newth. Effect of tidal volume in children with acute hypoxemic respiratory failure. Intensive care medicine, 35 (8):1428–1437, 2009.   
+Atanas Kiryakov, Borislav Popov, Ivan Terziev, Dimitar Manov, and Damyan Ognyanoff. Semantic annotation, indexing, and retrieval. Web Semantics: Science, Services and Agents on the World Wide Web, 2(1):49–79, 2004.   
+Atsuhiro Kojima, Takeshi Tamura, and Kunio Fukunaga. Natural language description of human activities from video images based on concept hierarchy of actions. International Journal of Computer Vision, 50(2):171–184, 2002.   
+Dimitrios Kotzias, Misha Denil, Phil Blunsom, and Nando de Freitas. Deep multi-instance transfer learning. arXiv preprint arXiv:1411.3128, 2014.   
+Oren Z Kraus, Jimmy Lei Ba, and Brendan J Frey. Classifying and segmenting microscopy images with deep multiple instance learning. Bioinformatics, 32(12):i52–i59, 2016.   
+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.   
+Stephen C Kurachek, Christopher J Newth, Michael W Quasney, Tom Rice, Ramesh C Sachdeva, Neal R Patel, Jeanne Takano, Larry Easterling, Mathew Scanlon, Ndidiamaka Musa, et al. Extubation failure in pediatric intensive care: a multiple-center study of risk factors and outcomes. Critical care medicine, 31(11):2657–2664, 2003.   
+Christoph H Lampert, Matthew B Blaschko, and Thomas Hofmann. Beyond sliding windows: Object localization by efficient subwindow search. In Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, pp. 1–8. IEEE, 2008.   
+Anna S Law, Yvonne Freer, Jim Hunter, Robert H Logie, Neil McIntosh, and John Quinn. A comparison of graphical and textual presentations of time series data to support medical decision making in the neonatal intensive care unit. Journal of clinical monitoring and computing, 19(3): 183–194, 2005.   
+Jessica Lin, Eamonn Keogh, Stefano Lonardi, and Bill Chiu. A symbolic representation of time series, with implications for streaming algorithms. In Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery, pp. 2–11. ACM, 2003.   
+Zachary C Lipton, David C Kale, Charles Elkan, and Randall Wetzell. Learning to diagnose with lstm recurrent neural networks. arXiv preprint arXiv:1511.03677, 2015.   
+Fenglong Ma, Radha Chitta, Jing Zhou, Quanzeng You, Tong Sun, and Jing Gao. Dipole: Diagnosis prediction in healthcare via attention-based bidirectional recurrent neural networks. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1903–1911. ACM, 2017.   
+Junhua Mao, Wei Xu, Yi Yang, Jiang Wang, Zhiheng Huang, and Alan Yuille. Deep captioning with multimodal recurrent neural networks (m-rnn). arXiv preprint arXiv:1412.6632, 2014.
+
+World Health Organization and Practice Management Information Corporation. ICD-9-CM: International Classification of Diseases, 9th Revision: Clinical Modification, volume 1. PMIC (Practice Management Information Corporation), 1998.
+
+Trang Pham, Truyen Tran, Dinh Phung, and Svetha Venkatesh. Deepcare: A deep dynamic memory model for predictive medicine. In Advances in Knowledge Discovery and Data Mining, pp. 30–41. Springer, 2016.
+
+Catherine Plaisant, Brett Milash, Anne Rose, Seth Widoff, and Ben Shneiderman. Lifelines: visualizing personal histories. In Proceedings of the SIGCHI conference on Human factors in computing systems, pp. 221–227. ACM, 1996.
+
+Catherine Plaisant, Rich Mushlin, Aaron Snyder, Jia Li, Daniel Heller, and Ben Shneiderman. Lifelines: using visualization to enhance navigation and analysis of patient records. In Proceedings of the AMIA Symposium, pp. 76. American Medical Informatics Association, 1998.
+
+Jan Ramon and Luc De Raedt. Multi instance neural networks. In Proceedings of the ICML-2000 workshop on attribute-value and relational learning, pp. 53–60, 2000.
+
+Adrienne G Randolph, David Wypij, Shekhar T Venkataraman, James H Hanson, Rainer G Gedeit, Kathleen L Meert, Peter M Luckett, Peter Forbes, Michelle Lilley, John Thompson, et al. Effect of mechanical ventilator weaning protocols on respiratory outcomes in infants and children: a randomized controlled trial. Jama, 288(20):2561–2568, 2002.
+
+Guergana Savova, Steven Bethard, Will Styler, James Martin, Martha Palmer, James Masanz, and Wayne Ward. Towards temporal relation discovery from the clinical narrative. In AMIA annual symposium proceedings, volume 2009, pp. 568. American Medical Informatics Association, 2009.
+
+Donia Scott, Catalina Hallett, and Rachel Fettiplace. Data-to-text summarisation of patient records: Using computer-generated summaries to access patient histories. Patient education and counseling, 92(2):153–159, 2013.
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.
+
+Hossein Soleimani and David J Miller. Semisupervised, multilabel, multi-instance learning for structured data. Neural computation, 2017.
+
+Somayajulu G Sripada, Ehud Reiter, Jim Hunter, and Jin Yu. Generating english summaries of time series data using the gricean maxims. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 187–196. ACM, 2003.
+
+Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. In Advances in neural information processing systems, pp. 3104–3112, 2014.
+
+Marian Van Der Meulen, Robert H Logie, Yvonne Freer, Cindy Sykes, Neil McIntosh, and Jim Hunter. When a graph is poorer than 100 words: A comparison of computerised natural language generation, human generated descriptions and graphical displays in neonatal intensive care. Applied Cognitive Psychology, 24(1):77–89, 2010.
+
+Veronika Vincze, György Szarvas, Richárd Farkas, György Móra, and János Csirik. The bioscope corpus: biomedical texts annotated for uncertainty, negation and their scopes. BMC bioinformatics, 9(11):S9, 2008.
+
+Xing Wang, Jessica Lin, Nital Patel, and Martin Braun. A self-learning and online algorithm for time series anomaly detection, with application in cpu manufacturing. In Proceedings of the 25th ACM International on Conference on Information and Knowledge Management, pp. 1823–1832. ACM, 2016.
+
+Nils Weidmann, Eibe Frank, and Bernhard Pfahringer. A two-level learning method for generalized multi-instance problems. In European Conference on Machine Learning, pp. 468–479. Springer, 2003.
+
+Brigham C Willis, Alan S Graham, Eunice Yoon, Randall C Wetzel, and Christopher JL Newth. Pressure-rate products and phase angles in children on minimal support ventilation and after extubation. Intensive care medicine, 31(12):1700–1705, 2005.
+
+Jiajun Wu, Yinan Yu, Chang Huang, and Kai Yu. Deep multiple instance learning for image classification and auto-annotation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3460–3469, 2015.
+
+Hua Xu, Shane P Stenner, Son Doan, Kevin B Johnson, Lemuel R Waitman, and Joshua C Denny. Medex: a medication information extraction system for clinical narratives. Journal of the American Medical Informatics Association, 17(1):19–24, 2010.   
+Zhennan Yan, Yiqiang Zhan, Zhigang Peng, Shu Liao, Yoshihisa Shinagawa, Shaoting Zhang, Dimitris N Metaxas, and Xiang Sean Zhou. Multi-instance deep learning: Discover discriminative local anatomies for bodypart recognition. IEEE transactions on medical imaging, 35(5):1332–1343, 2016.   
+Haonan Yu, Jiang Wang, Zhiheng Huang, Yi Yang, and Wei Xu. Video paragraph captioning using hierarchical recurrent neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4584–4593, 2016.   
+Cha Zhang, John C Platt, and Paul A Viola. Multiple instance boosting for object detection. In Advances in neural information processing systems, pp. 1417–1424, 2006.   
+Dan Zhang, Yan Liu, Luo Si, Jian Zhang, and Richard D Lawrence. Multiple instance learning on structured data. In Advances in Neural Information Processing Systems, pp. 145–153, 2011.   
+Li Zhou, Genevieve B Melton, Simon Parsons, and George Hripcsak. A temporal constraint structure for extracting temporal information from clinical narrative. Journal of biomedical informatics, 39 (4):424–439, 2006.   
+Zhi-Hua Zhou and Min-Ling Zhang. Multi-instance multi-label learning with application to scene classification. In Advances in neural information processing systems, pp. 1609–1616, 2007.   
+Zhi-Hua Zhou, Yu-Yin Sun, and Yu-Feng Li. Multi-instance learning by treating instances as non-iid samples. In Proceedings of the 26th annual international conference on machine learning, pp. 1249–1256. ACM, 2009.   
+Wentao Zhu, Qi Lou, Yeeleng Scott Vang, and Xiaohui Xie. Deep multi-instance networks with sparse label assignment for whole mammogram classification. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 603–611. Springer, 2017.   
+Pierre Zweigenbaum, Dina Demner-Fushman, Hong Yu, and Kevin B Cohen. Frontiers of biomedical text mining: current progress. Briefings in bioinformatics, 8(5):358–375, 2007.
+
+# 6 APPENDIX
+
+# 6.1 MIMIC-III CONCEPT LIST
+
+Table 6 show the concept list of the MIMIC-III RESP dataset used in our experiments.
+
+6.2 ANOMALY DETECTION FOR TIME SERIES
+
+Here, we will demonstrate anomaly detection from medical time series data as a use case application of our RMIL framework.
+
+# 6.2.1 MOTIVATION AND RELATED WORK
+
+Each year, more than 1,000,000 US adults and children are put on mechanical ventilation during their stays in ICU. However, lack of effective tools to aid with ventilator weaning and extubation (removal of the breathing tube) readiness assessment results in nearly half of the patients spending unnecessary days on ventilators (Randolph et al., 2002), and up to $20 \%$ of them having ventilators discontinued too soon (Kurachek et al., 2003). Spending unnecessary days on ventilators can lead to hospital-acquired infections while having ventilators discontinued too soon could require painful reintubation. A work-of-breathing measure called Pressure-Rate Product (PRP), calculated from esophageal pressure, has shown potential to be used as a guideline for ventilator weaning and extubation readiness assessment (Willis et al., 2005). However, PRP calculations are susceptible to sensor artifacts and breathing pattern anomalies. These anomalies limit realtime use of PRP for clinical decision making. Our goal is to automatically detect and remove these anomalies by using our proposed Relational Multi-instance learning models, thereby enabling real-time clinical decision making. Figure 3 shows the example of anomaly detection from the ventilator time series data. Here, anomalies appear due to patient factors (cough, movement) and instrument factors (probing, catherter drift), and should be automatically detected from the ventalitor monitoring time series data.
+
+Table 6: MIMIC-III respiratory concept list.   
+
+<table><tr><td>Concept Number</td><td>Concept Name</td></tr><tr><td>1</td><td>Respiratory rate</td></tr><tr><td>2</td><td>Biomedical tube device</td></tr><tr><td>3</td><td>Analysis of arterial blood gases and ph</td></tr><tr><td>4</td><td>Ventilation,function (observable entity)</td></tr><tr><td>5</td><td>Medical history</td></tr><tr><td>6</td><td>Injury wounds</td></tr><tr><td>7</td><td>Body weights</td></tr><tr><td>8</td><td>Tidal volume</td></tr><tr><td>9</td><td>Inspired fraction of oxygen</td></tr><tr><td>10</td><td>Intubation</td></tr><tr><td>11</td><td>Positive end-expiratory pressure</td></tr><tr><td>12</td><td>Artificial airways</td></tr><tr><td>13</td><td>Bicarbonates</td></tr><tr><td>14</td><td>Tracheal extubation</td></tr><tr><td>15</td><td>Chest radiograph</td></tr><tr><td>16</td><td>Mean blood pressure</td></tr><tr><td>17</td><td>Oxygen measurement, partial pressure,arterial</td></tr><tr><td>18</td><td>Cardiac arrest</td></tr><tr><td>19</td><td>Peak inspiratory pressure</td></tr><tr><td>20</td><td>Biomechanical compliance</td></tr><tr><td>21</td><td>Tachycardia ventricular</td></tr><tr><td>22</td><td>Scoliosis</td></tr><tr><td>23</td><td>Acute lung injury</td></tr><tr><td>24</td><td>Partial pressure of carbon dioxide in arterial blood</td></tr><tr><td>25</td><td>Resuscitate</td></tr><tr><td>26</td><td>Sudden onset</td></tr></table>
+
+# Ventilator Time Series
+
+![](images/2209e7f2a1d65bee1998c9a0dce4bdcba978b5baa13d09634dccf2306f2caa99.jpg)  
+Figure 3: Anomaly detection from ventilator time series data.
+
+There is a long history and a rich body of research work on anomaly detection in time series data. See Hodge & Austin (2004); Chandola et al. (2009) for a quick survey on generic anomaly detection algorithms. Gupta et al. (2014) also author a survey on anomaly detection for time series data. In their follow-up survey, Chandola et al. (2012) gave a summary on discrete sequence anomaly detection algorithms. Unfortunately, techniques for time series anomaly detection are quite domain-specific. This is due to highly-varied nature of time series characteristics. In Chan & Mahoney (2005), a set of $k$ minimum bounding rectangles between each time step is used as a model of normal data generating distribution. In Camerra et al. (2010); Lin et al. (2003), continuous time series are converted to ordinal symbolic representation, allowing faster approximation of Euclidean distance between time series windows and early pruning. Then, anomaly detection can be performed by setting a threshold on distance value. Recently, Wang et al. (2016) proposes a self-learning method that learns clusters of constrained grammars based on ordinal symbolic approximation of time series value. Guigou et al. (2017) proposes incorporating experts into anomaly detection in a method inspired by immune system. In Jones et al. (2016), the authors propose extracting exemplars from Euclidean pairwise distance as a way to speed up anomaly detection algorithm. Most of these techniques for time series anomaly detection are quite domain-specific, and few of them model the anomaly detection problem in Multi-instance learning setting.
+
+# 6.2.2 PROBLEM FORMULATION
+
+We formulate the anomaly detection as concept annotation problem in MIL setting, where the bag corresponds to the medical time series data $X _ { i }$ and an instance is the features at a time-stamp, and the anomaly corresponds to a concept. Thus, the prediction (predicting $Y _ { i }$ ) and localizaton tasks (predicting $C _ { i }$ ) correspond to predicting the presence and location of the anomalies in the time series data. For anomaly detection, we assume $\begin{array} { r } { Y _ { i } ^ { k } = \mathcal { T } \left( \sum _ { 1 \leq j \leq T } C _ { i } ^ { j k } \geq \eta \right) } \end{array}$ , where $\eta$ is chosen as $\eta = 0 . 6 T$ and $T$ is the number of time series windows within the bag.
+
+# 6.2.3 EXPERIMENTS
+
+We conduct anomaly detection experiments on a PICU dataset described below.
+
+PICU Dataset This dataset consists of 1,530 recorded 5- minute $2 0 0 – \mathrm { H z }$ sensor readings from 385 subjects collected at a leading children’s hospital by a team of research clinicians. Statistics of the dataset is shown in Table 7. The recordings are made on mechanically-ventilated patients in pediatrics ICU ward. Medical time series data collected from four sensors are used: flow volume spirometry, esophageal pressure sensor, and dual band respiratory inductance plethysmography. Each subject can be under one of four breathing conditions: ventilated with Continuous Positive Airway Pressure (CPAP), ventilated with Pressure Support (PS), 5 minutes after extubation, and 60 minutes after extubation. Along with the 4 sensor signals, clinicians verified binary anomaly label generated using hard-crafted state-of-the-art breathing anomaly detection algorithm is provided as ground truth.
+
+Table 7: PICU dataset statistics.   
+
+<table><tr><td></td><td>PICU</td></tr><tr><td># of samples (N)</td><td>1530</td></tr><tr><td># of variables (D)</td><td>80</td></tr><tr><td># of time steps</td><td>60</td></tr><tr><td># of concepts</td><td>1</td></tr></table>
+
+We annotate the concept of breathing anomaly in MIL framework by processing the dataset as follows. The sensor recordings are split into non-overlapping 5-second windows. In MIL framework, each window becomes an instance and each recording becomes a bag. For each sensor signal of each window, we extract 20 Mel-Frequency Cepstral Coefficients (MFCCs) to be used as features of each instance; thus, each instance has 80 features in total. For each instance, its anomaly label is set to positive if at least $20 \%$ of the window are labeled as anomaly. The bag anomaly label is set to positive if at least $60 \%$ of its instances are labeled as anomaly.
+
+Table 8: Anomaly detection results on PICU dataset with different attention mechanisms and with max pooling.   
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="2">Prediction</td><td colspan="2">Localization</td></tr><tr><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td rowspan="10">RMIL Models</td><td>S2S</td><td>0.741</td><td>0.797</td><td>0.532</td><td>0.707</td></tr><tr><td>S2S with Attention-I</td><td>0.732</td><td>0.787</td><td>0.511</td><td>0.694</td></tr><tr><td>S2S with Attention-F</td><td>0.730</td><td>0.789</td><td>0.519</td><td>0.689</td></tr><tr><td>LSTM</td><td>0.708</td><td>0.763</td><td>0.543</td><td>0.712</td></tr><tr><td>LSTM with Attention-I</td><td>0.690</td><td>0.767</td><td>0.513</td><td>0.682</td></tr><tr><td>LSTM with Attention-F</td><td>0.712</td><td>0.776</td><td>0.522</td><td>0.690</td></tr><tr><td>Bi-LSTM</td><td>0.729</td><td>0.784</td><td>0.562</td><td>0.725</td></tr><tr><td>Bi-LSTM with Attention-I</td><td>0.711</td><td>0.769</td><td>0.535</td><td>0.708</td></tr><tr><td>Bi-LSTM with Attention-F</td><td>0.736</td><td>0.790</td><td>0.539</td><td>0.704</td></tr><tr><td>Cluster-MIL*</td><td>0.590</td><td>0.620</td><td>0.530</td><td>0.670</td></tr><tr><td>MIL Models</td><td>MISVM</td><td>0.572</td><td>0.647</td><td>0.542</td><td>0.606</td></tr></table>
+
+Results Table 8 shows the anomaly detection results using our RMIL models. We observe that (i) RMIL models mostly outperform the non-deep MIL models on both tasks, (ii) LSTM and Bi-LSTM based RMIL models obtain better overall results compared to the sequence-to-sequence models, and (iii) the attention mechanism does not help for localization task but sometimes obtains better results for the prediction task.
+
+Remark: Cluster-MIL\* is a HDBSCAN clustering (Campello et al., 2015) based approach for anomaly detection. It uses both instance and bag labels for training, while other models only use bag labels for training.
+
+# 6.3 ADDITIONAL RESULTS ON MIMIC-III RESP DATASET
+
+We sampled the feature time series from MIMIC-III RESP dataset every 6 hours and generated a time series with more (12) time stamps. We call this MIMIC-III RESP-II dataset.
+
+Table 9 shows the results of RMIL models on this dataset. We observe that (i) All the RMIL models have similar performance for prediction task, (ii) Bi-LSTM RMIL has better localization results compared to other the models.
+
+Table 9: MIMIC-III RESP-II dataset with different attention mechanisms and with max pooling.   
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="2">Prediction</td><td colspan="2">Localization</td></tr><tr><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td rowspan="9">RMIL Models</td><td>S2S</td><td>0.874</td><td>0.746</td><td>0.787</td><td>0.185</td></tr><tr><td>S2S with Attention-I</td><td>0.875</td><td>0.749</td><td>0.794</td><td>0.186</td></tr><tr><td>S2S with Attention-F</td><td>0.869</td><td>0.739</td><td>0.777</td><td>0.188</td></tr><tr><td>LSTM</td><td>0.875</td><td>0.749</td><td>0.776</td><td>0.177</td></tr><tr><td>LSTM with Attention-I</td><td>0.875</td><td>0.750</td><td>0.781</td><td>0.175</td></tr><tr><td>LSTM with Attention-F</td><td>0.872</td><td>0.742</td><td>0.771</td><td>0.177</td></tr><tr><td>Bi-LSTM</td><td>0.876</td><td>0.749</td><td>0.801</td><td>0.206</td></tr><tr><td>Bi-LSTM with Attention-I</td><td>0.875</td><td>0.750</td><td>0.803</td><td>0.204</td></tr><tr><td>Bi-LSTM with Attention-F</td><td>0.871</td><td>0.741</td><td>0.769</td><td>0.171</td></tr></table>

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+        "text": "RELATIONAL MULTI-INSTANCE LEARNING FOR CONCEPT ANNOTATION FROM MEDICAL TIME SERIES ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "Recent advances in computing technology and sensor design have made it easier to collect longitudinal or time series data from patients, resulting in a gigantic amount of available medical data. Most of the medical time series lack annotations or even when the annotations are available they could be subjective and prone to human errors. Earlier works have developed natural language processing techniques to extract concept annotations and/or clinical narratives from doctor notes. However, these approaches are slow and do not use the accompanying medical time series data. To address this issue, we introduce the problem of concept annotation for the medical time series data, i.e., the task of predicting and localizing medical concepts by using the time series data as input. We propose Relational Multi-Instance Learning (RMIL) - a deep Multi Instance Learning framework based on recurrent neural networks, which uses pooling functions and attention mechanisms for the concept annotation tasks. Empirical results on medical datasets show that our proposed models outperform various multi-instance learning models. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Clinicians have limited time (e.g., only a few minutes (Howie et al., 1999)) to study and treat each patient. However, they are overloaded with a lot of patient data from multiple sources and in various formats, such as patient medical history and doctor’s notes in free-flowing text, vitals and monitoring data which are captured as time series, and prescriptions and drugs which appear as medical codes including ICD-9 (Organization & Corporation, 1998), LOINC codes (Forrey et al., 1996), etc. This rich information should be summarized and available to clinicians in easily digestible format for faster diagnosis and treatment. Graphical visualizations (Plaisant et al., 1998) are a popular approach to show patient data to doctors. However, recent studies have shown that graphical visualisations are not always helpful for clinicians’ decision-making (Law et al., 2005; Van Der Meulen et al., 2010). Text summaries on the other hand are widely embraced and are usually adopted in practice (Scott et al., 2013). Most existing systems use natural language processing techniques (Afantenos et al., 2005; Giordano et al., 2015) to generate summaries from doctor notes which include test results, discharge reports, observational notes, etc. While these systems are useful, they only use one source of data, i.e., doctor’s notes which might have noisy and erroneous entries, for text summarization. On the other hand, electronic health records have other sources of patient data such as vital signs, monitoring sensors, and lab results in the form of multivariate time-series, which can be more accurate and may contain rich information about patient’s conditions. Few existing patient summarization systems actually extract information directly from these time series for concept prediction and/or summarization. Generating simple text summaries such as trends from time series has been investigated before (Sripada et al., 2003) but is marginally useful since these trends are not mapped to the medical concepts which clinicians can quickly comprehend. Recent works (Pham et al., 2016; Choi et al., 2016a;b; Lipton et al., 2015; Che et al., 2016) have successfully shown that clinical events and outcomes can be predicted using medical codes or clinical time series data. However, directly obtaining medical concept annotations and summaries from the time series data is still an open question. ",
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+        "text": "In this work, we introduce the concept annotation task as the problem of predicting and localizing the medical concepts by modeling the related medical time series data. Figure 1 illustrates a concept annotation example where medical time series data such as heart rate, $\\mathrm { p H }$ and blood gas pressure are given, and the goal is to predict the time series of concepts such as intubation, extubation and resuscitate. To solve concept annotation problem, we formulate it as a Multi-Instance Learning (MIL) problem (Dietterich et al., 1997) and propose a deep learning based framework called Relational MultiInstance Learning (RMIL). RMIL uses Recurrent Neural Networks (RNNs) to model multivariate time series data, leverages instance relations via attention mechanisms, and provides concept predictions using pooling functions. ",
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+        "img_path": "images/d138111f217b2a0b650ab49fd7dfba33f87f2e1e94504c7e336ff367fad611f6.jpg",
+        "image_caption": [
+            "Figure 1: Medical time series to concept annotations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The main contributions of our work are the following. We present a unified view of the MIL approaches for time series data using RNN models with different pooling functions and attention mechanisms. We show that our RMIL model is capable of learning a good classifier for concept detection (bag label predictions) and concept localization tasks (instance label prediction), even though it is only trained using bag labels. We demonstrate that RMIL obtains promising results on real-world medical datasets and outperforms popular MIL approaches. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The rest of the paper is structured as follows. In the following section, we briefly discuss the related works. Afterwards, we describe MIL framework and describe how RNN can be combined with multi-instance learning framework to obtain our proposed RMIL. In Sections 4 and 5, we present experimental results and conclusions respectively. In the appendix, we demonstrate anomaly detection as another application of our RMIL framework. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Discovering concept annotations from the multivariate time series is a relatively new problem in medical domain with limited prior work. In this section, we will first highlight the related works on annotation tasks and then review related works on multi-instance learning. ",
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+        "type": "text",
+        "text": "Concept Annotation In medical domain, concept annotation is usually addressed in the clinical narrative mining and biomedical text mining literature (Aggarwal & Zhai, 2012; Cohen & Hersh, 2005; Zweigenbaum et al., 2007; Vincze et al., 2008). In other domains such as web-mining and computer vision, the concept annotation is usually analogous to semantic annotation (Kiryakov et al., 2004), image annotation (Jeon et al., 2003), object localization (Lampert et al., 2008) and image captioning (Karpathy et al., 2014). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Clinical Narratives Mining Automated discovery of temporal relations from clinical narratives (Savova et al., 2009; Zhou et al., 2006; Albright et al., 2013) and doctor notes (Plaisant et al., 1996) to uncover the patterns of disease progression is an important research problem in clinical informatics. Recent efforts such as SemEval competitions (Bethard et al., 2016) have been conducted to study this problem and evaluate/benchmark clinical information extraction systems $\\mathrm { X u }$ et al., 2010). These competitions focus on discrete, well-defined tasks which allow for rapid, reliable and repeatable evaluations. However, they only consider identifying and extracting temporal relations from clinical notes and do not use the accompanying medical time series data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Image Annotation and Captioning Successful object recognition systems have been developed in the past few decades for image annotation, object detection and localization in images and videos. ImageNet (Deng et al., 2009) and PASCAL challenges (Everingham et al., 2010) have greatly accelerated the research in this area. Image captioning and visual-to-text translation, which are more generalized image annotation tasks, have been recently studied in several works (Karpathy et al., 2014; Kojima et al., 2002; Mao et al., 2014; Yu et al., 2016) where the goal is to find a text caption for a given image. Deep learning models such as RNN and sequence-to-sequence models (Sutskever et al., 2014) have achieved excellent results for image annotation/captioning tasks. ",
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+    {
+        "type": "text",
+        "text": "Multi-Instance Learning Multi-Instance Learning (MIL), a well known researched topic in machine learning, was first introduced by Dietterich et al. (1997) as a form of weakly supervised learning for drug activity prediction. MIL frameworks have since been applied to many other domains including image and text annotations (Chen & Wang, 2004; Soleimani & Miller, 2017). Andrews et al. (2003) adapted Support Vector Machines (SVM) to the MIL framework and introduced miSVM and MISVM for optimizing instance-level and bag-level classifications respectively. Zhou & Zhang (2007) further extended MIL and proposed MIMLSVM for tackling multi-label problems. Zhou et al. (2009) introduced miGraph and MIGraph to model the structure in each bag. Zhang et al. (2011); Guan et al. (2016) also proposed MIL framework for structure data by leveraging the relational structures at the bag and instance levels. Generative model based MIL frameworks such as Multi-Instance Mixture Model (MIMM)(Foulds & Smyth, 2011), and Dirichlet Process Mixture of Guassians (DPMIL) (Kandemir & Hamprecht, 2014) have also been proposed for binary multi-instance classification. Guan et al. (2016) used an autoregressive hidden Markov model and proposed an MIL framework for activity recognition in time-series data. Garcez & Zaverucha (2012) used recurrent neural networks to combine instance-level preprocessing and bag-level classification in MIL setting. Comprehensive reviews of MIL approaches are provided in Amores (2013); Herrera et al. (2016); Soleimani & Miller (2017). Recently, deep learning models have been successfully applied for MIL framework (Zhu et al., 2017; Wu et al., 2015; Huang et al., 2013; Yan et al., 2016; Kotzias et al., 2014; Kraus et al., 2016) and these approaches are generally termed as deep multi-instance learning models. Most of these works use either convolutional neural networks or deep neural networks in their MIL framework for image annotation, labeling, segmentation or classification tasks. Despite the popularity of deep models for MIL, there are few works which have extended deep MIL models for multivariate time series data. The goal of this paper is to propose and study deep multi-instance learning models for multivariate time series data. ",
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+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
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+    {
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+        "text": "In this section, we will first describe the Multi-instance learning framework, and then present our problem formulation and our proposed relational multi-instance learning models. ",
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+        "type": "text",
+        "text": "3.1 MULTI-INSTANCE LEARNING",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Multi-instance learning (MIL) is a form of weakly supervised learning where the training data is arranged in sets called bags, and a label is provided for the entire bag. The data points inside a bag are referred to as instances. In the MIL framework, instance labels are not provided during training. The main goal of MIL is to learn a model based on the instances in the bag and the label of the bag - to make bag-level and instance-level predictions. In this work, we only focus on the classification task in MIL, leaving out other learning tasks such as regression. Generally, two broad assumptions can be used to model the relationship between instance label and bag label. In the standard MIL assumption (Dietterich et al., 1997), the bag label is negative if all the instances in the bag have a negative label, and the bag label is positive if at-least one of the instances in the bag has a positive label. Following the notations of Carbonneau et al. (2017), let $X$ denote a bag with $N$ feature vector instances i.e., $\\bar { X } = \\{ x _ { 1 } , . . . , x _ { N } \\}$ . Let each instance $x _ { i }$ in feature space $\\mathcal { X }$ be mapped to a class by some process $f : \\mathcal { X }  \\{ 0 , 1 \\}$ , where 0 and 1 correspond to negative and positive labels respectively. The bag classifier, also know as the aggregator function, $g ( X )$ is defined by: ",
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+    {
+        "type": "equation",
+        "img_path": "images/c6f4e28e9ae16c3c761c1ca3c3895089e8f82a177d3967b668a9c2f79d9a8b3b.jpg",
+        "text": "$$\ng ( X ) = { \\left\\{ \\begin{array} { l l } { 1 } & { \\quad { \\mathrm { i f } } \\exists x \\in X : f ( X ) = 1 ; } \\\\ { 0 } & { \\quad { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The standard assumption is quite restrictive for some problem settings, where the positive bags cannot be identified by a single instance. Thus, this assumption can be relaxed to a collective assumption ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "which says that several positive instances in a bag are necessary to assign a positive label to that bag. In this case, a bag classifier is given by: ",
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+    {
+        "type": "equation",
+        "img_path": "images/dafac0fb9423481b3ffef1689452e7af3c9b44571912fe7335fcf0d71b87b34a.jpg",
+        "text": "$$\ng ( X ) = { \\left\\{ \\begin{array} { l l } { 1 } & { \\quad { \\mathrm { i f } } \\ \\sum _ { x \\in X } f ( X ) \\geq \\theta ; } \\\\ { 0 } & { \\quad { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. }\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\theta$ is a threshold which indicates the minimum number of instances with positive labels that should be present in a bag to assign a positive label to that bag Weidmann et al. (2003). As discussed in section 2, a plethora of works have adapted machine learning models to the MIL setting to optimize instance-level and/or bag-level predictions. ",
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+    {
+        "type": "text",
+        "text": "3.2 PROBLEM FORMULATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We formulate the concept annotation task as the detection and localization of concepts given the medical time series data. Let each patient $i \\in \\{ 1 , . . , N \\}$ be associated with a medical time series (also referred to as feature time series) denoted by $\\pmb { X } _ { i } \\in \\mathbb { R } ^ { T \\times D }$ , where $D$ denotes the number of features (such as heart rate, blood pressure) and $T$ denotes the length of time series observations (i.e., amount of time a patient is monitored). Let $\\mathcal { C }$ denote the set of all the concepts associated with the $N$ patients, and $\\bar { Y _ { i } } \\in \\left\\{ 0 , 1 \\right\\} ^ { K }$ denote the concepts associated with $i ^ { t h }$ patient where $K = | \\mathcal { C } |$ . Let $C _ { i } \\in \\{ 0 , 1 \\} ^ { T \\times K }$ denote the concept time series of $X _ { i }$ with $C _ { i } ^ { j k } = 1$ when concept $k$ is present at time-stamp $j$ . In multi-instance learning settings, we treat each time series $X _ { i }$ as one bag, and the observation at each time step $j$ i.e. $\\boldsymbol { X } _ { i } ^ { j } \\in \\mathbb { R } ^ { D }$ as an instance in that bag. We are interested in the following tasks: ",
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+        "type": "text",
+        "text": "• Concept prediction task: for patient $i$ , given $X _ { i }$ , predict $\\mathbf { Y } _ { i }$ .   \n• Concept localization task: for patient $i$ , given $X _ { i }$ , predict $C _ { i }$ . ",
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+        "type": "text",
+        "text": "Notice that during training phase, only the input $X _ { i }$ and prediction label $\\mathbf { Y } _ { i }$ are available. Though $C _ { i }$ is not known, we usually have some assumptions about the relationship of prediction and localization labels. In this work, $\\begin{array} { r } { Y _ { i } ^ { k } = \\mathcal { T } \\left( \\sum _ { 1 \\leq j \\leq T } C _ { i } ^ { j k } \\geq \\eta \\right) } \\end{array}$ , where $\\mathcal { T }$ is an indicator function and $\\eta$ is a constant which depends on the MIL assumption. For example, in our concept annotation tasks we make the standard assumption, i.e we assume $\\eta = 1$ , i.e. the time series label (bag label) for a concept is positive if that concept is present at any one time-stamp (at-least one instance has positive label). ",
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+    {
+        "type": "text",
+        "text": "3.3 RELATIONAL MULTI-INSTANCE LEARNING FRAMEWORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Inspired by the recent success of recurrent neural networks in sequence modeling Bahdanau et al. (2014); Sutskever et al. (2014) and classification tasks Krizhevsky et al. (2012); Simonyan & Zisserman (2014), we adapt these models to the MIL framework to model multivariate time series data for concept annotation tasks. We denote all the variables at every time step as an instance and the entire multivariate time series as a bag. Unlike the traditional MIL setting, where the instances within a bag are independent of each other, in our case, the instances have relationships (namely temporal dependencies) among them. To model these dependencies, we propose to combine RNN models such as Long-Short Term Memory Neural Networks (LSTM) and Sequence-to-Sequence models with MIL, and propose our Relational Multi-Instance Learning framework, abbreviated as RMIL. RMIL takes in multivariate time series as input and outputs concept annotations. In RMIL, the outputs of RNN model provide the instance label predictions (i.e. solution for concept localization task) and the aggregation of the instance labels using aggregators such as pooling layer provides the bag label predictions (i.e. solution for concept prediction task). We propose different pooling functions and attention mechanisms which can be easily incorporated into our RMIL to improve the concept annotations. ",
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+        "text": "Pooling Layers for RMIL The bag-level prediction is obtained by using an aggregation gathering on all instance-level predictions. The aggregator function $g ( \\cdot ) : \\left[ 0 , 1 \\right] ^ { T } \\mapsto \\left[ 0 , 1 \\right]$ in RMIL can be modeled using the pooling layers. Without loss of generality we assume that RNN model computes a mapping from the feature time series to the concept time series for each of the concept $k \\in \\mathcal { C }$ . Let us denote the probability of an instance $j$ belonging to concept $k$ as $p _ { j k }$ . Then, the bag level probability for a concept $k$ is given by $P _ { k } = g ( p _ { 1 k } , p _ { 2 k } , . . . , p _ { T k } )$ . The role of aggregator function $g ( \\cdot )$ is to combine the instance probabilities from each class specific feature map $\\{ p _ { j k } \\}$ into a single bag probability $P _ { k }$ . Several pooling mechanisms shown in Table 1 have been introduced in MIL and deep learning literature which can be used in our RMIL. In Table $1 , r , a , b _ { k }$ , and $r _ { k }$ are parameters which can be fixed or are learned during training, and $\\sigma ( \\cdot )$ denotes the sigmoid function. ",
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+        "text": "",
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+        "img_path": "images/09ff2f20d78ca03524778d04a31c2a283e29ef8512e9dd22fe66cdb63bea739e.jpg",
+        "table_caption": [
+            "Table 1: Pooling functions for RMIL. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Pooling Functions</td></tr><tr><td rowspan=1 colspan=1>Noisy-OR poolingPk = g({𝑝jκ})=1-II(1- Pjk)(Zhang et al., 2006)</td></tr><tr><td rowspan=1 colspan=1>Log-sum-exponention (LSE)  Pk =g({𝑝jk}）)=1log(↓∑jexp(rPjκ））(Ramon &amp; De Raedt, 2000)</td></tr><tr><td rowspan=1 colspan=1>Generalized Mean (GM)      Pk =g({𝑝jx}）=（∑jjκ）1/r(Keeler et al., 1991)</td></tr><tr><td rowspan=1 colspan=1>Integrated segmentation       Px=g({pjx}）=（∑-x）Pjk）/(1+∑jr-)Pjkand recognition (ISR)(Keeler et al.,1991)</td></tr><tr><td rowspan=1 colspan=1>Noisy-AND pooling           P=g{pj}）=（（∑/-b）-ab）(Kraus et al., 2016)                                       σ(a(1-bk))-σ(-abk)</td></tr><tr><td rowspan=1 colspan=1>Max pooling                    Pk = g({pjk}) = maxj (Pjk)</td></tr><tr><td rowspan=1 colspan=1>Softmax pooling               Pk =g({𝑝jk})=(∑jPjκ exp(rkpjκ))/(∑,exp(rkPjk))</td></tr></table>",
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+        "text": "Attention Mechanism for RMIL Instances within each bag have temporal relations between them. We can use attention mechanism to focus on some of the instances and their relations to improve their instance-level predictions. In order to make predictions at time $j$ , the hidden state $\\boldsymbol { h } _ { j } \\in \\bar { \\mathbb { R } } ^ { Q }$ of RNN can be used, where $Q$ is the hidden state dimension. However, relevant information may be captured by hidden states at other time steps as well. Thus, we may want to introduce an attention vector or matrix $( a )$ to leverage information of hidden states $\\pmb { H } = \\left( \\hat { \\pmb { h } } _ { 1 } , \\cdot \\cdot \\cdot , \\pmb { h } _ { T } \\right) ^ { \\top } \\in \\mathbb { R } ^ { T \\times Q }$ from all time steps. Let us denote the output after attention as $\\tilde { H } \\in \\mathbb { R } ^ { T \\times Q }$ . The attention matrix can then be modeled using $\\tilde { \\cal H }$ in various ways as listed below. ",
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+        "type": "text",
+        "text": "Feature-based Attention One idea is to design the attention matrix based on the feature and its time-stamp. Let us define a feature-based attention matrix as $\\pmb { A } = \\left( \\pmb { a } _ { 1 } , \\cdot \\cdot \\cdot , \\pmb { a } _ { T } \\right) ^ { \\top } \\in \\mathbb { R } ^ { T \\times Q }$ . For each $j = 1 , \\cdots , T$ , we have ",
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+        "type": "equation",
+        "img_path": "images/0737ef9246fd50482f4049dc1946434a9c3a70b81796af6119282b7eab6a6a39.jpg",
+        "text": "$$\n\\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { } \\mathbf { }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "and $\\tilde { H } = A \\odot H$ , where $\\odot$ and $\\oslash$ are element-wise multiplication and division, respectively and $\\ b { W } = \\left( \\ b { w } _ { 1 } , \\cdot \\cdot \\cdot , \\ b { w } _ { T } \\right) ^ { \\top } \\in \\mathbb { R } ^ { T \\times T }$ is the weight matrix which can be learned during training. We call this Attention- $F$ mechanism. We can simplify this attention by averaging the attentions for all hidden dimensions by taking $\\begin{array} { r } { a _ { j q }  1 / Q \\cdot \\sum _ { 1 \\leq q ^ { \\prime } \\leq Q } { a _ { j q ^ { \\prime } } } } \\end{array}$ . We will denote this as Attention- $F S$ mechanism. ",
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+        "type": "text",
+        "text": "Time-based Attention Attention model (Ma et al., 2017) can be designed to capture the relation between the current time step $j$ and previous time steps $j ^ { \\prime } \\le j$ , by solely relying on previous hidden states $h _ { j ^ { \\prime } }$ . We can define a time-based attention matrix as $\\check { A } \\in \\mathbb { R } ^ { T \\times \\check { T } }$ . For each $j$ and $j ^ { \\prime }$ in $[ 1 , \\cdots , T ]$ , we have ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/417a9f156e775917a881f404b881b85b86e8689cac52aa5d7f9f714d35db223a.jpg",
+        "text": "$$\na _ { j , j ^ { \\prime } } = \\left\\{ \\begin{array} { l l } { \\left( \\exp \\left( \\boldsymbol { w ^ { \\intercal } } h _ { j ^ { \\prime } } \\right) \\right) / \\left( \\displaystyle \\sum _ { 1 \\leq j ^ { \\prime \\prime } \\leq j } \\exp \\left( \\boldsymbol { w ^ { \\intercal } } h _ { j ^ { \\prime \\prime } } \\right) \\right) , } & { j ^ { \\prime } \\leq j ; } \\\\ { 0 , } & { \\mathrm { o t h e r w i l } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "and ${ \\tilde { H } } = A \\cdot H$ , where ${ \\pmb w } \\in \\mathbb { R } ^ { D }$ is the weight vector to learn. We use Attention- $T$ to represent Time-based attention mechanism. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Interaction-based Attention The time-based attention can be further improved by considering both the previous and current hidden states $h _ { j ^ { \\prime } }$ and $h _ { j }$ (Ma et al., 2017). In this case, we have ",
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+        "type": "equation",
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+        "text": "$$\n\\begin{array} { r } { \\nu _ { j , j ^ { \\prime } } = \\left\\{ \\begin{array} { l l } { \\left( \\exp \\left( v ^ { \\top } \\operatorname { t a n h } ( W _ { 1 } h _ { j } + W _ { 2 } h _ { j ^ { \\prime } } ) \\right) \\right) / \\left( \\displaystyle \\sum _ { 1 \\leq j ^ { \\prime \\prime } \\leq j } \\exp \\left( v ^ { \\top } \\operatorname { t a n h } ( W _ { 1 } h _ { j } + W _ { 2 } h _ { j ^ { \\prime \\prime } } ) \\right) \\right) , } & { j ^ { \\prime } \\leq j ; } \\\\ { 0 , \\quad \\quad \\mathrm { o t h e r w i s e } } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "and similarly ${ \\tilde { H } } = A \\cdot H$ . Here, we need to learn $\\pmb { v } \\in \\mathbb { R } ^ { S } , \\pmb { W _ { 1 } } \\in \\mathbb { R } ^ { S \\times Q } , \\pmb { W _ { 2 } } \\in \\mathbb { R } ^ { S \\times Q }$ , and we choose $S = Q / 2$ . A simplified version of interaction-based attention can be obtained if we use vector ${ \\pmb w } _ { 1 } \\in \\mathbb { R } ^ { Q } , { \\pmb w } _ { 2 } \\in \\mathbb { R } ^ { Q }$ instead of matrices $W _ { 1 } , W _ { 2 }$ and by setting ${ \\pmb v } = { \\bf 1 }$ in the above equation. We use Attention- ${ \\bf \\nabla } \\cdot { \\cal I }$ and Attention- $. I S$ to represent Interaction-based attention mechanism and simplified version of interaction-based attention mechanism respectively. ",
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+    {
+        "type": "text",
+        "text": "The above attention mechanisms usually help both prediction and localization tasks. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "Here, we demonstrate the performance of our proposed RMIL models on concept annotation tasks i.e. concept prediction and localization tasks, using a real-world health-care dataset and compare its performance to the popular multi-instance learning approaches. In addition, we discuss the impact of using pooling functions and attention mechanism in our RMIL framework. ",
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+    {
+        "type": "text",
+        "text": "4.1 DATASET DESCRIPTIONS AND EXPERIMENTAL DESIGN ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To evaluate our RMIL, we ran experiments on MIMIC-III RESP datasets whose statistics is shown in Table 2. ",
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+        "text": "MIMIC-III RESP Dataset MIMIC-III is a public dataset (Johnson et al., 2016) which has deidentified clinical care data collected at Beth Israel Deaconess Medical Center from 2001 to 2012. It contains over 58,000 hospital admission records of 38,645 adults and 7,875 neonates. For our work, we extracted 21 feature time series from more than 2,000 adult patients who were diagnosed with a respiratory disorder such as Acute Hypoxemic Respiratory Failure (AHRF) (Khemani et al., 2009) at the time of admission. These 21 features are respiratory based features such as peak inspiratory pressure (PIP) and arterial partial pressure of oxygen (PaO2) and were collected during the first 3 days after admission. The feature time series has 4 time stamps and the first time stamp corresponds to the admission time. We denote this dataset as MIMIC-III RESP dataset. In addition, we also generated another feature time series with more time stamps whose results is shown in the appendix. ",
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+    {
+        "type": "table",
+        "img_path": "images/5e3174674a0bd893061845f168c9aff4aadc78b1526aff9d94726bbbc3e3d387.jpg",
+        "table_caption": [
+            "Table 2: MIMIC-III RESP dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MIMIC-IIIRESP</td></tr><tr><td># of samples (N)</td><td>2014</td></tr><tr><td># of variables (D)</td><td>21</td></tr><tr><td># of time steps</td><td>4</td></tr><tr><td># of concepts</td><td>26</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Concept Annotations The medical time series data of MIMIC-III dataset does not come with the concept annotations, however the medical concepts are available in the doctor notes of the MIMIC-III database. To obtain the concept annotations, we extract the concept time series from the doctor notes using the NOTEEVENTS table of MIMIC-III database. The total number of doctor notes is 2,083,180, out of which $9 8 . 1 5 \\%$ of notes (2,044,634 notes) have no timestamp and $1 . 8 5 \\%$ of notes (38,546 notes) have timestamps associated with them. The total number of unique concepts in the doctor notes in the first 3 days data is 6,197. To obtain concept time series for each patient with respiratory disorder such as AHRF, we first identified respiratory-related concepts from the medical literature (Antonelli et al., 2001; Khemani et al., 2009), and obtained their medical codes from the Unified Medical Language System (UMLS) dictionary (Bodenreider, 2004). Then, we mined the patient’s doctor notes from NOTEEVENTS table to extract all the possible medical concepts related to the respiratory system and its disorder. In total, we chose top 26 respiratory concepts to generate concept time series which has the same number of time stamps as feature time series. ",
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+    {
+        "type": "table",
+        "img_path": "images/b2d6132af1770d476537eb85de1715e3d7166edb0c680ccf0631f361e9ad94d1.jpg",
+        "table_caption": [
+            "Table 3: Concept annotation results on MIMIC-III RESP dataset. Max-pooling function was used in all RMIL models. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"2\">Prediction</td><td colspan=\"2\">Localization</td></tr><tr><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td rowspan=\"6\">RMIL Models</td><td>S2S</td><td>0.858</td><td>0.756</td><td>0.788</td><td>0.431</td></tr><tr><td>S2S with Attention I</td><td>0.858</td><td>0.758</td><td>0.793</td><td>0.433</td></tr><tr><td>LSTM</td><td>0.860</td><td>0.763</td><td>0.795</td><td>0.444</td></tr><tr><td>LSTM with Attention I</td><td>0.862</td><td>0.766</td><td>0.797</td><td>0.444</td></tr><tr><td>Bi-LSTM</td><td>0.864</td><td>0.769</td><td>0.789</td><td>0.415</td></tr><tr><td>Bi-LSTM with Attention I</td><td>0.864</td><td>0.769</td><td>0.796</td><td>0.420</td></tr><tr><td rowspan=\"4\">MIL Models</td><td>CNN</td><td>0.857</td><td>0.756</td><td>0.785</td><td>0.397</td></tr><tr><td>CNN with Attention I</td><td>0.855</td><td>0.755</td><td>0.785</td><td>0.409</td></tr><tr><td>DPMIL</td><td>0.531</td><td>0.222</td><td>0.516</td><td>0.148</td></tr><tr><td>MISVM</td><td>0.751</td><td>0.613</td><td>0.706</td><td>0.333</td></tr></table>",
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+        "text": "4.2 COMPARISON AND IMPLEMENTATION DETAILS ",
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+        "text": "We compare the performance of our proposed models to the popular MIL models such as MISVM (Andrews et al., 2003), DPMIL (Kandemir & Hamprecht, 2014) and Convolutional Neural Networks (CNN) Kraus et al. (2016), and CNN with attention. We categorize all the evaluated methods into two groups: ",
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+        "text": "1. Multi-Instance Learning models (MIL): We treat MISVM, DPMIL, Convolutional Neural Networks (CNN), and CNN with Attention I as our baseline MIL models.   \n2. Relational Multi-Instance Learning models (RMIL): We evaluate the following deep learning models as part of our RMIL framework: (a) Long Short-Term Memory neural networks (LSTM) (Hochreiter & Schmidhuber, 1997) (b) Bi-directional LSTM (Bi-LSTM) (Graves et al., 2013) (c) Sequence to Sequence models (S2S) (Sutskever et al., 2014) (d) The above three RMIL models with different attention mechanisms (e) The above RMIL models with different pooling functions ",
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+        "text": "For LSTM models, we use two LSTM layers and two dense layers. For S2S models, we use two LSTM layers for both the encoder and the decoder. For Bi-LSTM models, we use two bi-directional LSTM layers. All the models were constructed to have a comparable number of parameters. We train all the Deep learning models with the RMSProp optimization method and we use early stopping to find the best weights on the validation dataset. For baseline MIL models, we follow the suggestions of the corresponding papers to fine-tune the parameters. All the input variables in the training data are normalized to be 0 mean and 1 standard deviation. The inputs to all the models is the same feature time series data. We used Keras (Chollet, 2017) and Python to run the deep models and MISVM models. Matlab code from the original authors was used to obtain DPMIL results. We use the area under ROC (AUROC) and area under precision-recall curve (AUPRC) scores as our evaluation metrics and report the results from 5-fold cross validation for all the evaluated methods. ",
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+        "type": "text",
+        "text": "4.3 QUANTITATIVE RESULTS ",
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+        "text": "Table 3 shows the concept annotation results on the MIMIC-III RESP dataset. From this table, we see that RMIL models outperform the non-deep multi-instance learning models by at least $8 \\%$ for concept localization task, and by at least $10 \\mathrm { - } 1 5 \\%$ for concept prediction tasks in terms of AUROC and AUPRC. RMIL performs slightly better than CNN-based models on all the metrics. Among all the RMIL models, we find that LSTM model obtains slightly better overall results compared to the other models for localization task. ",
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+        "text": "To study the impact of pooling and attention, we trained and evaluated LSTM models with different pooling functions and different attention mechanisms, which are described in Section 3. Tables 4 and 5 show the comparison results. From these tables, we observed that (i) all the attention mechanisms except feature-based attention perform similar to each other especially for the prediction task, and (ii) all the pooling functions other than ISR and Noisy-OR obtain similar overall performance. This ",
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+        "table_caption": [
+            "Table 4: Results on RMIL LSTM models with different pooling functions and with Attention-I mechanism. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">Prediction</td><td colspan=\"2\">Localization</td></tr><tr><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td>LSTM with ISR Pooling</td><td>0.862</td><td>0.765</td><td>0.733</td><td>0.375</td></tr><tr><td>LSTM with Noisy-AND Pooling</td><td>0.863</td><td>0.767</td><td>0.779</td><td>0.431</td></tr><tr><td>LSTM with Generalized Mean Pooling</td><td>0.863</td><td>0.767</td><td>0.796</td><td>0.450</td></tr><tr><td>LSTM with LSE Pooling</td><td>0.863</td><td>0.767</td><td>0.792</td><td>0.452</td></tr><tr><td>LSTM with Softmax Pooling</td><td>0.863</td><td>0.767</td><td>0.790</td><td>0.450</td></tr><tr><td>LSTM with Noisy-OR Pooling</td><td>0.860</td><td>0.762</td><td>0.703</td><td>0.318</td></tr><tr><td>LSTM with Max Pooling</td><td>0.862</td><td>0.766</td><td>0.797</td><td>0.444</td></tr></table>",
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+        "img_path": "images/562a122ec127b761f3c67afdea40bd96e5326f2adc2f796ff16eeeca1fecd15e.jpg",
+        "image_caption": [],
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+    {
+        "type": "text",
+        "text": "(a) Ground truth annotations of intubation and extubation concepts. ",
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+        "image_caption": [
+            "(b) Intubation and extubation concept prediction probabilities identified by attention-based LSTM. ",
+            "Figure 2: Ground truth and predicted concept annotation comparison. In 2(a) white block corresponds to absence of a concept and black block corresponds to presence of a concept. In 2(b), darker gray value indicates higher chance for presence of a concept. X-axis represents time steps, Y-axis corresponds to different patients. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "demonstrates that choice of attention does not matter but choice of pooling has some impact in our RMIL framework. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 5: Results on RMIL LSTM models with different attention mechanisms and with max pooling. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">Prediction</td><td colspan=\"2\">Localization</td></tr><tr><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td>LSTM with Attention-T</td><td>0.864</td><td>0.768</td><td>0.796</td><td>0.445</td></tr><tr><td>LSTMwith Attention-IS</td><td>0.862</td><td>0.767</td><td>0.798</td><td>0.446</td></tr><tr><td>LSTM with Attention-I</td><td>0.862</td><td>0.766</td><td>0.797</td><td>0.444</td></tr><tr><td>LSTMwith Attention-FS</td><td>0.861</td><td>0.765</td><td>0.796</td><td>0.417</td></tr><tr><td>LSTM with Attention-F</td><td>0.861</td><td>0.764</td><td>0.780</td><td>0.408</td></tr><tr><td>LSTM with no attention</td><td>0.860</td><td>0.763</td><td>0.795</td><td>0.444</td></tr></table>",
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+        "text": "4.4 DISCUSSIONS ",
+        "text_level": 1,
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+        "text": "We can study the interpretability of concept localization by looking at the localization results of our RMIL models, even though the model is trained without the labels for localization. Figure 2 shows the ground truth annotations of two respiratory concepts - intubation and extubation concepts, and the prediction probabilities of these concepts obtained by our RMIL attention-based LSTM models. From figure 2(a) we can make the following observations, (i) intubation usually happens before extubation for the same patient, (ii) intubation and extubation could happen on the same day, and (iii) intubation and extubation occur commonly within the first 24 hours of admission. From the figure 2(b), we see that our RMIL attention based LSTM predicts that the probability of intubation happening on the first day of admission is higher (draker gray means higher probability of concept occurrence) and the probability of extubation happening within first day is lower. This indicates that the model has correctly learnt that intubation should appear before extubation. This also implicitly implies that the RMIL attention-based LSTM models have correctly learnt the instance-level relationships from the medical time series data with only bag-level labels. ",
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+        "type": "text",
+        "text": "5 SUMMARY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we presented Relational Multi-Instance Learning - a deep multi-instance learning framework using recurrent neural networks for concept annotation from the medical time series data. Empirical results on medical dataset demonstrated that our proposed models outperform the popular state-of-the-art multi-instance learning approaches. Experiments with different pooling and attention mechanisms showed that while attention mechanism does not have a significant impact on model’s performance, certain pooling functions such as ISR and Noisy-OR can negatively impact the instance prediction results. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
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+            174,
+            253,
+            287,
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+        "text": "Stergos Afantenos, Vangelis Karkaletsis, and Panagiotis Stamatopoulos. Summarization from medical documents: a survey. Artificial intelligence in medicine, 33(2):157–177, 2005.   \nCharu C Aggarwal and ChengXiang Zhai. Mining text data. Springer Science & Business Media, 2012.   \nDaniel Albright, Arrick Lanfranchi, Anwen Fredriksen, William F Styler IV, Colin Warner, Jena D Hwang, Jinho D Choi, Dmitriy Dligach, Rodney D Nielsen, James Martin, et al. Towards comprehensive syntactic and semantic annotations of the clinical narrative. Journal of the American Medical Informatics Association, 20(5):922–930, 2013.   \nJaume Amores. Multiple instance classification: Review, taxonomy and comparative study. Artificial Intelligence, 201:81–105, 2013.   \nStuart Andrews, Ioannis Tsochantaridis, and Thomas Hofmann. Support vector machines for multipleinstance learning. In Advances in neural information processing systems, pp. 577–584, 2003.   \nM Antonelli, G Conti, ML Moro, A Esquinas, G Gonzalez-Diaz, M Confalonieri, P Pelaia, T Principi, C Gregoretti, F Beltrame, et al. Predictors of failure of noninvasive positive pressure ventilation in patients with acute hypoxemic respiratory failure: a multi-center study. Intensive care medicine, 2001.   \nDzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014.   \nSteven Bethard, Guergana Savova, Wei-Te Chen, Leon Derczynski, James Pustejovsky, and Marc Verhagen. Semeval-2016 task 12: Clinical tempeval. Proceedings of SemEval, pp. 1052–1062, 2016.   \nOlivier Bodenreider. The unified medical language system (umls): integrating biomedical terminology. Nucleic acids research, 32(suppl_1):D267–D270, 2004.   \nAlessandro Camerra, Themis Palpanas, Jin Shieh, and Eamonn Keogh. isax 2.0: Indexing and mining one billion time series. In Data Mining (ICDM), 2010 IEEE 10th International Conference on, pp. 58–67. IEEE, 2010.   \nRicardo JGB Campello, Davoud Moulavi, Arthur Zimek, and Jörg Sander. Hierarchical density estimates for data clustering, visualization, and outlier detection. ACM Transactions on Knowledge Discovery from Data (TKDD), 10(1):5, 2015.   \nMarc-Andre Carbonneau, Veronika Cheplygina, Eric Granger, and Ghyslain Gagnon. Multiple instance learning: A survey of problem characteristics and applications. Pattern Recognition, 2017.   \nPhilip K Chan and Matthew V Mahoney. Modeling multiple time series for anomaly detection. In Data Mining, Fifth IEEE International Conference on, pp. 8–pp. IEEE, 2005.   \nVarun Chandola, Arindam Banerjee, and Vipin Kumar. Anomaly detection: A survey. ACM computing surveys (CSUR), 41(3):15, 2009.   \nVarun Chandola, Arindam Banerjee, and Vipin Kumar. Anomaly detection for discrete sequences: A survey. IEEE Transactions on Knowledge and Data Engineering, 24(5):823–839, 2012.   \nZhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, and Yan Liu. Recurrent neural networks for multivariate time series with missing values. arXiv preprint arXiv:1606.01865, 2016.   \nYixin Chen and James Z Wang. Image categorization by learning and reasoning with regions. Journal of Machine Learning Research, 5(Aug):913–939, 2004.   \nEdward Choi, Mohammad Taha Bahadori, Elizabeth Searles, Catherine Coffey, Michael Thompson, James Bost, Javier Tejedor-Sojo, and Jimeng Sun. Multi-layer representation learning for medical concepts. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1495–1504. ACM, 2016a.   \nYoungduck Choi, Chill Yi-I Chiu, and David Sontag. Learning low-dimensional representations of medical concepts. AMIA Summits on Translational Science Proceedings, 2016:41, 2016b.   \nFrançois Chollet. Keras (2015). URL http://keras. io, 2017.   \nAaron M Cohen and William R Hersh. A survey of current work in biomedical text mining. Briefings in bioinformatics, 6(1):57–71, 2005.   \nJia 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.   \nThomas G Dietterich, Richard H Lathrop, and Tomás Lozano-Pérez. Solving the multiple instance problem with axis-parallel rectangles. Artificial intelligence, 89(1):31–71, 1997.   \nMark Everingham, Luc Van Gool, Christopher KI Williams, John Winn, and Andrew Zisserman. The pascal visual object classes (voc) challenge. International journal of computer vision, 88(2): 303–338, 2010.   \nArden W Forrey, Clement J Mcdonald, Georges DeMoor, Stanley M Huff, Dennis Leavelle, Diane Leland, Tom Fiers, Linda Charles, Brian Griffin, Frank Stalling, et al. Logical observation identifier names and codes (loinc) database: a public use set of codes and names for electronic reporting of clinical laboratory test results. Clinical Chemistry, 42(1):81–90, 1996.   \nJames Foulds and Padhraic Smyth. Multi-instance mixture models and semi-supervised learning. In Proceedings of the 2011 SIAM International Conference on Data Mining, pp. 606–617. SIAM, 2011.   \nAS d’Avila Garcez and Gerson Zaverucha. Multi-instance learning using recurrent neural networks. In Neural Networks (IJCNN), The 2012 International Joint Conference on, pp. 1–6. IEEE, 2012.   \nDaniela Giordano, Isaak Kavasidis, and Concetto Spampinato. Automatic summary creation by applying natural language processing on unstructured medical records. In International Conference on Computer Analysis of Images and Patterns, pp. 383–393. Springer, 2015.   \nAlex Graves, Abdel-rahman Mohamed, and Geoffrey Hinton. Speech recognition with deep recurrent neural networks. In Acoustics, speech and signal processing (icassp), 2013 ieee international conference on, pp. 6645–6649. IEEE, 2013.   \nXinze Guan, Raviv Raich, and Weng-Keen Wong. Efficient multi-instance learning for activity recognition from time series data using an auto-regressive hidden markov model. In International Conference on Machine Learning, pp. 2330–2339, 2016.   \nFabio Guigou, Pierre Collet, and Pierre Parrend. The artificial immune ecosystem: a bio-inspired metaalgorithm for boosting time series anomaly detection with expert input. In European Conference on the Applications of Evolutionary Computation, pp. 573–588. Springer, 2017.   \nManish Gupta, Jing Gao, Charu C Aggarwal, and Jiawei Han. Outlier detection for temporal data: A survey. IEEE Transactions on Knowledge and Data Engineering, 26(9):2250–2267, 2014.   \nFrancisco Herrera, Sebastián Ventura, Rafael Bello, Chris Cornelis, Amelia Zafra, Dánel SánchezTarragó, and Sarah Vluymans. Introduction. In Multiple Instance Learning, pp. 1–16. Springer, 2016.   \nSepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8): 1735–1780, 1997.   \nVictoria Hodge and Jim Austin. A survey of outlier detection methodologies. Artificial intelligence review, 22(2):85–126, 2004.   \nJohn GR Howie, David J Heaney, Margaret Maxwell, Jeremy J Walker, George K Freeman, and Harbinder Rai. Quality at general practice consultations: cross sectional survey. Bmj, 319(7212): 738–743, 1999.   \nYan Huang, Wei Wang, Liang Wang, and Tieniu Tan. Multi-task deep neural network for multi-label learning. In Image Processing (ICIP), 2013 20th IEEE International Conference on, pp. 2897–2900. IEEE, 2013.   \nJiwoon Jeon, Victor Lavrenko, and Raghavan Manmatha. Automatic image annotation and retrieval using cross-media relevance models. In Proceedings of the 26th annual international ACM SIGIR conference on Research and development in informaion retrieval, pp. 119–126. ACM, 2003.   \nAEW Johnson, TJ Pollard, L Shen, L Lehman, M Feng, M Ghassemi, B Moody, P Szolovits, LA Celi, and RG Mark. Mimic-iii, a freely accessible critical care database. Scientific Data, 2016.   \nMichael Jones, Daniel Nikovski, Makoto Imamura, and Takahisa Hirata. Exemplar learning for extremely efficient anomaly detection in real-valued time series. Data Mining and Knowledge Discovery, 30(6):1427–1454, 2016.   \nMelih Kandemir and Fred A Hamprecht. Instance label prediction by dirichlet process multiple instance learning. In UAI, pp. 380–389, 2014.   \nAndrej Karpathy, George Toderici, Sanketh Shetty, Thomas Leung, Rahul Sukthankar, and Li Fei-Fei. Large-scale video classification with convolutional neural networks. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pp. 1725–1732, 2014.   \nJames D Keeler, David E Rumelhart, and Wee Kheng Leow. Integrated segmentation and recognition of hand-printed numerals. In Advances in neural information processing systems, pp. 557–563, 1991.   \nRobinder G Khemani, David Conti, Todd A Alonzo, Robert D Bart, and Christopher JL Newth. Effect of tidal volume in children with acute hypoxemic respiratory failure. Intensive care medicine, 35 (8):1428–1437, 2009.   \nAtanas Kiryakov, Borislav Popov, Ivan Terziev, Dimitar Manov, and Damyan Ognyanoff. Semantic annotation, indexing, and retrieval. Web Semantics: Science, Services and Agents on the World Wide Web, 2(1):49–79, 2004.   \nAtsuhiro Kojima, Takeshi Tamura, and Kunio Fukunaga. Natural language description of human activities from video images based on concept hierarchy of actions. International Journal of Computer Vision, 50(2):171–184, 2002.   \nDimitrios Kotzias, Misha Denil, Phil Blunsom, and Nando de Freitas. Deep multi-instance transfer learning. arXiv preprint arXiv:1411.3128, 2014.   \nOren Z Kraus, Jimmy Lei Ba, and Brendan J Frey. Classifying and segmenting microscopy images with deep multiple instance learning. Bioinformatics, 32(12):i52–i59, 2016.   \nAlex 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.   \nStephen C Kurachek, Christopher J Newth, Michael W Quasney, Tom Rice, Ramesh C Sachdeva, Neal R Patel, Jeanne Takano, Larry Easterling, Mathew Scanlon, Ndidiamaka Musa, et al. Extubation failure in pediatric intensive care: a multiple-center study of risk factors and outcomes. Critical care medicine, 31(11):2657–2664, 2003.   \nChristoph H Lampert, Matthew B Blaschko, and Thomas Hofmann. Beyond sliding windows: Object localization by efficient subwindow search. In Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, pp. 1–8. IEEE, 2008.   \nAnna S Law, Yvonne Freer, Jim Hunter, Robert H Logie, Neil McIntosh, and John Quinn. A comparison of graphical and textual presentations of time series data to support medical decision making in the neonatal intensive care unit. Journal of clinical monitoring and computing, 19(3): 183–194, 2005.   \nJessica Lin, Eamonn Keogh, Stefano Lonardi, and Bill Chiu. A symbolic representation of time series, with implications for streaming algorithms. In Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery, pp. 2–11. ACM, 2003.   \nZachary C Lipton, David C Kale, Charles Elkan, and Randall Wetzell. Learning to diagnose with lstm recurrent neural networks. arXiv preprint arXiv:1511.03677, 2015.   \nFenglong Ma, Radha Chitta, Jing Zhou, Quanzeng You, Tong Sun, and Jing Gao. Dipole: Diagnosis prediction in healthcare via attention-based bidirectional recurrent neural networks. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1903–1911. ACM, 2017.   \nJunhua Mao, Wei Xu, Yi Yang, Jiang Wang, Zhiheng Huang, and Alan Yuille. Deep captioning with multimodal recurrent neural networks (m-rnn). arXiv preprint arXiv:1412.6632, 2014. ",
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+            171,
+            275,
+            828,
+            925
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            89,
+            828,
+            926
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            70,
+            828,
+            928
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "World Health Organization and Practice Management Information Corporation. ICD-9-CM: International Classification of Diseases, 9th Revision: Clinical Modification, volume 1. PMIC (Practice Management Information Corporation), 1998. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Trang Pham, Truyen Tran, Dinh Phung, and Svetha Venkatesh. Deepcare: A deep dynamic memory model for predictive medicine. In Advances in Knowledge Discovery and Data Mining, pp. 30–41. Springer, 2016. ",
+        "bbox": [
+            176,
+            150,
+            821,
+            191
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Catherine Plaisant, Brett Milash, Anne Rose, Seth Widoff, and Ben Shneiderman. Lifelines: visualizing personal histories. In Proceedings of the SIGCHI conference on Human factors in computing systems, pp. 221–227. ACM, 1996. ",
+        "bbox": [
+            174,
+            196,
+            823,
+            239
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Catherine Plaisant, Rich Mushlin, Aaron Snyder, Jia Li, Daniel Heller, and Ben Shneiderman. Lifelines: using visualization to enhance navigation and analysis of patient records. In Proceedings of the AMIA Symposium, pp. 76. American Medical Informatics Association, 1998. ",
+        "bbox": [
+            174,
+            242,
+            825,
+            285
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jan Ramon and Luc De Raedt. Multi instance neural networks. In Proceedings of the ICML-2000 workshop on attribute-value and relational learning, pp. 53–60, 2000. ",
+        "bbox": [
+            169,
+            290,
+            823,
+            319
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Adrienne G Randolph, David Wypij, Shekhar T Venkataraman, James H Hanson, Rainer G Gedeit, Kathleen L Meert, Peter M Luckett, Peter Forbes, Michelle Lilley, John Thompson, et al. Effect of mechanical ventilator weaning protocols on respiratory outcomes in infants and children: a randomized controlled trial. Jama, 288(20):2561–2568, 2002. ",
+        "bbox": [
+            174,
+            321,
+            825,
+            378
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Guergana Savova, Steven Bethard, Will Styler, James Martin, Martha Palmer, James Masanz, and Wayne Ward. Towards temporal relation discovery from the clinical narrative. In AMIA annual symposium proceedings, volume 2009, pp. 568. American Medical Informatics Association, 2009. ",
+        "bbox": [
+            178,
+            382,
+            825,
+            426
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Donia Scott, Catalina Hallett, and Rachel Fettiplace. Data-to-text summarisation of patient records: Using computer-generated summaries to access patient histories. Patient education and counseling, 92(2):153–159, 2013. ",
+        "bbox": [
+            174,
+            429,
+            823,
+            472
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014. ",
+        "bbox": [
+            173,
+            476,
+            823,
+            505
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hossein Soleimani and David J Miller. Semisupervised, multilabel, multi-instance learning for structured data. Neural computation, 2017. ",
+        "bbox": [
+            173,
+            508,
+            823,
+            537
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Somayajulu G Sripada, Ehud Reiter, Jim Hunter, and Jin Yu. Generating english summaries of time series data using the gricean maxims. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 187–196. ACM, 2003. ",
+        "bbox": [
+            174,
+            541,
+            821,
+            584
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. In Advances in neural information processing systems, pp. 3104–3112, 2014. ",
+        "bbox": [
+            173,
+            588,
+            821,
+            617
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Marian Van Der Meulen, Robert H Logie, Yvonne Freer, Cindy Sykes, Neil McIntosh, and Jim Hunter. When a graph is poorer than 100 words: A comparison of computerised natural language generation, human generated descriptions and graphical displays in neonatal intensive care. Applied Cognitive Psychology, 24(1):77–89, 2010. ",
+        "bbox": [
+            173,
+            621,
+            825,
+            678
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Veronika Vincze, György Szarvas, Richárd Farkas, György Móra, and János Csirik. The bioscope corpus: biomedical texts annotated for uncertainty, negation and their scopes. BMC bioinformatics, 9(11):S9, 2008. ",
+        "bbox": [
+            173,
+            681,
+            823,
+            723
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Xing Wang, Jessica Lin, Nital Patel, and Martin Braun. A self-learning and online algorithm for time series anomaly detection, with application in cpu manufacturing. In Proceedings of the 25th ACM International on Conference on Information and Knowledge Management, pp. 1823–1832. ACM, 2016. ",
+        "bbox": [
+            173,
+            728,
+            826,
+            784
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Nils Weidmann, Eibe Frank, and Bernhard Pfahringer. A two-level learning method for generalized multi-instance problems. In European Conference on Machine Learning, pp. 468–479. Springer, 2003. ",
+        "bbox": [
+            173,
+            787,
+            825,
+            830
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Brigham C Willis, Alan S Graham, Eunice Yoon, Randall C Wetzel, and Christopher JL Newth. Pressure-rate products and phase angles in children on minimal support ventilation and after extubation. Intensive care medicine, 31(12):1700–1705, 2005. ",
+        "bbox": [
+            176,
+            834,
+            825,
+            877
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jiajun Wu, Yinan Yu, Chang Huang, and Kai Yu. Deep multiple instance learning for image classification and auto-annotation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3460–3469, 2015. ",
+        "bbox": [
+            174,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hua Xu, Shane P Stenner, Son Doan, Kevin B Johnson, Lemuel R Waitman, and Joshua C Denny. Medex: a medication information extraction system for clinical narratives. Journal of the American Medical Informatics Association, 17(1):19–24, 2010.   \nZhennan Yan, Yiqiang Zhan, Zhigang Peng, Shu Liao, Yoshihisa Shinagawa, Shaoting Zhang, Dimitris N Metaxas, and Xiang Sean Zhou. Multi-instance deep learning: Discover discriminative local anatomies for bodypart recognition. IEEE transactions on medical imaging, 35(5):1332–1343, 2016.   \nHaonan Yu, Jiang Wang, Zhiheng Huang, Yi Yang, and Wei Xu. Video paragraph captioning using hierarchical recurrent neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4584–4593, 2016.   \nCha Zhang, John C Platt, and Paul A Viola. Multiple instance boosting for object detection. In Advances in neural information processing systems, pp. 1417–1424, 2006.   \nDan Zhang, Yan Liu, Luo Si, Jian Zhang, and Richard D Lawrence. Multiple instance learning on structured data. In Advances in Neural Information Processing Systems, pp. 145–153, 2011.   \nLi Zhou, Genevieve B Melton, Simon Parsons, and George Hripcsak. A temporal constraint structure for extracting temporal information from clinical narrative. Journal of biomedical informatics, 39 (4):424–439, 2006.   \nZhi-Hua Zhou and Min-Ling Zhang. Multi-instance multi-label learning with application to scene classification. In Advances in neural information processing systems, pp. 1609–1616, 2007.   \nZhi-Hua Zhou, Yu-Yin Sun, and Yu-Feng Li. Multi-instance learning by treating instances as non-iid samples. In Proceedings of the 26th annual international conference on machine learning, pp. 1249–1256. ACM, 2009.   \nWentao Zhu, Qi Lou, Yeeleng Scott Vang, and Xiaohui Xie. Deep multi-instance networks with sparse label assignment for whole mammogram classification. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 603–611. Springer, 2017.   \nPierre Zweigenbaum, Dina Demner-Fushman, Hong Yu, and Kevin B Cohen. Frontiers of biomedical text mining: current progress. Briefings in bioinformatics, 8(5):358–375, 2007. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "6 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6.1 MIMIC-III CONCEPT LIST ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Table 6 show the concept list of the MIMIC-III RESP dataset used in our experiments. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "6.2 ANOMALY DETECTION FOR TIME SERIES ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here, we will demonstrate anomaly detection from medical time series data as a use case application of our RMIL framework. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "6.2.1 MOTIVATION AND RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Each year, more than 1,000,000 US adults and children are put on mechanical ventilation during their stays in ICU. However, lack of effective tools to aid with ventilator weaning and extubation (removal of the breathing tube) readiness assessment results in nearly half of the patients spending unnecessary days on ventilators (Randolph et al., 2002), and up to $20 \\%$ of them having ventilators discontinued too soon (Kurachek et al., 2003). Spending unnecessary days on ventilators can lead to hospital-acquired infections while having ventilators discontinued too soon could require painful reintubation. A work-of-breathing measure called Pressure-Rate Product (PRP), calculated from esophageal pressure, has shown potential to be used as a guideline for ventilator weaning and extubation readiness assessment (Willis et al., 2005). However, PRP calculations are susceptible to sensor artifacts and breathing pattern anomalies. These anomalies limit realtime use of PRP for clinical decision making. Our goal is to automatically detect and remove these anomalies by using our proposed Relational Multi-instance learning models, thereby enabling real-time clinical decision making. Figure 3 shows the example of anomaly detection from the ventilator time series data. Here, anomalies appear due to patient factors (cough, movement) and instrument factors (probing, catherter drift), and should be automatically detected from the ventalitor monitoring time series data. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/558e3bf479ad6bd1453438acdc06d9c4e2811657a0439d37e62b5c022677c3ba.jpg",
+        "table_caption": [
+            "Table 6: MIMIC-III respiratory concept list. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Concept Number</td><td>Concept Name</td></tr><tr><td>1</td><td>Respiratory rate</td></tr><tr><td>2</td><td>Biomedical tube device</td></tr><tr><td>3</td><td>Analysis of arterial blood gases and ph</td></tr><tr><td>4</td><td>Ventilation,function (observable entity)</td></tr><tr><td>5</td><td>Medical history</td></tr><tr><td>6</td><td>Injury wounds</td></tr><tr><td>7</td><td>Body weights</td></tr><tr><td>8</td><td>Tidal volume</td></tr><tr><td>9</td><td>Inspired fraction of oxygen</td></tr><tr><td>10</td><td>Intubation</td></tr><tr><td>11</td><td>Positive end-expiratory pressure</td></tr><tr><td>12</td><td>Artificial airways</td></tr><tr><td>13</td><td>Bicarbonates</td></tr><tr><td>14</td><td>Tracheal extubation</td></tr><tr><td>15</td><td>Chest radiograph</td></tr><tr><td>16</td><td>Mean blood pressure</td></tr><tr><td>17</td><td>Oxygen measurement, partial pressure,arterial</td></tr><tr><td>18</td><td>Cardiac arrest</td></tr><tr><td>19</td><td>Peak inspiratory pressure</td></tr><tr><td>20</td><td>Biomechanical compliance</td></tr><tr><td>21</td><td>Tachycardia ventricular</td></tr><tr><td>22</td><td>Scoliosis</td></tr><tr><td>23</td><td>Acute lung injury</td></tr><tr><td>24</td><td>Partial pressure of carbon dioxide in arterial blood</td></tr><tr><td>25</td><td>Resuscitate</td></tr><tr><td>26</td><td>Sudden onset</td></tr></table>",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Ventilator Time Series ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/2209e7f2a1d65bee1998c9a0dce4bdcba978b5baa13d09634dccf2306f2caa99.jpg",
+        "image_caption": [
+            "Figure 3: Anomaly detection from ventilator time series data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "There is a long history and a rich body of research work on anomaly detection in time series data. See Hodge & Austin (2004); Chandola et al. (2009) for a quick survey on generic anomaly detection algorithms. Gupta et al. (2014) also author a survey on anomaly detection for time series data. In their follow-up survey, Chandola et al. (2012) gave a summary on discrete sequence anomaly detection algorithms. Unfortunately, techniques for time series anomaly detection are quite domain-specific. This is due to highly-varied nature of time series characteristics. In Chan & Mahoney (2005), a set of $k$ minimum bounding rectangles between each time step is used as a model of normal data generating distribution. In Camerra et al. (2010); Lin et al. (2003), continuous time series are converted to ordinal symbolic representation, allowing faster approximation of Euclidean distance between time series windows and early pruning. Then, anomaly detection can be performed by setting a threshold on distance value. Recently, Wang et al. (2016) proposes a self-learning method that learns clusters of constrained grammars based on ordinal symbolic approximation of time series value. Guigou et al. (2017) proposes incorporating experts into anomaly detection in a method inspired by immune system. In Jones et al. (2016), the authors propose extracting exemplars from Euclidean pairwise distance as a way to speed up anomaly detection algorithm. Most of these techniques for time series anomaly detection are quite domain-specific, and few of them model the anomaly detection problem in Multi-instance learning setting. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "6.2.2 PROBLEM FORMULATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We formulate the anomaly detection as concept annotation problem in MIL setting, where the bag corresponds to the medical time series data $X _ { i }$ and an instance is the features at a time-stamp, and the anomaly corresponds to a concept. Thus, the prediction (predicting $Y _ { i }$ ) and localizaton tasks (predicting $C _ { i }$ ) correspond to predicting the presence and location of the anomalies in the time series data. For anomaly detection, we assume $\\begin{array} { r } { Y _ { i } ^ { k } = \\mathcal { T } \\left( \\sum _ { 1 \\leq j \\leq T } C _ { i } ^ { j k } \\geq \\eta \\right) } \\end{array}$ , where $\\eta$ is chosen as $\\eta = 0 . 6 T$ and $T$ is the number of time series windows within the bag. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6.2.3 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We conduct anomaly detection experiments on a PICU dataset described below. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "PICU Dataset This dataset consists of 1,530 recorded 5- minute $2 0 0 – \\mathrm { H z }$ sensor readings from 385 subjects collected at a leading children’s hospital by a team of research clinicians. Statistics of the dataset is shown in Table 7. The recordings are made on mechanically-ventilated patients in pediatrics ICU ward. Medical time series data collected from four sensors are used: flow volume spirometry, esophageal pressure sensor, and dual band respiratory inductance plethysmography. Each subject can be under one of four breathing conditions: ventilated with Continuous Positive Airway Pressure (CPAP), ventilated with Pressure Support (PS), 5 minutes after extubation, and 60 minutes after extubation. Along with the 4 sensor signals, clinicians verified binary anomaly label generated using hard-crafted state-of-the-art breathing anomaly detection algorithm is provided as ground truth. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5cf9000360a2ddf4a8157748b3866f2c0136721da0c9ac791b4114f93c6f2238.jpg",
+        "table_caption": [
+            "Table 7: PICU dataset statistics. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>PICU</td></tr><tr><td># of samples (N)</td><td>1530</td></tr><tr><td># of variables (D)</td><td>80</td></tr><tr><td># of time steps</td><td>60</td></tr><tr><td># of concepts</td><td>1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "We annotate the concept of breathing anomaly in MIL framework by processing the dataset as follows. The sensor recordings are split into non-overlapping 5-second windows. In MIL framework, each window becomes an instance and each recording becomes a bag. For each sensor signal of each window, we extract 20 Mel-Frequency Cepstral Coefficients (MFCCs) to be used as features of each instance; thus, each instance has 80 features in total. For each instance, its anomaly label is set to positive if at least $20 \\%$ of the window are labeled as anomaly. The bag anomaly label is set to positive if at least $60 \\%$ of its instances are labeled as anomaly. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/6fdeacd0684a8489a08f05755fc94a90fe01d511316eed91652224bbb79ea93c.jpg",
+        "table_caption": [
+            "Table 8: Anomaly detection results on PICU dataset with different attention mechanisms and with max pooling. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"2\">Prediction</td><td colspan=\"2\">Localization</td></tr><tr><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td rowspan=\"10\">RMIL Models</td><td>S2S</td><td>0.741</td><td>0.797</td><td>0.532</td><td>0.707</td></tr><tr><td>S2S with Attention-I</td><td>0.732</td><td>0.787</td><td>0.511</td><td>0.694</td></tr><tr><td>S2S with Attention-F</td><td>0.730</td><td>0.789</td><td>0.519</td><td>0.689</td></tr><tr><td>LSTM</td><td>0.708</td><td>0.763</td><td>0.543</td><td>0.712</td></tr><tr><td>LSTM with Attention-I</td><td>0.690</td><td>0.767</td><td>0.513</td><td>0.682</td></tr><tr><td>LSTM with Attention-F</td><td>0.712</td><td>0.776</td><td>0.522</td><td>0.690</td></tr><tr><td>Bi-LSTM</td><td>0.729</td><td>0.784</td><td>0.562</td><td>0.725</td></tr><tr><td>Bi-LSTM with Attention-I</td><td>0.711</td><td>0.769</td><td>0.535</td><td>0.708</td></tr><tr><td>Bi-LSTM with Attention-F</td><td>0.736</td><td>0.790</td><td>0.539</td><td>0.704</td></tr><tr><td>Cluster-MIL*</td><td>0.590</td><td>0.620</td><td>0.530</td><td>0.670</td></tr><tr><td>MIL Models</td><td>MISVM</td><td>0.572</td><td>0.647</td><td>0.542</td><td>0.606</td></tr></table>",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Results Table 8 shows the anomaly detection results using our RMIL models. We observe that (i) RMIL models mostly outperform the non-deep MIL models on both tasks, (ii) LSTM and Bi-LSTM based RMIL models obtain better overall results compared to the sequence-to-sequence models, and (iii) the attention mechanism does not help for localization task but sometimes obtains better results for the prediction task. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Remark: Cluster-MIL\\* is a HDBSCAN clustering (Campello et al., 2015) based approach for anomaly detection. It uses both instance and bag labels for training, while other models only use bag labels for training. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "6.3 ADDITIONAL RESULTS ON MIMIC-III RESP DATASET ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We sampled the feature time series from MIMIC-III RESP dataset every 6 hours and generated a time series with more (12) time stamps. We call this MIMIC-III RESP-II dataset. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Table 9 shows the results of RMIL models on this dataset. We observe that (i) All the RMIL models have similar performance for prediction task, (ii) Bi-LSTM RMIL has better localization results compared to other the models. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/1e5b158c796b52d1f85fbafbbd6c8719948b4e792dcfe75111a91bd6ebfc472e.jpg",
+        "table_caption": [
+            "Table 9: MIMIC-III RESP-II dataset with different attention mechanisms and with max pooling. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"2\">Prediction</td><td colspan=\"2\">Localization</td></tr><tr><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td rowspan=\"9\">RMIL Models</td><td>S2S</td><td>0.874</td><td>0.746</td><td>0.787</td><td>0.185</td></tr><tr><td>S2S with Attention-I</td><td>0.875</td><td>0.749</td><td>0.794</td><td>0.186</td></tr><tr><td>S2S with Attention-F</td><td>0.869</td><td>0.739</td><td>0.777</td><td>0.188</td></tr><tr><td>LSTM</td><td>0.875</td><td>0.749</td><td>0.776</td><td>0.177</td></tr><tr><td>LSTM with Attention-I</td><td>0.875</td><td>0.750</td><td>0.781</td><td>0.175</td></tr><tr><td>LSTM with Attention-F</td><td>0.872</td><td>0.742</td><td>0.771</td><td>0.177</td></tr><tr><td>Bi-LSTM</td><td>0.876</td><td>0.749</td><td>0.801</td><td>0.206</td></tr><tr><td>Bi-LSTM with Attention-I</td><td>0.875</td><td>0.750</td><td>0.803</td><td>0.204</td></tr><tr><td>Bi-LSTM with Attention-F</td><td>0.871</td><td>0.741</td><td>0.769</td><td>0.171</td></tr></table>",
+        "bbox": [
+            222,
+            381,
+            777,
+            541
+        ],
+        "page_idx": 15
+    }
+]

parse/train/GhMZNcr54zt/GhMZNcr54zt.md

@@ -0,0 +1,201 @@
+# Learning to Predict Vehicle Trajectories with Model-based Planning
+
+Haoran Song Di Luan Wenchao Ding Michael Yu Wang Qifeng Chen The Hong Kong University of Science and Technology {hsongad, wdingae, dluan, mywang, cqf}@ust.hk
+
+Abstract: Predicting the future trajectories of on-road vehicles is critical for autonomous driving. In this paper, we introduce a novel prediction framework called PRIME, which stands for Prediction with Model-based Planning. Unlike recent prediction works that utilize neural networks to model scene context and produce unconstrained trajectories, PRIME is designed to generate accurate and feasibility-guaranteed future trajectory predictions. PRIME guarantees the trajectory feasibility by exploiting a model-based generator to produce future trajectories under explicit constraints and enables accurate multimodal prediction by utilizing a learning-based evaluator to select future trajectories. We conduct experiments on the large-scale Argoverse Motion Forecasting Benchmark, where PRIME outperforms the state-of-the-art methods in prediction accuracy, feasibility, and robustness under imperfect tracking.
+
+Keywords: trajectory prediction, autonomous driving
+
+# 1 Introduction
+
+In the architecture of autonomous driving, prediction serves as the bridging module that reasons future states based on the perceived information from upstream detection and tracking and provides the predicted future states to facilitate the downstream planning. Therefore, making accurate and reasonable trajectory predictions for on-road vehicles is vital for planning safe, efficient, and comfortable motion for self-driving vehicles (SDVs).
+
+The widely known challenge of trajectory prediction lies in modeling multi-agent interaction and inferring multimodal future states under driving scenarios. Traditional methods [1, 2, 3, 4, 5] produce motion forecasting by handcrafted rules or models with embedded physical and environmental features, which are insufficient for modeling interactive agents in complex scenes. Learning-based approaches [6, 7, 8], with deep neural networks to fuse scene context information and generate future trajectories, significantly promote the prediction accuracy and dominate the recent motion forecasting benchmarks for autonomous driving [9, 10].
+
+Despite achieving steady improvement in accuracy, much less attention has been paid to the feasibility and robustness of learning-based trajectory prediction. Indeed, most traffic participants operate under their inherent kinematic constraints (e.g., non-holonomic motion constraints for vehicles) while in compliance with the road structure (e.g., lane connectivity, static obstacles) and semantic information (e.g., traffic lights, speed limits). All these kinematic and environmental constraints explicitly regularize the trajectory space. However, most existing approaches model traffic agents as points and generate future trajectories without imposing constraints. Such constraint-free predictions may be incompliant with kinematic or environmental characteristics and thus give rise to massive uncertainty in the predicted future states. As a result, the downstream planning module would inevitably undergo some extra burdens, and even the “freezing robot problem” [11]. Furthermore, the trajectory predictions typically generated by neural network regression have high dependences on long-term tracking. For some dense driving scenarios where the target would be momently occluded or suddenly appears within the sensing range, tracking results are discontinuous or not accumulated enough. The prediction accuracy would thereby degrade under such imperfect tracking cases.
+
+![](images/3ef52fc9ad6bfaa96c99ea75f99e746cc7a8cf749dde7edcdbf7c3f79115ea68.jpg)  
+Figure 1: Illustration of the PRIME framework. The model-based generator (left) samples feasible future trajectories $\tau$ for the target agent by taking its real-time state $\mathbf { \bar { s } } _ { t a r } ^ { 0 }$ and HD map $\mathcal { M }$ , while imposing explicit constraints $\mathcal { C }$ to guarantee trajectory feasibility. The learning-based evaluator (right) receives the feasible trajectory set $\tau$ and all observed tracks $s$ to model the implicit interactions in scene context and then selects a final set of trajectories $\mathcal { T } _ { t a r } \subset \mathcal { T }$ as the prediction result.
+
+Toward overcoming these challenges, we propose PRIME, a novel architecture for vehicle trajectory prediction, as illustrated in Fig. 1. The core idea is to exploit a model-based motion planner as the prediction generator to produce feasibility-guaranteed future trajectories under explicit physical constraints, together with a deep neural network as the prediction evaluator to enable accurate multimodal prediction by learning complex implicit interactions. To the best of our knowledge, PRIME is the first to incorporate an interpretable motion planner in prediction learning and also the only method that ensures kinematic and environmental feasibility in data-driven trajectory prediction. We conduct experiments on the large-scale Argoverse motion forecasting benchmark and achieves better prediction accuracy over the state-of-the-art. Furthermore, PRIME shows significant superiority in trajectory feasibility guarantee and prediction robustness under imperfect tracking. These attributes would facilitate more flexible and safe motion planning for SDVs.
+
+# 2 Related Work
+
+Prediction and Planning are closely intertwined in autonomous driving [12, 13, 14, 15]. Planning is to generate constraint-compliant trajectory candidates and, after considering safety, comfort, path progress, etc., select the best trajectory for execution by the SDV (ego agent). Prediction facilitates the trajectory selection in planning by inferring future trajectories of the surrounding vehicles (target agents). Their different focuses make the corresponding mainstream solutions diverge. Model-based approaches [16, 17, 18, 19] are preferred in planning due to their interpretability and reliability in computing safe trajectories under explicit constraints. Learning-based methods [6, 7, 20, 21], in contrast, prevail in prediction by utilizing its advantage in modeling implicit interactions.
+
+Some learning-based prediction works incorporate the goal-directed idea from planning to infer the possible goals and then produce goal-conditioned trajectories [22, 23, 24, 25]. Moreover, the novel planning-prediction-coupled frameworks are introduced to make predictions conditioned on ego intentions [26] or motion plans [27, 28]. Although much emphasis on improving the point-level prediction accuracy, the data-driven frameworks cannot ensure the given constraints are indeed imposed on trajectory generation. Despite DKM [29] embeds the two-axle vehicle kinematics [30] in the output layer to ensure kinematic feasibility, yet still no guarantee on environmental compliance. Inspired by the popular sampling-based paradigm in vehicle motion planning [13, 17], we employ a model-based planner for providing feasibility-guaranteed trajectory sets, and thereby the learningbased part is reduced to evaluate future trajectories. With making the most of model-based planning and learning-based prediction, PRIME handles complex agent-map interactions while fulfilling environmental and kinematic constraints.
+
+Modeling agent-map interactions is fundamental for capturing information from scene map and dynamic agents. The rasterized representation [20, 31, 32] is proposed for learning-based methods, which renders traffic entities into images by different colors or intensities and then encodes rasters with Convolutional Neural Networks. As an alternative, the vectorized representation [33, 34, 25] vectorizes scene context as nodes to construct a graph, which exploits High Definition (HD) maps more explicitly and improves prediction accuracy. By contrast, we address the agent-map modeling with a hierarchical structure that incorporates the lane-association ideas from [3] while extends to learn global scene context. To be specific, our prediction generator acts locally in a planning manner to generate path-conditioned trajectory sets, and the prediction evaluator learns a global understanding of the scene context by aggregating all trajectory and map features.
+
+Generating multimodal trajectories is essential for handling the intrinsic multimodal prediction distributions. Stochastic models are mostly built upon conditional variational autoencoder [7, 35, 36, 37, 38] and generative adversarial network [39, 40, 41, 42], while sampling with uncontrollable latent variables at inference may impede their deployment on safety-critical driving systems. Deterministic models are mainly based on multi-mode trajectory regression [43, 44, 20, 34]. To alleviate mode collapse in prediction learning, recent works decompose the task into classification over anchor trajectories [45] or goal-conditioned trajectories [25], followed by trajectory offset regression. However, no feasibility could be ensured for the regressed results. CoverNet [32] formulates multimodal prediction by directly classifying on a pre-constructed trajectory set, but still, its predictions may violate the agent kinematics or scene constraints. For our framework, leveraging model-based planning as the prediction generator brings superiorities in 1) maintaining multimodal distributions by generating trajectories on diverse reachable paths, 2) ensuring trajectory feasibility by imposing real-time constraints, 3) mitigating the high reliance on long-term tracking, and 4) producing trajectories with continuous information rather than just discrete locations.
+
+# 3 Overview
+
+Problem formulation. Assume the self-driving vehicle is equipped with detection and tracking modules to provide observed states $s$ of on-road agents $\mathcal { A }$ and has access to HD map $\mathcal { M }$ . Let $\mathbf { s } _ { i } ^ { \bar { t } }$ denote the state of agent $a _ { i } \in { \mathcal { A } }$ at frame $t$ , including position, heading, velocity, turning rate and actor type, and $\mathbf { s } _ { i } = \left\{ \mathbf { s } _ { i } ^ { - T _ { P } + 1 } , \mathbf { s } _ { i } ^ { - T _ { P } + 2 } , . . . , \mathbf { s } _ { i } ^ { 0 } \right\}$ denotes the state sequence in the observed period $T _ { P }$ . Given any agent as the prediction target, we denote it by $a _ { t a r }$ and its surrounding agents by $\mathcal { A } _ { n b r s } = \{ a _ { 1 } , a _ { 2 } , . . . , a _ { m } \}$ for differentiation, with their state sequence correspondingly given as $\mathbf { s } _ { t a r }$ and $S _ { n b r s } = \{ \mathbf { s } _ { 1 } , \mathbf { s } _ { 2 } , . . . , \mathbf { s } _ { m } \}$ . Accordingly, $S = \{ \mathbf { s } _ { t a r } \} \cup S _ { n b r s }$ and $\mathcal { A } = \{ a _ { t a r } \} \cup \mathcal { A } _ { n b r s }$ . Our objective is to predict multimodal future trajectories $\mathcal { T } _ { t a r } = \{ \mathcal { T } _ { k } | k = 1 , 2 , . . . , K \}$ together with corresponding trajectory probability $\left\{ p _ { k } \right\}$ , where $\mathcal { T } _ { k }$ denotes a predicted trajectory for target agent $a _ { t a r }$ with continuous state information up to the prediction horizon $T _ { F }$ , $K$ is the number of predicted trajectories. Additionally, it is required to ensure each prediction $\mathcal { T } _ { k } \in \mathcal { T } _ { t a r }$ is feasible with existing constraints $\mathcal { C }$ , which includes environmental constraints $\mathcal { C } _ { \mathcal { M } }$ and the kinematic constraints $\mathcal { C } _ { t a r }$ .
+
+Our framework. The two-stage architecture of PRIME consists of model-based generator $G$ and learning-based evaluator $E$ . Concretely, the generator $G : ( \mathbf { s } _ { t a r } ^ { 0 } , \mathcal { M } , \mathcal { C } ) \mapsto ( \mathcal { P } , \mathcal { T } )$ is tasked to produce the trajectory space for the target, which is approximated by a finite set of feasible trajectories $\tau$ . This part starts with searching a set of reachable paths $\mathcal { P } = \{ \mathcal { P } _ { j } | j = 1 , 2 , . . . , l \}$ from HD map $\mathcal { M }$ , which provides reference path for trajectory generation. Then a classical sampling-based planner is utilized to generate trajectory samples under constraints in $\mathcal { C }$ , and thus provide the feasible future trajectory set $\mathcal { T } = \cup _ { j = 1 } ^ { l } \{ \bar { \mathcal { T } _ { j , k } } | k = 1 , 2 , . . . , n _ { j } \}$ for the target. $\mathcal { T } _ { j , k }$ denotes the $k$ -th feasible trajectory generated from path $\mathcal { P } _ { j }$ , and the total number of trajectories is $\begin{array} { r } { n = \sum _ { j = 1 } ^ { l } n _ { j } } \end{array}$ . The model-based actions. The evaluator $E : ( \mathcal { P } , \mathcal { T } , \overset { \cdot } { S } ) \mapsto ( \mathcal { T } _ { t a r } , \{ p _ { k } \} )$ takes charge of learning implicit interactions, which features with a dual representation for spatial information and with the attention mechanism to process the varying sizes of $l$ reachable paths, $m$ surrounding agents, and $n$ feasible trajectories. Notably, the evaluator $E$ is reduced to score trajectories and select prediction results $\bar { \mathcal { T } _ { t a r } } \subset \mathcal { T }$ , rather than regressing position or displacement as most learning-based frameworks do.
+
+# 4 Model-based Generator
+
+# 4.1 Path Search
+
+Unlike motion planning, where the reference path for the controllable ego agent is given, the future paths of uncontrollable targets in prediction are unknown. Therefore, we conduct the path search in advance of trajectory generation such that any prediction target could be associated with a set of potential paths ${ \mathcal { P } } ^ { + }$ . Our path search algorithm $\dot { G } _ { p a t h } : ( \mathcal { M } , \mathbf { s } _ { t a r } ^ { 0 } ) \mapsto \mathcal { P } ^ { + }$ is implemented by Depth
+
+First-Search on HD map $\mathcal { M }$ , with more details described in the supplementary material. Yielding a potential path $\mathcal { P } _ { j } \in \mathcal { P } ^ { \bar { + } }$ with the centerline coordinates of each lane segment sequence, we expect all the paths of ${ \bar { \mathcal { P } } } ^ { + }$ to provide sufficient coverage for the future trajectory space of $a _ { t a r }$ . As no dynamic constraint is imposed in this phase, for target with current state $\bar { \bf s } _ { t a r } ^ { 0 }$ , some paths in ${ \mathcal { P } } ^ { + }$ may not be reachable at frame $t = T _ { F }$ . For instance, a high-speed vehicle cannot change to the opposite lane with a U-turn in few seconds. Such unreachable paths could be recognized in the following trajectory generation phase as no trajectories samples towards them are feasible. Finally, a set of reachable paths ${ \mathcal { P } } \subseteq { \mathcal { P } } ^ { + }$ would be reserved.
+
+# 4.2 Trajectory Generation
+
+Given the potential paths in ${ \mathcal { P } } ^ { + }$ as dynamic references, we choose to generate future trajectories in a planning manner. For SDV, motion planning typically aims at finding an optimal trajectory to connect the current state and a goal state, essentially different from prediction that infers probable trajectories for vehicles with unknown intentions. Despite this, the modelbased generator in planning, which computes a large number of trajectory samples for the follow-up selection, could also be exploited in prediction.
+
+![](images/d9cefb7d5e84b74f18b436febf332ffc0f7ebc613ab7fb9942c04939003a11eb.jpg)  
+Figure 2: Trajectory generation in a Frenet Frame ´
+
+We adopt the trajectory generation phase of Frenet plan- ´ ner [17] in our trajectory generator $G _ { t r a j } ^ { \bf { \bar { \alpha } } } : ( \mathcal { P } ^ { + } , { \bf s } _ { t a r } ^ { 0 } , \mathcal { \bar { C } } ) \mapsto $ $\tau$ . Given a reference path in ${ \mathcal { P } } ^ { + }$ , a dynamic curvilinear frame is given by the tangential vector $\vec { t _ { r } }$ and normal vector $\vec { n } _ { r }$ at a certain point $r$ on the path centerline. The Cartesian coordinate $\vec { x } = ( x , y )$ could be converted to the Frenet coordinate ´ $( s , d )$ , with the relation
+
+$$
+\vec { x } ( s ( t ) , d ( t ) ) = \vec { r } ( s ( t ) ) + d ( t ) \vec { n } _ { r } ( s ( t ) ) ,
+$$
+
+in which $\vec { r }$ represents a vector pointing from the path root, $s$ and $d$ denote the covered arc length and the perpendicular offset, as illustrated in Fig. 2. The trajectory generation phase first projects the current state $\mathbf { s } _ { t a r } ^ { 0 }$ onto the Frenet frame and obtains the state tuple ´ $[ s _ { 0 } , \dot { s } _ { 0 } , \bar { \dot { s } } _ { 0 } , d _ { 0 } , \dot { d } _ { 0 } , \ddot { d } _ { 0 } ]$ . The longitudinal movement $s ( t )$ and lateral movement $d ( t )$ within the prediction horizon $T _ { F }$ are then generated independently by connecting the fixed start state with different end states using parametric curves to cover different driving maneuvers. Compared with planning, prediction receives less accurate state estimation for targets and does not need fine-grained trajectories. In our trajectory generator, therefore, some high-order state variables are simplified to zero. For longitudinal movement, we sample the target velocity $\dot { s } ( T _ { F } )  \dot { s } _ { i }$ in the range of $[ \mathrm { m a x } ( 0 , \dot { s } _ { 0 } - \delta ^ { - } T _ { F } ) , \mathrm { m i n } ( \hat { \dot { s } } , \dot { s } _ { 0 } + \delta ^ { + } T _ { F } ) ]$ while leaving $s ( T _ { F } )$ unconstrained. The constants $\delta ^ { - }$ , $\delta ^ { + }$ and $\hat { s }$ are given by considering the actor type of $a _ { t a r }$ and speed limit in $\mathcal { M }$ , to control the longitudinal velocity $\dot { s } _ { i }$ in a reasonable range. Each longitudinal trajectory $s _ { i } ( t )$ is calculated using a quartic polynomial
+
+$$
+\mathrm { t . } \quad [ s ( 0 ) , \dot { s } ( 0 ) , \ddot { s } ( 0 ) , \dot { s } ( T _ { F } ) , \ddot { s } ( T _ { F } ) ] = [ s _ { 0 } , \dot { s } _ { 0 } , 0 , \dot { s } _ { i } , 0 ] .
+$$
+
+For lateral movement, we sample the target offset $d ( T _ { F } )  d _ { j }$ in the range of $[ - d _ { l a n e } / 2 , d _ { l a n e } / 2 ]$ where $d _ { l a n e }$ denotes lane width. Each lateral trajectory $d _ { j } ( t )$ is calculated using a quintic polynomial
+
+$$
+[ d ( 0 ) , \dot { d } ( 0 ) , \ddot { d } ( 0 ) , d ( T _ { F } ) , \dot { d } ( T _ { F } ) , \ddot { d } ( T _ { F } ) ] = [ d _ { 0 } , \dot { d } _ { 0 } , 0 , d _ { j } , 0 , 0 ] .
+$$
+
+With the resulted longitudinal and lateral trajectory set $\mathcal { T } _ { l o n }$ and $\mathcal { T } _ { l a t }$ , a full trajectory $\vec { x } ( s ( t ) , d ( t ) )$ is formed by every combinations in $\mathcal { T } _ { l o n } \times \mathcal { T } _ { l a t }$ . Next, the trajectories incompliant with given constraints $\mathcal { C }$ would be filtered out. We first project the Frenet coordinates ´ $( s , d )$ back to global coordinates $( x , y )$ to check if the trajectory collides with static obstacles given in $\mathcal { C } _ { \mathcal { M } }$ . For collision-free trajectories, their high-order state variables are then converted by the Frenet-Cartesian-transfomation ´
+
+$$
+\ [ s , \dot { s } , \ddot { s } , d , \dot { d } , \ddot { d } ] \longmapsto [ \vec { x } , v , \kappa , \alpha ]
+$$
+
+to check if any velocity $v$ , acceleration $\alpha$ or curvature $\kappa$ exceeds the kinematic constraints given in $\mathcal { C } _ { t a r }$ . Finally, each reference path $\mathcal { P } _ { j } \in \mathcal { P }$ would generate a set of $n _ { j }$ feasibility-guaranteed future trajectories $\mathbf { \bar { \{ } }  T _ { j , k } | k = 1 , 2 , . . . , n _ { j } \}$ , and all the trajectories together form an overall trajectory space $\tau$ . Although the constraints are set conservatively with leaving some margin for the learning-based evaluator, our model-based generator effectively narrows down the trajectory space $\tau$ by imposing constraints. This unique advantage would set our framework with higher accuracy and robustness.
+
+![](images/71c32cfcbf3bda2747de8beba9bccc66a157f5ccb8c8e59d18af4fd85e48ab69.jpg)  
+Figure 3: PRIME framework overview. The model-based generator searches reachable paths $\mathcal { P }$ through the map and produces feasible future trajectories $\tau$ . The learning-based evaluator encodes the traffic entities in $( \mathcal { P } , \mathcal { T } , \mathcal { S } )$ and learns implicit interactions in the subsequent attention modules. Afterwards, each future trajectory $\mathcal { T } _ { j , k }$ could query its track tensor $\mathbf { X } _ { j } ( \mathbf { s } _ { t a r } )$ from P2T, interaction tensor $\mathbf { Y } _ { j } ( \mathbf { s } _ { t a r } )$ from A2A and future tensor $\mathbf { Z } ( T _ { j , k } )$ from F2F, and it is scored by feeding the concatenation of these tensors to fully-connected layers. Finally, the evaluator ranks all feasible future trajectories in $\tau$ by scoring and outputs a final set of $K$ predicted trajectories.
+
+# 5 Learning-based Evaluator
+
+# 5.1 State Representation
+
+The prediction evaluator aggregates scene context, including observed state sequences $s$ , path set $\mathcal { P }$ , and future trajectory set $\tau$ . To make it compatible with most existing trajectory prediction datasets, state sequence $\mathbf { s } _ { i }$ is reduced to history track in the learning part. Before feeding to the network, we discretize each history track $\mathbf { s } _ { i }$ and future trajectory $\mathcal { T } _ { j , k }$ as a location sequence with time interval $\Delta T$ , and each reference path $\mathcal { P } _ { j }$ as a waypoint sequence with distance interval $\Delta D$ . Since the longitudinal movement $s$ and lateral offset $d$ indicate how an agent moves relative to a reference path, they represent the local spatial relationship more straightforwardly. For this reason, we use the Frenet coordinates ´ $( s , d )$ in addition to the Cartesian coordinates $( x , y )$ to form a dual spatial representation. Here, the spatial information $( x , y , s , d )$ of future trajectories in $\tau$ is given by the generator, while the $( s , d )$ coordinates of history tracks in $s$ are obtained by projecting $( x , y )$ coordinates on the corresponding reference path. Additionally, we adopt the approach of [34] to add a binary mask $b$ to history track’s representation $( x , y , s , d , b )$ to indicate if the location is padded.
+
+# 5.2 Encoding Scene Context
+
+Prior to capture interrelationships between traffic entities, we first encode each kind of entity in the scene. All encoders are structured with a temporal convolutional layer followed by a long short-term memory (LSTM) layer. The track encoder and the future encoder employ a unidirectional LSTM and make the last hidden state $h ( \cdot )$ as the motion encoding for history track and future trajectory, while the path encoder uses a bidirectional LSTM and provides the sequence of hidden states $H ( \cdot )$ as the path spatial encoding. Given the scene context description $( \mathcal { S } , \mathcal { P } , \mathcal { T } )$ , each reachable path $\mathcal { P } _ { j } \in \mathcal { P }$ is encoded as a $H ( \mathcal { P } _ { j } )$ , where $j = 1 , 2 , . . . , l$ . Considering that the Frenet representation ´ is dependent with the path frame, we encode all history tracks with respect to each reference path $\mathcal { P } _ { j }$ , which results in $l$ groups of track encodings $\{ h ( \mathbf { s } _ { t a r } ) , h ( \mathbf { s } _ { 1 } ) , . . . , h ( \mathbf { s } _ { m } ) \} _ { j }$ . Each future trajectory $\mathcal { T } _ { j , k } \in \mathcal { T }$ is relative to its reference path $\mathcal { P } _ { j }$ , so all future trajectories are encoded correspondingly to form $l$ groups of future encodings $\{ h ( \mathcal { T } _ { j , k } ^ { \setminus } ) | k = 1 , 2 , . . . , n _ { j } \}$ .
+
+# 5.3 Modeling Interactions
+
+Next is to capture the implicit interactions resulted from the static environment and multiple dynamic agents. To fuse the spatial-temporal information from varying numbers of entities in the scene context, the attention mechanism [46] is adopted to construct four modules, namely, path to track (P2T), path to future (P2F), agent to agent (A2A), and future to future (F2F). They are implemented in the same way of scaled dot-product attention and use linear layers for mapping key, query and value. The overall workflow is shown in Fig. 3. In the upper branch, P2T brings the spatial information of each path encoding $\mathcal { P } _ { j }$ into the corresponding track encodings $\{ h ( \mathbf { s } _ { t a r } ) , h ( \mathbf { s } _ { 1 } ) , . . . , h ( \mathbf { s } _ { m } ) \} _ { j }$ . The track encodings are further processed by a self-attention structure in A2A, aiming to capture the interactions between agents in the past time domain. The lower branch lays emphasis on updating the features contained in future encodings. P2F brings the spatial information of path encoding $H ( \mathcal P _ { j } )$ into the corresponding future encodings $\{ h ( \mathcal { T } _ { j , k } ) | k = 1 , 2 , . . . , n _ { j } \}$ . It is followed by F2F that fuses all future encodings $\bigcup _ { j = 1 } ^ { l } \left\{ h ( \mathcal T _ { j , k } ) | k = 1 , 2 , . . . , n _ { j } \right\}$ from different paths $\mathcal { P } _ { j } ( j = 1 , 2 , . . . , l )$ using self-attention. In particular, F2F obtains a global understanding of the reachable space given by $\mathcal { P }$ and, by this way, attempts to further perceive the differences between different trajectories in $\tau$ . For any future trajectories $\mathcal { T } _ { j , k } \in \mathcal { T }$ , the corresponding track tensor $\mathbf { X } _ { j } ( \mathbf { s } _ { t a r } )$ , interaction tensor $\mathbf { Y } _ { j } ( \mathbf { s } _ { t a r } )$ and future tensor $\bar { \bf Z } ( \mathcal { T } _ { j , k } )$ could be obtained from P2T, A2A and F2F modules, which are then concatenated together to form a full description $\mathbf { U } _ { j , k } = \operatorname { C o n c a t } ( \mathbf { X } _ { j } ( \mathbf { s } _ { t a r } ) , \mathbf { Y } _ { j } ( \mathbf { s } _ { t a r } ) , \mathbf { Z } ( { \mathcal { T } } _ { j , k } ) )$ .
+
+# 5.4 Trajectory Scoring, Learning, and Inference
+
+With $\mathbf { U } _ { j , k }$ as a full description, we score all the $n$ trajectories $\mathcal { T } _ { j , k }$ using a maximum entropy model:
+
+$$
+\gamma ( \mathcal { T } _ { j , k } ) = \frac { \exp ( f ( \mathbf { U } _ { j , k } ) ) } { \sum _ { j = 1 } ^ { l } \sum _ { k = 1 } ^ { n _ { j } } \exp ( f ( \mathbf { U } _ { j , k } ) ) } ,
+$$
+
+in which $f ( \cdot )$ is implemented using a 3-layer MLP at the end of the evaluation network $E$ . The score label $\psi ( T _ { j , k } )$ is resulted from calculating the accumulated squared distance error $\mathrm { D i s t } ( \cdot )$ between the future trajectory $\mathcal { T } _ { j , k }$ and the ground truth trajectory $\mathcal { T } _ { G T }$ within the prediction horizon $T _ { F }$ :
+
+$$
+\psi ( \mathcal { T } _ { j , k } ) = \frac { \exp ( - \mathrm { D i s t } ( \mathbf { T } _ { j , k } , \mathbf { T } _ { G T } ) / \tau ) } { \sum _ { j = 1 } ^ { l } \sum _ { k = 1 } ^ { n _ { j } } \exp ( - \mathrm { D i s t } ( \mathbf { T } _ { j , k } , \mathbf { T } _ { G T } ) / \tau ) } ,
+$$
+
+where $\tau$ is set as a temperature factor. The overall network is trained by cross entropy between the evaluated scores and the labeled scores L = CrossEntropy $( \gamma ( \mathcal { T } _ { j , k } ) , \psi ( \mathcal { T } _ { j , k } ) )$ . For the inference stage that requires $K$ predicted trajectories, we adopt the non-maximum suppression (NMS) algorithm to remove near-duplicate trajectories, as did in [25]. According to the predicted scores, this method greedily picks trajectories from $\tau$ and excludes the lower scored trajectory between very close ones. Finally, $K$ trajectories with descending order of scores form the prediction result $\begin{array} { r l } { \mathcal { T } _ { t a r } } & { { } = } \end{array}$ $\{ \mathcal { T } _ { i } | k = 1 , 2 , . . . , K \}$ , and the prediction probability $p _ { k }$ is derived by $p _ { k } = \gamma ( \mathcal T _ { k } ) / { \sum _ { k = 1 } ^ { K } \gamma ( \mathcal T _ { k } ) }$ .
+
+# 6 Experiments
+
+Dataset. Argoverse [10] is one of the largest publicly available motion forecasting datasets, which contains over 324K data sequence collected from complex urban driving scenarios. The training, validation, and test sets are taken from disjoint parts of the cities. Each sequence lasts for 5 seconds, containing the centroid locations of each tracked agent sampled at $1 0 \ : \mathrm { H z }$ , in which one vehicle with relatively complex motion is marked as the prediction target. The objective is to predict its locations 3 seconds into the future, given an initial 2-second observation.
+
+Metrics. We follow the Argoverse evaluation criteria under $K = 1$ and $K = 6$ . Minimum Average Displacement Error $( \mathrm { m i n A D E } _ { K }$ ) is the average L2 distance error of the best predicted trajectory. Minimum Final Displacement Error $( \mathrm { m i n F D E } _ { K } )$ ) is the L2 distance error of the best predicted trajectory at the final timestamp. Miss Rate $( \mathrm { M R } _ { K } )$ is the ratio of scenarios that none of $K$ predicted trajectories has less than 2 meters L2 final displacement error. For multimodal prediction, the probability-based metrics ${ \mathsf { p } } { \cdot } { \mathrm { m i n A D E } } _ { K }$ and ${ \mathsf { p } } { \cdot } { \mathrm { m i n F D E } } _ { K }$ are calculated by adding $- l o g ( p )$ to $\mathrm { m i n A D E } _ { K }$ and $\mathrm { m i n F D E } _ { K }$ , where $p$ corresponds to the probability of the best predicted trajectory. In the Argoverse benchmark, best refers to the predicted trajectory with the minimum endpoint error.
+
+Implementation Details. Our implementation is detailed in the supplementary material. Among the state-of-the-art methods, only LaneGCN [34] is open-source. So we use its official implementation and Argoverse baselines [10] for additional tests about trajectory feasibility and imperfect tracking.
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">K=1</td><td colspan="5">K=6</td><td rowspan="2">Infeasibility (%)</td></tr><tr><td>minADE</td><td>minFDE</td><td>MR (%)</td><td>minADE</td><td>minFDE</td><td>p-minADE</td><td>p-minFDE</td><td>MR (%)</td></tr><tr><td>Argo-CV</td><td>3.53</td><td>7.89</td><td>83.48</td><td>3.39</td><td>7.57</td><td>5.18</td><td>9.36</td><td>81.68</td><td>0.00</td></tr><tr><td>Argo-LSTM+map</td><td>2.96</td><td>6.81</td><td>81.22</td><td>2.34</td><td>5.44</td><td>4.14</td><td>7.23</td><td>69.16</td><td>43.53</td></tr><tr><td>Argo-NN+map</td><td>3.65</td><td>8.12</td><td>83.55</td><td>2.08</td><td>4.03</td><td>3.87</td><td>5.82</td><td>58.21</td><td>86.39</td></tr><tr><td>LaneGCN [34]</td><td>1.71</td><td>3.78</td><td>59.05</td><td>0.87</td><td>1.36</td><td>2.66</td><td>3.16</td><td>16.34</td><td>16.52</td></tr><tr><td>Alibaba-ADLab</td><td>1.97</td><td>4.35</td><td>63.76</td><td>0.92</td><td>1.48</td><td>2.67</td><td>3.23</td><td>15.86</td><td>1</td></tr><tr><td>TNT [25]</td><td>1.78</td><td>3.91</td><td>59.72</td><td>0.94</td><td>1.54</td><td>2.73</td><td>3.33</td><td>13.28</td><td>1</td></tr><tr><td>Jean [21]</td><td>1.74</td><td>4.24</td><td>68.56</td><td>1.00</td><td>1.42</td><td>2.79</td><td>3.21</td><td>13.08</td><td>1</td></tr><tr><td>Poly</td><td>1.70</td><td>3.82</td><td>58.80</td><td>0.87</td><td>1.47</td><td>2.67</td><td>3.26</td><td>12.02</td><td>1</td></tr><tr><td>PRIME (Ours)</td><td>1.91</td><td>3.82</td><td>58.67</td><td>1.22</td><td>1.56</td><td>2.71</td><td>3.05</td><td>11.50</td><td>0.00</td></tr></table>
+
+Table 1: Comparison with the Argoverse baselines and the state-of-the-art methods on the Argoverse test set. All metrics are lower the better and Miss Rate (MR, $\mathrm { K } { = } 6$ ) is the key metric.
+
+# 6.1 Comparison with State-of-the-art
+
+We compare our proposed PRIME against the Argoverse baselines [10] (CV, LSTM+map, $\mathrm { N N + m a p } ,$ ), the top-3 methods in the Argoverse Motion Forecasting Competition 2020 (Jean [21], Poly, Alibaba-ADLab), and the recently published state-of-the-art, LaneGCN [34] and TNT [25]. The performance comparison under Argoverse test set is shown in Table 1. It could be noted that PRIME outperforms all other methods on Miss Rate $K = 6$ ), which is the official ranking metric in Argoverse Competition 2020. It reflects that our method produces accurate multimodal predictions consistently in diverse scenarios. We also achieve the best on the probability-based metric p${ \mathrm { m i n F D E } } _ { 6 }$ , which would be highly beneficial to weigh between multiple predictions in making decisions and motion plans. From the methods with public details, including LaneGCN [34], TNT [25], and Jean [21], we can find they all perform the learning-based paradigm that utilizes neural networks to model traffic entities and generates future trajectories, while PRIME is the only one that integrates a model-based motion generator into prediction learning. Notably, due to the lack of more detailed on-road information in the Argoverse dataset, such as vehicle types, bounding box, static obstacles, etc., the quantitative result is achieved by imposing general constraints on the model-based generator. This indicates there exists more space to improve when deploying our framework in a real autonomous driving system. Furthermore, handling environmental and dynamic constraints in an interpretable model-based manner and generating trajectories with continuous state information is significant for real-world deployment, which could not be reflected from the evaluation metrics.
+
+# 6.2 Ablation Studies
+
+We ablate the F2F module and Frenet´ representation (denoted by SD) from the complete evaluation network to study their impacts. Table 2 reports the results on the Argoverse validation set. With P2T, P2A, and A2A attention modules capturing the basic interactions between map and agents, the base model performs at the same level with TNT $\mathbf { \bar { ( M R _ { 6 } } = 9 \% }$ reported in [25]), indicat
+
+<table><tr><td>Modules</td><td colspan="3"> p-minADE6 p-minFDE6 MR6(%)</td><td># Params</td></tr><tr><td>Base</td><td>2.33</td><td>2.63</td><td>8.52</td><td>0.69 M</td></tr><tr><td>Base+F2F</td><td>2.31</td><td>2.61</td><td>8.23</td><td>0.72M</td></tr><tr><td>Base+SD</td><td>2.29</td><td>2.58</td><td>7.81</td><td>0.99M</td></tr><tr><td>Base+F2F+SD</td><td>2.29</td><td>2.57</td><td>7.51</td><td>1.02 M</td></tr></table>
+
+Table 2: Ablation studies on the Argoverse validation set.
+
+ing that these modules are effective in capturing agent-map interactions. As for the Frenet represen- ´ tation providing the local spatial relationship and the F2F module fusing all feasible trajectories to get a global understanding of the reachable space, they both promote the performance. By comparison, the inclusion of Frenet representation is more effective. Additionally, the complete network ´ makes the best performance with only 1.02M parameters, which indicates that separating the function of trajectory generation would reduce the learning burden while achieving high performance.
+
+# 6.3 Trajectory Feasibility
+
+As a typical non-holonomic motion system, vehicles are constrained by inherent kinematic characteristics. So we investigate the ratio of infeasible trajectories produced by prediction models. Since the high-order states (velocity, acceleration, or turning rate) cannot be estimated accurately from discrete locations predicted by common learning-based models, we evaluate the trajectory feasibility only using curvature. By interpolating the predicted positions with pairwise cubic splines, we get the curvature at each point. A trajectory is labeled as infeasible if the curvature $\kappa > 1 / 3$ (i.e., the minimum turning radius is 3 meters) at any of its points. The ratio of infeasible trajectories is shown in the last column of Table 1. Except for the physical baseline Argo-CV (Constant Velocity), the others, as representatives of the unconstrained learning-based models, have at least $1 6 . 5 \%$ infeasible predictions. Although we only use curvature for judgment and set a fairly conservative threshold (the minimal turning radius for a regular sedan is around $4 . 5 \sim 6 . 0$ meters), the infeasible predictions still take up a considerable proportion, which would cause redundant burdens for SDVs to make decisions and plans. By contrast, the model-based generator in our framework can handle any kinematic and environmental constraints, thereby ensuring trajectory feasibility.
+
+# 6.4 Imperfect Tracking
+
+While most motion forecasting datasets provide tracking results of a certain duration for prediction targets, a self-driving vehicle would inevitably encounter realworld situations where the target is lost in some timestamps or not tracked long enough yet. Then the prediction model is required to robustly handle imperfect tracks rather than being restricted to fixedduration tracking inputs. To let the models (ours, LaneGCN, and $\mathrm { N N } +$ map baseline) be aware of imperfect tracks, we retrained them by randomly dropping out tracked locations. For processing such inputs while keeping network structures, we pad the locations of dropped timestamps with the nearest tracked location and add a dimension of the binary mask to denote the padded location. The drop rate is randomly sampled from $0 \sim 0 . 6$ for each data
+
+![](images/d237d2e351ad915855ab26005ded27ee62528a58ccb22c0b893306afe21cac66.jpg)  
+Figure 4: Comparison of prediction robustness under imperfect tracking.
+
+sequence in training but fixed in testing. The drop rate is pointwise applied, i.e., 0.6 drop rate may drop more or less than $6 0 \%$ of locations on a track. The last timestamp is always kept to ensure the prediction target could be detected at inference. Fig. 4 shows how the miss rate varies with track drop rate, we observe that our model performs stably, with only $3 . 6 \%$ relative increase on $\mathrm { M R _ { 6 } }$ , while the relative increase is around $3 0 \% \sim 4 0 \%$ for the others. The result indicates that the learning-based prediction models rely on long-term tracked results to regress trajectories, while our framework design relieves that to a certain extent, thereby improving the prediction robustness.
+
+# 7 Conclusion
+
+We present the prediction framework PRIME that learns to predict vehicle trajectories with modelbased planning. PRIME guarantees the trajectory feasibility by exploiting a model-based generator to produce future trajectories under explicit constraints. It makes accurate trajectory predictions by employing a learning-based evaluator to capture implicit interactions in scene context and select future trajectories by scoring. With the novel framework design, PRIME outperforms the state-ofthe-art in prediction accuracy, feasibility, and robustness. Moreover, the advantages of reasonably regularizing trajectory space, predicting trajectories with continuous state, and the compatibility with on-road information would set our framework highly useful in real system deployment.
+
+# Acknowledgments
+
+This research work was supported in part by The Hong Kong University of Science and Technology under the project “LDSERF-Autonomy Through Learning”.
+
+# References
+
+[1] D. Helbing and P. Molnar. Social force model for pedestrian dynamics. Physical review E, 51 (5):4282, 1995. [2] A. Houenou, P. Bonnifait, V. Cherfaoui, and W. Yao. Vehicle trajectory prediction based on motion model and maneuver recognition. In IROS, 2013.   
+[3] J. Ziegler et al. Making bertha drive—an autonomous journey on a historic route. IEEE Intelligent transportation systems magazine, 6(2):8–20, 2014. [4] S. Lefevre, D. Vasquez, and C. Laugier. A survey on motion prediction and risk assessment \` for intelligent vehicles. ROBOMECH journal, 1(1):1–14, 2014. [5] W. Schwarting, A. Pierson, J. Alonso-Mora, S. Karaman, and D. Rus. Social behavior for autonomous vehicles. PNAS, 116(50):24972–24978, 2019. [6] A. Alahi, K. Goel, V. Ramanathan, A. Robicquet, L. Fei-Fei, and S. Savarese. Social lstm: Human trajectory prediction in crowded spaces. In CVPR, 2016.   
+[7] N. Lee, W. Choi, P. Vernaza, C. B. Choy, P. H. Torr, and M. Chandraker. Desire: Distant future prediction in dynamic scenes with interacting agents. In CVPR, 2017. [8] M. Bansal, A. Krizhevsky, and A. Ogale. Chauffeurnet: Learning to drive by imitating the best and synthesizing the worst. In CVPR, 2018.   
+[9] H. Caesar, V. Bankiti, A. H. Lang, S. Vora, V. E. Liong, Q. Xu, A. Krishnan, Y. Pan, G. Baldan, and O. Beijbom. nuscenes: A multimodal dataset for autonomous driving. In CVPR, 2020.   
+[10] M.-F. Chang, J. Lambert, P. Sangkloy, J. Singh, S. Bak, A. Hartnett, D. Wang, P. Carr, S. Lucey, D. Ramanan, et al. Argoverse: 3d tracking and forecasting with rich maps. In CVPR, 2019.   
+[11] P. Trautman and A. Krause. Unfreezing the robot: Navigation in dense, interacting crowds. In IROS, 2010.   
+[12] D. Ferguson, T. M. Howard, and M. Likhachev. Motion planning in urban environments. Journal of Field Robotics, 25(11-12):939–960, 2008.   
+[13] J. Wei, J. M. Dolan, and B. Litkouhi. A prediction-and cost function-based algorithm for robust autonomous freeway driving. In IEEE Intelligent Vehicles Symposium, 2010.   
+[14] C. Katrakazas, M. Quddus, W.-H. Chen, and L. Deka. Real-time motion planning methods for autonomous on-road driving: State-of-the-art and future research directions. Transportation Research Part C: Emerging Technologies, 60:416–442, 2015.   
+[15] W. Schwarting, J. Alonso-Mora, and D. Rus. Planning and decision-making for autonomous vehicles. Annual Review of Control, Robotics, and Autonomous Systems, 2018.   
+[16] M. Pivtoraiko, R. A. Knepper, and A. Kelly. Differentially constrained mobile robot motion planning in state lattices. Journal of Field Robotics, 26(3):308–333, 2009.   
+[17] M. Werling, J. Ziegler, S. Kammel, and S. Thrun. Optimal trajectory generation for dynamic street scenarios in a frenet frame. In ICRA, 2010.   
+[18] M. McNaughton, C. Urmson, J. M. Dolan, and J.-W. Lee. Motion planning for autonomous driving with a conformal spatiotemporal lattice. In ICRA, 2011.   
+[19] E. Galceran, A. G. Cunningham, R. M. Eustice, and E. Olson. Multipolicy decision-making for autonomous driving via changepoint-based behavior prediction. In RSS, 2015.   
+[20] H. Cui, V. Radosavljevic, F.-C. Chou, T.-H. Lin, T. Nguyen, T.-K. Huang, J. Schneider, and N. Djuric. Multimodal trajectory predictions for autonomous driving using deep convolutional networks. In ICRA, 2019.   
+[21] J. Mercat, T. Gilles, N. El Zoghby, G. Sandou, D. Beauvois, and G. P. Gil. Multi-head attention for multi-modal joint vehicle motion forecasting. In ICRA, 2020.   
+[22] B. D. Ziebart, N. Ratliff, G. Gallagher, C. Mertz, K. Peterson, J. A. Bagnell, M. Hebert, A. K. Dey, and S. Srinivasa. Planning-based prediction for pedestrians. In IROS, 2009.   
+[23] E. Rehder, F. Wirth, M. Lauer, and C. Stiller. Pedestrian prediction by planning using deep neural networks. In ICRA, 2018.   
+[24] K. Mangalam, H. Girase, S. Agarwal, K.-H. Lee, E. Adeli, J. Malik, and A. Gaidon. It is not the journey but the destination: Endpoint conditioned trajectory prediction. In ECCV, 2020.   
+[25] H. Zhao, J. Gao, T. Lan, C. Sun, B. Sapp, B. Varadarajan, Y. Shen, Y. Shen, Y. Chai, C. Schmid, et al. Tnt: Target-driven trajectory prediction. In CoRL, 2020.   
+[26] N. Rhinehart, R. McAllister, K. Kitani, and S. Levine. Precog: Prediction conditioned on goals in visual multi-agent settings. In ICCV, 2019.   
+[27] H. Song, W. Ding, Y. Chen, S. Shen, M. Y. Wang, and Q. Chen. Pip: Planning-informed trajectory prediction for autonomous driving. In ECCV, 2020.   
+[28] T. Salzmann, B. Ivanovic, P. Chakravarty, and M. Pavone. Trajectron $^ { + + }$ : Dynamically-feasible trajectory forecasting with heterogeneous data. In ECCV, 2020.   
+[29] H. Cui, T. Nguyen, F.-C. Chou, T.-H. Lin, J. Schneider, D. Bradley, and N. Djuric. Deep kinematic models for kinematically feasible vehicle trajectory predictions. In ICRA, 2020.   
+[30] R. Rajamani. Vehicle dynamics and control. Springer Science & Business Media, 2011.   
+[31] N. Djuric, V. Radosavljevic, H. Cui, T. Nguyen, F.-C. Chou, T.-H. Lin, N. Singh, and J. Schneider. Uncertainty-aware short-term motion prediction of traffic actors for autonomous driving. In WACV, 2020.   
+[32] T. Phan-Minh, E. C. Grigore, F. A. Boulton, O. Beijbom, and E. M. Wolff. Covernet: Multimodal behavior prediction using trajectory sets. In CVPR, 2020.   
+[33] J. Gao, C. Sun, H. Zhao, Y. Shen, D. Anguelov, C. Li, and C. Schmid. Vectornet: Encoding hd maps and agent dynamics from vectorized representation. In CVPR, 2020.   
+[34] M. Liang, B. Yang, R. Hu, Y. Chen, R. Liao, S. Feng, and R. Urtasun. Learning lane graph representations for motion forecasting. In ECCV, 2020.   
+[35] N. Rhinehart, K. M. Kitani, and P. Vernaza. R2p2: A reparameterized pushforward policy for diverse, precise generative path forecasting. In ECCV, 2018.   
+[36] J. Hong, B. Sapp, and J. Philbin. Rules of the road: Predicting driving behavior with a convolutional model of semantic interactions. In CVPR, 2019.   
+[37] Y. C. Tang and R. Salakhutdinov. Multiple futures prediction. In NeurIPS, 2019.   
+[38] S. Casas, C. Gulino, S. Suo, K. Luo, R. Liao, and R. Urtasun. Implicit latent variable model for scene-consistent motion forecasting. In ECCV, 2020.   
+[39] A. Gupta, J. Johnson, L. Fei-Fei, S. Savarese, and A. Alahi. Social gan: Socially acceptable trajectories with generative adversarial networks. In CVPR, 2018.   
+[40] A. Sadeghian, V. Kosaraju, A. Sadeghian, N. Hirose, H. Rezatofighi, and S. Savarese. Sophie: An attentive gan for predicting paths compliant to social and physical constraints. In CVPR, 2019.   
+[41] T. Zhao, Y. Xu, M. Monfort, W. Choi, C. Baker, Y. Zhao, Y. Wang, and Y. N. Wu. Multi-agent tensor fusion for contextual trajectory prediction. In CVPR, 2019.   
+[42] X. Li, G. Rosman, I. Gilitschenski, C.-I. Vasile, J. A. DeCastro, S. Karaman, and D. Rus. Vehicle trajectory prediction using generative adversarial network with temporal logic syntax tree features. IEEE Robotics and Automation Letters, 2021.   
+[43] N. Deo and M. M. Trivedi. Convolutional social pooling for vehicle trajectory prediction. In CVPR Workshops, 2018.   
+[44] S. Casas, W. Luo, and R. Urtasun. Intentnet: Learning to predict intention from raw sensor data. In CoRL, 2018.   
+[45] Y. Chai, B. Sapp, M. Bansal, and D. Anguelov. Multipath: Multiple probabilistic anchor trajectory hypotheses for behavior prediction. In CoRL, 2019.   
+[46] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. In NeurIPS, 2017.

parse/train/GhMZNcr54zt/GhMZNcr54zt_content_list.json

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+    {
+        "type": "text",
+        "text": "Learning to Predict Vehicle Trajectories with Model-based Planning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Haoran Song Di Luan Wenchao Ding Michael Yu Wang Qifeng Chen The Hong Kong University of Science and Technology {hsongad, wdingae, dluan, mywang, cqf}@ust.hk ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract: Predicting the future trajectories of on-road vehicles is critical for autonomous driving. In this paper, we introduce a novel prediction framework called PRIME, which stands for Prediction with Model-based Planning. Unlike recent prediction works that utilize neural networks to model scene context and produce unconstrained trajectories, PRIME is designed to generate accurate and feasibility-guaranteed future trajectory predictions. PRIME guarantees the trajectory feasibility by exploiting a model-based generator to produce future trajectories under explicit constraints and enables accurate multimodal prediction by utilizing a learning-based evaluator to select future trajectories. We conduct experiments on the large-scale Argoverse Motion Forecasting Benchmark, where PRIME outperforms the state-of-the-art methods in prediction accuracy, feasibility, and robustness under imperfect tracking. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Keywords: trajectory prediction, autonomous driving ",
+        "bbox": [
+            233,
+            449,
+            586,
+            464
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            488,
+            310,
+            506
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the architecture of autonomous driving, prediction serves as the bridging module that reasons future states based on the perceived information from upstream detection and tracking and provides the predicted future states to facilitate the downstream planning. Therefore, making accurate and reasonable trajectory predictions for on-road vehicles is vital for planning safe, efficient, and comfortable motion for self-driving vehicles (SDVs). ",
+        "bbox": [
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+            523,
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+            592
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The widely known challenge of trajectory prediction lies in modeling multi-agent interaction and inferring multimodal future states under driving scenarios. Traditional methods [1, 2, 3, 4, 5] produce motion forecasting by handcrafted rules or models with embedded physical and environmental features, which are insufficient for modeling interactive agents in complex scenes. Learning-based approaches [6, 7, 8], with deep neural networks to fuse scene context information and generate future trajectories, significantly promote the prediction accuracy and dominate the recent motion forecasting benchmarks for autonomous driving [9, 10]. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Despite achieving steady improvement in accuracy, much less attention has been paid to the feasibility and robustness of learning-based trajectory prediction. Indeed, most traffic participants operate under their inherent kinematic constraints (e.g., non-holonomic motion constraints for vehicles) while in compliance with the road structure (e.g., lane connectivity, static obstacles) and semantic information (e.g., traffic lights, speed limits). All these kinematic and environmental constraints explicitly regularize the trajectory space. However, most existing approaches model traffic agents as points and generate future trajectories without imposing constraints. Such constraint-free predictions may be incompliant with kinematic or environmental characteristics and thus give rise to massive uncertainty in the predicted future states. As a result, the downstream planning module would inevitably undergo some extra burdens, and even the “freezing robot problem” [11]. Furthermore, the trajectory predictions typically generated by neural network regression have high dependences on long-term tracking. For some dense driving scenarios where the target would be momently occluded or suddenly appears within the sensing range, tracking results are discontinuous or not accumulated enough. The prediction accuracy would thereby degrade under such imperfect tracking cases. ",
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+    {
+        "type": "image",
+        "img_path": "images/3ef52fc9ad6bfaa96c99ea75f99e746cc7a8cf749dde7edcdbf7c3f79115ea68.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of the PRIME framework. The model-based generator (left) samples feasible future trajectories $\\tau$ for the target agent by taking its real-time state $\\mathbf { \\bar { s } } _ { t a r } ^ { 0 }$ and HD map $\\mathcal { M }$ , while imposing explicit constraints $\\mathcal { C }$ to guarantee trajectory feasibility. The learning-based evaluator (right) receives the feasible trajectory set $\\tau$ and all observed tracks $s$ to model the implicit interactions in scene context and then selects a final set of trajectories $\\mathcal { T } _ { t a r } \\subset \\mathcal { T }$ as the prediction result. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Toward overcoming these challenges, we propose PRIME, a novel architecture for vehicle trajectory prediction, as illustrated in Fig. 1. The core idea is to exploit a model-based motion planner as the prediction generator to produce feasibility-guaranteed future trajectories under explicit physical constraints, together with a deep neural network as the prediction evaluator to enable accurate multimodal prediction by learning complex implicit interactions. To the best of our knowledge, PRIME is the first to incorporate an interpretable motion planner in prediction learning and also the only method that ensures kinematic and environmental feasibility in data-driven trajectory prediction. We conduct experiments on the large-scale Argoverse motion forecasting benchmark and achieves better prediction accuracy over the state-of-the-art. Furthermore, PRIME shows significant superiority in trajectory feasibility guarantee and prediction robustness under imperfect tracking. These attributes would facilitate more flexible and safe motion planning for SDVs. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            505,
+            321,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Prediction and Planning are closely intertwined in autonomous driving [12, 13, 14, 15]. Planning is to generate constraint-compliant trajectory candidates and, after considering safety, comfort, path progress, etc., select the best trajectory for execution by the SDV (ego agent). Prediction facilitates the trajectory selection in planning by inferring future trajectories of the surrounding vehicles (target agents). Their different focuses make the corresponding mainstream solutions diverge. Model-based approaches [16, 17, 18, 19] are preferred in planning due to their interpretability and reliability in computing safe trajectories under explicit constraints. Learning-based methods [6, 7, 20, 21], in contrast, prevail in prediction by utilizing its advantage in modeling implicit interactions. ",
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+    {
+        "type": "text",
+        "text": "Some learning-based prediction works incorporate the goal-directed idea from planning to infer the possible goals and then produce goal-conditioned trajectories [22, 23, 24, 25]. Moreover, the novel planning-prediction-coupled frameworks are introduced to make predictions conditioned on ego intentions [26] or motion plans [27, 28]. Although much emphasis on improving the point-level prediction accuracy, the data-driven frameworks cannot ensure the given constraints are indeed imposed on trajectory generation. Despite DKM [29] embeds the two-axle vehicle kinematics [30] in the output layer to ensure kinematic feasibility, yet still no guarantee on environmental compliance. Inspired by the popular sampling-based paradigm in vehicle motion planning [13, 17], we employ a model-based planner for providing feasibility-guaranteed trajectory sets, and thereby the learningbased part is reduced to evaluate future trajectories. With making the most of model-based planning and learning-based prediction, PRIME handles complex agent-map interactions while fulfilling environmental and kinematic constraints. ",
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+        "type": "text",
+        "text": "Modeling agent-map interactions is fundamental for capturing information from scene map and dynamic agents. The rasterized representation [20, 31, 32] is proposed for learning-based methods, which renders traffic entities into images by different colors or intensities and then encodes rasters with Convolutional Neural Networks. As an alternative, the vectorized representation [33, 34, 25] vectorizes scene context as nodes to construct a graph, which exploits High Definition (HD) maps more explicitly and improves prediction accuracy. By contrast, we address the agent-map modeling with a hierarchical structure that incorporates the lane-association ideas from [3] while extends to learn global scene context. To be specific, our prediction generator acts locally in a planning manner to generate path-conditioned trajectory sets, and the prediction evaluator learns a global understanding of the scene context by aggregating all trajectory and map features. ",
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+        "type": "text",
+        "text": "Generating multimodal trajectories is essential for handling the intrinsic multimodal prediction distributions. Stochastic models are mostly built upon conditional variational autoencoder [7, 35, 36, 37, 38] and generative adversarial network [39, 40, 41, 42], while sampling with uncontrollable latent variables at inference may impede their deployment on safety-critical driving systems. Deterministic models are mainly based on multi-mode trajectory regression [43, 44, 20, 34]. To alleviate mode collapse in prediction learning, recent works decompose the task into classification over anchor trajectories [45] or goal-conditioned trajectories [25], followed by trajectory offset regression. However, no feasibility could be ensured for the regressed results. CoverNet [32] formulates multimodal prediction by directly classifying on a pre-constructed trajectory set, but still, its predictions may violate the agent kinematics or scene constraints. For our framework, leveraging model-based planning as the prediction generator brings superiorities in 1) maintaining multimodal distributions by generating trajectories on diverse reachable paths, 2) ensuring trajectory feasibility by imposing real-time constraints, 3) mitigating the high reliance on long-term tracking, and 4) producing trajectories with continuous information rather than just discrete locations. ",
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+    {
+        "type": "text",
+        "text": "3 Overview ",
+        "text_level": 1,
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+        "text": "Problem formulation. Assume the self-driving vehicle is equipped with detection and tracking modules to provide observed states $s$ of on-road agents $\\mathcal { A }$ and has access to HD map $\\mathcal { M }$ . Let $\\mathbf { s } _ { i } ^ { \\bar { t } }$ denote the state of agent $a _ { i } \\in { \\mathcal { A } }$ at frame $t$ , including position, heading, velocity, turning rate and actor type, and $\\mathbf { s } _ { i } = \\left\\{ \\mathbf { s } _ { i } ^ { - T _ { P } + 1 } , \\mathbf { s } _ { i } ^ { - T _ { P } + 2 } , . . . , \\mathbf { s } _ { i } ^ { 0 } \\right\\}$ denotes the state sequence in the observed period $T _ { P }$ . Given any agent as the prediction target, we denote it by $a _ { t a r }$ and its surrounding agents by $\\mathcal { A } _ { n b r s } = \\{ a _ { 1 } , a _ { 2 } , . . . , a _ { m } \\}$ for differentiation, with their state sequence correspondingly given as $\\mathbf { s } _ { t a r }$ and $S _ { n b r s } = \\{ \\mathbf { s } _ { 1 } , \\mathbf { s } _ { 2 } , . . . , \\mathbf { s } _ { m } \\}$ . Accordingly, $S = \\{ \\mathbf { s } _ { t a r } \\} \\cup S _ { n b r s }$ and $\\mathcal { A } = \\{ a _ { t a r } \\} \\cup \\mathcal { A } _ { n b r s }$ . Our objective is to predict multimodal future trajectories $\\mathcal { T } _ { t a r } = \\{ \\mathcal { T } _ { k } | k = 1 , 2 , . . . , K \\}$ together with corresponding trajectory probability $\\left\\{ p _ { k } \\right\\}$ , where $\\mathcal { T } _ { k }$ denotes a predicted trajectory for target agent $a _ { t a r }$ with continuous state information up to the prediction horizon $T _ { F }$ , $K$ is the number of predicted trajectories. Additionally, it is required to ensure each prediction $\\mathcal { T } _ { k } \\in \\mathcal { T } _ { t a r }$ is feasible with existing constraints $\\mathcal { C }$ , which includes environmental constraints $\\mathcal { C } _ { \\mathcal { M } }$ and the kinematic constraints $\\mathcal { C } _ { t a r }$ . ",
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+    {
+        "type": "text",
+        "text": "Our framework. The two-stage architecture of PRIME consists of model-based generator $G$ and learning-based evaluator $E$ . Concretely, the generator $G : ( \\mathbf { s } _ { t a r } ^ { 0 } , \\mathcal { M } , \\mathcal { C } ) \\mapsto ( \\mathcal { P } , \\mathcal { T } )$ is tasked to produce the trajectory space for the target, which is approximated by a finite set of feasible trajectories $\\tau$ . This part starts with searching a set of reachable paths $\\mathcal { P } = \\{ \\mathcal { P } _ { j } | j = 1 , 2 , . . . , l \\}$ from HD map $\\mathcal { M }$ , which provides reference path for trajectory generation. Then a classical sampling-based planner is utilized to generate trajectory samples under constraints in $\\mathcal { C }$ , and thus provide the feasible future trajectory set $\\mathcal { T } = \\cup _ { j = 1 } ^ { l } \\{ \\bar { \\mathcal { T } _ { j , k } } | k = 1 , 2 , . . . , n _ { j } \\}$ for the target. $\\mathcal { T } _ { j , k }$ denotes the $k$ -th feasible trajectory generated from path $\\mathcal { P } _ { j }$ , and the total number of trajectories is $\\begin{array} { r } { n = \\sum _ { j = 1 } ^ { l } n _ { j } } \\end{array}$ . The model-based actions. The evaluator $E : ( \\mathcal { P } , \\mathcal { T } , \\overset { \\cdot } { S } ) \\mapsto ( \\mathcal { T } _ { t a r } , \\{ p _ { k } \\} )$ takes charge of learning implicit interactions, which features with a dual representation for spatial information and with the attention mechanism to process the varying sizes of $l$ reachable paths, $m$ surrounding agents, and $n$ feasible trajectories. Notably, the evaluator $E$ is reduced to score trajectories and select prediction results $\\bar { \\mathcal { T } _ { t a r } } \\subset \\mathcal { T }$ , rather than regressing position or displacement as most learning-based frameworks do. ",
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+    {
+        "type": "text",
+        "text": "4 Model-based Generator ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 Path Search ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Unlike motion planning, where the reference path for the controllable ego agent is given, the future paths of uncontrollable targets in prediction are unknown. Therefore, we conduct the path search in advance of trajectory generation such that any prediction target could be associated with a set of potential paths ${ \\mathcal { P } } ^ { + }$ . Our path search algorithm $\\dot { G } _ { p a t h } : ( \\mathcal { M } , \\mathbf { s } _ { t a r } ^ { 0 } ) \\mapsto \\mathcal { P } ^ { + }$ is implemented by Depth",
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+    {
+        "type": "text",
+        "text": "First-Search on HD map $\\mathcal { M }$ , with more details described in the supplementary material. Yielding a potential path $\\mathcal { P } _ { j } \\in \\mathcal { P } ^ { \\bar { + } }$ with the centerline coordinates of each lane segment sequence, we expect all the paths of ${ \\bar { \\mathcal { P } } } ^ { + }$ to provide sufficient coverage for the future trajectory space of $a _ { t a r }$ . As no dynamic constraint is imposed in this phase, for target with current state $\\bar { \\bf s } _ { t a r } ^ { 0 }$ , some paths in ${ \\mathcal { P } } ^ { + }$ may not be reachable at frame $t = T _ { F }$ . For instance, a high-speed vehicle cannot change to the opposite lane with a U-turn in few seconds. Such unreachable paths could be recognized in the following trajectory generation phase as no trajectories samples towards them are feasible. Finally, a set of reachable paths ${ \\mathcal { P } } \\subseteq { \\mathcal { P } } ^ { + }$ would be reserved. ",
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+    {
+        "type": "text",
+        "text": "4.2 Trajectory Generation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Given the potential paths in ${ \\mathcal { P } } ^ { + }$ as dynamic references, we choose to generate future trajectories in a planning manner. For SDV, motion planning typically aims at finding an optimal trajectory to connect the current state and a goal state, essentially different from prediction that infers probable trajectories for vehicles with unknown intentions. Despite this, the modelbased generator in planning, which computes a large number of trajectory samples for the follow-up selection, could also be exploited in prediction. ",
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+    {
+        "type": "image",
+        "img_path": "images/d9cefb7d5e84b74f18b436febf332ffc0f7ebc613ab7fb9942c04939003a11eb.jpg",
+        "image_caption": [
+            "Figure 2: Trajectory generation in a Frenet Frame ´ "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "We adopt the trajectory generation phase of Frenet plan- ´ ner [17] in our trajectory generator $G _ { t r a j } ^ { \\bf { \\bar { \\alpha } } } : ( \\mathcal { P } ^ { + } , { \\bf s } _ { t a r } ^ { 0 } , \\mathcal { \\bar { C } } ) \\mapsto $ $\\tau$ . Given a reference path in ${ \\mathcal { P } } ^ { + }$ , a dynamic curvilinear frame is given by the tangential vector $\\vec { t _ { r } }$ and normal vector $\\vec { n } _ { r }$ at a certain point $r$ on the path centerline. The Cartesian coordinate $\\vec { x } = ( x , y )$ could be converted to the Frenet coordinate ´ $( s , d )$ , with the relation ",
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+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/25b84e68494c2ccb1431677e9ae7ae8717086dd2a234c37a715010afc3f8bcc7.jpg",
+        "text": "$$\n\\vec { x } ( s ( t ) , d ( t ) ) = \\vec { r } ( s ( t ) ) + d ( t ) \\vec { n } _ { r } ( s ( t ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "in which $\\vec { r }$ represents a vector pointing from the path root, $s$ and $d$ denote the covered arc length and the perpendicular offset, as illustrated in Fig. 2. The trajectory generation phase first projects the current state $\\mathbf { s } _ { t a r } ^ { 0 }$ onto the Frenet frame and obtains the state tuple ´ $[ s _ { 0 } , \\dot { s } _ { 0 } , \\bar { \\dot { s } } _ { 0 } , d _ { 0 } , \\dot { d } _ { 0 } , \\ddot { d } _ { 0 } ]$ . The longitudinal movement $s ( t )$ and lateral movement $d ( t )$ within the prediction horizon $T _ { F }$ are then generated independently by connecting the fixed start state with different end states using parametric curves to cover different driving maneuvers. Compared with planning, prediction receives less accurate state estimation for targets and does not need fine-grained trajectories. In our trajectory generator, therefore, some high-order state variables are simplified to zero. For longitudinal movement, we sample the target velocity $\\dot { s } ( T _ { F } )  \\dot { s } _ { i }$ in the range of $[ \\mathrm { m a x } ( 0 , \\dot { s } _ { 0 } - \\delta ^ { - } T _ { F } ) , \\mathrm { m i n } ( \\hat { \\dot { s } } , \\dot { s } _ { 0 } + \\delta ^ { + } T _ { F } ) ]$ while leaving $s ( T _ { F } )$ unconstrained. The constants $\\delta ^ { - }$ , $\\delta ^ { + }$ and $\\hat { s }$ are given by considering the actor type of $a _ { t a r }$ and speed limit in $\\mathcal { M }$ , to control the longitudinal velocity $\\dot { s } _ { i }$ in a reasonable range. Each longitudinal trajectory $s _ { i } ( t )$ is calculated using a quartic polynomial ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/7e851b88f93c257e64be33ec621d7298d8ec995152505af5845536f19f697c37.jpg",
+        "text": "$$\n\\mathrm { t . } \\quad [ s ( 0 ) , \\dot { s } ( 0 ) , \\ddot { s } ( 0 ) , \\dot { s } ( T _ { F } ) , \\ddot { s } ( T _ { F } ) ] = [ s _ { 0 } , \\dot { s } _ { 0 } , 0 , \\dot { s } _ { i } , 0 ] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "For lateral movement, we sample the target offset $d ( T _ { F } )  d _ { j }$ in the range of $[ - d _ { l a n e } / 2 , d _ { l a n e } / 2 ]$ where $d _ { l a n e }$ denotes lane width. Each lateral trajectory $d _ { j } ( t )$ is calculated using a quintic polynomial ",
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+        "img_path": "images/a641b872cf9db78d808fb93364693da8ae76a40a9e8d8cd4c88423f1084e6864.jpg",
+        "text": "$$\n[ d ( 0 ) , \\dot { d } ( 0 ) , \\ddot { d } ( 0 ) , d ( T _ { F } ) , \\dot { d } ( T _ { F } ) , \\ddot { d } ( T _ { F } ) ] = [ d _ { 0 } , \\dot { d } _ { 0 } , 0 , d _ { j } , 0 , 0 ] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "With the resulted longitudinal and lateral trajectory set $\\mathcal { T } _ { l o n }$ and $\\mathcal { T } _ { l a t }$ , a full trajectory $\\vec { x } ( s ( t ) , d ( t ) )$ is formed by every combinations in $\\mathcal { T } _ { l o n } \\times \\mathcal { T } _ { l a t }$ . Next, the trajectories incompliant with given constraints $\\mathcal { C }$ would be filtered out. We first project the Frenet coordinates ´ $( s , d )$ back to global coordinates $( x , y )$ to check if the trajectory collides with static obstacles given in $\\mathcal { C } _ { \\mathcal { M } }$ . For collision-free trajectories, their high-order state variables are then converted by the Frenet-Cartesian-transfomation ´ ",
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+        "type": "equation",
+        "img_path": "images/cbd3e7f605738393ca24f955ece0b8647a8fb95b84efbdb19cd2badadf2a46f0.jpg",
+        "text": "$$\n\\ [ s , \\dot { s } , \\ddot { s } , d , \\dot { d } , \\ddot { d } ] \\longmapsto [ \\vec { x } , v , \\kappa , \\alpha ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "to check if any velocity $v$ , acceleration $\\alpha$ or curvature $\\kappa$ exceeds the kinematic constraints given in $\\mathcal { C } _ { t a r }$ . Finally, each reference path $\\mathcal { P } _ { j } \\in \\mathcal { P }$ would generate a set of $n _ { j }$ feasibility-guaranteed future trajectories $\\mathbf { \\bar { \\{ } }  T _ { j , k } | k = 1 , 2 , . . . , n _ { j } \\}$ , and all the trajectories together form an overall trajectory space $\\tau$ . Although the constraints are set conservatively with leaving some margin for the learning-based evaluator, our model-based generator effectively narrows down the trajectory space $\\tau$ by imposing constraints. This unique advantage would set our framework with higher accuracy and robustness. ",
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+    {
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+        "image_caption": [
+            "Figure 3: PRIME framework overview. The model-based generator searches reachable paths $\\mathcal { P }$ through the map and produces feasible future trajectories $\\tau$ . The learning-based evaluator encodes the traffic entities in $( \\mathcal { P } , \\mathcal { T } , \\mathcal { S } )$ and learns implicit interactions in the subsequent attention modules. Afterwards, each future trajectory $\\mathcal { T } _ { j , k }$ could query its track tensor $\\mathbf { X } _ { j } ( \\mathbf { s } _ { t a r } )$ from P2T, interaction tensor $\\mathbf { Y } _ { j } ( \\mathbf { s } _ { t a r } )$ from A2A and future tensor $\\mathbf { Z } ( T _ { j , k } )$ from F2F, and it is scored by feeding the concatenation of these tensors to fully-connected layers. Finally, the evaluator ranks all feasible future trajectories in $\\tau$ by scoring and outputs a final set of $K$ predicted trajectories. "
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+    {
+        "type": "text",
+        "text": "5 Learning-based Evaluator ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 State Representation ",
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+    {
+        "type": "text",
+        "text": "The prediction evaluator aggregates scene context, including observed state sequences $s$ , path set $\\mathcal { P }$ , and future trajectory set $\\tau$ . To make it compatible with most existing trajectory prediction datasets, state sequence $\\mathbf { s } _ { i }$ is reduced to history track in the learning part. Before feeding to the network, we discretize each history track $\\mathbf { s } _ { i }$ and future trajectory $\\mathcal { T } _ { j , k }$ as a location sequence with time interval $\\Delta T$ , and each reference path $\\mathcal { P } _ { j }$ as a waypoint sequence with distance interval $\\Delta D$ . Since the longitudinal movement $s$ and lateral offset $d$ indicate how an agent moves relative to a reference path, they represent the local spatial relationship more straightforwardly. For this reason, we use the Frenet coordinates ´ $( s , d )$ in addition to the Cartesian coordinates $( x , y )$ to form a dual spatial representation. Here, the spatial information $( x , y , s , d )$ of future trajectories in $\\tau$ is given by the generator, while the $( s , d )$ coordinates of history tracks in $s$ are obtained by projecting $( x , y )$ coordinates on the corresponding reference path. Additionally, we adopt the approach of [34] to add a binary mask $b$ to history track’s representation $( x , y , s , d , b )$ to indicate if the location is padded. ",
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+    {
+        "type": "text",
+        "text": "5.2 Encoding Scene Context ",
+        "text_level": 1,
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+        "text": "Prior to capture interrelationships between traffic entities, we first encode each kind of entity in the scene. All encoders are structured with a temporal convolutional layer followed by a long short-term memory (LSTM) layer. The track encoder and the future encoder employ a unidirectional LSTM and make the last hidden state $h ( \\cdot )$ as the motion encoding for history track and future trajectory, while the path encoder uses a bidirectional LSTM and provides the sequence of hidden states $H ( \\cdot )$ as the path spatial encoding. Given the scene context description $( \\mathcal { S } , \\mathcal { P } , \\mathcal { T } )$ , each reachable path $\\mathcal { P } _ { j } \\in \\mathcal { P }$ is encoded as a $H ( \\mathcal { P } _ { j } )$ , where $j = 1 , 2 , . . . , l$ . Considering that the Frenet representation ´ is dependent with the path frame, we encode all history tracks with respect to each reference path $\\mathcal { P } _ { j }$ , which results in $l$ groups of track encodings $\\{ h ( \\mathbf { s } _ { t a r } ) , h ( \\mathbf { s } _ { 1 } ) , . . . , h ( \\mathbf { s } _ { m } ) \\} _ { j }$ . Each future trajectory $\\mathcal { T } _ { j , k } \\in \\mathcal { T }$ is relative to its reference path $\\mathcal { P } _ { j }$ , so all future trajectories are encoded correspondingly to form $l$ groups of future encodings $\\{ h ( \\mathcal { T } _ { j , k } ^ { \\setminus } ) | k = 1 , 2 , . . . , n _ { j } \\}$ . ",
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+    {
+        "type": "text",
+        "text": "5.3 Modeling Interactions ",
+        "text_level": 1,
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+        "text": "Next is to capture the implicit interactions resulted from the static environment and multiple dynamic agents. To fuse the spatial-temporal information from varying numbers of entities in the scene context, the attention mechanism [46] is adopted to construct four modules, namely, path to track (P2T), path to future (P2F), agent to agent (A2A), and future to future (F2F). They are implemented in the same way of scaled dot-product attention and use linear layers for mapping key, query and value. The overall workflow is shown in Fig. 3. In the upper branch, P2T brings the spatial information of each path encoding $\\mathcal { P } _ { j }$ into the corresponding track encodings $\\{ h ( \\mathbf { s } _ { t a r } ) , h ( \\mathbf { s } _ { 1 } ) , . . . , h ( \\mathbf { s } _ { m } ) \\} _ { j }$ . The track encodings are further processed by a self-attention structure in A2A, aiming to capture the interactions between agents in the past time domain. The lower branch lays emphasis on updating the features contained in future encodings. P2F brings the spatial information of path encoding $H ( \\mathcal P _ { j } )$ into the corresponding future encodings $\\{ h ( \\mathcal { T } _ { j , k } ) | k = 1 , 2 , . . . , n _ { j } \\}$ . It is followed by F2F that fuses all future encodings $\\bigcup _ { j = 1 } ^ { l } \\left\\{ h ( \\mathcal T _ { j , k } ) | k = 1 , 2 , . . . , n _ { j } \\right\\}$ from different paths $\\mathcal { P } _ { j } ( j = 1 , 2 , . . . , l )$ using self-attention. In particular, F2F obtains a global understanding of the reachable space given by $\\mathcal { P }$ and, by this way, attempts to further perceive the differences between different trajectories in $\\tau$ . For any future trajectories $\\mathcal { T } _ { j , k } \\in \\mathcal { T }$ , the corresponding track tensor $\\mathbf { X } _ { j } ( \\mathbf { s } _ { t a r } )$ , interaction tensor $\\mathbf { Y } _ { j } ( \\mathbf { s } _ { t a r } )$ and future tensor $\\bar { \\bf Z } ( \\mathcal { T } _ { j , k } )$ could be obtained from P2T, A2A and F2F modules, which are then concatenated together to form a full description $\\mathbf { U } _ { j , k } = \\operatorname { C o n c a t } ( \\mathbf { X } _ { j } ( \\mathbf { s } _ { t a r } ) , \\mathbf { Y } _ { j } ( \\mathbf { s } _ { t a r } ) , \\mathbf { Z } ( { \\mathcal { T } } _ { j , k } ) )$ . ",
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+        "type": "text",
+        "text": "5.4 Trajectory Scoring, Learning, and Inference ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "With $\\mathbf { U } _ { j , k }$ as a full description, we score all the $n$ trajectories $\\mathcal { T } _ { j , k }$ using a maximum entropy model: ",
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+        "type": "equation",
+        "img_path": "images/2169557d652aece8972e5defb0b269604a4d6879817caa1c5a6641edb7df2ce5.jpg",
+        "text": "$$\n\\gamma ( \\mathcal { T } _ { j , k } ) = \\frac { \\exp ( f ( \\mathbf { U } _ { j , k } ) ) } { \\sum _ { j = 1 } ^ { l } \\sum _ { k = 1 } ^ { n _ { j } } \\exp ( f ( \\mathbf { U } _ { j , k } ) ) } ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "in which $f ( \\cdot )$ is implemented using a 3-layer MLP at the end of the evaluation network $E$ . The score label $\\psi ( T _ { j , k } )$ is resulted from calculating the accumulated squared distance error $\\mathrm { D i s t } ( \\cdot )$ between the future trajectory $\\mathcal { T } _ { j , k }$ and the ground truth trajectory $\\mathcal { T } _ { G T }$ within the prediction horizon $T _ { F }$ : ",
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+        "type": "equation",
+        "img_path": "images/1e92cf53c885bf711c527f4a6004115413c3a71bd5a733019c8e90faaec1e9c0.jpg",
+        "text": "$$\n\\psi ( \\mathcal { T } _ { j , k } ) = \\frac { \\exp ( - \\mathrm { D i s t } ( \\mathbf { T } _ { j , k } , \\mathbf { T } _ { G T } ) / \\tau ) } { \\sum _ { j = 1 } ^ { l } \\sum _ { k = 1 } ^ { n _ { j } } \\exp ( - \\mathrm { D i s t } ( \\mathbf { T } _ { j , k } , \\mathbf { T } _ { G T } ) / \\tau ) } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\tau$ is set as a temperature factor. The overall network is trained by cross entropy between the evaluated scores and the labeled scores L = CrossEntropy $( \\gamma ( \\mathcal { T } _ { j , k } ) , \\psi ( \\mathcal { T } _ { j , k } ) )$ . For the inference stage that requires $K$ predicted trajectories, we adopt the non-maximum suppression (NMS) algorithm to remove near-duplicate trajectories, as did in [25]. According to the predicted scores, this method greedily picks trajectories from $\\tau$ and excludes the lower scored trajectory between very close ones. Finally, $K$ trajectories with descending order of scores form the prediction result $\\begin{array} { r l } { \\mathcal { T } _ { t a r } } & { { } = } \\end{array}$ $\\{ \\mathcal { T } _ { i } | k = 1 , 2 , . . . , K \\}$ , and the prediction probability $p _ { k }$ is derived by $p _ { k } = \\gamma ( \\mathcal T _ { k } ) / { \\sum _ { k = 1 } ^ { K } \\gamma ( \\mathcal T _ { k } ) }$ . ",
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+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Dataset. Argoverse [10] is one of the largest publicly available motion forecasting datasets, which contains over 324K data sequence collected from complex urban driving scenarios. The training, validation, and test sets are taken from disjoint parts of the cities. Each sequence lasts for 5 seconds, containing the centroid locations of each tracked agent sampled at $1 0 \\ : \\mathrm { H z }$ , in which one vehicle with relatively complex motion is marked as the prediction target. The objective is to predict its locations 3 seconds into the future, given an initial 2-second observation. ",
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+        "type": "text",
+        "text": "Metrics. We follow the Argoverse evaluation criteria under $K = 1$ and $K = 6$ . Minimum Average Displacement Error $( \\mathrm { m i n A D E } _ { K }$ ) is the average L2 distance error of the best predicted trajectory. Minimum Final Displacement Error $( \\mathrm { m i n F D E } _ { K } )$ ) is the L2 distance error of the best predicted trajectory at the final timestamp. Miss Rate $( \\mathrm { M R } _ { K } )$ is the ratio of scenarios that none of $K$ predicted trajectories has less than 2 meters L2 final displacement error. For multimodal prediction, the probability-based metrics ${ \\mathsf { p } } { \\cdot } { \\mathrm { m i n A D E } } _ { K }$ and ${ \\mathsf { p } } { \\cdot } { \\mathrm { m i n F D E } } _ { K }$ are calculated by adding $- l o g ( p )$ to $\\mathrm { m i n A D E } _ { K }$ and $\\mathrm { m i n F D E } _ { K }$ , where $p$ corresponds to the probability of the best predicted trajectory. In the Argoverse benchmark, best refers to the predicted trajectory with the minimum endpoint error. ",
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+        "type": "text",
+        "text": "Implementation Details. Our implementation is detailed in the supplementary material. Among the state-of-the-art methods, only LaneGCN [34] is open-source. So we use its official implementation and Argoverse baselines [10] for additional tests about trajectory feasibility and imperfect tracking. ",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Comparison with the Argoverse baselines and the state-of-the-art methods on the Argoverse test set. All metrics are lower the better and Miss Rate (MR, $\\mathrm { K } { = } 6$ ) is the key metric. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">K=1</td><td colspan=\"5\">K=6</td><td rowspan=\"2\">Infeasibility (%)</td></tr><tr><td>minADE</td><td>minFDE</td><td>MR (%)</td><td>minADE</td><td>minFDE</td><td>p-minADE</td><td>p-minFDE</td><td>MR (%)</td></tr><tr><td>Argo-CV</td><td>3.53</td><td>7.89</td><td>83.48</td><td>3.39</td><td>7.57</td><td>5.18</td><td>9.36</td><td>81.68</td><td>0.00</td></tr><tr><td>Argo-LSTM+map</td><td>2.96</td><td>6.81</td><td>81.22</td><td>2.34</td><td>5.44</td><td>4.14</td><td>7.23</td><td>69.16</td><td>43.53</td></tr><tr><td>Argo-NN+map</td><td>3.65</td><td>8.12</td><td>83.55</td><td>2.08</td><td>4.03</td><td>3.87</td><td>5.82</td><td>58.21</td><td>86.39</td></tr><tr><td>LaneGCN [34]</td><td>1.71</td><td>3.78</td><td>59.05</td><td>0.87</td><td>1.36</td><td>2.66</td><td>3.16</td><td>16.34</td><td>16.52</td></tr><tr><td>Alibaba-ADLab</td><td>1.97</td><td>4.35</td><td>63.76</td><td>0.92</td><td>1.48</td><td>2.67</td><td>3.23</td><td>15.86</td><td>1</td></tr><tr><td>TNT [25]</td><td>1.78</td><td>3.91</td><td>59.72</td><td>0.94</td><td>1.54</td><td>2.73</td><td>3.33</td><td>13.28</td><td>1</td></tr><tr><td>Jean [21]</td><td>1.74</td><td>4.24</td><td>68.56</td><td>1.00</td><td>1.42</td><td>2.79</td><td>3.21</td><td>13.08</td><td>1</td></tr><tr><td>Poly</td><td>1.70</td><td>3.82</td><td>58.80</td><td>0.87</td><td>1.47</td><td>2.67</td><td>3.26</td><td>12.02</td><td>1</td></tr><tr><td>PRIME (Ours)</td><td>1.91</td><td>3.82</td><td>58.67</td><td>1.22</td><td>1.56</td><td>2.71</td><td>3.05</td><td>11.50</td><td>0.00</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "6.1 Comparison with State-of-the-art ",
+        "text_level": 1,
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+        "text": "We compare our proposed PRIME against the Argoverse baselines [10] (CV, LSTM+map, $\\mathrm { N N + m a p } ,$ ), the top-3 methods in the Argoverse Motion Forecasting Competition 2020 (Jean [21], Poly, Alibaba-ADLab), and the recently published state-of-the-art, LaneGCN [34] and TNT [25]. The performance comparison under Argoverse test set is shown in Table 1. It could be noted that PRIME outperforms all other methods on Miss Rate $K = 6$ ), which is the official ranking metric in Argoverse Competition 2020. It reflects that our method produces accurate multimodal predictions consistently in diverse scenarios. We also achieve the best on the probability-based metric p${ \\mathrm { m i n F D E } } _ { 6 }$ , which would be highly beneficial to weigh between multiple predictions in making decisions and motion plans. From the methods with public details, including LaneGCN [34], TNT [25], and Jean [21], we can find they all perform the learning-based paradigm that utilizes neural networks to model traffic entities and generates future trajectories, while PRIME is the only one that integrates a model-based motion generator into prediction learning. Notably, due to the lack of more detailed on-road information in the Argoverse dataset, such as vehicle types, bounding box, static obstacles, etc., the quantitative result is achieved by imposing general constraints on the model-based generator. This indicates there exists more space to improve when deploying our framework in a real autonomous driving system. Furthermore, handling environmental and dynamic constraints in an interpretable model-based manner and generating trajectories with continuous state information is significant for real-world deployment, which could not be reflected from the evaluation metrics. ",
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+    {
+        "type": "text",
+        "text": "6.2 Ablation Studies ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We ablate the F2F module and Frenet´ representation (denoted by SD) from the complete evaluation network to study their impacts. Table 2 reports the results on the Argoverse validation set. With P2T, P2A, and A2A attention modules capturing the basic interactions between map and agents, the base model performs at the same level with TNT $\\mathbf { \\bar { ( M R _ { 6 } } = 9 \\% }$ reported in [25]), indicat",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Ablation studies on the Argoverse validation set. "
+        ],
+        "table_body": "<table><tr><td>Modules</td><td colspan=\"3\"> p-minADE6 p-minFDE6 MR6(%)</td><td># Params</td></tr><tr><td>Base</td><td>2.33</td><td>2.63</td><td>8.52</td><td>0.69 M</td></tr><tr><td>Base+F2F</td><td>2.31</td><td>2.61</td><td>8.23</td><td>0.72M</td></tr><tr><td>Base+SD</td><td>2.29</td><td>2.58</td><td>7.81</td><td>0.99M</td></tr><tr><td>Base+F2F+SD</td><td>2.29</td><td>2.57</td><td>7.51</td><td>1.02 M</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "ing that these modules are effective in capturing agent-map interactions. As for the Frenet represen- ´ tation providing the local spatial relationship and the F2F module fusing all feasible trajectories to get a global understanding of the reachable space, they both promote the performance. By comparison, the inclusion of Frenet representation is more effective. Additionally, the complete network ´ makes the best performance with only 1.02M parameters, which indicates that separating the function of trajectory generation would reduce the learning burden while achieving high performance. ",
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+    {
+        "type": "text",
+        "text": "6.3 Trajectory Feasibility ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As a typical non-holonomic motion system, vehicles are constrained by inherent kinematic characteristics. So we investigate the ratio of infeasible trajectories produced by prediction models. Since the high-order states (velocity, acceleration, or turning rate) cannot be estimated accurately from discrete locations predicted by common learning-based models, we evaluate the trajectory feasibility only using curvature. By interpolating the predicted positions with pairwise cubic splines, we get the curvature at each point. A trajectory is labeled as infeasible if the curvature $\\kappa > 1 / 3$ (i.e., the minimum turning radius is 3 meters) at any of its points. The ratio of infeasible trajectories is shown in the last column of Table 1. Except for the physical baseline Argo-CV (Constant Velocity), the others, as representatives of the unconstrained learning-based models, have at least $1 6 . 5 \\%$ infeasible predictions. Although we only use curvature for judgment and set a fairly conservative threshold (the minimal turning radius for a regular sedan is around $4 . 5 \\sim 6 . 0$ meters), the infeasible predictions still take up a considerable proportion, which would cause redundant burdens for SDVs to make decisions and plans. By contrast, the model-based generator in our framework can handle any kinematic and environmental constraints, thereby ensuring trajectory feasibility. ",
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+    {
+        "type": "text",
+        "text": "6.4 Imperfect Tracking ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "While most motion forecasting datasets provide tracking results of a certain duration for prediction targets, a self-driving vehicle would inevitably encounter realworld situations where the target is lost in some timestamps or not tracked long enough yet. Then the prediction model is required to robustly handle imperfect tracks rather than being restricted to fixedduration tracking inputs. To let the models (ours, LaneGCN, and $\\mathrm { N N } +$ map baseline) be aware of imperfect tracks, we retrained them by randomly dropping out tracked locations. For processing such inputs while keeping network structures, we pad the locations of dropped timestamps with the nearest tracked location and add a dimension of the binary mask to denote the padded location. The drop rate is randomly sampled from $0 \\sim 0 . 6$ for each data ",
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+        "page_idx": 7
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+    {
+        "type": "image",
+        "img_path": "images/d237d2e351ad915855ab26005ded27ee62528a58ccb22c0b893306afe21cac66.jpg",
+        "image_caption": [
+            "Figure 4: Comparison of prediction robustness under imperfect tracking. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "sequence in training but fixed in testing. The drop rate is pointwise applied, i.e., 0.6 drop rate may drop more or less than $6 0 \\%$ of locations on a track. The last timestamp is always kept to ensure the prediction target could be detected at inference. Fig. 4 shows how the miss rate varies with track drop rate, we observe that our model performs stably, with only $3 . 6 \\%$ relative increase on $\\mathrm { M R _ { 6 } }$ , while the relative increase is around $3 0 \\% \\sim 4 0 \\%$ for the others. The result indicates that the learning-based prediction models rely on long-term tracked results to regress trajectories, while our framework design relieves that to a certain extent, thereby improving the prediction robustness. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We present the prediction framework PRIME that learns to predict vehicle trajectories with modelbased planning. PRIME guarantees the trajectory feasibility by exploiting a model-based generator to produce future trajectories under explicit constraints. It makes accurate trajectory predictions by employing a learning-based evaluator to capture implicit interactions in scene context and select future trajectories by scoring. With the novel framework design, PRIME outperforms the state-ofthe-art in prediction accuracy, feasibility, and robustness. Moreover, the advantages of reasonably regularizing trajectory space, predicting trajectories with continuous state, and the compatibility with on-road information would set our framework highly useful in real system deployment. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This research work was supported in part by The Hong Kong University of Science and Technology under the project “LDSERF-Autonomy Through Learning”. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "[1] D. Helbing and P. Molnar. Social force model for pedestrian dynamics. Physical review E, 51 (5):4282, 1995. [2] A. Houenou, P. Bonnifait, V. Cherfaoui, and W. Yao. Vehicle trajectory prediction based on motion model and maneuver recognition. In IROS, 2013.   \n[3] J. Ziegler et al. Making bertha drive—an autonomous journey on a historic route. IEEE Intelligent transportation systems magazine, 6(2):8–20, 2014. [4] S. Lefevre, D. Vasquez, and C. Laugier. A survey on motion prediction and risk assessment \\` for intelligent vehicles. ROBOMECH journal, 1(1):1–14, 2014. [5] W. Schwarting, A. Pierson, J. Alonso-Mora, S. Karaman, and D. Rus. Social behavior for autonomous vehicles. PNAS, 116(50):24972–24978, 2019. [6] A. Alahi, K. Goel, V. Ramanathan, A. Robicquet, L. Fei-Fei, and S. Savarese. Social lstm: Human trajectory prediction in crowded spaces. In CVPR, 2016.   \n[7] N. Lee, W. Choi, P. Vernaza, C. B. Choy, P. H. Torr, and M. Chandraker. Desire: Distant future prediction in dynamic scenes with interacting agents. In CVPR, 2017. [8] M. Bansal, A. Krizhevsky, and A. Ogale. Chauffeurnet: Learning to drive by imitating the best and synthesizing the worst. In CVPR, 2018.   \n[9] H. Caesar, V. Bankiti, A. H. Lang, S. Vora, V. E. Liong, Q. Xu, A. Krishnan, Y. Pan, G. Baldan, and O. Beijbom. nuscenes: A multimodal dataset for autonomous driving. In CVPR, 2020.   \n[10] M.-F. Chang, J. Lambert, P. Sangkloy, J. Singh, S. Bak, A. Hartnett, D. Wang, P. Carr, S. Lucey, D. Ramanan, et al. Argoverse: 3d tracking and forecasting with rich maps. In CVPR, 2019.   \n[11] P. Trautman and A. Krause. Unfreezing the robot: Navigation in dense, interacting crowds. In IROS, 2010.   \n[12] D. Ferguson, T. M. Howard, and M. Likhachev. Motion planning in urban environments. Journal of Field Robotics, 25(11-12):939–960, 2008.   \n[13] J. Wei, J. M. Dolan, and B. Litkouhi. A prediction-and cost function-based algorithm for robust autonomous freeway driving. In IEEE Intelligent Vehicles Symposium, 2010.   \n[14] C. Katrakazas, M. Quddus, W.-H. Chen, and L. Deka. Real-time motion planning methods for autonomous on-road driving: State-of-the-art and future research directions. Transportation Research Part C: Emerging Technologies, 60:416–442, 2015.   \n[15] W. Schwarting, J. Alonso-Mora, and D. Rus. Planning and decision-making for autonomous vehicles. Annual Review of Control, Robotics, and Autonomous Systems, 2018.   \n[16] M. Pivtoraiko, R. A. Knepper, and A. Kelly. Differentially constrained mobile robot motion planning in state lattices. Journal of Field Robotics, 26(3):308–333, 2009.   \n[17] M. Werling, J. Ziegler, S. Kammel, and S. Thrun. Optimal trajectory generation for dynamic street scenarios in a frenet frame. In ICRA, 2010.   \n[18] M. McNaughton, C. Urmson, J. M. Dolan, and J.-W. Lee. Motion planning for autonomous driving with a conformal spatiotemporal lattice. In ICRA, 2011.   \n[19] E. Galceran, A. G. Cunningham, R. M. Eustice, and E. Olson. Multipolicy decision-making for autonomous driving via changepoint-based behavior prediction. In RSS, 2015.   \n[20] H. Cui, V. Radosavljevic, F.-C. Chou, T.-H. Lin, T. Nguyen, T.-K. Huang, J. Schneider, and N. Djuric. Multimodal trajectory predictions for autonomous driving using deep convolutional networks. In ICRA, 2019.   \n[21] J. Mercat, T. Gilles, N. El Zoghby, G. Sandou, D. Beauvois, and G. P. Gil. Multi-head attention for multi-modal joint vehicle motion forecasting. In ICRA, 2020.   \n[22] B. D. Ziebart, N. Ratliff, G. Gallagher, C. Mertz, K. Peterson, J. A. Bagnell, M. Hebert, A. K. Dey, and S. Srinivasa. Planning-based prediction for pedestrians. In IROS, 2009.   \n[23] E. Rehder, F. Wirth, M. Lauer, and C. Stiller. Pedestrian prediction by planning using deep neural networks. In ICRA, 2018.   \n[24] K. Mangalam, H. Girase, S. Agarwal, K.-H. Lee, E. Adeli, J. Malik, and A. Gaidon. It is not the journey but the destination: Endpoint conditioned trajectory prediction. In ECCV, 2020.   \n[25] H. Zhao, J. Gao, T. Lan, C. Sun, B. Sapp, B. Varadarajan, Y. Shen, Y. Shen, Y. Chai, C. Schmid, et al. Tnt: Target-driven trajectory prediction. In CoRL, 2020.   \n[26] N. Rhinehart, R. McAllister, K. Kitani, and S. Levine. Precog: Prediction conditioned on goals in visual multi-agent settings. In ICCV, 2019.   \n[27] H. Song, W. Ding, Y. Chen, S. Shen, M. Y. Wang, and Q. Chen. Pip: Planning-informed trajectory prediction for autonomous driving. In ECCV, 2020.   \n[28] T. Salzmann, B. Ivanovic, P. Chakravarty, and M. Pavone. Trajectron $^ { + + }$ : Dynamically-feasible trajectory forecasting with heterogeneous data. In ECCV, 2020.   \n[29] H. Cui, T. Nguyen, F.-C. Chou, T.-H. Lin, J. Schneider, D. Bradley, and N. Djuric. Deep kinematic models for kinematically feasible vehicle trajectory predictions. In ICRA, 2020.   \n[30] R. Rajamani. Vehicle dynamics and control. Springer Science & Business Media, 2011.   \n[31] N. Djuric, V. Radosavljevic, H. Cui, T. Nguyen, F.-C. Chou, T.-H. Lin, N. Singh, and J. Schneider. Uncertainty-aware short-term motion prediction of traffic actors for autonomous driving. In WACV, 2020.   \n[32] T. Phan-Minh, E. C. Grigore, F. A. Boulton, O. Beijbom, and E. M. Wolff. Covernet: Multimodal behavior prediction using trajectory sets. In CVPR, 2020.   \n[33] J. Gao, C. Sun, H. Zhao, Y. Shen, D. Anguelov, C. Li, and C. Schmid. Vectornet: Encoding hd maps and agent dynamics from vectorized representation. In CVPR, 2020.   \n[34] M. Liang, B. Yang, R. Hu, Y. Chen, R. Liao, S. Feng, and R. Urtasun. Learning lane graph representations for motion forecasting. In ECCV, 2020.   \n[35] N. Rhinehart, K. M. Kitani, and P. Vernaza. R2p2: A reparameterized pushforward policy for diverse, precise generative path forecasting. In ECCV, 2018.   \n[36] J. Hong, B. Sapp, and J. Philbin. Rules of the road: Predicting driving behavior with a convolutional model of semantic interactions. In CVPR, 2019.   \n[37] Y. C. Tang and R. Salakhutdinov. Multiple futures prediction. In NeurIPS, 2019.   \n[38] S. Casas, C. Gulino, S. Suo, K. Luo, R. Liao, and R. Urtasun. Implicit latent variable model for scene-consistent motion forecasting. In ECCV, 2020.   \n[39] A. Gupta, J. Johnson, L. Fei-Fei, S. Savarese, and A. Alahi. Social gan: Socially acceptable trajectories with generative adversarial networks. In CVPR, 2018.   \n[40] A. Sadeghian, V. Kosaraju, A. Sadeghian, N. Hirose, H. Rezatofighi, and S. Savarese. Sophie: An attentive gan for predicting paths compliant to social and physical constraints. In CVPR, 2019.   \n[41] T. Zhao, Y. Xu, M. Monfort, W. Choi, C. Baker, Y. Zhao, Y. Wang, and Y. N. Wu. Multi-agent tensor fusion for contextual trajectory prediction. In CVPR, 2019.   \n[42] X. Li, G. Rosman, I. Gilitschenski, C.-I. Vasile, J. A. DeCastro, S. Karaman, and D. Rus. Vehicle trajectory prediction using generative adversarial network with temporal logic syntax tree features. IEEE Robotics and Automation Letters, 2021.   \n[43] N. Deo and M. M. Trivedi. Convolutional social pooling for vehicle trajectory prediction. In CVPR Workshops, 2018.   \n[44] S. Casas, W. Luo, and R. Urtasun. Intentnet: Learning to predict intention from raw sensor data. In CoRL, 2018.   \n[45] Y. Chai, B. Sapp, M. Bansal, and D. Anguelov. Multipath: Multiple probabilistic anchor trajectory hypotheses for behavior prediction. In CoRL, 2019.   \n[46] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. In NeurIPS, 2017. ",
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parse/train/HJlQfnCqKX/HJlQfnCqKX.md

@@ -0,0 +1,335 @@
+# PREDICTING THE GENERALIZATION GAP IN DEEP NETWORKS WITH MARGIN DISTRIBUTIONS
+
+Yiding Jiang ∗, Dilip Krishnan, Hossein Mobahi, Samy Bengio Google AI {ydjiang, dilipkay, hmobahi, bengio}@google.com
+
+# ABSTRACT
+
+Recent research has demonstrated that deep neural networks can perfectly fit randomly labeled data, but with very poor accuracy on held out data. This phenomenon indicates that loss functions such as cross-entropy are not a reliable indicator of generalization. This leads to the crucial question of how generalization gap can be predicted from training data and network parameters. In this paper, we propose such a measure, and conduct extensive empirical studies on how well it can predict the generalization gap. Our measure is based on the concept of margin distribution, which are the distances of training points to the decision boundary. We find that it is necessary to use margin distributions at multiple layers of a deep network. On the CIFAR-10 and the CIFAR-100 datasets, our proposed measure correlates very strongly with the generalization gap. In addition, we find the following other factors to be of importance: normalizing margin values for scale independence, using characterizations of margin distribution rather than just the margin (closest distance to decision boundary), and working in log space instead of linear space (effectively using a product of margins rather than a sum). Our measure can be easily applied to feedforward deep networks with any architecture and may point towards new training loss functions that could enable better generalization.
+
+# 1 INTRODUCTION
+
+Generalization, the ability of a classifier to perform well on unseen examples, is a desideratum for progress towards real-world deployment of deep neural networks in domains such as autonomous cars and healthcare. Until recently, it was commonly believed that deep networks generalize well to unseen examples. This was based on empirical evidence about performance on held-out dataset. However, new research has started to question this assumption. Adversarial examples cause networks to misclassify even slightly perturbed images at very high rates (Goodfellow et al., 2014; Papernot et al., 2016). In addition, deep networks can overfit to arbitrarily corrupted data (Zhang et al., 2016), and they are sensitive to small geometric transformations (Azulay & Weiss, 2018; Engstrom et al., 2017). These results have led to the important question about how the generalization gap (difference between train and test accuracy) of a deep network can be predicted using the training data and network parameters. Since in all of the above cases, the training loss is usually very small, it is clear that existing losses such as cross-entropy cannot serve that purpose. It has also been shown (e.g. in Zhang et al. (2016)) that regularizers such as weight decay cannot solve this problem either.
+
+Consequently, a number of recent works (Neyshabur et al., 2017b; Kawaguchi et al., 2017; Bartlett et al., 2017; Poggio et al., 2017; Arora et al., 2018) have started to address this question, proposing generalization bounds based on analyses of network complexity or noise stability properties. However, a thorough empirical assessment of these bounds in terms of how accurately they can predict the generalization gap across various practical settings is not yet available.
+
+In this work, we propose a new quantity for predicting generalization gap of a feedforward neural network. Using the notion of margin in support vector machines (Vapnik, 1995) and extension to deep networks (Elsayed et al., 2018), we develop a measure that shows a strong correlation with generalization gap and significantly outperforms recently developed theoretical bounds on generalization2. This is empirically shown by studying a wide range of deep networks trained on the CIFAR-10 and CIFAR-100 datasets. The measure presented in this paper may be useful for a constructing new loss functions with better generalization. Besides improvement in the prediction of the generalization gap, our work is distinct from recently developed bounds and margin definitions in a number of ways:
+
+![](images/03562c10d8f9584eed1851d990037f1622fd9b6e3ec3e568694582843569898c.jpg)  
+Figure 1: (Best seen as PDF) Density plots (top) and box plots (bottom) of normalized margin of three convolutional networks trained with cross-entropy loss on CIFAR-10 with varying test accuracy: left: $5 5 . 2 \%$ , middle: $7 0 . 6 \%$ , right: $8 5 . 1 \%$ . The left network was trained with $20 \%$ corrupted labels. Train accuracy of all above networks are close to $100 \%$ , and training losses close to zero. The densities and box plots are computed on the training set. Normalized margin distributions are strongly correlated with test accuracy (moving to the right as accuracy increases). This motivates our use of normalized margins at all layers. The (Tukey) box plots show the median and other order statistics (see section 3.2 for details), and motivates their use as features to summarize the distributions.
+
+1. These recently developed bounds are typically functions of weight norms (such as the spectral, Frobenius or various mixed norms). Consequently, they cannot capture variations in network topology that are not reflected in the weight norms, e.g. adding residual connections (He et al., 2016) without careful additional engineering based on the topology changes. Furthermore, some of the bounds require specific treatment for nonlinear activations. Our proposed measure can handle any feedforward deep network.   
+2. Although some of these bounds involve margin, the margin is only defined and measured at the output layer (Bartlett et al., 2017; Neyshabur et al., 2017b). For a deep network, however, margin can be defined at any layer (Elsayed et al., 2018). We show that measuring margin at a single layer does not suffice to capture generalization gap. We argue that it is crucial to use margin information across layers and show that this significantly improves generalization gap prediction.   
+3. The common definition of margin, as used in the recent bounds e.g. Neyshabur et al. (2017b), or as extended to deep networks, is based on the closest distance of the training points to the decision boundary. However, this notion is brittle and sensitive to outliers. In contrast, we adopt margin distribution (Garg et al., 2002; Langford & Shawe-Taylor, 2002; Zhang & Zhou, 2017; 2018) by looking at the entire distribution of distances. This is shown to have far better prediction power.
+
+4. We argue that the direct extension of margin definition to deep networks (Elsayed et al.,   
+2018), although allowing margin to be defined on all layers of the model, is unable to capture
+
+generalization gap without proper normalization. We propose a simple normalization scheme that significantly boosts prediction accuracy.
+
+# 2 RELATED WORK
+
+The recent seminal work of Zhang et al. (2016) has brought into focus the question of how generalization can be measured from training data. They showed that deep networks can easily learn to fit randomly labeled data with extremely high accuracy, but with arbitrarily low generalization capability. This overfitting is not countered by deploying commonly used regularizers. The work of Bartlett et al. (2017) proposes a measure based on the ratio of two quantities: the margin distribution measured at the output layer of the network; and a spectral complexity measure related to the network’s Lipschitz constant. Their normalized margin distribution provides a strong indication of the complexity of the learning task, e.g. the distribution is skewed towards the origin (lower normalized margin) for training with random labels. Neyshabur et al. (2017b;a) also develop bounds based on the product of norms of the weights across layers. Arora et al. (2018) develop bounds based on noise stability properties of networks: more stability implies better generalization. Using these criteria, they are able to derive stronger generalization bounds than previous works.
+
+The margin distribution (specifically, boosting of margins across the training set) has been shown to correspond to generalization properties in the literature on linear models (Schapire et al., 1998): they used this connection to explain the effectiveness of boosting and bagging techniques. Reyzin & Schapire (2006) showed that it was important to control the complexity of a classifier when measuring margin, which calls for some type of normalization. In the linear case (SVM), margin is naturally defined as a function of norm of the weights Vapnik (1995). In the case of deep networks, true margin is intractable. Recent work (Elsayed et al., 2018) proposed a linearization to approximate the margin, and defined the margin at any layer of the network. (Sokolic et al., 2016) provide another approximation to the margin based on the norm of the Jacobian with respect to the input layer. They show that maximizing their approximations to the margin leads to improved generalization. However, their analysis was restricted to margin at the input layer. Poggio et al. (2017) and Liao et al. (2018) propose a normalized cross-entropy measure that correlates well with test loss. Their proposed normalized loss trades off confidence of predictions with stability, which leads to better correlation with test accuracy, leading to a significant lowering of output margin.
+
+# 3 PREDICTION OF GENERALIZATION GAP
+
+In this section, we introduce our margin-based measure. We first explain the construction scheme for obtaining the margin distribution. We then squeeze the distributional information of the margin to a small number of statistics. Finally, we regress these statistics to the value of the generalization gap. We assess prediction quality by applying the learned regression coefficients to predict the generalization gap of unseen models. We will start with providing a motivation for using the margins at the hidden layers which is supported by our empirical findings. SVM owes a large part of its success to the kernel that allows for inner product in a higher and richer feature space. At its crux, the primal kernel SVM problem is separated into the feature extractor and the classifier on the extracted features. We can separate any feed forward network at any given hidden layer and treat the hidden representation as a feature map. From this view, the layers that precede this hidden layer can be treated as a learned feature extractor and then the layers that come after are naturally the classifier. If the margins at the input layers or the output layers play important roles in generalization of the classifier, it is a natural conjecture that the margins at these hidden representations are also important in generalization. In fact, if we ignore the optimization procedure and focus on a converged network, generalization theories developed on the input such as Lv et al. (2019) can be easily extended to the hidden layers or the extracted features.
+
+# 3.1 MARGIN APPROXIMATION
+
+First, we establish some notation. Consider a classification problem with $n$ classes. We assume a classifier $f$ consists of non-linear functions $f _ { i } : \mathcal { X }  \mathbb { R }$ , for $i = 1 , \ldots , n$ that generate a prediction score for classifying the input vector $\mathbf { \boldsymbol { x } } \in \mathcal { X }$ to class $i$ . The predicted label is decided by the class with maximal score, i.e. $i ^ { * } = \arg \operatorname* { m a x } _ { i } f _ { i } ( \pmb { x } _ { }$ . Define the decision boundary for each class pair $( i , j )$
+
+as:
+
+$$
+{ \mathcal { D } } _ { ( i , j ) } \triangleq \{ { \pmb x } | f _ { i } ( { \pmb x } ) = f _ { j } ( { \pmb x } ) \}
+$$
+
+Under this definition, the $l _ { p }$ distance of a point $_ { \textbf { \em x } }$ to the decision boundary $\mathcal { D } _ { ( i , j ) }$ can be expressed as the smallest displacement of the point that results in a score tie:
+
+$$
+d _ { f , \pmb { x } , ( i , j ) } \triangleq \operatorname* { m i n } _ { \pmb { \delta } } \| \pmb { \delta } \| _ { p } \quad \mathrm { s . t . } \quad f _ { i } ( \pmb { x } + \pmb { \delta } ) = f _ { j } ( \pmb { x } + \pmb { \delta } )
+$$
+
+Unlike an SVM, computing the “exact” distance of a point to the decision boundary (Eq. 2) for a deep network is intractable3. In this work, we adopt the approximation scheme from Elsayed et al. (2018) to capture the distance of a point to the decision boundary. This a first-order Taylor approximation to the true distance Eq. 2. Formally, given an input $_ { \textbf { \em x } }$ to a network, denote its representation at the $l ^ { t h }$ layer (the layer activation vector) by $\mathbf { \Delta } _ { \mathbf { \boldsymbol { x } } } l$ . For the input layer, let $l = 0$ and thus $\mathbf { x } ^ { 0 } = \pmb { x }$ . Then for $p = 2$ , the distance of the representation vector $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } } l$ to the decision boundary for class pair $( i , j )$ is given by the following approximation:
+
+$$
+d _ { f , ( i , j ) } ( \pmb { x } ^ { l } ) = \frac { f _ { i } ( \pmb { x } ^ { l } ) - f _ { j } ( \pmb { x } ^ { l } ) } { \| \nabla _ { \pmb { x } ^ { l } } f _ { i } ( \pmb { x } ^ { l } ) - \nabla _ { \pmb { x } ^ { l } } f _ { j } ( \pmb { x } ^ { l } ) \| _ { 2 } }
+$$
+
+Here $f _ { i } ( { \pmb x } ^ { l } )$ represents the output (logit) of the network logit $i$ given $\mathbf { \Delta } _ { \mathbf { \boldsymbol { x } } } l$ . Note that this distance can be positive or negative, denoting whether the training sample is on the “correct” or “wrong” side of the decision boundary respectively. This distance is well defined for all $( i , j )$ pairs, but in this work we assume that $i$ always refers to the ground truth label and $j$ refers to the second highest or highest class (if the point is misclassified). The training data $_ { \textbf { \em x } }$ induces a distribution of distances at each layer $l$ which, following earlier naming convention (Garg et al., 2002; Langford $\&$ Shawe-Taylor, 2002), we refer to as margin distribution (at layer $l$ ). For margin distribution, we only consider distances with positive sign (we ignore all misclassified training points). Such design choice facilitates our empirical analysis when we transform our features (e.g. log transform); further, it has also been suggested that it may be possible to obtain a better generalization bound by only considering the correct examples when the classifier classifies a significant proportion of the training examples correctly, which is usually the case for neural networks (Bartlett, 1998). For completeness, the results with negative margins are included in appendix Sec. 7.
+
+A problem with plain distances and their associated distribution is that they can be trivially boosted without any significant change in the way classifier separates the classes. For example, consider multiplying weights at a layer by a constant and dividing weights in the following layer by the same constant. In a ReLU network, due to positive homogeneity property (Liao et al., 2018), this operation does not affect how the network classifies a point, but it changes the distances to the decision boundary4. To offset the scaling effect, we normalize the margin distribution. Consider margin distribution at some layer $l$ , and let $\mathbf { \Delta } _ { \mathbf { \boldsymbol { x } } _ { k } ^ { l } }$ be the representation vector for training sample $k$ . We compute the variance of each coordinate of $\{ \pmb { x } _ { k } ^ { l } \}$ separately, and then sum these individual variances. This quantity is called total variation of $\mathbf { \Delta } _ { \mathbf { \boldsymbol { x } } } l$ . The square root of this quantity relates to the scale of the distribution. That is, if $\mathbf { \Delta } _ { \mathbf { \boldsymbol { x } } } l$ is scaled by a factor, so is the square root of the total variation. Thus, by dividing distances by the square root of total variation, we can construct a margin distribution invariant to scaling. More concretely, the total variation is computed as:
+
+$$
+\nu ( { \pmb x } ^ { l } ) = \mathrm { t r } \Big ( \frac { 1 } { n } \sum _ { k = 1 } ^ { n } ( { \pmb x } _ { k } ^ { l } - \bar { { \pmb x } } ^ { l } ) ( { \pmb x } _ { k } ^ { l } - \bar { { \pmb x } } ^ { l } ) ^ { T } \Big ) \quad , \quad \bar { { \pmb x } } ^ { l } = \frac { 1 } { n } \sum _ { k = 1 } ^ { n } { \pmb x } _ { k } ^ { l } ,
+$$
+
+i.e. the trace of the empirical covariance matrix of activations. Using the total variation, the normalized margin is specified by:
+
+$$
+\hat { d } _ { f , ( i , j ) } ( \pmb { x } _ { k } ^ { l } ) = \frac { d _ { f , ( i , j ) } ( \pmb { x } _ { k } ^ { l } ) } { \sqrt { \nu ( \pmb { x } ^ { l } ) } }
+$$
+
+While the quantity is relatively primitive and easy to compute, Fig. 1 (top) shows that the normalizedmargin distributions based on Eq. 5 have the desirable effect of becoming heavier tailed and shifting
+
+to the right (increasing margin) as generalization gap decreases. We find that this effect holds across a range of networks trained with different hyper-parameters.
+
+# 3.2 SUMMARIZING THE MARGIN DISTRIBUTION
+
+Instead of working directly with the (normalized) margin distribution, it is easier to analyze a compact signature of that. The moments of a distribution are a natural criterion for this purpose. Perhaps the most standard way of doing this is computing the empirical moments from the samples and then take the $n ^ { t h }$ root of the $n ^ { t h }$ moment. In our experiments, we used the first five moments. However, it is a well-known phenomenon that the estimation of higher order moments based on samples can be unreliable. Therefore, we also consider an alternate way to construct the distribution’s signature. Given a set of distances $\mathcal { D } = \{ \hat { d } _ { m } \} _ { m = 1 } ^ { n }$ , which constitute the margin distribution. We use the median $Q _ { 2 }$ , first quartile $Q _ { 1 }$ and third quartile $Q _ { 3 }$ of the normalized margin distribution, along with the two fences that indicate variability outside the upper and lower quartiles. There are many variations for fences, but in this work, with $I Q R = Q _ { 3 } - Q _ { 1 }$ , we define the upper fence to be $\operatorname* { m a x } ( \{ \hat { d } _ { m } : \hat { d } _ { m } \in$ ${ \mathcal { D } } \wedge { \hat { d } } _ { m } \leq Q _ { 3 } + 1 . 5 I Q R \} ,$ ) and the lower fence to be $\operatorname* { m i n } ( \{ \hat { d } _ { m } : \hat { d } _ { m } \in \mathcal { D } \wedge \hat { d } _ { m } \geq Q _ { 1 } - 1 . 5 I Q R \} )$ (McGill et al., 1978). These 5 statistics form the quartile description that summarizes the normalized margin distribution at a specific layer, as shown in the box plots of Fig. 1. We will later see that both signature representations are able to predict the generalization gap, with the second signature working slightly better.
+
+A number of prior works such as Bartlett et al. (2017), Neyshabur et al. (2017b), Liu et al. (2016), Sun et al. (2015), Sokolic et al. (2016), and Liang et al. (2017) have focused on analyzing or maximizing the margin at either the input or the output layer of a deep network. Since a deep network has many hidden layers with evolving representations, it is not immediately clear which of the layer margins is of importance for improving generalization. Our experiments reveal that margin distribution from all of the layers of the network contribute to prediction of generalization gap. This is also clear from Fig. 1 (top): comparing the input layer (layer 0) margin distributions between the left and right plots, the input layer distribution shifts slightly left, but the other layer distributions shift the other way. For example, if we use quartile signature, we have $5 L$ components in this vector, where $L$ is the total number of layers in the network. We incorporate dependence on all layers simply by concatenating margin signatures of all layers into a single combined vector $\pmb { \theta }$ that we refer to as total signature. Empirically, we found constructing the total signature based on four evenly-spaced layers (input, and 3 hidden layers) sufficiently captures the variation in the distributions and generalization gap, and also makes the signature agnostic to the depth of the network.
+
+# 3.3 EVALUATION METRICS
+
+Our goal is to predict the generalization gap, i.e. the difference between training and test accuracy at the end of training, based on total signature $\pmb \theta$ of a trained model. We use the simplest prediction model, i.e. a linear form $\hat { \boldsymbol g } = \pmb { a } ^ { T } \phi ( \pmb \theta ) + \boldsymbol b$ , where $\pmb { a } \in \mathbb { R } ^ { \mathrm { d i m } ( \pmb { \theta } ) }$ and $b \in \mathbb { R }$ are parameters of the predictor, and $\phi : \mathbb { R }  \mathbb { R }$ is a function applied element-wise to $\pmb { \theta }$ . Specifically, we will explore two choices of $\phi$ : the identity $\phi ( x ) = x$ and entry-wise log transform $\phi ( x ) = \log ( x )$ , which correspond to additive and multiplicative combination of margin statistics respectively. We do not claim this model is the true relation, but rather it is a simple model for prediction; and our results suggest that it is a surprisingly good approximation.
+
+In order to estimate predictor parameters $a , b$ , we generate a pool of $n$ pretrained models (covering different datasets, architectures, regularization schemes, etc. as explained in Sec. 4) each of which gives one instance of the pair $\theta , g$ $\boldsymbol { \mathscr { g } }$ being the generalization gap for that model). We then find $\mathbf { \delta } _ { \mathbf { \alpha } ^ { a , b } }$ by minimizing mean squared error: $\begin{array} { r } { ( { \pmb a } ^ { * } , { \pmb b } ^ { * } ) = \arg \operatorname* { m i n } _ { { \pmb a } , { \boldsymbol b } } \sum _ { i } \left( { \pmb a } ^ { T } \phi ( { \pmb \theta } _ { i } ) + { \boldsymbol b } - g _ { i } \right) ^ { 2 } } \end{array}$ , where $i$ indexes the $i ^ { t h }$ model in the pool. The next step is to assess the prediction quality. We consider two metrics for this. The first metric examines quality of predictions on unseen models. For that, we consider a held-out pool of $m$ models, different from those used to estimate $( a , b )$ , and compute the value of $\hat { g }$ on them via $\hat { g } = { \pmb a } ^ { T } \phi ( { \pmb \theta } ) + b$ . In order to quantify the discrepancy between predicted gap $\hat { g }$ and ground truth gap $g$ we use the notion of coefficient of determination $\dot { ( R ^ { 2 } ) }$ (Glantz et al., 1990):
+
+$$
+R ^ { 2 } = 1 - \frac { \sum _ { j = 1 } ^ { n } ( \hat { g } _ { j } - g _ { j } ) ^ { 2 } } { \sum _ { j = 1 } ^ { n } ( g _ { j } - \frac { 1 } { n } \sum _ { j = 1 } ^ { n } g _ { j } ) ^ { 2 } }
+$$
+
+$R ^ { 2 }$ measures what fraction of data variance can be explained by the linear model5 (it ranges from 0 to 1 on training points but can be outside that range on unseen points). To be precise, we use $\mathbf { k }$ -fold validation to study how the predictor can perform on held out pool of trained deep networks. We use 90/10 split, fit the linear model with the training pool, and measure $R ^ { 2 }$ on the held out pool. The performance is averaged over the 10 splits. Since $\bar { R ^ { 2 } }$ is now not measured on the training pool, it does not suffer from high data dimension and can be negative. In all of our experiments, we use $k = 1 0$ . We provide a subset of residual plots and corresponding univariate F-Test for the experiments in the appendix (Sec. 8). The F-score also indicates how important each individual variable is. The second metric examines how well the model fits based on the provided training pool; it does not require a test pool. To characterize this, we use adjusted $\bar { R } ^ { 2 }$ (Glantz et al., 1990) defined as:
+
+$$
+\bar { R } ^ { 2 } = 1 - ( 1 - R ^ { 2 } ) \frac { n - 1 } { n - \mathrm { d i m } ( \theta ) - 1 } .
+$$
+
+The $\bar { R } ^ { 2 }$ can be negative when the data is non-linear. Note that $\bar { R } ^ { 2 }$ is always smaller than $R ^ { 2 }$ . Intuitively, $\bar { R } ^ { 2 }$ penalizes the model if the number of features is high relative to the available data points. The closer $\bar { R } ^ { 2 }$ is to 1, the better the model fits. Using $\bar { R } ^ { 2 }$ is a simple yet effective method to test the fitness of linear model and is independent of the scale of the target, making it a more illustrative metric than residuals.
+
+# 4 EXPERIMENTS
+
+We tested our measure of generalization gap $\hat { g }$ , along with baseline measures, on a number of deep networks and architectures: nine-layer convolutional networks on CIFAR-10 (10 with input layer), and 32-layer residual networks on both CIFAR-10 and CIFAR-100 datasets. The trained models and relevant Tensorflow Abadi et al. (2016) code to compute margin distributions are released at https://github.com/google-research/google-research/tree/master/demogen
+
+# 4.1 CONVOLUTIONAL NEURAL NETWORKS ON CIFAR-10
+
+Using the CIFAR-10 dataset, we train 216 nine-layer convolutional networks with different settings of hyperparameters and training techniques. We apply weight decay and dropout with different strengths; we use networks with and without batch norm and data augmentation; we change the number of hidden units in the hidden layers. Finally, we also include training with and without corrupted labels, as introduced in Zhang et al. (2016); we use a fixed amount of $2 0 \%$ corruption of the true labels. The accuracy on the test set ranges from $6 0 \%$ to $9 0 . 5 \%$ and the generalization gap ranges from $1 \%$ to $3 5 \%$ . In standard settings, creating neural network models with small generalization gap is difficult; in order to create sufficiently diverse generalization behaviors, we limit some models’ capacities by large weight regularization which decreases generalization gap by lowering the training accuracy. All networks are trained by SGD with momentum. Further details are provided in the supplementary material (Sec. 6).
+
+For each trained network, we compute the depth-agnostic signature of the normalized margin distribution (see Sec. 3). This results in a 20-dimensional signature vector. We estimate the parameters of the linear predictor $( a , b )$ with the log transform $\phi ( x ) \bar { { } } = \log ( x )$ and using the 20-dimensional signature vector $\pmb \theta$ . Fig. 2 (left) shows the resulting scatter plot of the predicted generalization gap $\hat { g }$ and the true generalization gap $g$ . As it can be seen, it is very close to being linear across the range of generalization gaps, and this is also supported by the $\bar { R } ^ { 2 }$ of the model, which is 0.96 (max is 1).
+
+As a first baseline method, we compare against the work of Bartlett et al. (2017) which provides one of the best generalization bounds currently known for deep networks. This work also constructs a margin distribution for the network, but in a different way. To make a fair comparison, we extract the same signature $\pmb { \theta }$ from their margin distribution. Since their margin distribution can only be defined for the output layer, their $\pmb \theta$ is 5-dimensional for any network. The resulting fit is shown in Fig. 2(right). It is clearly a poorer fit than that of our signature, with a significantly lower $\bar { R } ^ { 2 }$ of 0.72.
+
+For a fairer comparison, we also reduced our signature $\pmb \theta$ from 20 dimensions to the best performing 4 dimensions (even one dimension less than what we used for Bartlett’s) by dropping 16 components in our $\pmb \theta$ . This is shown in Fig. 2 (middle) and has a $\bar { R } ^ { 2 }$ of 0.89, which is poorer than our complete
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>CNN+CIFAR10</td><td rowspan=1 colspan=3>ResNet+CIFAR10</td><td rowspan=1 colspan=3>ResNet+CIFAR100</td></tr><tr><td rowspan=1 colspan=1>Experiment Settings</td><td rowspan=1 colspan=1>A R²</td><td rowspan=1 colspan=1>kf R²</td><td rowspan=1 colspan=1>mse</td><td rowspan=1 colspan=1>A R²</td><td rowspan=1 colspan=1>kf R²</td><td rowspan=1 colspan=1>mse</td><td rowspan=1 colspan=1>A R²</td><td rowspan=1 colspan=1>kf R²</td><td rowspan=1 colspan=1>mse</td></tr><tr><td rowspan=1 colspan=1>qrt+log</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>0.90</td><td rowspan=1 colspan=1>1.5</td><td rowspan=1 colspan=1>0.87</td><td rowspan=1 colspan=1>0.81</td><td rowspan=1 colspan=1>0.40</td><td rowspan=1 colspan=1>0.97</td><td rowspan=1 colspan=1>0.96</td><td rowspan=1 colspan=1>1.6</td></tr><tr><td rowspan=1 colspan=1>qrt+log+unnorm</td><td rowspan=1 colspan=1>0.89</td><td rowspan=1 colspan=1>0.86</td><td rowspan=1 colspan=1>2.0</td><td rowspan=1 colspan=1>0.82</td><td rowspan=1 colspan=1>0.74</td><td rowspan=1 colspan=1>0.48</td><td rowspan=1 colspan=1>0.95</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>1.9</td></tr><tr><td rowspan=1 colspan=1>qrt+linear</td><td rowspan=1 colspan=1>0.88</td><td rowspan=1 colspan=1>0.84</td><td rowspan=1 colspan=1>2.2</td><td rowspan=1 colspan=1>0.77</td><td rowspan=1 colspan=1>0.66</td><td rowspan=1 colspan=1>0.54</td><td rowspan=1 colspan=1>0.91</td><td rowspan=1 colspan=1>0.87</td><td rowspan=1 colspan=1>2.6</td></tr><tr><td rowspan=1 colspan=1>sf+log</td><td rowspan=1 colspan=1>0.73</td><td rowspan=1 colspan=1>0.69</td><td rowspan=1 colspan=1>3.5</td><td rowspan=1 colspan=1>0.53</td><td rowspan=1 colspan=1>0.41</td><td rowspan=1 colspan=1>0.80</td><td rowspan=1 colspan=1>0.80</td><td rowspan=1 colspan=1>0.78</td><td rowspan=1 colspan=1>4.0</td></tr><tr><td rowspan=1 colspan=1>sl+log</td><td rowspan=1 colspan=1>0.86</td><td rowspan=1 colspan=1>0.84</td><td rowspan=1 colspan=1>2.2</td><td rowspan=1 colspan=1>0.44</td><td rowspan=1 colspan=1>0.33</td><td rowspan=1 colspan=1>0.87</td><td rowspan=1 colspan=1>0.95</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>1.8</td></tr><tr><td rowspan=1 colspan=1>moment+log</td><td rowspan=1 colspan=1>0.93</td><td rowspan=1 colspan=1>0.87</td><td rowspan=1 colspan=1>1.6</td><td rowspan=1 colspan=1>0.83</td><td rowspan=1 colspan=1>0.74</td><td rowspan=1 colspan=1>0.45</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>0.92</td><td rowspan=1 colspan=1>2.0</td></tr><tr><td rowspan=1 colspan=1>best4+log</td><td rowspan=1 colspan=1>0.89</td><td rowspan=1 colspan=1>0.88</td><td rowspan=1 colspan=1>2.1</td><td rowspan=1 colspan=1>0.54</td><td rowspan=1 colspan=1>0.43</td><td rowspan=1 colspan=1>0.80</td><td rowspan=1 colspan=1>0.93</td><td rowspan=1 colspan=1>0.92</td><td rowspan=1 colspan=1>2.4</td></tr><tr><td rowspan=1 colspan=1>spectral+log</td><td rowspan=1 colspan=1>0.73</td><td rowspan=1 colspan=1>0.70</td><td rowspan=1 colspan=1>3.5</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>Jacobian+log</td><td rowspan=1 colspan=1>0.42</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>5.0</td><td rowspan=1 colspan=1>0.20</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>1.0</td><td rowspan=1 colspan=1>0.47</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>6.0</td></tr><tr><td rowspan=1 colspan=1>LM+linear</td><td rowspan=1 colspan=1>0.35</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>5.2</td><td rowspan=1 colspan=1>0.68</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>0.66</td><td rowspan=1 colspan=1>0.74</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>4.0</td></tr></table>
+
+Table 1: Ablation experiments considering different scenarios (see text for details). The last 3 rows are baselines from: (Bartlett et al., 2017; Sokolic et al., 2016; Elsayed et al., 2018). A indicates adjusted; kf indicates $\mathbf { k }$ -fold; mse indicates mean squared error in $1 0 ^ { - 3 }$ ; N indicates negative.
+
+$\pmb \theta$ but still significantly higher than that of Bartlett et al. (2017). In addition, we considered two other baseline comparisons: Sokolic et al. (2016), where margin at input is defined as a function of the Jacobian of output (logits) with respect to input; and Elsayed et al. (2018) where the linearized approximation to margin is derived (for the same layers where we use our normalized margin approximation).
+
+![](images/df87a04a36071b6827e52d466dfa23c93144269f5868e78f522b5de0c57fb5be.jpg)  
+Figure 2: (Best seen as PDF) Regression models to predict generalization gap. Left: regression model fit in log space for the full 20-dimensional feature space ( $\ddot { R ^ { 2 } } = 0 . 9 4 )$ ; Middle: fit for a subset of only 4 features, 2 each from 2 of the hidden layers $\bar { R } ^ { 2 } = 0 . 8 \dot { 9 } )$ ; Right: fit for features extracted from the normalized margin distribution as used in Bartlett et al. (2017) ( $\bar { R } ^ { 2 } = 0 . 7 2 $ ).
+
+To quantify the effect of the normalization, different layers, feature transformation etc., we conduct a number of ablation experiments with the following configuration: 1. linear/log: Use signature transform of $\phi ( x ) = { \dot { x } }$ or $\phi ( x ) = \log ( x )$ ; 2. sl: Use signature from the single best layer $\mathcal { O } \in \mathbb { R } ^ { 5 } .$ ); 3. sf: Use only the single best statistic from the total signature for all the layers $\pmb { \theta } \in \dot { \mathbb { R } } ^ { 4 }$ , individual layer result can be found in Sec. 7); 4. moment: Use the first 5 moments of the normalized margin distribution as signature instead of quartile statistics $\pmb \theta \in \mathbb { R } ^ { 2 0 }$ (Sec. 3); 5. spectral: Use signature of spectrally normalized margins from Bartlett et al. (2017) $( \pmb \theta \in \mathbb R ^ { 5 } )$ ); 6. qrt: Use all the quartile statistics as total signature $\pmb { \theta } \in \mathbb { R } ^ { 2 0 }$ (Sec. 3); 7. best4: Use the 4 best statistics from the total signature $( \pmb { \theta } \in \mathbb { R } ^ { 4 } )$ ); 8. Jacobian: Use the Jacobian-based margin defined in Eq (39) of Sokolic et al. (2016) $( \pmb \theta \in \mathbb R ^ { 5 } )$ ); 9. LM: Use the large margin loss from Elsayed et al. (2018) at the same four layers where the statistics are measured $( \pmb \theta \in \mathbb R ^ { 4 }$ ); 10. unnorm indicates no normalization. In Table 1, we list the $\bar { R } ^ { 2 }$ from fitting models based on each of these scenarios. We see that, both quartile and moment signatures perform similarly, lending support to our thesis that the margin distribution, rather than the smallest or largest margin, is of importance in the context of generalization.
+
+# 4.2 RESIDUAL NETWORKS ON CIFAR-10
+
+On the CIFAR-10 dataset, we train 216 convolutional networks with residual connections; these networks are 32 layers deep with standard ResNet 32 topology (He et al., 2016). Since it is difficult to train ResNet without activation normalization, we created generalization gap variation with batch normalization (Ioffe & Szegedy, 2015) and group normalization $\mathrm { W u }$ & He, 2018). We further use different initial learning rates. The range of accuracy on the test set ranges from $8 3 \%$ to $9 3 . 5 \%$ and generalization gap from $6 \%$ to $1 3 . 5 \%$ . The residual networks were much deeper, and so we only chose 4 layers for feature-length compatibility with the shallower convoluational networks. This design choice also facilitates ease of analysis and circumvents the dependency on depth of the models. Table 1 shows the $\bar { R } ^ { 2 }$ . Note in the presence of residual connections that use convolution instead of identity and identity blocks that span more than one convolutional layers, it is not immediately clear how to properly apply the bounds of Bartlett et al. (2017) (third from last row) without morphing the topology of the architecture and careful design of reference matrices. As such, we omit them for ResNet. Fig. 3 (left) shows the fit for the resnet models, with $\bar { R } ^ { 2 } = 0 . 8 7$ . Fig. 3 (middle) and Fig. 3 (right) compare the log normalized density plots of a CIFAR-10 resnet and CIFAR-10 CNN. The plots show that the Resnet achieves a better margin distribution, correlated with greater test accuracy, even though it was trained without data augmentation.
+
+![](images/97b0d34c4c39da0138c436c1bd4ceb57b5391d0fd52066495a3876f277762826.jpg)  
+Figure 3: (Best seen as PDF) Left: Regression model fit in log space for the full 20-dimensional feature space for 216 residual networks $\bar { R } ^ { 2 } = 0 . 8 7 )$ on CIFAR-10; Middle: Log density plot of normalized margins of a particular residual network that achieves $9 1 . 7 \%$ test accuracy without data augmentation; Right: Log density plot of normalized margins of a CNN that achieves $8 7 . 2 \%$ with data augmentation. We see that the resnet achieves larger margins, especially at the hidden layers, and this is reflected in the higher test accuracy.
+
+# 4.3 RESNET ON CIFAR-100
+
+On the CIFAR-100 dataset, we trained 324 ResNet-32 with the same variation in hyperparameter settings as for the networks for CIFAR-10 with one additional initial learning rate. The range of accuracy on the test set ranges from $1 2 \%$ to $7 3 \%$ and the generalization gap ranges from $1 \%$ to $7 5 \%$ . Table 1 shows $\bar { R } ^ { 2 }$ for a number of ablation experiments and the full feature set. Fig. 4 (left) shows the fit of predicted and true generalization gaps over the networks $\bar { R } ^ { 2 } = 0 . 9 7 )$ . Fig. 4 (middle) and Fig. 4 (right) compare a CIFAR-100 residual network and a CIFAR-10 residual network with the same architecture and hyperparameters. Under these settings, the CIFAR-100 network achieves $4 4 \%$ test accuracy, whereas CIFAR-10 achieves $6 1 \%$ . The resulting normalized margin density plots clearly reflect the better generalization achieved by CIFAR-10: the densities at all layers are wider and shifted to the right. Thus, the normalized margin distributions reflect the relative “difficulty” of a particular dataset for a given architecture.
+
+# 5 DISCUSSION
+
+We have presented a predictor for generalization gap based on margin distribution in deep networks and conducted extensive experiments to assess it. Our results show that our scheme achieves a high adjusted coefficient of determination (a linear regression predicts generalization gap accurately). Specifically, the predictor uses normalized margin distribution across multiple layers of the network. The best predictor uses quartiles of the distribution combined in multiplicative way (additive in log transform). Compared to the strong baseline of spectral complexity normalized output margin (Bartlett et al., 2017), our scheme exhibits much higher predictive power and can be applied to any feedforward network (including ResNets, unlike generalization bounds such as (Bartlett et al., 2017; Neyshabur et al., 2017b; Arora et al., 2018)). We also find that using hidden layers is crucial for the predictive power. Our findings could be a stepping stone for studying new generalization theories and new loss functions with better generalization properties. We include the results on cross architecture and cross data comparison as well as some final thoughts in Appendix Sec. 9.
+
+![](images/201a3c0415e2f8911996417d27a7e079e5f3a5c7d9a1166ff7cc7a882780befd.jpg)  
+Figure 4: (Best seen as PDF) Left: Regression model fit in log space for the full 20-dimensional feature space for 300 residual networks $\bar { R } ^ { 2 } = 0 . 9 7 )$ on CIFAR-100; Middle: density plot of normalized margins of a particular residual network trained on CIFAR-100 that achieves $4 4 \%$ test accuracy; Right: Density plot of normalized margins of a residual network trained on CIFAR-10 that achieves $6 1 \%$ .
+
+# ACKNOWLEDGMENTS
+
+We are thankful to Gamaleldin Elsayed (Google), Tomer Koren (Google), Sergey Ioffe (Google), Vighnesh Birodkar (Google), Shraman Ray Chaudhuri (Google), Kevin Regan (Google), Behnam Neyshabur (NYU), and Dylan Foster (Cornell), for discussions and helpful feedbacks.
+
+# REFERENCES
+
+Mart´ın Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. Tensorflow: A system for large-scale machine learning. In OSDI, volume 16, pp. 265–283, 2016.
+
+Sanjeev Arora, Rong Ge, Behnam Neyshabur, and Yi Zhang. Stronger generalization bounds for deep nets via a compression approach. arXiv preprint arXiv:1802.05296, 2018.
+
+Aharon Azulay and Yair Weiss. Why do deep convolutional networks generalize so poorly to small image transformations? arXiv preprint arXiv:1805.12177, 2018.
+
+P. L. Bartlett. The sample complexity of pattern classification with neural networks: the size of the weights is more important than the size of the network. IEEE Transactions on Information Theory, 44(2):525–536, March 1998. ISSN 0018-9448. doi: 10.1109/18.661502.
+
+Peter L Bartlett, Dylan J Foster, and Matus J Telgarsky. Spectrally-normalized margin bounds for neural networks. In Advances in Neural Information Processing Systems, pp. 6240–6249, 2017.
+
+Gamaleldin F Elsayed, Dilip Krishnan, Hossein Mobahi, Kevin Regan, and Samy Bengio. Large margin deep networks for classification. arXiv preprint arXiv:1803.05598, 2018.
+
+Logan Engstrom, Dimitris Tsipras, Ludwig Schmidt, and Aleksander Madry. A rotation and a translation suffice: Fooling cnns with simple transformations. arXiv preprint arXiv:1712.02779, 2017.
+
+Ashutosh Garg, Sariel Har-Peled, and Dan Roth. On generalization bounds, projection profile, and margin distribution. In Machine Learning, Proceedings of the Nineteenth International Conference (ICML 2002), University of New South Wales, Sydney, Australia, July 8-12, 2002, pp. 171–178, 2002.
+
+Stanton A Glantz, Bryan K Slinker, and Torsten B Neilands. Primer of applied regression and analysis of variance, volume 309. McGraw-Hill New York, 1990.
+
+Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. arXiv preprint arXiv:1412.6572, 2014.
+
+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.
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015.
+
+Kenji Kawaguchi, Leslie Pack Kaelbling, and Yoshua Bengio. Generalization in deep learning. arXiv preprint arXiv:1710.05468, 2017.
+
+John Langford and John Shawe-Taylor. Pac-bayes margins. In Proceedings of the 15th International Conference on Neural Information Processing Systems, NIPS’02, pp. 439–446, Cambridge, MA, USA, 2002. MIT Press.
+
+Xuezhi Liang, Xiaobo Wang, Zhen Lei, Shengcai Liao, and Stan Z Li. Soft-margin softmax for deep classification. In International Conference on Neural Information Processing, pp. 413–421. Springer, 2017.
+
+Qianli Liao, Brando Miranda, Andrzej Banburski, Jack Hidary, and Tomaso Poggio. A surprising linear relationship predicts test performance in deep networks. arXiv preprint arXiv:1807.09659, 2018.
+
+Min Lin, Qiang Chen, and Shuicheng Yan. Network in network. arXiv preprint arXiv:1312.4400, 2013.
+
+Weiyang Liu, Yandong Wen, Zhiding Yu, and Meng Yang. Large-margin softmax loss for convolutional neural networks. In ICML, pp. 507–516, 2016.
+
+Shen-Huan Lv, Lu Wang, and Zhi-Hua Zhou. Optimal margin distribution network, 2019. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ HygcvsAcFX.
+
+Robert McGill, John W Tukey, and Wayne A Larsen. Variations of box plots. The American Statistician, 32(1):12–16, 1978.
+
+Behnam Neyshabur, Srinadh Bhojanapalli, David McAllester, and Nathan Srebro. A pacbayesian approach to spectrally-normalized margin bounds for neural networks. arXiv preprint arXiv:1707.09564, 2017a.
+
+Behnam Neyshabur, Srinadh Bhojanapalli, David McAllester, and Nati Srebro. Exploring generalization in deep learning. In Advances in Neural Information Processing Systems, pp. 5947–5956, 2017b.
+
+Nicolas Papernot, Patrick McDaniel, Ian Goodfellow, Somesh Jha, Z Berkay Celik, and Ananthram Swami. Practical black-box attacks against deep learning systems using adversarial examples. arXiv preprint arXiv:1602.02697, 2016.
+
+Tomaso Poggio, Kenji Kawaguchi, Qianli Liao, Brando Miranda, Lorenzo Rosasco, Xavier Boix, Jack Hidary, and Hrushikesh Mhaskar. Theory of deep learning iii: explaining the non-overfitting puzzle. arXiv preprint arXiv:1801.00173, 2017.
+
+Lev Reyzin and Robert E Schapire. How boosting the margin can also boost classifier complexity. In Proceedings of the 23rd international conference on Machine learning, pp. 753–760. ACM, 2006.
+
+Robert E Schapire, Yoav Freund, Peter Bartlett, Wee Sun Lee, et al. Boosting the margin: A new explanation for the effectiveness of voting methods. The annals of statistics, 26(5):1651–1686, 1998.
+
+Jure Sokolic, Raja Giryes, Guillermo Sapiro, and Miguel R. D. Rodrigues. Robust large margin deep neural networks. CoRR, abs/1605.08254, 2016. URL http://arxiv.org/abs/1605. 08254.
+
+Shizhao Sun, Wei Chen, Liwei Wang, and Tie-Yan Liu. Large margin deep neural networks: Theory and algorithms. CoRR, abs/1506.05232, 2015. URL http://arxiv.org/abs/ 1506.05232.
+
+Vladimir N. Vapnik. The Nature of Statistical Learning Theory. Springer-Verlag New York, Inc., New York, NY, USA, 1995. ISBN 0-387-94559-8.
+
+Yuxin Wu and Kaiming He. Group normalization. arXiv preprint arXiv:1803.08494, 2018.
+
+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.
+
+Teng Zhang and Zhi-Hua Zhou. Multi-class optimal margin distribution machine. In Doina Precup and Yee Whye Teh (eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp. 4063–4071, International Convention Centre, Sydney, Australia, 06–11 Aug 2017. PMLR. URL http://proceedings.mlr. press/v70/zhang17h.html.
+
+Teng Zhang and Zhi-Hua Zhou. Optimal margin distribution clustering, 2018. URL https: //aaai.org/ocs/index.php/AAAI/AAAI18/paper/view/16895.
+
+# 6 APPENDIX: EXPERIMENTAL DETAILS
+
+# 6.1 CNN $^ +$ CIFAR-10
+
+We use an architecture very similar to Network in Network (Lin et al. (2013)), but we remove all dropout and max pool from the network.
+
+Table 2: Architecture of base CNN model.   
+
+<table><tr><td rowspan=1 colspan=1>Layer Index</td><td rowspan=1 colspan=1>Layer Type</td><td rowspan=1 colspan=1>Output Shape</td></tr><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>Input</td><td rowspan=1 colspan=1>32 × 32×3</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>3 ×3 convolution + stride 2</td><td rowspan=1 colspan=1>16×16×192</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>1 ×1 convolution + stride 1</td><td rowspan=1 colspan=1>16 ×16×192</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>1 ×1 convolution + stride 1</td><td rowspan=1 colspan=1>16 ×16×192</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>3 ×3 convolution + stride 2</td><td rowspan=1 colspan=1>8×8×192</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>1 ×1 convolution+ stride1</td><td rowspan=1 colspan=1>8×8×192</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>1 ×1 convolution + stride 1</td><td rowspan=1 colspan=1>8×8×192</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>3 × 3 convolution + stride 2</td><td rowspan=1 colspan=1>4×4×192</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>1 × 1 convolution + stride 1</td><td rowspan=1 colspan=1>4×4×192</td></tr><tr><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>1 ×1 convolution + stride 1</td><td rowspan=1 colspan=1>4×4×192</td></tr><tr><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>4 × 4 convolution + stride 1</td><td rowspan=1 colspan=1>1×1×10</td></tr></table>
+
+To create generalization gap in this model, we make the following modification to the base architecture:
+
+1. Use channel size of 192, 288, and 384 to create different width   
+2. Train with and without batch norm at all convolutional layers   
+3. Apply dropout at layer 3 and 6 with $p = 0 . 0 , 0 . 2 , 0 . 5$   
+4. Apply $l _ { 2 }$ regularization with $\lambda = 0 . 0 , 0 . 0 0 1 , 0 . 0 0 5$   
+5. Train with and without data augmentation with random cropping, flipping and shifting   
+6. Train each configuration twice
+
+In total this gives us $3 \times 2 \times 3 \times 3 \times 2 \times 2 = 2 1 6$ different network architectures. The models are trained with SGD with momentum $\alpha = 0 . 9$ ) at minibatch size of 128 and intial learning rate of 0.01. All networks are trained for 380 epoch with $1 0 \times$ learning rate decay at interval of 100 epoch.
+
+# 6.2 RESNET $^ { 3 2 + }$ CIFAR-10
+
+For this experiments, we use the standard ResNet 32 architectures. We consider down sampling to the marker of a stage, so there are in total 3 stages in the ResNet 32 architecture. To create generalization gap in this model, we make the following modifications to the architecture:
+
+1. Use network width that are $1 \times , 2 \times , 4 \times$ wider in number of channels.   
+2. Train with batch norm or group norm (Wu & He, 2018)   
+3. Train with initial learning rate of 0.01, 0.001   
+4. Apply $l _ { 2 }$ regularization with $\lambda = 0 . 0 , 0 . 0 2 , 0 . 0 0 2$   
+5. Trian with and without data augmentation with random cropping, flipping and shifting   
+6. Train each configuration 3 times
+
+In total this gives us $3 \times 2 \times 2 \times 3 \times 2 \times 3 = 2 1 6$ different network architectures. The models are trained with SGD with momentum $\dot { \alpha } = 0 . 9$ ) at minibatch size of 128. All networks are trained for 380 epoch with $1 0 \times$ learning rate decay at interval of 100 epoch.
+
+# 6.3 RESNET $^ { 3 2 + }$ CIFAR-100
+
+For this experiments, we use the standard ResNet 32 architectures. We consider down sampling to the marker of a stage, so there are in total 3 stages in the ResNet 32 architecture. To create generalization gap in this model, we make the following modifications to the architecture:
+
+1. Use network width that are $1 \times , 2 \times , 4 \times$ wider in number of channels.   
+2. Train with batch norm or group norm (Wu & He, 2018)   
+3. Train with initial learning rate of $0 . 1 , 0 . 0 1 , 0 . 0 0 1$   
+4. Apply $l _ { 2 }$ regularization with $\lambda = 0 . 0 , 0 . 0 2 , 0 . 0 0 2$   
+5. Trian with and without data augmentation with random cropping, flipping and shifting   
+6. Train each configuration 3 times
+
+In total this gives us $3 \times 2 \times 3 \times 3 \times 2 \times 3 = 3 2 4$ different network architectures. The models are trained with SGD with momentum $\alpha = 0 . 9$ ) at minibatch size of 128. All networks are trained for 380 epoch with $1 0 \times$ learning rate decay at interval of 100 epoch.
+
+# 7 APPENDIX: MORE REGRESSION RESULTS
+
+# 7.1 ANALYSIS WITH NEGATIVE MARGINS
+
+The last two rows contain the results of including the negative margins and regress against both the gap (generalization gap) and against acc (test accuracy). We see that including when negative margin is included, it is in general easier to predict the accuracy of the models rather than the gap itself. For convenience, we have reproduced Table 1.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>CNN+CIFAR10</td><td rowspan=1 colspan=3>ResNet+CIFAR10</td><td rowspan=1 colspan=3>ResNet+CIFAR100</td></tr><tr><td rowspan=1 colspan=1>Experiment Settings</td><td rowspan=1 colspan=1>AR</td><td rowspan=1 colspan=1>kf R²</td><td rowspan=1 colspan=1>mse</td><td rowspan=1 colspan=1>AR</td><td rowspan=1 colspan=1>kf R²</td><td rowspan=1 colspan=1>mse</td><td rowspan=1 colspan=1>A R²</td><td rowspan=1 colspan=1>kf R²</td><td rowspan=1 colspan=1>mse</td></tr><tr><td rowspan=1 colspan=1>qrt+log</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>0.90</td><td rowspan=1 colspan=1>1.5</td><td rowspan=1 colspan=1>0.87</td><td rowspan=1 colspan=1>0.81</td><td rowspan=1 colspan=1>0.40</td><td rowspan=1 colspan=1>0.97</td><td rowspan=1 colspan=1>0.96</td><td rowspan=1 colspan=1>1.6</td></tr><tr><td rowspan=1 colspan=1>qrt+log+unnorm</td><td rowspan=1 colspan=1>0.89</td><td rowspan=1 colspan=1>0.86</td><td rowspan=1 colspan=1>2.0</td><td rowspan=1 colspan=1>0.82</td><td rowspan=1 colspan=1>0.74</td><td rowspan=1 colspan=1>0.48</td><td rowspan=1 colspan=1>0.95</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>1.9</td></tr><tr><td rowspan=1 colspan=1>qrt+linear</td><td rowspan=1 colspan=1>0.88</td><td rowspan=1 colspan=1>0.84</td><td rowspan=1 colspan=1>2.2</td><td rowspan=1 colspan=1>0.77</td><td rowspan=1 colspan=1>0.66</td><td rowspan=1 colspan=1>0.54</td><td rowspan=1 colspan=1>0.91</td><td rowspan=1 colspan=1>0.87</td><td rowspan=1 colspan=1>2.6</td></tr><tr><td rowspan=1 colspan=1>sf+log</td><td rowspan=1 colspan=1>0.73</td><td rowspan=1 colspan=1>0.69</td><td rowspan=1 colspan=1>3.5</td><td rowspan=1 colspan=1>0.53</td><td rowspan=1 colspan=1>0.41</td><td rowspan=1 colspan=1>0.80</td><td rowspan=1 colspan=1>0.80</td><td rowspan=1 colspan=1>0.78</td><td rowspan=1 colspan=1>4.0</td></tr><tr><td rowspan=1 colspan=1>sl+log</td><td rowspan=1 colspan=1>0.86</td><td rowspan=1 colspan=1>0.84</td><td rowspan=1 colspan=1>2.2</td><td rowspan=1 colspan=1>0.44</td><td rowspan=1 colspan=1>0.33</td><td rowspan=1 colspan=1>0.87</td><td rowspan=1 colspan=1>0.95</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>1.8</td></tr><tr><td rowspan=1 colspan=1>moment+log</td><td rowspan=1 colspan=1>0.93</td><td rowspan=1 colspan=1>0.87</td><td rowspan=1 colspan=1>1.6</td><td rowspan=1 colspan=1>0.83</td><td rowspan=1 colspan=1>0.74</td><td rowspan=1 colspan=1>0.45</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>0.92</td><td rowspan=1 colspan=1>2.0</td></tr><tr><td rowspan=1 colspan=1>best4+log</td><td rowspan=1 colspan=1>0.89</td><td rowspan=1 colspan=1>0.88</td><td rowspan=1 colspan=1>2.1</td><td rowspan=1 colspan=1>0.54</td><td rowspan=1 colspan=1>0.43</td><td rowspan=1 colspan=1>0.80</td><td rowspan=1 colspan=1>0.93</td><td rowspan=1 colspan=1>0.92</td><td rowspan=1 colspan=1>2.4</td></tr><tr><td rowspan=1 colspan=1>spectral+log</td><td rowspan=1 colspan=1>0.73</td><td rowspan=1 colspan=1>0.70</td><td rowspan=1 colspan=1>3.5</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>Jacobian+log</td><td rowspan=1 colspan=1>0.42</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>5.0</td><td rowspan=1 colspan=1>0.20</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>1.0</td><td rowspan=1 colspan=1>0.47</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>6.0</td></tr><tr><td rowspan=1 colspan=1>LM+linear</td><td rowspan=1 colspan=1>0.35</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>5.2</td><td rowspan=1 colspan=1>0.68</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>0.66</td><td rowspan=1 colspan=1>0.74</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>4.0</td></tr><tr><td rowspan=1 colspan=1>qrt+linear+gap</td><td rowspan=1 colspan=1>0.90</td><td rowspan=1 colspan=1>0.86</td><td rowspan=1 colspan=1>2.0</td><td rowspan=1 colspan=1>0.71</td><td rowspan=1 colspan=1>0.61</td><td rowspan=1 colspan=1>0.6</td><td rowspan=1 colspan=1>0.93</td><td rowspan=1 colspan=1>0.89</td><td rowspan=1 colspan=1>2.0</td></tr><tr><td rowspan=1 colspan=1>qrt+linear+acc</td><td rowspan=1 colspan=1>0.91</td><td rowspan=1 colspan=1>0.88</td><td rowspan=1 colspan=1>1.6</td><td rowspan=1 colspan=1>0.98</td><td rowspan=1 colspan=1>0.96</td><td rowspan=1 colspan=1>0.2</td><td rowspan=1 colspan=1>0.92</td><td rowspan=1 colspan=1>0.89</td><td rowspan=1 colspan=1>2.0</td></tr></table>
+
+Table 3: Table 1 with two additional row. The second to last row shows prediction of generalization gap when negative margins are used; the last row is for prediction of test accuracy rather than gap. Note that negative margins can only be used with linear features.
+
+# 7.2 ANALYSIS FOR INDIVIDUAL LAYER’S MARGIN DISTRIBUTIONS
+
+This is a comparison of different individual layer’s predictive power by using only the margin distribution at that layer. This results illustrates the importance of margins in the hidden layers.
+
+Table 4: Single layer comparison, all with qrt+log   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=3>CNN+CIFAR10</td><td rowspan=1 colspan=3>ResNet+CIFAR10</td><td rowspan=1 colspan=3>ResNet+CIFAR100</td></tr><tr><td rowspan=1 colspan=1>Experiment Settings</td><td rowspan=1 colspan=1>A R²</td><td rowspan=1 colspan=1>kf R²</td><td rowspan=1 colspan=1>mse</td><td rowspan=1 colspan=1>A R²</td><td rowspan=1 colspan=1>kf R²</td><td rowspan=1 colspan=1>mse</td><td rowspan=1 colspan=1>A R²</td><td rowspan=1 colspan=1>kf R²</td><td rowspan=1 colspan=1>mse</td></tr><tr><td rowspan=1 colspan=1>input</td><td rowspan=1 colspan=1>0.77</td><td rowspan=1 colspan=1>0.73</td><td rowspan=1 colspan=1>3.1</td><td rowspan=1 colspan=1>0.16</td><td rowspan=1 colspan=1>0.02</td><td rowspan=1 colspan=1>1.0</td><td rowspan=1 colspan=1>0.82</td><td rowspan=1 colspan=1>0.81</td><td rowspan=1 colspan=1>3.9</td></tr><tr><td rowspan=1 colspan=1>h1</td><td rowspan=1 colspan=1>0.77</td><td rowspan=1 colspan=1>0.74</td><td rowspan=1 colspan=1>3.1</td><td rowspan=1 colspan=1>0.36</td><td rowspan=1 colspan=1>0.26</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>0.95</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>1.8</td></tr><tr><td rowspan=1 colspan=1>h2</td><td rowspan=1 colspan=1>0.80</td><td rowspan=1 colspan=1>0.77</td><td rowspan=1 colspan=1>3.0</td><td rowspan=1 colspan=1>0.41</td><td rowspan=1 colspan=1>0.31</td><td rowspan=1 colspan=1>0.90</td><td rowspan=1 colspan=1>0.77</td><td rowspan=1 colspan=1>0.73</td><td rowspan=1 colspan=1>4.4</td></tr><tr><td rowspan=1 colspan=1>h3</td><td rowspan=1 colspan=1>0.86</td><td rowspan=1 colspan=1>0.84</td><td rowspan=1 colspan=1>2.2</td><td rowspan=1 colspan=1>0.44</td><td rowspan=1 colspan=1>0.33</td><td rowspan=1 colspan=1>0.87</td><td rowspan=1 colspan=1>0.54</td><td rowspan=1 colspan=1>0.53</td><td rowspan=1 colspan=1>6.3</td></tr><tr><td rowspan=1 colspan=1>alllayer</td><td rowspan=1 colspan=1>0.94</td><td rowspan=1 colspan=1>0.90</td><td rowspan=1 colspan=1>1.5</td><td rowspan=1 colspan=1>0.87</td><td rowspan=1 colspan=1>0.81</td><td rowspan=1 colspan=1>0.40</td><td rowspan=1 colspan=1>0.97</td><td rowspan=1 colspan=1>0.96</td><td rowspan=1 colspan=1>1.6</td></tr></table>
+
+# 8 APPENDIX: FURTHER ANALYSIS OF REGRESSION
+
+# 8.1 CNN $^ +$ CIFAR- $^ { 1 0 + }$ ALL QUARTILE SIGNATURE
+
+![](images/a5e3730957998e5b76910f316281e715a04adc5b1a122ff0ea3c31f772d19bc5.jpg)  
+Figure 5: Residual plots for all explanatory variables, row: h0, h1, h2, h3, column: lower fence, $Q _ { 1 }$ , $Q _ { 2 }$ , $Q _ { 3 }$ , upper fence. lower fence is clipped because distance cannot be smaller than 0. The residual is fairly evenly distributed around 0.   
+Table 5: F score (top) and $\mathsf { p }$ -values (bottom) for all 20 variables. Using $p = 0 . 0 5$ , the null hypotheses are rejected for every variable.
+
+<table><tr><td rowspan=5 colspan=1></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>lower fence</td><td rowspan=1 colspan=2>Q1</td><td rowspan=1 colspan=1>Q2</td><td rowspan=1 colspan=2>Q3</td><td rowspan=1 colspan=2>upper fence</td><td rowspan=5 colspan=1></td></tr><tr><td rowspan=1 colspan=2>h0</td><td rowspan=1 colspan=2>306.40</td><td rowspan=1 colspan=2>114.41</td><td rowspan=1 colspan=1>39.56</td><td rowspan=1 colspan=2>12.54</td><td rowspan=1 colspan=2>5.07</td></tr><tr><td rowspan=1 colspan=2>h1</td><td rowspan=1 colspan=2>286.53</td><td rowspan=1 colspan=2>9.42</td><td rowspan=1 colspan=1>5.16</td><td rowspan=1 colspan=2>17.29</td><td rowspan=1 colspan=2>38.57</td></tr><tr><td rowspan=1 colspan=2>h2</td><td rowspan=1 colspan=2>259.68</td><td rowspan=1 colspan=2>6.95</td><td rowspan=1 colspan=1>77.03</td><td rowspan=1 colspan=2>110.40</td><td rowspan=1 colspan=2>152.20</td></tr><tr><td rowspan=1 colspan=2>h3</td><td rowspan=1 colspan=2>188.59</td><td rowspan=1 colspan=2>10.29</td><td rowspan=1 colspan=1>49.76</td><td rowspan=1 colspan=2>83.40</td><td rowspan=1 colspan=2>143.23</td></tr><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>lower fence</td><td rowspan=1 colspan=2>Q1</td><td rowspan=1 colspan=2>Q2</td><td></td><td rowspan=1 colspan=2>Q3</td><td rowspan=1 colspan=2>upper fence</td></tr><tr><td rowspan=1 colspan=2>h0</td><td rowspan=1 colspan=2>3.59e-43</td><td rowspan=1 colspan=2>1.13e-21</td><td rowspan=1 colspan=2>1.76e-9</td><td></td><td rowspan=1 colspan=2>4.87e-4</td><td rowspan=1 colspan=2>2.52e-2</td></tr><tr><td rowspan=1 colspan=2>h1</td><td rowspan=1 colspan=2>2.34e-41</td><td rowspan=1 colspan=2>2.41e-3</td><td rowspan=1 colspan=2>2.40e-2</td><td></td><td rowspan=1 colspan=2>4.64e-5</td><td rowspan=1 colspan=2>2.70e-09</td></tr><tr><td rowspan=1 colspan=2>h2</td><td rowspan=1 colspan=2>8.76e-39</td><td rowspan=1 colspan=2>8.95e-3</td><td rowspan=1 colspan=2>5.38e-16</td><td></td><td rowspan=1 colspan=2>4.30e-21</td><td rowspan=1 colspan=2>9.12e-27</td></tr><tr><td rowspan=1 colspan=2>h3</td><td rowspan=1 colspan=2>3.40e-31</td><td rowspan=1 colspan=2>1.54e-3</td><td rowspan=1 colspan=2>2.37e-11</td><td></td><td rowspan=1 colspan=2>5.17e-17</td><td rowspan=1 colspan=2>1.31e-25</td></tr></table>
+
+![](images/873f307696948cfb69c17cc91021df2acc85f4c43b54a5e1b08290cdf997adc2.jpg)  
+Figure 6: Residual plots for all explanatory variables, row: h0, h1, h2, h3, column: lower fence, $Q _ { 1 }$ , $Q _ { 2 }$ , $Q _ { 3 }$ , upper fence. lower fence is clipped because distance cannot be smaller than 0. The residual is less evenly distributed as are in other two settings; this fact is well reflected in the cluster along the $\mathbf { X }$ axis and in the $\bar { R } ^ { 2 }$ ; we speculate that this is due to not having diverse enough generalization gap in the models trained to cover the entire space of the “model” unlike in the other two settings.
+
+<table><tr><td rowspan=5 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>lower fence</td><td rowspan=1 colspan=1>Q1</td><td rowspan=1 colspan=1>Q2</td><td rowspan=1 colspan=2>Q3</td><td rowspan=1 colspan=2>upper fence</td><td rowspan=5 colspan=1></td></tr><tr><td rowspan=1 colspan=1>h0</td><td rowspan=1 colspan=2>45.67</td><td rowspan=1 colspan=1>16.67</td><td rowspan=1 colspan=1>6.97</td><td rowspan=1 colspan=2>1.71</td><td rowspan=1 colspan=2>0.68</td></tr><tr><td rowspan=1 colspan=1>h1</td><td rowspan=1 colspan=2>58.84</td><td rowspan=1 colspan=1>88.14</td><td rowspan=1 colspan=1>44.15</td><td rowspan=1 colspan=2>15.59</td><td rowspan=1 colspan=2>9.36</td></tr><tr><td rowspan=1 colspan=1>h2</td><td rowspan=1 colspan=2>60.20</td><td rowspan=1 colspan=1>78.57</td><td rowspan=1 colspan=1>35.76</td><td rowspan=1 colspan=2>12.89</td><td rowspan=1 colspan=2>7.52</td></tr><tr><td rowspan=1 colspan=1>h3</td><td rowspan=1 colspan=2>59.75</td><td rowspan=1 colspan=1>0.27</td><td rowspan=1 colspan=1>1.192</td><td rowspan=1 colspan=2>7.37</td><td rowspan=1 colspan=2>44.22</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>lower fence</td><td rowspan=1 colspan=2>Q1</td><td rowspan=1 colspan=2>Q2</td><td rowspan=1 colspan=2>Q3</td><td rowspan=1 colspan=2>upper fence</td></tr><tr><td rowspan=1 colspan=1>h0</td><td rowspan=1 colspan=2>1.30e-10</td><td rowspan=1 colspan=2>6.25e-5</td><td rowspan=1 colspan=2>8.88e-3</td><td rowspan=1 colspan=2>0.192</td><td rowspan=1 colspan=2>0.40</td></tr><tr><td rowspan=1 colspan=1>h1</td><td rowspan=1 colspan=2>5.94e-13</td><td rowspan=1 colspan=2>9.33e-18</td><td rowspan=1 colspan=2>2.47e-10</td><td rowspan=1 colspan=2>1.06e-4</td><td rowspan=1 colspan=2>2.49e-3</td></tr><tr><td rowspan=1 colspan=1>h2</td><td rowspan=1 colspan=2>3.45e-13</td><td rowspan=1 colspan=2>3.04e-16</td><td rowspan=1 colspan=2>9.21e-9</td><td rowspan=1 colspan=2>4.07e-4</td><td rowspan=1 colspan=2>6.59e-3</td></tr><tr><td rowspan=1 colspan=1>h3</td><td rowspan=1 colspan=2>4.14e-13</td><td rowspan=1 colspan=2>0.60</td><td rowspan=1 colspan=2>0.27</td><td rowspan=1 colspan=2>7.14e-3</td><td rowspan=1 colspan=2>2.4e-10</td></tr></table>
+
+Table 6: F score (top) and $\mathfrak { p }$ -values (bottom) for all 20 variables. Using $p = 0 . 0 5$ , we see that the null hypotheses are not rejected for 4 of the variables. We believe having a more diverse generalization behavior in the study will solve this problem.
+
+![](images/da5d8d2a8d342790d5d5cea5682a85757fdeebda1da3530fa0c5644e0b23b7b8.jpg)  
+Figure 7: Residual plots for all explanatory variables, row: h0, h1, h2, h3, column: lower fence, $Q _ { 1 }$ , $Q _ { 2 }$ , $Q _ { 3 }$ , upper fence. lower fence is clipped because distance cannot be smaller than 0. The residual is fairly evenly distributed around 0. There is one outlier in this experimental setting as shown in the plots.
+
+Table 7: F score (top) and $\mathsf { p }$ -values (bottom) for all 20 variables. Using $p = 0 . 0 5$ , the null hypotheses are rejected for every variable except for h3 upper fence.   
+
+<table><tr><td rowspan=5 colspan=1></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>lower fence</td><td rowspan=1 colspan=1>Q1</td><td rowspan=1 colspan=1>Q2</td><td rowspan=1 colspan=1>Q3</td><td rowspan=1 colspan=2>upper fence</td><td rowspan=5 colspan=1></td></tr><tr><td rowspan=1 colspan=2>h0</td><td rowspan=1 colspan=2>80.12</td><td rowspan=1 colspan=1>8.40</td><td rowspan=1 colspan=1>59.62</td><td rowspan=1 colspan=1>141.56</td><td rowspan=1 colspan=2>248.77</td></tr><tr><td rowspan=1 colspan=2>h1</td><td rowspan=1 colspan=2>65.24</td><td rowspan=1 colspan=1>109.86</td><td rowspan=1 colspan=1>343.57</td><td rowspan=1 colspan=1>700.91</td><td rowspan=1 colspan=2>1124.43</td></tr><tr><td rowspan=1 colspan=2>h2</td><td rowspan=1 colspan=2>99.06</td><td rowspan=1 colspan=1>15.47</td><td rowspan=1 colspan=1>122.36</td><td rowspan=1 colspan=1>305.88</td><td rowspan=1 colspan=2>512.69</td></tr><tr><td rowspan=1 colspan=2>h3</td><td rowspan=1 colspan=2>244.07</td><td rowspan=1 colspan=1>128.45</td><td rowspan=1 colspan=1>65.58</td><td rowspan=1 colspan=1>28.10</td><td rowspan=1 colspan=2>2.34</td></tr><tr><td rowspan=1 colspan=2></td><td rowspan=1 colspan=2>lower fence</td><td rowspan=1 colspan=2>Q1</td><td rowspan=1 colspan=1>Q2</td><td rowspan=1 colspan=2>Q3</td><td rowspan=1 colspan=2>upper fence</td></tr><tr><td rowspan=1 colspan=2>h0</td><td rowspan=1 colspan=2>2.85e-17</td><td rowspan=1 colspan=2>4.00e-3</td><td rowspan=1 colspan=1>1.46e-13</td><td rowspan=1 colspan=2>2.65e-27</td><td rowspan=1 colspan=2>6.32e-42</td></tr><tr><td rowspan=1 colspan=2>h1</td><td rowspan=1 colspan=2>1.34e-14</td><td rowspan=1 colspan=2>2.60e-22</td><td rowspan=1 colspan=1>1.04e-52</td><td rowspan=1 colspan=2>8.12e-83</td><td rowspan=1 colspan=2>4.55e-107</td></tr><tr><td rowspan=1 colspan=2>h2</td><td rowspan=1 colspan=2>1.59e-20</td><td rowspan=1 colspan=2>1.03e-4</td><td rowspan=1 colspan=1>2.53e-24</td><td rowspan=1 colspan=2>1.29e-48</td><td rowspan=1 colspan=2>1.42e-68</td></tr><tr><td rowspan=1 colspan=2>h3</td><td rowspan=1 colspan=2>2.40e-41</td><td rowspan=1 colspan=2>2.78e-25</td><td rowspan=1 colspan=1>1.16e-14</td><td rowspan=1 colspan=2>2.13e-7</td><td rowspan=1 colspan=2>0.127</td></tr></table>
+
+# 9 APPENDIX: SOME OBSERVATIONS AND CONJECTURES
+
+Everythig here uses the full quartile description.
+
+# 9.1 CROSS ARCHITECTURE COMPARISON
+
+We perform regression analysis with both base CNN and ResNet32 on CIFAR-10. The resulting $\bar { R } ^ { 2 } = 0 . 9 1$ and the $\mathbf { k }$ -fold $R ^ { \bar { 2 } } = 0 . 8 8$ . This suggests that the same coefficient works generally well across architectures provided they are trained on the same data. Somehow, the distribution at the 3 locations of the networks are comparable even though the depths are vastly different.
+
+![](images/93c04551a2fb9d1884665e95b0a3660af68f90ac784c77d1461876ef35c2e991.jpg)  
+Figure 8: Scatter Plots
+
+# 9.2 CROSS DATASET COMPARISON
+
+We perform regression analysis with ResNet32 on both CIFAR-10 and CIFAR-100. The resulting $\bar { R } ^ { 2 } = 0 . 9 6$ and the $\mathbf { k }$ -fold $R ^ { \bar { 2 } } = 0 . 9 5$ . This suggests that the same coefficient works generally well across dataset of the same architecture.
+
+![](images/eac6a4cce419b2ae2242bf39139b079e7e15c0c0f2871c84710fc7687d7a94ad.jpg)  
+Figure 9: Scatter Plots
+
+# 9.3 CROSS EVERYTHING
+
+We join all our experiment data and the resulting The resulting $\bar { R } ^ { 2 } = 0 . 9 3$ and the $\mathrm { k }$ -fold $R ^ { 2 } = 0 . 9 3$ .   
+It is perhaps surprising that a set of coefficient exists across both datasets and architectures.
+
+![](images/6ceb7f85c9b353a206b2a4e6cf3ba78f40f013dab77f433c22ed4d4c46c80162.jpg)  
+Figure 10: Scatter Plots
+
+# 9.4 IMPLICATIONS ON GENERALIZATION BOUNDS
+
+We believe that the method developed here can be used in complementary with existing generalization bound; more sophisticated engineering of the predictor may be used to actually verify what kind of function the generalization bound should look like up to constant factor or exponents; it may be helpful for developing generalization bound tighter than the existing ones.

parse/train/WEKfdiZYZi-/WEKfdiZYZi-.md

@@ -0,0 +1,445 @@
+# LSB: Local Self-Balancing MCMC in Discrete Spaces
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Markov Chain Monte Carlo (MCMC) methods are promising solutions to sample   
+2 from target distributions in high dimensions. While MCMC methods enjoy nice   
+3 theoretical properties, like guaranteed convergence and mixing to the true target, in   
+4 practice their sampling efficiency depends on the choice of the proposal distribution   
+5 and the target at hand. This work considers using machine learning to adapt the   
+6 proposal distribution to the target, in order to improve the sampling efficiency in   
+7 the purely discrete domain. Specifically, (i) it proposes a new parametrization for a   
+8 family of proposal distributions, called locally balanced proposals, (ii) it defines   
+9 an objective function based on mutual information and (iii) it devises a learning   
+10 procedure to adapt the parameters of the proposal to the target, thus achieving fast   
+11 convergence and fast mixing. We call the resulting sampler as the Locally Self  
+12 Balancing Sampler (LSB). We show through experimental analysis on the Ising   
+13 model and Bayesian networks that LSB is indeed able to improve the efficiency   
+14 over a state-of-the-art sampler based on locally balanced proposals, thus reducing   
+15 the number of iterations required to converge, while achieving comparable mixing   
+16 performance.
+
+# 17 1 Introduction
+
+18 Sampling from complex and intractable probability distributions is of fundamental importance for   
+19 learning and inference [16]. MCMC algorithms are promising solutions to handle the intractability   
+20 of sampling in high dimensions and they have found numerous applications, in Bayesian statistics   
+21 and statistical physics [17, 23], bioinformatics and computational biology [3, 2] as well as machine   
+22 learning [4, 14, 18].   
+23 Although, MCMC can be applied to sample from any target distribution, in practice its efficiency   
+24 strongly depends on the choice of the proposal. Indeed, common phenomena, like slow convergence   
+25 and slow mixing, are typically the result of wrong choices of the proposal distribution. Therefore, it’s   
+26 extremely important to devise strategies enabling the tuning of the proposal to target distributions [5,   
+27 13]. While there has been a lot of work focusing on designing machine learning-based strategies to   
+28 improve the efficiency of MCMC in the continuous domain [29, 20, 1, 6, 19], less effort has been   
+29 devoted to the discrete counterpart. Most common solutions consider continuous relaxations of   
+30 the problem by using embeddings and then leverage existing sampling strategies designed for the   
+31 continuous case. These strategies are suboptimal, either because they consider limited settings, where   
+32 the target distribution has specific analytic forms [29], or because they make strong assumptions on   
+33 the properties of the embeddings, thus not having guarantees of preserving the topological properties   
+34 of the original discrete domain [20, 1, 6, 19].   
+35 This work focuses on MCMC strategies for the purely discrete domain. Specifically, (i) we introduce   
+36 a new parametrization for a family of proposal distributions, called locally balanced proposals, which   
+37 have been recently studied in [28], (ii) we define an objective function based on mutual information,   
+38 which reduces the distance between the proposal and the target distribution and also reduces the   
+39 statistical dependence between consecutive samples, and (iii) we devise a learning procedure to adapt   
+40 the parameters of the proposal to the target distribution using our objective. The resulting procedure,   
+41 called the Local Self-Balancing sampler (LSB), automatically discovers an optimal locally balanced   
+42 proposal, with the advantage of reducing the amount of user intervention and of improving the overall   
+43 sampling efficiency, both in terms of convergence speed and mixing time.   
+44 We provide some empirical analysis of sampling from the 2D Ising model and from Bayesian   
+45 networks and show that in some cases LSB is able to halve the number of iterations required to   
+46 converge, while achieving similar mixing performance to [28].   
+47 We start by providing some background on locally balanced proposal distributions (Section 2),   
+48 we introduce LSB by describing the parametrizations, the objective and the learning procedure   
+49 (Section 3), we discuss the related work (Section 4) and the experiments (Section 5), and finally we   
+50 conclude by highlighting the main limitations of LSB and possible future directions (Section 6).
+
+# 51 2 Background
+
+52 We consider the problem of sampling from a distribution $p$ with a support defined over a large finite   
+53 discrete sample space $\mathcal { X }$ , i.e. $\begin{array} { r } { p ( \bar { \mathbf { x } } ) \stackrel { - } { = } \tilde { p } ( \mathbf { x } ) / \sum _ { \mathbf { x } ^ { \prime \prime } \in \mathcal { X } } \tilde { p } ( \mathbf { x } ^ { \prime \prime } ) } \end{array}$ , where the normalization term cannot be   
+54 tractably computed and only $\tilde { p }$ can be evaluated. One solution to the problem consists of sampling   
+55 using MCMC [17]. The main idea of MCMC is to sequentially sample from a tractable surrogate   
+56 distribution, alternatively called proposal, and to use an acceptance criterion to ensure that generated   
+57 samples are distributed according to the original distribution. More formally, MCMC is a Markov   
+58 chain with a transition probability of the form:
+
+$$
+T ( \mathbf { x } ^ { \prime } | \mathbf { x } ) = A ( \mathbf { x } ^ { \prime } , \mathbf { x } ) Q ( \mathbf { x } ^ { \prime } | \mathbf { x } ) + 1 [ \mathbf { x } ^ { \prime } = \mathbf { x } ] \sum _ { \mathbf { x } ^ { \prime \prime } \in \mathcal { X } } \big ( 1 - A ( \mathbf { x } ^ { \prime \prime } , \mathbf { x } ) \big ) Q ( \mathbf { x } ^ { \prime \prime } | \mathbf { x } ) \big )
+$$
+
+59 where $Q ( \mathbf { x } ^ { \prime } | \mathbf { x } )$ is the probability of sampling $\mathbf { x } ^ { \prime }$ given a previously sampled $\mathbf { x }$ , namely the proposal   
+60 distribution, $1 [ \cdot ]$ is the Kronecker delta function and $A ( \mathbf { x } ^ { \prime } , \mathbf { x } )$ is the probability of accepting sample   
+61 $\mathbf { x } ^ { \prime }$ given $\mathbf { x }$ , e.g. $\begin{array} { r } { A ( \mathbf { x } ^ { \prime } , \mathbf { x } ) = \operatorname* { m i n } \left. 1 , \frac { \tilde { p } ( \mathbf { x } ^ { \prime } ) Q ( \mathbf { x } | \mathbf { x } ^ { \prime } ) } { \tilde { p } ( \mathbf { x } ) Q ( \mathbf { x } ^ { \prime } | \mathbf { x } ) } \right. } \end{array}$ , p˜(x0)Q(x|x0)p˜(x)Q(x0|x) 	.1 In this work, we consider the family of locally   
+62 informed proposals [28], which are characterized by the following expression:
+
+$$
+Q ( \mathbf { x } ^ { \prime } | \mathbf { x } ) = \frac { g \big ( \frac { \tilde { p } ( \mathbf { x } ^ { \prime } ) } { \tilde { p } ( \mathbf { x } ) } \big ) 1 [ \mathbf { x } ^ { \prime } \in N ( \mathbf { x } ) ] } { Z ( \mathbf { x } ) }
+$$
+
+where 63 $N ( \mathbf { x } )$ is the neighborhood of $\mathbf { x }$ based on the Hamming metric.2
+
+64 Note that the choice of $g$ has a dramatic impact on the performance of the Markov chain, as   
+65 investigated in [28]. In fact, there is a family of functions called balancing functions, satisfying the   
+66 relation $g ( t ) = t g ( 1 / t )$ (for all $t > 0$ ), which have extremely desirable properties, briefly recalled   
+67 hereunder.   
+68 Acceptance rform, namely 69 $\begin{array} { r } { A ( \mathbf { x } ^ { \prime } , \mathbf { x } ) = \operatorname* { m i n } \left. 1 , \frac { Z ( \mathbf { x } ) } { Z ( \mathbf { x } ^ { \prime } ) } \right. } \end{array}$ ty allows to rewrite the acceptan. Therefore, a proper choice of $g$ e function in a simplercan increase the ratio   
+70 between the normalization constants $Z ( \mathbf { x } )$ and $Z ( \mathbf { x } ^ { \prime } )$ with consequent increase of the acceptance   
+71 rate even in high dimensional spaces.   
+72 Detailed balance. Note that for all $\mathbf { x } ^ { \prime } = \mathbf { x }$ , detailed balance trivially holds, viz. $p ( \mathbf { x } ) T ( \mathbf { x } ^ { \prime } | \mathbf { x } ) =$   
+73 $p ( \mathbf { x } ^ { \prime } ) T ( \mathbf { x } | \mathbf { x } ^ { \prime } )$ . In all other cases, detailed balance can be proved, by exploiting the fact that $T ( \mathbf { x } ^ { \prime } | \mathbf { x } ) =$   
+74 $A ( \mathbf { x } ^ { \prime } , \mathbf { x } ) Q ( \mathbf { x } ^ { \prime } | \mathbf { x } )$ and by using the balancing property (see the Supplementary material for more   
+75 details). Detailed balance is a sufficient condition for invariance. Consequently, the target $p$ is a fixed   
+76 point of the Markov chain.   
+77 Ergodicity. Under mild assumptions, we have also ergodicity (we leave more detailed discussion   
+78 to the Supplementary material). In other words, the Markov chain converges to the fixed point $p$   
+79 independently from its initialization.   
+80 Efficiency. The efficiency of MCMC is generally measured in terms of the resulting asymptotic   
+81 variance for sample mean estimators. This is indeed a proxy to quantify the level of correlation   
+82 between samples generated through MCMC. Higher levels of asymptotic variance correspond to   
+83 higher levels of correlation, meaning that the Markov chain produces more dependent samples and   
+84 it is therefore less efficient. Balancing functions are asymptotically optimal according to Peskun   
+85 ordering [28].   
+86 The work in [28] proposes a pool of balancing functions with closed-form expression together with   
+87 some general guidelines to choose one. However, this pool is only a subset of the whole family of   
+88 balancing functions and several cases do not even have an analytical expression. Consequently, it is   
+89 not clear which function to use in order to sample efficiently from the target distribution. Indeed, we   
+90 will see in the experimental section that (i) the optimality of the balancing function depends on the   
+91 target distribution and that (ii) in some cases the optimal balancing function may be different from the   
+92 ones proposed in [28]. In the next sections, we propose a strategy to automatically learn the balancing   
+93 function from the target distribution, thus achieving fast convergence (burn-in) and fast mixing.
+
+# 94 3 LSB: Local Self-Balancing Strategy
+
+We start by introducing two different parametrizations for the family of balancing functions in increasing order of functional expressiveness. Then, we propose an objective criterion based on mutual information, that allows us to learn the parametrization with fast convergence and fast mixing on the target distribution.
+
+# 3.1 Parametrizations
+
+We state the following proposition and then use it to devise the first parametrization.
+
+Proposition 1. Given n balancing functions $\mathbf { g } ( t ) = [ g _ { 1 } ( t ) , \ldots , g _ { n } ( t ) ] ^ { T }$ and a vector of scalar positive weights $\mathbf { w } = [ w _ { 1 } , \ldots , x _ { n } ] ^ { \tilde { T } }$ , the linear combination $\mathbf { \dot { \mathbf { \rho } } } _ { g ( t ) } \doteq \mathbf { w } ^ { T } \mathbf { \dot { \mathbf { g } } } ( t )$ satisfies the balancing property.
+
+$$
+\begin{array} { r } { g ( t ) = \mathbf { w } ^ { T } \mathbf { g } ( t ) = \sum _ { i = 1 } ^ { n } w _ { i } g _ { i } ( t ) = t \sum _ { i = 1 } ^ { n } w _ { i } g _ { i } ( 1 / t ) = t \mathbf { w } ^ { T } \mathbf { g } ( 1 / t ) = t g ( 1 / t ) } \end{array}
+$$
+
+105 Despite its simplicity, the proposition has important implications. First of all, it allows to convert the   
+106 problem of choosing the optimal balancing function into a learning problem. Secondly, the linear   
+107 combination introduces only few parameters (in the experiments we consider $n = 4$ ) and therefore   
+108 the learning problem can be solved in an efficient way. The requirement about positive weights is   
+109 necessary to guarantee ergodicity (see Supplementary material on ergodicity for further details).
+
+The first parametrization (LSB 1) consists of the relations 110 $\begin{array} { r } { w _ { i } = e ^ { \theta _ { i } } / \sum _ { j = 1 } ^ { n } e ^ { \theta _ { j } } } \end{array}$ for all $i = 1 , \ldots , n$ 111 where $\pmb \theta = [ \theta _ { 1 } , \dots , \theta _ { n } ] \in \mathbb { R } ^ { n }$ . Note that the softmax is used to smoothly select one among the $n$ 112 balancing functions. Therefore, we refer to this parametrization as learning to select among existing 113 balancing functions.
+
+The second parametrization (LSB 2) is obtained from the following proposition.
+
+115 Proposition 2. Given $g _ { \pmb { \theta } } ( t ) = \operatorname* { m i n } \{ h _ { \pmb { \theta } } ( t ) , t h _ { \pmb { \theta } } ( 1 / t ) \}$ , where hθ is a universal real valued function   
+116 approximator parameterized by vector $\pmb \theta \in \mathbb { R } ^ { k }$ (e.g. a neural network), and any balancing function \`,   
+117 there always exists $\widetilde { \pmb { \theta } } \in \mathbb { R } ^ { k }$ such that $g _ { \tilde { \pmb { \theta } } } ( t ) = \ell ( t )$ for all $t > 0$ .   
+118 Proof. Given any balancing function $\ell$ , we can always find a $\tilde { \pmb { \theta } }$ such that $h _ { \tilde { \pmb { \theta } } } ( t ) = \ell ( t )$ for all $t > 0$   
+119 (because $h _ { \pmb { \theta } }$ is a universal function approximator). This implies that $h _ { \tilde { \pmb { \theta } } }$ satisfies the balancing property,   
+120 i.e. $h _ { \tilde { \pmb { \theta } } } ( t ) = t h _ { \tilde { \pmb { \theta } } } ( 1 / t )$ for all $t > 0$ . Consequently, by definition of $g _ { \pmb { \theta } }$ , we have that $g _ { \pmb { \theta } } ( t ) \stackrel { - } { = } \bar { h } _ { \tilde { \pmb { \theta } } } ( \bar { t } )$ .   
+121 And finally we can conclude that $g _ { \pmb { \theta } } ( t ) = \ell \bar { ( } t )$ for all $t > 0$ . □   
+122 In theory, LSB 2 parameterizes the whole family of balancing functions and it allows to find the   
+123 optimal one from the whole set.3 In practice, it is better to restrict the analysis only to monotonic   
+124 increasing functions, as we prefer to choose a proposal distribution sampling from regions of higher   
+25 probability mass. In the following proposition, we provide sufficient conditions to ensure the   
+26 monotonicity of LSB 2.   
+127 Proposition 3. Define gθ as in LSB 2. Assume that $h _ { \pmb \theta } ( t )$ is a differentiable and monotonic increasing   
+128 function with respect to variable $t$ , satisfying the relation $\begin{array} { r } { h _ { \pmb \theta } ( 1 / t ) \geq \frac { 1 } { t } \frac { d h _ { \pmb \theta } ( 1 / t ) } { d t } } \end{array}$ for all $t > 0$ . Then,   
+129 gθ is a monotonic increasing function in $t$ .   
+130 Proof. We have that $h _ { \pmb \theta } ( 1 / t ) - \frac { 1 } { t } \frac { d h _ { \pmb \theta } ( 1 / t ) } { d t } \geq 0$ for all $t ~ > ~ 0$ . Also, note that $\begin{array} { r l } { \frac { d t h _ { \pmb \theta } ( 1 / t ) } { d t } } & { = } \end{array}$   
+131 $h _ { \pmb \theta } ( 1 / t ) - \frac { 1 } { t } \frac { d h _ { \pmb \theta } ( 1 / t ) } { d t } > 0$ $t ~ > ~ 0$ 1t dhθ (1/t)dt > 0. Therefore, thθ (1/t) is a monotonic increasing function in t for $t _ { 1 } , t _ { 2 }$ $t _ { 1 } ~ \ge ~ t _ { 2 } ~ > ~ 0$ $g _ { \pmb \theta } ( t _ { 1 } ) \ = \ \operatorname* { m i n } \{ \bar { h } _ { \pmb \theta } ( t _ { 1 } ) , t _ { 1 } h _ { \pmb \theta } ( 1 / t _ { 1 } ) \}$   
+133 and $g _ { \pmb \theta } ( t _ { 2 } ) = \mathrm { m i n } \{ h _ { \pmb \theta } ( t _ { 2 } ) , t _ { 2 } h _ { \pmb \theta } ( 1 / t _ { 2 } ) \}$ . By monotonicity of $h _ { \pmb \theta } ( t )$ and $t h _ { \pmb { \theta } } ( 1 / t )$ , we have that   
+134 $h _ { \pmb \theta } ( t _ { 1 } ) \geq h _ { \pmb \theta } ( t _ { 2 } )$ and $t _ { 1 } h _ { \pmb \theta } ( 1 / t _ { 1 } ) \geq t _ { 2 } h _ { \pmb \theta } ( 1 / t _ { 2 } )$ . Therefore, we can conclude that $g _ { \pmb { \theta } } ( t _ { 1 } ) \geq g _ { \pmb { \theta } } ( t _ { 2 } )$ for   
+135 all $t _ { 1 } , t _ { 2 }$ with $t _ { 1 } \geq t _ { 2 } > 0$ . □   
+136 Therefore, to build $g _ { \pmb { \theta } }$ , we need a monotonic function $h _ { \pmb \theta } ( t )$ . Specifically, we can choose a monotonic   
+137 network [24] and constrain $h _ { \pmb { \theta } }$ to satisfy the condition $\begin{array} { r } { h _ { \pmb \theta } ( 1 / t ) \geq \frac { 1 } { t } \frac { d h _ { \pmb \theta } ( 1 / t ) } { d t } } \end{array}$ for all $t > 0$ (see the   
+138 Supplementary material for further information on how to impose the condition on $h _ { \pmb { \theta } }$ ). In the   
+139 next paragraphs, we propose an objective and a learning strategy to train the parameters of the two   
+140 parametrizations.
+
+# 3.2 Objective and Learning Algorithm
+
+142 The goal here is to devise a criterion to find the balancing function with the fastest speed of conver  
+143 gence/mixing on the target distribution $p$ . Note that the ideal case would be to sample from $p$ in an   
+144 independent fashion. We have already mentioned that this operation is computationally expensive   
+145 due to the intractability of computing the normalizing constant. In our case, we have to consider   
+146 the agnostic case, because the proposal distribution is a tractable surrogate for $p$ . In this regard, we   
+147 define a criterion taking into account the distance from this ideal case. Specifically, we measure the   
+148 distance of the transition probability of the Markov chain in Eq. 1 from the target $p$ and the amount   
+149 of dependence between consecutive samples generated through it. In other words, we introduce the   
+150 following criterion, which is indeed a form of mutual information objective:
+
+$$
+\mathcal { T } _ { \pmb { \theta } } = K L \big \{ p ( \pmb { x } ) \tilde { T } _ { \pmb { \theta } } ( \pmb { x } ^ { \prime } | \pmb { x } ) \| p ( \pmb { x } ) p ( \pmb { x } ^ { \prime } ) \big \} = E _ { p } \big \{ K L \big \{ \tilde { T } _ { \pmb { \theta } } ( \pmb { x } ^ { \prime } | \pmb { x } ) \| p ( \pmb { x } ^ { \prime } ) \big \} \big \}
+$$
+
+151 where $K L$ is the Kullback Leibler divergence and $E _ { p }$ is the expected value of random vector $\pmb { x }$   
+152 distributed according to $p$ and
+
+$$
+\tilde { T } _ { \pmb { \theta } } ( \pmb { x } ^ { \prime } | \pmb { x } ) \doteq \left\{ \begin{array} { c c } { \frac { T _ { \pmb { \theta } } ( \pmb { x } ^ { \prime } | \pmb { x } ) } { Z _ { T } } } & { \forall \pmb { x } ^ { \prime } \neq \pmb { x } } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+153 is a conditional distribution defined over the transition probability $T _ { \pmb \theta } ( \pmb x ^ { \prime } | \pmb x )$ in Eq. 1, where we have   
+154 explicited the dependence on $\pmb \theta$ and we have introduced the normalizing constant $Z _ { T }$ , to ensure   
+155 that $\tilde { T } _ { \pmb { \theta } } ( \pmb { x } ^ { \prime } | \pmb { x } )$ is a proper probability distribution. Note also that $\tilde { T } _ { \pmb { \theta } } ( \pmb { x } ^ { \prime } | \pmb { x } )$ discards all pair of equal   
+156 samples, i.e. ${ \pmb x } ^ { \prime } = { \pmb x }$ , as they are samples rejected by the Markov chain.   
+157 Minimizing Eq. 3 allows us to find the configuration of parameters bringing us "closer" in terms   
+158 of Kullback Leibler to the ideal case, namely $T _ { \pmb \theta } ( \pmb x ^ { \prime } | \pmb x ) = \bar { p } ( \pmb x ^ { \prime } )$ for all ${ \pmb x } ^ { \prime } \neq { \pmb x }$ . The expectation in   
+159 Eq. 3 requires access to samples from $p$ and therefore cannot be computed. Nevertheless, note that   
+160 the KL term in Eq. 3 can be rewritten in an equivalent form (see the Supplementary material for the   
+161 derivation):
+
+$$
+K L \{ \tilde { T } _ { \theta } ( { \pmb x } ^ { \prime } | { \pmb x } ) \| p ( { \pmb x } ^ { \prime } ) \} \propto \mathcal { I } ( \pmb \theta , { \pmb x } ) \doteq E _ { Q _ { \theta _ { 0 } } } \Bigl \{ \omega _ { \pmb \theta , \pmb \theta _ { 0 } } A _ { \theta } ( { \pmb x } ^ { \prime } , { \pmb x } ) \log \frac { A _ { \theta } ( { \pmb x } ^ { \prime } , { \pmb x } ) Q _ { \pmb \theta } ( { \pmb x } ^ { \prime } | { \pmb x } ) } { \tilde { p } ( { \pmb x } ^ { \prime } ) } \Bigr \}
+$$
+
+162 where $\begin{array} { r } { \omega _ { \pmb { \theta } , \pmb { \theta } _ { 0 } } = \frac { Q _ { \pmb { \theta } } ( \pmb { x } ^ { \prime } | \pmb { x } ) } { Q _ { \pmb { \theta } _ { 0 } } ( \pmb { x } ^ { \prime } | \pmb { x } ) } } \end{array}$ and $\pmb { \theta } _ { 0 }$ is the reference parameter vector for the proposal distribution. Alternatively to Eq. 3, we can minimize the following quantity:4163
+
+$$
+J ( \pmb { \theta } ) = E _ { Q _ { i n i t } } \{ \mathcal { I } ( \pmb { \theta } , \pmb { x } ) \} + E _ { Q _ { \pmb { \theta } _ { 0 } } } \{ \mathcal { I } ( \pmb { \theta } , \pmb { x } ) \}
+$$
+
+164 where $Q _ { i n i t }$ is the distribution used at initialization, typically uniform on the support $\mathcal { X }$ . Note that   
+165 the first and the second terms in Eq. 6 encourage fast burn-in and fast mixing, respectively. Therefore,   
+166 $\theta$ can be learnt using the procedure described in Algorithm 1.
+
+Learning rate $\eta = 1 e - 4$ , initial parameter $\pmb { \theta } _ { 0 }$ , burn-in iterations $K$ and batch of samples $N$ .   
+$\{ \pmb { x } _ { 0 } ^ { ( i ) } \} _ { i = 1 } ^ { N } \sim Q _ { i n i t }$ ;   
+while $k { = } I { : } K$ do $\{ \pmb { x } _ { i n i t } ^ { ( i ) } \} _ { i = 1 } ^ { N } \sim Q _ { i n i t }$ ; while $i { = } I { : } N$ do Jb(i)0 ← Estimate of J (θ, x(i)init) using one sample from $Q _ { \pmb { \theta } _ { 0 } } ( \pmb { x } | \pmb { x } _ { i n i t } ^ { ( i ) } )$ ; $\widehat { \mathcal { I } } ^ { ( i ) } \gets$ Estimate of $\mathcal { I } ( \pmb \theta , \pmb x _ { 0 } ^ { ( i ) } )$ using one sample from $Q _ { \pmb \theta _ { 0 } } ( \pmb x | \pmb x _ { 0 } ^ { ( i ) } )$ ; end $\begin{array} { r } { \widehat { J } ( \pmb { \theta } ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \widehat { \mathcal { I } } ^ { ( i ) \prime } + \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \widehat { \mathcal { I } } ^ { ( i ) } \ ; } \end{array}$ $\begin{array} { r } { \pmb { \theta }  \pmb { \theta } - \frac { \eta } { N } \nabla _ { \pmb { \theta } } \widehat { J } ( \pmb { \theta } ) } \end{array}$ ; Update $\{ \pmb { x } _ { 0 } ^ { ( i ) } \} _ { i = 1 } ^ { N }$ with accepted samples ; $\pmb \theta _ { 0 } \gets \pmb \theta$ ;   
+end
+
+# 167 4 Related Work
+
+8 It’s important to devise strategies, which enable the automatic adaption of proposals to target distributions, not only to reduce user intervention, but also to increase the efficiency of MCMC samplers [5, 13]. Recently, there has been a surge of interest in using machine learning and in particular deep learning to learn proposals directly from data, especially in the continuous domain. Here, we provide a brief overview of recent integrations of machine learning and MCMC samplers according to different parametrizations and training objectives.
+
+174 Parametrizations and objectives in the continuous domain. The work in [27] proposes a strategy   
+175 based on block Gibbs sampling, where blocks are large motifs of the underlying probabilistic graphical   
+176 structure. It parameterizes the conditional distributions of each block using mixture density networks   
+177 and trains them using meta-learning on a log-likelihood-based objective. The work in [25] considers   
+178 a global sampling strategy, where the proposal is parameterized by a deep generative model. The   
+179 model is learnt through adversarial training, where a neural discriminator is used to detect whether or   
+180 not generated samples are distributed according to the target distribution. Authors in [10] propose a   
+181 global sampling strategy based on MCMC with auxiliary variables [12]. The proposals are modelled   
+182 as Gaussian distributions parameterized by neural networks and are trained on a variational bound   
+183 of a log-likelihood-based objective. The works in [15, 8] propose a gradient-based MCMC [7, 9],   
+184 where neural models are used to learn the hyperparameters of the equations governing the dynamics   
+185 of the sampler. Different objectives are used during training. In particular, the work in [8] uses   
+186 a log-likelihood based objective, whereas the work in [15] considers the expected squared jump   
+187 distance, namely a tractable proxy for the lag-1 autocorrelation function [21]. The work in [30]   
+188 proposes a global two-stage strategy, which consists of (i) sampling according to a Gaussian proposal   
+189 and (ii) updating its parameters using the first- and second-order statistics computed from a properly   
+190 maintained pool of samples. The parameter update can be equivalently seen as finding the solution   
+191 maximizing a log-likelihood function defined over the pool of samples. Finally, the work in [22]   
+192 extends this last strategy to the case of Gaussian mixture proposals. All these works differ from the   
+193 current one in at least two aspects. Firstly, it is not clear how these parametrizations can be applied   
+194 to sampling in the discrete domain. Secondly, the proposed objectives compute either a distance   
+195 between the proposal distribution and the target one, namely using an adversarial objective or a   
+196 variational bound on the log-likelihood, or a proxy on the correlation between consecutive generated   
+197 samples, namely the expected squared jump distance. Instead, our proposed objective is more general   
+198 in the sense that it allows to (i) reduce the distance between the proposal and the target distribution as   
+199 well as to (ii) reduce the statistical dependence between consecutive samples, as being closely related   
+200 to mutual information.   
+201 Sampling in the discrete domain. Less efforts have been devoted to devise sampling strategies   
+202 for a purely discrete domain. Most of the works consider problem relaxations by embedding   
+203 the discrete domain into a continuous one, applying existing strategies like Hamiltonian Monte   
+204 Carlo [29, 20, 1, 6, 19] on it and then moving back to the original domain. These strategies are   
+205 suboptimal, either because they consider limited settings, where the target distribution has specific   
+206 analytic forms [29], or because they make strong assumptions on the properties of the embeddings,   
+207 thus not preserving the topological properties of the discrete domain [20, 1, 6, 19].5. The work   
+208 in [28] provides an extensive experimental comparison between several discrete sampling strategies,   
+209 including the ones based on embeddings, based on stochastic local search [11] and the Hamming ball   
+210 sampler [26], which can be regarded as a more efficient version of block Gibbs sampling. Notably,   
+211 the sampling strategy based on locally informed proposals and balancing functions proposed in [28]   
+212 can be considered as the current state of the art for discrete MCMC. Our work builds and extends   
+213 upon this sampler by integrating it with a machine learning strategy. To the best of our knowledge,   
+214 this is the first attempt to consider the integration of machine learning and MCMC in the discrete   
+215 setting.
+
+![](images/ddb8a6056fe8518e66f63e0da28045c52cd3f6600642cf00248ba1c0689a390b.jpg)  
+Figure 1: Examples of $\pmb { \alpha }$ in different settings of the Ising model $( 3 0 \times 3 0 )$ ), i.e noisy $\mu = 1 , \sigma = 3$ and clean $\mu = 3 , \sigma = 3$ .
+
+# 16 5 Experiments
+
+Firstly, we analyze samplers’ performance on the 2D Ising model. Then, we perform experiments on additional UAI benchmarks. Code to replicate the experiments in this section is available in the Supplementary material. All experiments are performed on a laptop provided with 4 Intel i5 cores (2 GHz) and 16 GB of RAM memory.
+
+# 5.1 2D Ising Model
+
+The Ising model has been introduced in statistical mechanics in 1920 and it has been applied in several domains since then. In this section, we consider an application to image analysis, where the goal is to segment an image to identify an object from its background. Consider a binary state space $\mathbf { \bar { \mathcal { X } } } = \{ - 1 , 1 \} ^ { V }$ , where $( V , E )$ defines a square lattice graph of the same size of the analyzed image, namely $n \times n$ . For each state configuration $\mathbf { x } = ( x _ { i } ) _ { i \in V } \in \mathcal { X }$ , define a prior distribution
+
+$$
+p _ { p r i o r } ( \mathbf { x } ) \propto \exp \left\{ \lambda \sum _ { ( i , j ) \in E } x _ { i } x _ { j } \right\}
+$$
+
+227 where $\lambda$ is a non-negative scalar used to weight the dependence among neighboring variables in the   
+228 lattice. Then, consider that each pixel $y _ { i }$ is influenced only by the corresponding hidden variable   
+229 $x _ { i }$ and generated according to a Gaussian density with mean $\mu x _ { i }$ and variance $\sigma ^ { \bar { 2 } }$ . Note that each   
+230 variable in the lattice tells whether the corresponding pixel belongs to the object or to the background   
+231 (1 or $^ { - 1 }$ , respectively). The corresponding posterior distribution of a hidden state $\mathbf { x }$ given an observed   
+232 image is defined as follows:
+
+$$
+p ( \mathbf { x } ) = \frac { 1 } { Z } \exp \bigg \{ \sum _ { i \in V } \alpha _ { i } x _ { i } + \lambda \sum _ { ( i , j ) \in E } x _ { i } x _ { j } \bigg \}
+$$
+
+where $\alpha _ { i } = y _ { i } \mu / \sigma ^ { 2 }$ is a coefficient biasing $x _ { i }$ towards either 1 or $- 1$ . Therefore, ${ \pmb { \alpha } } = ( \alpha _ { i } ) _ { i \in V }$ contains information about the observed image. Figure 1 shows two synthetically generated examples of $\pmb { \alpha }$ . Our goal is to analyze the sampling performance on the distribution defined in Eq. 7.
+
+Universally optimal balancing function. We start by comparing the balancing functions proposed�� in [28], namely $g ( t ) = t / ( 1 + t )$ (a.k.a Barker function), $\sqrt { t } , \operatorname* { m i n } \{ 1 , t \}$ and $\operatorname* { m a x } \{ 1 , t \}$ , in order to test
+
+![](images/2778fc2876168809545d3c206b1df0e2b364b56c5a40a3446e0d5adc7156b071.jpg)  
+Figure 2: Samplers’ performance on noisy and clean cases of the Ising model $\left( 3 0 \times 3 0 \right)$ . (a)-(b) are the traceplots for the burn-in phase, (c)-(d) are the autocorrelation functions for the mixing one.
+
+Table 1: Quantitative performance for mixing measured by effective sample size on the noisy and clean cases of the Ising model $\left( 3 0 \times 3 0 \right)$ . max $\{ 1 , t \}$ is performing significantly worse in statistical terms than the other functions.   
+
+<table><tr><td>Setting</td><td>t 1+t</td><td>Vt</td><td>min{1,t}</td><td>max{1,t}</td></tr><tr><td>Noisy</td><td>2.48±0.21</td><td>2.30±0.22</td><td>2.42±0.19</td><td>1.75± 0.17</td></tr><tr><td>Clean</td><td>2.58± 0.73</td><td>1.99 ± 0.43</td><td>2.56± 0.62</td><td>1.26 ± 0.12</td></tr></table>
+
+238 whether there is a universally optimal balancing function among this subset. In particular, we run the   
+239 samplers over two instances of the Ising model, viz. a setting with independent $( \lambda , \mu , \sigma ) = ( 0 , 1 , 3 )$   
+240 and another one with dependent $( \lambda , \mu , \bar { \sigma } ) = ( 1 , 1 , 3 )$ variables. We evaluate the performance over 30   
+241 repeated trials by analyzing the convergence speed during the burn-in phase (using traceplots) and   
+242 the mixing time (computing the autocorrelation function and the effective sample size). We visualize   
+243 the corresponding results in Figure 2 and report the quantitative performance in Table 1. Further   
+244 details about the simulations are available in the Supplementary Material.   
+245 By comparing the performance of convergence speed and mixing time in Figure 2, we observe that√   
+246 unbounded functions, like $\operatorname* { m a x } \{ 1 , t \}$ and $\sqrt { t }$ , tend to converge faster while having slower mixing   
+247 compared to the other two functions, thus being in line with the empirical findings of [28]. This is due   
+248 to the fact that unbounded functions have an intrinsic preference for visiting more likely regions at the   
+249 cost of a reduced amount of exploration. Moving a step further, we compare Figure 2a and Figure 2b   
+250 and observe that the optimal function is different for the two cases (i.e. $\operatorname* { m a x } \{ 1 , t \}$ in the independent   
+251 case and $\sqrt { t }$ in the dependent one). These results suggest that optimality not only depends on the   
+252 performance of burn-in and mixing but also on the distribution we are sampling from. This allows   
+253 us to reject the hypothesis about the existence of a universal optimum among the pool of balancing   
+254 functions proposed in [28] and to motivate our next set of experiments, where the aim is to learn to   
+255 adapt the balancing function to the target distribution.
+
+Learning the balancing function. We compare the four balancing functions used in the previous set of experiments with our two parametrizations on four different settings of the Ising model, namely independent and noisy $( \lambda , \mu , \sigma ) \overset { } { = } ( 0 , 1 , 3 )$ , independent and clean $( \lambda , { \bar { \mu } } , \sigma ) = ( 0 , { \bar { 3 } } , 3 )$ , dependent and noisy $( \lambda , \mu , \sigma ) = ( 1 , 1 , 3 )$ and dependent and clean $( \lambda , \mu , \sigma ) = ( 1 , 3 , 3 )$ cases and show the corresponding performance in Figure 3 and Table 2. We leave additional details and results to the Supplementary Material.
+
+![](images/338a8499adc828d15f481ffa113483aa902e4ce36c1bc209ecb9bcd67c7ddaf8.jpg)  
+Figure 3: Samplers’ performance on four cases of the Ising model $\left( 3 0 \times 3 0 \right)$ for the burn-in phase. (a) Case 1: Independent-noisy, (b) case 2: Independent-clean, (c) case 3: Dependent-noisy, (d) case 4: Dependent-clean
+
+Table 2: Quantitative performance for mixing measured by effective sample size on the four cases of the Ising model $\left( 3 0 \times 3 0 \right)$ . $\operatorname* { m a x } \{ 1 , t \}$ is performing significantly worse in statistical terms than the other functions.   
+
+<table><tr><td>Setting</td><td>t 1+t</td><td>√t</td><td>min{1,t}</td><td>max{1,t}</td><td>LSB 1</td><td>LSB 2</td></tr><tr><td>Case 1</td><td>2.48± 0.21</td><td>2.30± 0.22</td><td>2.42±0.19</td><td>1.75 ± 0.17</td><td>2.50±0.28</td><td>2.46 ± 0.28</td></tr><tr><td>Case 2</td><td>3.33 ± 0.32</td><td>2.94 ± 0.36</td><td>3.33 ± 0.33</td><td>1.72 ± 0.18</td><td>2.98± 0.24</td><td>3.33 ± 0.43</td></tr><tr><td>Case 3</td><td>2.58± 0.73</td><td>1.99 ± 0.43</td><td>2.56±0.62</td><td>1.26 ± 0.12</td><td>2.48± 0.61</td><td>2.67± 0.84</td></tr><tr><td>Case 4</td><td>32.8± 9.2</td><td>18.5 ± 6.8</td><td>31.8 ± 10.0</td><td>2.60 ± 1.46</td><td>18.4± 8.0</td><td>30.8± 9.2</td></tr></table>
+
+![](images/042d53adb2e2fb390d08e4c9fd57939312bc3cf49971cbf91ea79dc348e96c83.jpg)  
+Figure 4: Realizations obtained after 500 (Case 3) and 300 (Case 4) burn-in iterations on the Ising model.
+
+262 From Figure 3, we can see that our first parametrization LSB 1 is able to always "select" an unbounded   
+263 balancing function during burn-in, while when approaching convergence it is able to adapt to preserve   
+264 fast mixing, as measured by the effective sample size in Table 2. It’s interesting to mention also that   
+265 the softmax nonlinearity used in LSB 1 can sometimes slow down the adaptation due to vanishing   
+266 gradients. This can be observed by looking at the case 4 of Figure 3, where for a large part of the√   
+267 burn-in period the strategy prefers $\operatorname* { m a x } \{ 1 , t \}$ over $\sqrt { t }$ . Nevertheless, it is still able to recover a   
+268 solution different from $\operatorname* { m a x } \{ 1 , t \}$ at the end of burn-in, as confirmed by the larger effective sample   
+269 size in Table 2 compared to the one achieved by max $\{ 1 , t \}$ .   
+70 Furthermore, we observe that our second parametrization LSB 2, which is functionally more expres  
+71 sive compared to LSB 1, allows to outperform all previous cases in terms of convergence speed, while   
+72 preserving optimal mixing, as shown in Figure 3 and Table 2. This provides further evidence that   
+73 the optimality of the balancing function is influenced by the target distribution and that exploiting   
+74 such information can dramatically boost the sampling performance (e.g. in case 3 of Figure 3, LSB 2   
+75 converges twice time faster as the best balancing function $\sqrt { t } )$ ). Figure 4 provides some realizations   
+76 obtained by the samplers for the cases with dependent variables $\lambda = 1$ . We clearly see from these   
+77 pictures that convergence for LSB 2 occurs at an earlier stage than the other balancing functions and   
+78 therefore the latent variables in the Ising model converge faster to their ground truth configuration.
+
+![](images/6a9b3b0d0556dcfdea293bc9dfa3d89873051262f20c7d68601dcfab54e07433.jpg)  
+Figure 5: Samplers’ performance on Bayesian networks from UAI competition (100 variables). (a)-(b) are the traceplots for the burn-in phase, (c)-(d) are the autocorrelation functions for the mixing one.
+
+Table 3: Quantitative performance for mixing measured by effective sample size on two Bayesian networks from UAI competition.   
+
+<table><tr><td>Dataset</td><td></td><td>Vt</td><td>min{1,t}</td><td>max{1,t}</td><td>LSB1</td><td>LSB 2</td></tr><tr><td>BN1</td><td>2.90±0.76</td><td>3.41 ± 0.77</td><td>2.54± 0.32</td><td>2.70 ± 0.63</td><td>3.19 ± 0.46</td><td>3.22 ± 0.38</td></tr><tr><td>BN2</td><td>3.43 ± 0.75</td><td>3.92 ± 0.94</td><td>3.78 ± 0.50</td><td>3.63 ± 0.67</td><td>3.52 ± 0.42</td><td>3.44 ± 0.44</td></tr></table>
+
+# 279 5.2 Bayesian Networks: UAI data
+
+We evaluate how our strategy generalizes to different graph topologies compared to the one of the Ising model. In particular, we consider two Bayesian networks, with 100 discrete variables each and near-deterministic dependencies, from the 2006 UAI competition.6 Similarly to the previous experiments for the Ising model, we measure the performance over 5 repeated trials by analyzing the convergence speed during the burn-in phase (using traceplots) and the mixing time (computing the autocorrelation function and the effective sample size). Further details about the simulations are available in the Supplementary Material.
+
+We observe that the proposed strategy is able to adapt to the target distribution, thus achieving fast convergence (Figure 5) while preserving fast mixing (both Figure 5 and Table 3) compared to existing balancing functions.
+
+# 6 Conclusion
+
+We have presented a strategy to learn locally informed proposals for MCMC in discrete spaces. The strategy consists of (i) a new parametrization of balancing functions and (ii) a learning procedure adapting the proposal to the target distribution, in order to improve the sampling performance, both in terms of convergence speed and mixing.
+
+Note that the LSB sampler belongs to the family of local sampling strategies, thus inheriting their limitations. The locality assumption can be quite restrictive, for example when sampling from discrete distributions with deterministic dependencies among variables. In such situations, local sampling might fail to correctly sample from the target in a finite amount of time, as being required to cross regions with zero probability mass. This remains an open challenge to be investigated in future work.
+
+It’s important to mention that this work is foundational and general. The proposed strategy reduces the amount of user intervention and improves the sampling efficiency. However, these results could have potential impacts on society. At first glance, one could argue that the automation of the sampling procedure could have a negative impact on society as reducing the amount of human labour required to run the sampling strategy. However, we think that the benefits are of a far greater number compared to the negative aspects. In fact, the procedure reduces the costs of domain knowledge, with the advantage of democratizing the sampling strategy and reducing suboptimal configurations of the algorithm resulting from possibly wrong human decisions. Furthermore, the proposed strategy introduces a small amount of additional computation, which is used to reduce the amount of queries to the target distribution, thus improving the query and the sampling efficiency. We think that this last aspect could contribute positively towards devising more energy-efficient algorithms and therefore being more environment friendly.
+
+# References
+
+[1] H. Mohasel Afshar, J. Domke, et al. Reflection, Refraction, and Hamiltonian Monte Carlo. In NeurIPS, pages 3007–3015, 2015.
+
+[2] N. Alexeev, J. Isomurodov, V. Sukhov, G. Korotkevich, and A. Sergushichev. Markov Chain Monte Carlo for Active Module Identification Problem. BMC Bioinformatics, 21(6):1–20, 2020.
+
+[3] G. Altekar, S. Dwarkadas, J. P. Huelsenbeck, and F. Ronquist. Parallel Metropolis Coupled Markov Chain Monte Carlo for Bayesian Phylogenetic Inference. Bioinformatics, 20(3):407– 415, 2004.
+
+[4] C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan. An Introduction to MCMC for Machine Learning. Machine learning, 50(1):5–43, 2003.   
+[5] C. Andrieu and J. Thoms. A Tutorial on Adaptive MCMC. Statistics and Computing, 18(4):343– 373, 2008.   
+[6] V. Dinh, A. Bilge, C. Zhang, and Frederick A. Matsen I. V. Probabilistic Path Hamiltonian Monte Carlo. In ICML, pages 1009–1018, 2017.   
+[7] S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth. Hybrid Monte Carlo. Physics letters B, 195(2):216–222, 1987.   
+[8] W. Gong, Y. Li, and J. M. Hernández-Lobato. Meta-Learning for Stochastic Gradient MCMC. In ICLR, 2019.   
+[9] U. Grenander and M. I. Miller. Representations of Knowledge in Complex Systems. Journal of the Royal Statistical Society: Series B (Methodological), 56(4):549–581, 1994.   
+[10] R. Habib and D. Barber. Auxiliary Variational MCMC. In ICLR, 2018.   
+[11] C. Hans, A. Dobra, and M. West. Shotgun Stochastic Search for “Large P” Regression. Journal of the American Statistical Association, 102(478):507–516, 2007.   
+[12] D. M. Higdon. Auxiliary Variable Methods for Markov Chain Monte Carlo with Applications. Journal of the American statistical Association, 93(442):585–595, 1998.   
+[13] M. D. Hoffman and A. Gelman. The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res., 15(1):1593–1623, 2014.   
+[14] D. Koller and N. Friedman. Probabilistic Graphical Models: Principles and Techniques. MIT press, 2009.   
+[15] D. Levy, Matt D. H., and J. Sohl-Dickstein. Generalizing Hamiltonian Monte Carlo with Neural Networks. In ICLR, 2018.   
+[16] D. J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge university press, 2003.   
+[17] R. Neal. Probabilistic Inference Using Markov Chain Monte Carlo Methods. Department of Computer Science, University of Toronto Toronto, Ontario, Canada, 1993.   
+[18] E. Nijkamp, M. Hill, T. Han, S. C. Zhu, and Y. N. Wu. On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models. In AAAI, pages 5272–5280, 2020.   
+[19] A. Nishimura, D. B. Dunson, and J. Lu. Discontinuous Hamiltonian Monte Carlo for Discrete Parameters and Discontinuous Likelihoods. Biometrika, 107(2):365–380, 2020.   
+[20] A. Pakman and L. Paninski. Auxiliary-Variable Exact Hamiltonian Monte Carlo Samplers for Binary Distributions. In NeurIPS, 2013.   
+[21] C. Pasarica and A. Gelman. Adaptively Scaling the Metropolis Algorithm Using Expected Squared Jumped Distance. Statistica Sinica, pages 343–364, 2010.   
+[22] Pompe, E. and Holmes, C. and Łatuszynski, K. and others. A Framework for Adaptive MCMC ´ Targeting Multimodal Distributions. Annals of Statistics, 48(5):2930–2952, 2020.   
+[23] C. Robert and G. Casella. Monte Carlo Statistical Methods. Springer Science & Business Media, 2013.   
+[24] J. Sill. Monotonic Networks. In NeurIPS, pages 661–667, 1997.   
+[25] J. Song, S. Zhao, and S. Ermon. A-NICE-MC: Adversarial Training for MCMC. In NeurIPS, 2017.   
+[26] M. K. Titsias and C. Yau. The Hamming Ball Sampler. Journal of the American Statistical Association, 112(520):1598–1611, 2017.   
+[27] T. Wang, Y. Wu, D. A. Moore, and S. J. Russell. Meta-Learning MCMC Proposals. In NeurIPS, pages 4150–4160, 2018.   
+[28] G. Zanella. Informed Proposals for Local MCMC in Discrete Spaces. Journal of the American Statistical Association, 115(530):852–865, 2020.   
+[29] Y. Zhang, Z. Ghahramani, A. J. Storkey, and C. Sutton. Continuous Relaxations for Discrete Hamiltonian Monte Carlo. NeurIPS, 25:3194–3202, 2012.
+
+[30] M. Zhu. Sample Adaptive MCMC. In NeurIPS, volume 32, 2019.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] We included a paragraph in the conclusions (Section 6) discussing the main limitations of our strategy.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] We included a paragraph in the conclusions (Section 6) discussing both pros and cons of the proposed work from the societal perspective.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] Please check the three propositions in Section 3.   
+(b) Did you include complete proofs of all theoretical results? [Yes] Proofs are shown in the main paper in Section 3.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Code and instructions to replicate the experiments in the paper are available in the Supplementary Material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] The Supplementary Material provide all details about the experiments. Furthermore, we are sharing the scripts to run all experiments.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] At the beginning of Section 5, we have described the resources used to run the experiments.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] The only external asset used in this work is the UAI benchmark data, which is publicly available at http://sli.ics.uci.edu/\~ihler/uai-data/. We have explicitly mentioned the source of the data in the main paper.   
+(b) Did you mention the license of the assets? [Yes] UAI data is freely accessible online without any explicit license, that means that copyright law applies in this case. We will not share the data, but only the source code and include a README file with a pointer to the main website and credit the main author.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We have included the URL of the UAI dataset in the main paper.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] See answer (b).
+
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] This doesn’t apply to UAI data.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+    {
+        "type": "text",
+        "text": "LSB: Local Self-Balancing MCMC in Discrete Spaces ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "1 Markov Chain Monte Carlo (MCMC) methods are promising solutions to sample   \n2 from target distributions in high dimensions. While MCMC methods enjoy nice   \n3 theoretical properties, like guaranteed convergence and mixing to the true target, in   \n4 practice their sampling efficiency depends on the choice of the proposal distribution   \n5 and the target at hand. This work considers using machine learning to adapt the   \n6 proposal distribution to the target, in order to improve the sampling efficiency in   \n7 the purely discrete domain. Specifically, (i) it proposes a new parametrization for a   \n8 family of proposal distributions, called locally balanced proposals, (ii) it defines   \n9 an objective function based on mutual information and (iii) it devises a learning   \n10 procedure to adapt the parameters of the proposal to the target, thus achieving fast   \n11 convergence and fast mixing. We call the resulting sampler as the Locally Self  \n12 Balancing Sampler (LSB). We show through experimental analysis on the Ising   \n13 model and Bayesian networks that LSB is indeed able to improve the efficiency   \n14 over a state-of-the-art sampler based on locally balanced proposals, thus reducing   \n15 the number of iterations required to converge, while achieving comparable mixing   \n16 performance. ",
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+    {
+        "type": "text",
+        "text": "17 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            568,
+            312,
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+        ],
+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "18 Sampling from complex and intractable probability distributions is of fundamental importance for   \n19 learning and inference [16]. MCMC algorithms are promising solutions to handle the intractability   \n20 of sampling in high dimensions and they have found numerous applications, in Bayesian statistics   \n21 and statistical physics [17, 23], bioinformatics and computational biology [3, 2] as well as machine   \n22 learning [4, 14, 18].   \n23 Although, MCMC can be applied to sample from any target distribution, in practice its efficiency   \n24 strongly depends on the choice of the proposal. Indeed, common phenomena, like slow convergence   \n25 and slow mixing, are typically the result of wrong choices of the proposal distribution. Therefore, it’s   \n26 extremely important to devise strategies enabling the tuning of the proposal to target distributions [5,   \n27 13]. While there has been a lot of work focusing on designing machine learning-based strategies to   \n28 improve the efficiency of MCMC in the continuous domain [29, 20, 1, 6, 19], less effort has been   \n29 devoted to the discrete counterpart. Most common solutions consider continuous relaxations of   \n30 the problem by using embeddings and then leverage existing sampling strategies designed for the   \n31 continuous case. These strategies are suboptimal, either because they consider limited settings, where   \n32 the target distribution has specific analytic forms [29], or because they make strong assumptions on   \n33 the properties of the embeddings, thus not having guarantees of preserving the topological properties   \n34 of the original discrete domain [20, 1, 6, 19].   \n35 This work focuses on MCMC strategies for the purely discrete domain. Specifically, (i) we introduce   \n36 a new parametrization for a family of proposal distributions, called locally balanced proposals, which   \n37 have been recently studied in [28], (ii) we define an objective function based on mutual information,   \n38 which reduces the distance between the proposal and the target distribution and also reduces the   \n39 statistical dependence between consecutive samples, and (iii) we devise a learning procedure to adapt   \n40 the parameters of the proposal to the target distribution using our objective. The resulting procedure,   \n41 called the Local Self-Balancing sampler (LSB), automatically discovers an optimal locally balanced   \n42 proposal, with the advantage of reducing the amount of user intervention and of improving the overall   \n43 sampling efficiency, both in terms of convergence speed and mixing time.   \n44 We provide some empirical analysis of sampling from the 2D Ising model and from Bayesian   \n45 networks and show that in some cases LSB is able to halve the number of iterations required to   \n46 converge, while achieving similar mixing performance to [28].   \n47 We start by providing some background on locally balanced proposal distributions (Section 2),   \n48 we introduce LSB by describing the parametrizations, the objective and the learning procedure   \n49 (Section 3), we discuss the related work (Section 4) and the experiments (Section 5), and finally we   \n50 conclude by highlighting the main limitations of LSB and possible future directions (Section 6). ",
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+    {
+        "type": "text",
+        "text": "51 2 Background ",
+        "text_level": 1,
+        "bbox": [
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+            279,
+            308,
+            296
+        ],
+        "page_idx": 1
+    },
+    {
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+        "text": "52 We consider the problem of sampling from a distribution $p$ with a support defined over a large finite   \n53 discrete sample space $\\mathcal { X }$ , i.e. $\\begin{array} { r } { p ( \\bar { \\mathbf { x } } ) \\stackrel { - } { = } \\tilde { p } ( \\mathbf { x } ) / \\sum _ { \\mathbf { x } ^ { \\prime \\prime } \\in \\mathcal { X } } \\tilde { p } ( \\mathbf { x } ^ { \\prime \\prime } ) } \\end{array}$ , where the normalization term cannot be   \n54 tractably computed and only $\\tilde { p }$ can be evaluated. One solution to the problem consists of sampling   \n55 using MCMC [17]. The main idea of MCMC is to sequentially sample from a tractable surrogate   \n56 distribution, alternatively called proposal, and to use an acceptance criterion to ensure that generated   \n57 samples are distributed according to the original distribution. More formally, MCMC is a Markov   \n58 chain with a transition probability of the form: ",
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+        "text": "$$\nT ( \\mathbf { x } ^ { \\prime } | \\mathbf { x } ) = A ( \\mathbf { x } ^ { \\prime } , \\mathbf { x } ) Q ( \\mathbf { x } ^ { \\prime } | \\mathbf { x } ) + 1 [ \\mathbf { x } ^ { \\prime } = \\mathbf { x } ] \\sum _ { \\mathbf { x } ^ { \\prime \\prime } \\in \\mathcal { X } } \\big ( 1 - A ( \\mathbf { x } ^ { \\prime \\prime } , \\mathbf { x } ) \\big ) Q ( \\mathbf { x } ^ { \\prime \\prime } | \\mathbf { x } ) \\big )\n$$",
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+        "text": "59 where $Q ( \\mathbf { x } ^ { \\prime } | \\mathbf { x } )$ is the probability of sampling $\\mathbf { x } ^ { \\prime }$ given a previously sampled $\\mathbf { x }$ , namely the proposal   \n60 distribution, $1 [ \\cdot ]$ is the Kronecker delta function and $A ( \\mathbf { x } ^ { \\prime } , \\mathbf { x } )$ is the probability of accepting sample   \n61 $\\mathbf { x } ^ { \\prime }$ given $\\mathbf { x }$ , e.g. $\\begin{array} { r } { A ( \\mathbf { x } ^ { \\prime } , \\mathbf { x } ) = \\operatorname* { m i n } \\left. 1 , \\frac { \\tilde { p } ( \\mathbf { x } ^ { \\prime } ) Q ( \\mathbf { x } | \\mathbf { x } ^ { \\prime } ) } { \\tilde { p } ( \\mathbf { x } ) Q ( \\mathbf { x } ^ { \\prime } | \\mathbf { x } ) } \\right. } \\end{array}$ , p˜(x0)Q(x|x0)p˜(x)Q(x0|x) \t.1 In this work, we consider the family of locally   \n62 informed proposals [28], which are characterized by the following expression: ",
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+        "text": "$$\nQ ( \\mathbf { x } ^ { \\prime } | \\mathbf { x } ) = \\frac { g \\big ( \\frac { \\tilde { p } ( \\mathbf { x } ^ { \\prime } ) } { \\tilde { p } ( \\mathbf { x } ) } \\big ) 1 [ \\mathbf { x } ^ { \\prime } \\in N ( \\mathbf { x } ) ] } { Z ( \\mathbf { x } ) }\n$$",
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+        "type": "text",
+        "text": "where 63 $N ( \\mathbf { x } )$ is the neighborhood of $\\mathbf { x }$ based on the Hamming metric.2 ",
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+        "text": "64 Note that the choice of $g$ has a dramatic impact on the performance of the Markov chain, as   \n65 investigated in [28]. In fact, there is a family of functions called balancing functions, satisfying the   \n66 relation $g ( t ) = t g ( 1 / t )$ (for all $t > 0$ ), which have extremely desirable properties, briefly recalled   \n67 hereunder.   \n68 Acceptance rform, namely 69 $\\begin{array} { r } { A ( \\mathbf { x } ^ { \\prime } , \\mathbf { x } ) = \\operatorname* { m i n } \\left. 1 , \\frac { Z ( \\mathbf { x } ) } { Z ( \\mathbf { x } ^ { \\prime } ) } \\right. } \\end{array}$ ty allows to rewrite the acceptan. Therefore, a proper choice of $g$ e function in a simplercan increase the ratio   \n70 between the normalization constants $Z ( \\mathbf { x } )$ and $Z ( \\mathbf { x } ^ { \\prime } )$ with consequent increase of the acceptance   \n71 rate even in high dimensional spaces.   \n72 Detailed balance. Note that for all $\\mathbf { x } ^ { \\prime } = \\mathbf { x }$ , detailed balance trivially holds, viz. $p ( \\mathbf { x } ) T ( \\mathbf { x } ^ { \\prime } | \\mathbf { x } ) =$   \n73 $p ( \\mathbf { x } ^ { \\prime } ) T ( \\mathbf { x } | \\mathbf { x } ^ { \\prime } )$ . In all other cases, detailed balance can be proved, by exploiting the fact that $T ( \\mathbf { x } ^ { \\prime } | \\mathbf { x } ) =$   \n74 $A ( \\mathbf { x } ^ { \\prime } , \\mathbf { x } ) Q ( \\mathbf { x } ^ { \\prime } | \\mathbf { x } )$ and by using the balancing property (see the Supplementary material for more   \n75 details). Detailed balance is a sufficient condition for invariance. Consequently, the target $p$ is a fixed   \n76 point of the Markov chain.   \n77 Ergodicity. Under mild assumptions, we have also ergodicity (we leave more detailed discussion   \n78 to the Supplementary material). In other words, the Markov chain converges to the fixed point $p$   \n79 independently from its initialization.   \n80 Efficiency. The efficiency of MCMC is generally measured in terms of the resulting asymptotic   \n81 variance for sample mean estimators. This is indeed a proxy to quantify the level of correlation   \n82 between samples generated through MCMC. Higher levels of asymptotic variance correspond to   \n83 higher levels of correlation, meaning that the Markov chain produces more dependent samples and   \n84 it is therefore less efficient. Balancing functions are asymptotically optimal according to Peskun   \n85 ordering [28].   \n86 The work in [28] proposes a pool of balancing functions with closed-form expression together with   \n87 some general guidelines to choose one. However, this pool is only a subset of the whole family of   \n88 balancing functions and several cases do not even have an analytical expression. Consequently, it is   \n89 not clear which function to use in order to sample efficiently from the target distribution. Indeed, we   \n90 will see in the experimental section that (i) the optimality of the balancing function depends on the   \n91 target distribution and that (ii) in some cases the optimal balancing function may be different from the   \n92 ones proposed in [28]. In the next sections, we propose a strategy to automatically learn the balancing   \n93 function from the target distribution, thus achieving fast convergence (burn-in) and fast mixing. ",
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+        "type": "text",
+        "text": "94 3 LSB: Local Self-Balancing Strategy ",
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+        "text": "We start by introducing two different parametrizations for the family of balancing functions in increasing order of functional expressiveness. Then, we propose an objective criterion based on mutual information, that allows us to learn the parametrization with fast convergence and fast mixing on the target distribution. ",
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+        "text": "3.1 Parametrizations ",
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+        "text": "We state the following proposition and then use it to devise the first parametrization. ",
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+        "text": "Proposition 1. Given n balancing functions $\\mathbf { g } ( t ) = [ g _ { 1 } ( t ) , \\ldots , g _ { n } ( t ) ] ^ { T }$ and a vector of scalar positive weights $\\mathbf { w } = [ w _ { 1 } , \\ldots , x _ { n } ] ^ { \\tilde { T } }$ , the linear combination $\\mathbf { \\dot { \\mathbf { \\rho } } } _ { g ( t ) } \\doteq \\mathbf { w } ^ { T } \\mathbf { \\dot { \\mathbf { g } } } ( t )$ satisfies the balancing property. ",
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+        "text": "$$\n\\begin{array} { r } { g ( t ) = \\mathbf { w } ^ { T } \\mathbf { g } ( t ) = \\sum _ { i = 1 } ^ { n } w _ { i } g _ { i } ( t ) = t \\sum _ { i = 1 } ^ { n } w _ { i } g _ { i } ( 1 / t ) = t \\mathbf { w } ^ { T } \\mathbf { g } ( 1 / t ) = t g ( 1 / t ) } \\end{array}\n$$",
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+        "text": "105 Despite its simplicity, the proposition has important implications. First of all, it allows to convert the   \n106 problem of choosing the optimal balancing function into a learning problem. Secondly, the linear   \n107 combination introduces only few parameters (in the experiments we consider $n = 4$ ) and therefore   \n108 the learning problem can be solved in an efficient way. The requirement about positive weights is   \n109 necessary to guarantee ergodicity (see Supplementary material on ergodicity for further details). ",
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+        "text": "The first parametrization (LSB 1) consists of the relations 110 $\\begin{array} { r } { w _ { i } = e ^ { \\theta _ { i } } / \\sum _ { j = 1 } ^ { n } e ^ { \\theta _ { j } } } \\end{array}$ for all $i = 1 , \\ldots , n$ 111 where $\\pmb \\theta = [ \\theta _ { 1 } , \\dots , \\theta _ { n } ] \\in \\mathbb { R } ^ { n }$ . Note that the softmax is used to smoothly select one among the $n$ 112 balancing functions. Therefore, we refer to this parametrization as learning to select among existing 113 balancing functions. ",
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+        "type": "text",
+        "text": "The second parametrization (LSB 2) is obtained from the following proposition. ",
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+        "text": "115 Proposition 2. Given $g _ { \\pmb { \\theta } } ( t ) = \\operatorname* { m i n } \\{ h _ { \\pmb { \\theta } } ( t ) , t h _ { \\pmb { \\theta } } ( 1 / t ) \\}$ , where hθ is a universal real valued function   \n116 approximator parameterized by vector $\\pmb \\theta \\in \\mathbb { R } ^ { k }$ (e.g. a neural network), and any balancing function \\`,   \n117 there always exists $\\widetilde { \\pmb { \\theta } } \\in \\mathbb { R } ^ { k }$ such that $g _ { \\tilde { \\pmb { \\theta } } } ( t ) = \\ell ( t )$ for all $t > 0$ .   \n118 Proof. Given any balancing function $\\ell$ , we can always find a $\\tilde { \\pmb { \\theta } }$ such that $h _ { \\tilde { \\pmb { \\theta } } } ( t ) = \\ell ( t )$ for all $t > 0$   \n119 (because $h _ { \\pmb { \\theta } }$ is a universal function approximator). This implies that $h _ { \\tilde { \\pmb { \\theta } } }$ satisfies the balancing property,   \n120 i.e. $h _ { \\tilde { \\pmb { \\theta } } } ( t ) = t h _ { \\tilde { \\pmb { \\theta } } } ( 1 / t )$ for all $t > 0$ . Consequently, by definition of $g _ { \\pmb { \\theta } }$ , we have that $g _ { \\pmb { \\theta } } ( t ) \\stackrel { - } { = } \\bar { h } _ { \\tilde { \\pmb { \\theta } } } ( \\bar { t } )$ .   \n121 And finally we can conclude that $g _ { \\pmb { \\theta } } ( t ) = \\ell \\bar { ( } t )$ for all $t > 0$ . □   \n122 In theory, LSB 2 parameterizes the whole family of balancing functions and it allows to find the   \n123 optimal one from the whole set.3 In practice, it is better to restrict the analysis only to monotonic   \n124 increasing functions, as we prefer to choose a proposal distribution sampling from regions of higher   \n25 probability mass. In the following proposition, we provide sufficient conditions to ensure the   \n26 monotonicity of LSB 2.   \n127 Proposition 3. Define gθ as in LSB 2. Assume that $h _ { \\pmb \\theta } ( t )$ is a differentiable and monotonic increasing   \n128 function with respect to variable $t$ , satisfying the relation $\\begin{array} { r } { h _ { \\pmb \\theta } ( 1 / t ) \\geq \\frac { 1 } { t } \\frac { d h _ { \\pmb \\theta } ( 1 / t ) } { d t } } \\end{array}$ for all $t > 0$ . Then,   \n129 gθ is a monotonic increasing function in $t$ .   \n130 Proof. We have that $h _ { \\pmb \\theta } ( 1 / t ) - \\frac { 1 } { t } \\frac { d h _ { \\pmb \\theta } ( 1 / t ) } { d t } \\geq 0$ for all $t ~ > ~ 0$ . Also, note that $\\begin{array} { r l } { \\frac { d t h _ { \\pmb \\theta } ( 1 / t ) } { d t } } & { = } \\end{array}$   \n131 $h _ { \\pmb \\theta } ( 1 / t ) - \\frac { 1 } { t } \\frac { d h _ { \\pmb \\theta } ( 1 / t ) } { d t } > 0$ $t ~ > ~ 0$ 1t dhθ (1/t)dt > 0. Therefore, thθ (1/t) is a monotonic increasing function in t for $t _ { 1 } , t _ { 2 }$ $t _ { 1 } ~ \\ge ~ t _ { 2 } ~ > ~ 0$ $g _ { \\pmb \\theta } ( t _ { 1 } ) \\ = \\ \\operatorname* { m i n } \\{ \\bar { h } _ { \\pmb \\theta } ( t _ { 1 } ) , t _ { 1 } h _ { \\pmb \\theta } ( 1 / t _ { 1 } ) \\}$   \n133 and $g _ { \\pmb \\theta } ( t _ { 2 } ) = \\mathrm { m i n } \\{ h _ { \\pmb \\theta } ( t _ { 2 } ) , t _ { 2 } h _ { \\pmb \\theta } ( 1 / t _ { 2 } ) \\}$ . By monotonicity of $h _ { \\pmb \\theta } ( t )$ and $t h _ { \\pmb { \\theta } } ( 1 / t )$ , we have that   \n134 $h _ { \\pmb \\theta } ( t _ { 1 } ) \\geq h _ { \\pmb \\theta } ( t _ { 2 } )$ and $t _ { 1 } h _ { \\pmb \\theta } ( 1 / t _ { 1 } ) \\geq t _ { 2 } h _ { \\pmb \\theta } ( 1 / t _ { 2 } )$ . Therefore, we can conclude that $g _ { \\pmb { \\theta } } ( t _ { 1 } ) \\geq g _ { \\pmb { \\theta } } ( t _ { 2 } )$ for   \n135 all $t _ { 1 } , t _ { 2 }$ with $t _ { 1 } \\geq t _ { 2 } > 0$ . □   \n136 Therefore, to build $g _ { \\pmb { \\theta } }$ , we need a monotonic function $h _ { \\pmb \\theta } ( t )$ . Specifically, we can choose a monotonic   \n137 network [24] and constrain $h _ { \\pmb { \\theta } }$ to satisfy the condition $\\begin{array} { r } { h _ { \\pmb \\theta } ( 1 / t ) \\geq \\frac { 1 } { t } \\frac { d h _ { \\pmb \\theta } ( 1 / t ) } { d t } } \\end{array}$ for all $t > 0$ (see the   \n138 Supplementary material for further information on how to impose the condition on $h _ { \\pmb { \\theta } }$ ). In the   \n139 next paragraphs, we propose an objective and a learning strategy to train the parameters of the two   \n140 parametrizations. ",
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+        "type": "text",
+        "text": "3.2 Objective and Learning Algorithm ",
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+        "text": "142 The goal here is to devise a criterion to find the balancing function with the fastest speed of conver  \n143 gence/mixing on the target distribution $p$ . Note that the ideal case would be to sample from $p$ in an   \n144 independent fashion. We have already mentioned that this operation is computationally expensive   \n145 due to the intractability of computing the normalizing constant. In our case, we have to consider   \n146 the agnostic case, because the proposal distribution is a tractable surrogate for $p$ . In this regard, we   \n147 define a criterion taking into account the distance from this ideal case. Specifically, we measure the   \n148 distance of the transition probability of the Markov chain in Eq. 1 from the target $p$ and the amount   \n149 of dependence between consecutive samples generated through it. In other words, we introduce the   \n150 following criterion, which is indeed a form of mutual information objective: ",
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+        "text": "$$\n\\mathcal { T } _ { \\pmb { \\theta } } = K L \\big \\{ p ( \\pmb { x } ) \\tilde { T } _ { \\pmb { \\theta } } ( \\pmb { x } ^ { \\prime } | \\pmb { x } ) \\| p ( \\pmb { x } ) p ( \\pmb { x } ^ { \\prime } ) \\big \\} = E _ { p } \\big \\{ K L \\big \\{ \\tilde { T } _ { \\pmb { \\theta } } ( \\pmb { x } ^ { \\prime } | \\pmb { x } ) \\| p ( \\pmb { x } ^ { \\prime } ) \\big \\} \\big \\}\n$$",
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+        "type": "text",
+        "text": "151 where $K L$ is the Kullback Leibler divergence and $E _ { p }$ is the expected value of random vector $\\pmb { x }$   \n152 distributed according to $p$ and ",
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+        "text": "$$\n\\tilde { T } _ { \\pmb { \\theta } } ( \\pmb { x } ^ { \\prime } | \\pmb { x } ) \\doteq \\left\\{ \\begin{array} { c c } { \\frac { T _ { \\pmb { \\theta } } ( \\pmb { x } ^ { \\prime } | \\pmb { x } ) } { Z _ { T } } } & { \\forall \\pmb { x } ^ { \\prime } \\neq \\pmb { x } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "153 is a conditional distribution defined over the transition probability $T _ { \\pmb \\theta } ( \\pmb x ^ { \\prime } | \\pmb x )$ in Eq. 1, where we have   \n154 explicited the dependence on $\\pmb \\theta$ and we have introduced the normalizing constant $Z _ { T }$ , to ensure   \n155 that $\\tilde { T } _ { \\pmb { \\theta } } ( \\pmb { x } ^ { \\prime } | \\pmb { x } )$ is a proper probability distribution. Note also that $\\tilde { T } _ { \\pmb { \\theta } } ( \\pmb { x } ^ { \\prime } | \\pmb { x } )$ discards all pair of equal   \n156 samples, i.e. ${ \\pmb x } ^ { \\prime } = { \\pmb x }$ , as they are samples rejected by the Markov chain.   \n157 Minimizing Eq. 3 allows us to find the configuration of parameters bringing us \"closer\" in terms   \n158 of Kullback Leibler to the ideal case, namely $T _ { \\pmb \\theta } ( \\pmb x ^ { \\prime } | \\pmb x ) = \\bar { p } ( \\pmb x ^ { \\prime } )$ for all ${ \\pmb x } ^ { \\prime } \\neq { \\pmb x }$ . The expectation in   \n159 Eq. 3 requires access to samples from $p$ and therefore cannot be computed. Nevertheless, note that   \n160 the KL term in Eq. 3 can be rewritten in an equivalent form (see the Supplementary material for the   \n161 derivation): ",
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+        "text": "$$\nK L \\{ \\tilde { T } _ { \\theta } ( { \\pmb x } ^ { \\prime } | { \\pmb x } ) \\| p ( { \\pmb x } ^ { \\prime } ) \\} \\propto \\mathcal { I } ( \\pmb \\theta , { \\pmb x } ) \\doteq E _ { Q _ { \\theta _ { 0 } } } \\Bigl \\{ \\omega _ { \\pmb \\theta , \\pmb \\theta _ { 0 } } A _ { \\theta } ( { \\pmb x } ^ { \\prime } , { \\pmb x } ) \\log \\frac { A _ { \\theta } ( { \\pmb x } ^ { \\prime } , { \\pmb x } ) Q _ { \\pmb \\theta } ( { \\pmb x } ^ { \\prime } | { \\pmb x } ) } { \\tilde { p } ( { \\pmb x } ^ { \\prime } ) } \\Bigr \\}\n$$",
+        "text_format": "latex",
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+        "text": "162 where $\\begin{array} { r } { \\omega _ { \\pmb { \\theta } , \\pmb { \\theta } _ { 0 } } = \\frac { Q _ { \\pmb { \\theta } } ( \\pmb { x } ^ { \\prime } | \\pmb { x } ) } { Q _ { \\pmb { \\theta } _ { 0 } } ( \\pmb { x } ^ { \\prime } | \\pmb { x } ) } } \\end{array}$ and $\\pmb { \\theta } _ { 0 }$ is the reference parameter vector for the proposal distribution. Alternatively to Eq. 3, we can minimize the following quantity:4163 ",
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+        "text": "$$\nJ ( \\pmb { \\theta } ) = E _ { Q _ { i n i t } } \\{ \\mathcal { I } ( \\pmb { \\theta } , \\pmb { x } ) \\} + E _ { Q _ { \\pmb { \\theta } _ { 0 } } } \\{ \\mathcal { I } ( \\pmb { \\theta } , \\pmb { x } ) \\}\n$$",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "164 where $Q _ { i n i t }$ is the distribution used at initialization, typically uniform on the support $\\mathcal { X }$ . Note that   \n165 the first and the second terms in Eq. 6 encourage fast burn-in and fast mixing, respectively. Therefore,   \n166 $\\theta$ can be learnt using the procedure described in Algorithm 1. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Learning rate $\\eta = 1 e - 4$ , initial parameter $\\pmb { \\theta } _ { 0 }$ , burn-in iterations $K$ and batch of samples $N$ .   \n$\\{ \\pmb { x } _ { 0 } ^ { ( i ) } \\} _ { i = 1 } ^ { N } \\sim Q _ { i n i t }$ ;   \nwhile $k { = } I { : } K$ do $\\{ \\pmb { x } _ { i n i t } ^ { ( i ) } \\} _ { i = 1 } ^ { N } \\sim Q _ { i n i t }$ ; while $i { = } I { : } N$ do Jb(i)0 ← Estimate of J (θ, x(i)init) using one sample from $Q _ { \\pmb { \\theta } _ { 0 } } ( \\pmb { x } | \\pmb { x } _ { i n i t } ^ { ( i ) } )$ ; $\\widehat { \\mathcal { I } } ^ { ( i ) } \\gets$ Estimate of $\\mathcal { I } ( \\pmb \\theta , \\pmb x _ { 0 } ^ { ( i ) } )$ using one sample from $Q _ { \\pmb \\theta _ { 0 } } ( \\pmb x | \\pmb x _ { 0 } ^ { ( i ) } )$ ; end $\\begin{array} { r } { \\widehat { J } ( \\pmb { \\theta } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\widehat { \\mathcal { I } } ^ { ( i ) \\prime } + \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\widehat { \\mathcal { I } } ^ { ( i ) } \\ ; } \\end{array}$ $\\begin{array} { r } { \\pmb { \\theta }  \\pmb { \\theta } - \\frac { \\eta } { N } \\nabla _ { \\pmb { \\theta } } \\widehat { J } ( \\pmb { \\theta } ) } \\end{array}$ ; Update $\\{ \\pmb { x } _ { 0 } ^ { ( i ) } \\} _ { i = 1 } ^ { N }$ with accepted samples ; $\\pmb \\theta _ { 0 } \\gets \\pmb \\theta$ ;   \nend ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "167 4 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "8 It’s important to devise strategies, which enable the automatic adaption of proposals to target distributions, not only to reduce user intervention, but also to increase the efficiency of MCMC samplers [5, 13]. Recently, there has been a surge of interest in using machine learning and in particular deep learning to learn proposals directly from data, especially in the continuous domain. Here, we provide a brief overview of recent integrations of machine learning and MCMC samplers according to different parametrizations and training objectives. ",
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+    },
+    {
+        "type": "text",
+        "text": "174 Parametrizations and objectives in the continuous domain. The work in [27] proposes a strategy   \n175 based on block Gibbs sampling, where blocks are large motifs of the underlying probabilistic graphical   \n176 structure. It parameterizes the conditional distributions of each block using mixture density networks   \n177 and trains them using meta-learning on a log-likelihood-based objective. The work in [25] considers   \n178 a global sampling strategy, where the proposal is parameterized by a deep generative model. The   \n179 model is learnt through adversarial training, where a neural discriminator is used to detect whether or   \n180 not generated samples are distributed according to the target distribution. Authors in [10] propose a   \n181 global sampling strategy based on MCMC with auxiliary variables [12]. The proposals are modelled   \n182 as Gaussian distributions parameterized by neural networks and are trained on a variational bound   \n183 of a log-likelihood-based objective. The works in [15, 8] propose a gradient-based MCMC [7, 9],   \n184 where neural models are used to learn the hyperparameters of the equations governing the dynamics   \n185 of the sampler. Different objectives are used during training. In particular, the work in [8] uses   \n186 a log-likelihood based objective, whereas the work in [15] considers the expected squared jump   \n187 distance, namely a tractable proxy for the lag-1 autocorrelation function [21]. The work in [30]   \n188 proposes a global two-stage strategy, which consists of (i) sampling according to a Gaussian proposal   \n189 and (ii) updating its parameters using the first- and second-order statistics computed from a properly   \n190 maintained pool of samples. The parameter update can be equivalently seen as finding the solution   \n191 maximizing a log-likelihood function defined over the pool of samples. Finally, the work in [22]   \n192 extends this last strategy to the case of Gaussian mixture proposals. All these works differ from the   \n193 current one in at least two aspects. Firstly, it is not clear how these parametrizations can be applied   \n194 to sampling in the discrete domain. Secondly, the proposed objectives compute either a distance   \n195 between the proposal distribution and the target one, namely using an adversarial objective or a   \n196 variational bound on the log-likelihood, or a proxy on the correlation between consecutive generated   \n197 samples, namely the expected squared jump distance. Instead, our proposed objective is more general   \n198 in the sense that it allows to (i) reduce the distance between the proposal and the target distribution as   \n199 well as to (ii) reduce the statistical dependence between consecutive samples, as being closely related   \n200 to mutual information.   \n201 Sampling in the discrete domain. Less efforts have been devoted to devise sampling strategies   \n202 for a purely discrete domain. Most of the works consider problem relaxations by embedding   \n203 the discrete domain into a continuous one, applying existing strategies like Hamiltonian Monte   \n204 Carlo [29, 20, 1, 6, 19] on it and then moving back to the original domain. These strategies are   \n205 suboptimal, either because they consider limited settings, where the target distribution has specific   \n206 analytic forms [29], or because they make strong assumptions on the properties of the embeddings,   \n207 thus not preserving the topological properties of the discrete domain [20, 1, 6, 19].5. The work   \n208 in [28] provides an extensive experimental comparison between several discrete sampling strategies,   \n209 including the ones based on embeddings, based on stochastic local search [11] and the Hamming ball   \n210 sampler [26], which can be regarded as a more efficient version of block Gibbs sampling. Notably,   \n211 the sampling strategy based on locally informed proposals and balancing functions proposed in [28]   \n212 can be considered as the current state of the art for discrete MCMC. Our work builds and extends   \n213 upon this sampler by integrating it with a machine learning strategy. To the best of our knowledge,   \n214 this is the first attempt to consider the integration of machine learning and MCMC in the discrete   \n215 setting. ",
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+        "type": "text",
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+    {
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+        "img_path": "images/ddb8a6056fe8518e66f63e0da28045c52cd3f6600642cf00248ba1c0689a390b.jpg",
+        "image_caption": [
+            "Figure 1: Examples of $\\pmb { \\alpha }$ in different settings of the Ising model $( 3 0 \\times 3 0 )$ ), i.e noisy $\\mu = 1 , \\sigma = 3$ and clean $\\mu = 3 , \\sigma = 3$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "16 5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Firstly, we analyze samplers’ performance on the 2D Ising model. Then, we perform experiments on additional UAI benchmarks. Code to replicate the experiments in this section is available in the Supplementary material. All experiments are performed on a laptop provided with 4 Intel i5 cores (2 GHz) and 16 GB of RAM memory. ",
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+    {
+        "type": "text",
+        "text": "5.1 2D Ising Model ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The Ising model has been introduced in statistical mechanics in 1920 and it has been applied in several domains since then. In this section, we consider an application to image analysis, where the goal is to segment an image to identify an object from its background. Consider a binary state space $\\mathbf { \\bar { \\mathcal { X } } } = \\{ - 1 , 1 \\} ^ { V }$ , where $( V , E )$ defines a square lattice graph of the same size of the analyzed image, namely $n \\times n$ . For each state configuration $\\mathbf { x } = ( x _ { i } ) _ { i \\in V } \\in \\mathcal { X }$ , define a prior distribution ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2511714f160334a85c400cfbe9a4dde5be13521955a7021c6d2576c2fd0c010.jpg",
+        "text": "$$\np _ { p r i o r } ( \\mathbf { x } ) \\propto \\exp \\left\\{ \\lambda \\sum _ { ( i , j ) \\in E } x _ { i } x _ { j } \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "227 where $\\lambda$ is a non-negative scalar used to weight the dependence among neighboring variables in the   \n228 lattice. Then, consider that each pixel $y _ { i }$ is influenced only by the corresponding hidden variable   \n229 $x _ { i }$ and generated according to a Gaussian density with mean $\\mu x _ { i }$ and variance $\\sigma ^ { \\bar { 2 } }$ . Note that each   \n230 variable in the lattice tells whether the corresponding pixel belongs to the object or to the background   \n231 (1 or $^ { - 1 }$ , respectively). The corresponding posterior distribution of a hidden state $\\mathbf { x }$ given an observed   \n232 image is defined as follows: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7794159153763018051eae5b9625dcdfcd353a9f25b178a933a9f4f74defe7ec.jpg",
+        "text": "$$\np ( \\mathbf { x } ) = \\frac { 1 } { Z } \\exp \\bigg \\{ \\sum _ { i \\in V } \\alpha _ { i } x _ { i } + \\lambda \\sum _ { ( i , j ) \\in E } x _ { i } x _ { j } \\bigg \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            742,
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+            785
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha _ { i } = y _ { i } \\mu / \\sigma ^ { 2 }$ is a coefficient biasing $x _ { i }$ towards either 1 or $- 1$ . Therefore, ${ \\pmb { \\alpha } } = ( \\alpha _ { i } ) _ { i \\in V }$ contains information about the observed image. Figure 1 shows two synthetically generated examples of $\\pmb { \\alpha }$ . Our goal is to analyze the sampling performance on the distribution defined in Eq. 7. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Universally optimal balancing function. We start by comparing the balancing functions proposed√ in [28], namely $g ( t ) = t / ( 1 + t )$ (a.k.a Barker function), $\\sqrt { t } , \\operatorname* { m i n } \\{ 1 , t \\}$ and $\\operatorname* { m a x } \\{ 1 , t \\}$ , in order to test ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/2778fc2876168809545d3c206b1df0e2b364b56c5a40a3446e0d5adc7156b071.jpg",
+        "image_caption": [
+            "Figure 2: Samplers’ performance on noisy and clean cases of the Ising model $\\left( 3 0 \\times 3 0 \\right)$ . (a)-(b) are the traceplots for the burn-in phase, (c)-(d) are the autocorrelation functions for the mixing one. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/9d52cd7dd1c396c1f6636243ad90bf7c93f9277161a09e3075668030d8758275.jpg",
+        "table_caption": [
+            "Table 1: Quantitative performance for mixing measured by effective sample size on the noisy and clean cases of the Ising model $\\left( 3 0 \\times 3 0 \\right)$ . max $\\{ 1 , t \\}$ is performing significantly worse in statistical terms than the other functions. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Setting</td><td>t 1+t</td><td>Vt</td><td>min{1,t}</td><td>max{1,t}</td></tr><tr><td>Noisy</td><td>2.48±0.21</td><td>2.30±0.22</td><td>2.42±0.19</td><td>1.75± 0.17</td></tr><tr><td>Clean</td><td>2.58± 0.73</td><td>1.99 ± 0.43</td><td>2.56± 0.62</td><td>1.26 ± 0.12</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "238 whether there is a universally optimal balancing function among this subset. In particular, we run the   \n239 samplers over two instances of the Ising model, viz. a setting with independent $( \\lambda , \\mu , \\sigma ) = ( 0 , 1 , 3 )$   \n240 and another one with dependent $( \\lambda , \\mu , \\bar { \\sigma } ) = ( 1 , 1 , 3 )$ variables. We evaluate the performance over 30   \n241 repeated trials by analyzing the convergence speed during the burn-in phase (using traceplots) and   \n242 the mixing time (computing the autocorrelation function and the effective sample size). We visualize   \n243 the corresponding results in Figure 2 and report the quantitative performance in Table 1. Further   \n244 details about the simulations are available in the Supplementary Material.   \n245 By comparing the performance of convergence speed and mixing time in Figure 2, we observe that√   \n246 unbounded functions, like $\\operatorname* { m a x } \\{ 1 , t \\}$ and $\\sqrt { t }$ , tend to converge faster while having slower mixing   \n247 compared to the other two functions, thus being in line with the empirical findings of [28]. This is due   \n248 to the fact that unbounded functions have an intrinsic preference for visiting more likely regions at the   \n249 cost of a reduced amount of exploration. Moving a step further, we compare Figure 2a and Figure 2b   \n250 and observe that the optimal function is different for the two cases (i.e. $\\operatorname* { m a x } \\{ 1 , t \\}$ in the independent   \n251 case and $\\sqrt { t }$ in the dependent one). These results suggest that optimality not only depends on the   \n252 performance of burn-in and mixing but also on the distribution we are sampling from. This allows   \n253 us to reject the hypothesis about the existence of a universal optimum among the pool of balancing   \n254 functions proposed in [28] and to motivate our next set of experiments, where the aim is to learn to   \n255 adapt the balancing function to the target distribution. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Learning the balancing function. We compare the four balancing functions used in the previous set of experiments with our two parametrizations on four different settings of the Ising model, namely independent and noisy $( \\lambda , \\mu , \\sigma ) \\overset { } { = } ( 0 , 1 , 3 )$ , independent and clean $( \\lambda , { \\bar { \\mu } } , \\sigma ) = ( 0 , { \\bar { 3 } } , 3 )$ , dependent and noisy $( \\lambda , \\mu , \\sigma ) = ( 1 , 1 , 3 )$ and dependent and clean $( \\lambda , \\mu , \\sigma ) = ( 1 , 3 , 3 )$ cases and show the corresponding performance in Figure 3 and Table 2. We leave additional details and results to the Supplementary Material. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/338a8499adc828d15f481ffa113483aa902e4ce36c1bc209ecb9bcd67c7ddaf8.jpg",
+        "image_caption": [
+            "Figure 3: Samplers’ performance on four cases of the Ising model $\\left( 3 0 \\times 3 0 \\right)$ for the burn-in phase. (a) Case 1: Independent-noisy, (b) case 2: Independent-clean, (c) case 3: Dependent-noisy, (d) case 4: Dependent-clean "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2aa700630cc5103b1d20d15139e2d44e3254d9bf3128750955af1699f6679e3e.jpg",
+        "table_caption": [
+            "Table 2: Quantitative performance for mixing measured by effective sample size on the four cases of the Ising model $\\left( 3 0 \\times 3 0 \\right)$ . $\\operatorname* { m a x } \\{ 1 , t \\}$ is performing significantly worse in statistical terms than the other functions. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Setting</td><td>t 1+t</td><td>√t</td><td>min{1,t}</td><td>max{1,t}</td><td>LSB 1</td><td>LSB 2</td></tr><tr><td>Case 1</td><td>2.48± 0.21</td><td>2.30± 0.22</td><td>2.42±0.19</td><td>1.75 ± 0.17</td><td>2.50±0.28</td><td>2.46 ± 0.28</td></tr><tr><td>Case 2</td><td>3.33 ± 0.32</td><td>2.94 ± 0.36</td><td>3.33 ± 0.33</td><td>1.72 ± 0.18</td><td>2.98± 0.24</td><td>3.33 ± 0.43</td></tr><tr><td>Case 3</td><td>2.58± 0.73</td><td>1.99 ± 0.43</td><td>2.56±0.62</td><td>1.26 ± 0.12</td><td>2.48± 0.61</td><td>2.67± 0.84</td></tr><tr><td>Case 4</td><td>32.8± 9.2</td><td>18.5 ± 6.8</td><td>31.8 ± 10.0</td><td>2.60 ± 1.46</td><td>18.4± 8.0</td><td>30.8± 9.2</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/042d53adb2e2fb390d08e4c9fd57939312bc3cf49971cbf91ea79dc348e96c83.jpg",
+        "image_caption": [
+            "Figure 4: Realizations obtained after 500 (Case 3) and 300 (Case 4) burn-in iterations on the Ising model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "262 From Figure 3, we can see that our first parametrization LSB 1 is able to always \"select\" an unbounded   \n263 balancing function during burn-in, while when approaching convergence it is able to adapt to preserve   \n264 fast mixing, as measured by the effective sample size in Table 2. It’s interesting to mention also that   \n265 the softmax nonlinearity used in LSB 1 can sometimes slow down the adaptation due to vanishing   \n266 gradients. This can be observed by looking at the case 4 of Figure 3, where for a large part of the√   \n267 burn-in period the strategy prefers $\\operatorname* { m a x } \\{ 1 , t \\}$ over $\\sqrt { t }$ . Nevertheless, it is still able to recover a   \n268 solution different from $\\operatorname* { m a x } \\{ 1 , t \\}$ at the end of burn-in, as confirmed by the larger effective sample   \n269 size in Table 2 compared to the one achieved by max $\\{ 1 , t \\}$ .   \n70 Furthermore, we observe that our second parametrization LSB 2, which is functionally more expres  \n71 sive compared to LSB 1, allows to outperform all previous cases in terms of convergence speed, while   \n72 preserving optimal mixing, as shown in Figure 3 and Table 2. This provides further evidence that   \n73 the optimality of the balancing function is influenced by the target distribution and that exploiting   \n74 such information can dramatically boost the sampling performance (e.g. in case 3 of Figure 3, LSB 2   \n75 converges twice time faster as the best balancing function $\\sqrt { t } )$ ). Figure 4 provides some realizations   \n76 obtained by the samplers for the cases with dependent variables $\\lambda = 1$ . We clearly see from these   \n77 pictures that convergence for LSB 2 occurs at an earlier stage than the other balancing functions and   \n78 therefore the latent variables in the Ising model converge faster to their ground truth configuration. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/6a9b3b0d0556dcfdea293bc9dfa3d89873051262f20c7d68601dcfab54e07433.jpg",
+        "image_caption": [
+            "Figure 5: Samplers’ performance on Bayesian networks from UAI competition (100 variables). (a)-(b) are the traceplots for the burn-in phase, (c)-(d) are the autocorrelation functions for the mixing one. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            768,
+            794,
+            858
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/9244bd924c834329e6badebf9b0d140d0f6361efb5fb6d79f2d38cb0878f71d7.jpg",
+        "table_caption": [
+            "Table 3: Quantitative performance for mixing measured by effective sample size on two Bayesian networks from UAI competition. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td></td><td>Vt</td><td>min{1,t}</td><td>max{1,t}</td><td>LSB1</td><td>LSB 2</td></tr><tr><td>BN1</td><td>2.90±0.76</td><td>3.41 ± 0.77</td><td>2.54± 0.32</td><td>2.70 ± 0.63</td><td>3.19 ± 0.46</td><td>3.22 ± 0.38</td></tr><tr><td>BN2</td><td>3.43 ± 0.75</td><td>3.92 ± 0.94</td><td>3.78 ± 0.50</td><td>3.63 ± 0.67</td><td>3.52 ± 0.42</td><td>3.44 ± 0.44</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "279 5.2 Bayesian Networks: UAI data ",
+        "text_level": 1,
+        "bbox": [
+            147,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We evaluate how our strategy generalizes to different graph topologies compared to the one of the Ising model. In particular, we consider two Bayesian networks, with 100 discrete variables each and near-deterministic dependencies, from the 2006 UAI competition.6 Similarly to the previous experiments for the Ising model, we measure the performance over 5 repeated trials by analyzing the convergence speed during the burn-in phase (using traceplots) and the mixing time (computing the autocorrelation function and the effective sample size). Further details about the simulations are available in the Supplementary Material. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We observe that the proposed strategy is able to adapt to the target distribution, thus achieving fast convergence (Figure 5) while preserving fast mixing (both Figure 5 and Table 3) compared to existing balancing functions. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have presented a strategy to learn locally informed proposals for MCMC in discrete spaces. The strategy consists of (i) a new parametrization of balancing functions and (ii) a learning procedure adapting the proposal to the target distribution, in order to improve the sampling performance, both in terms of convergence speed and mixing. ",
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+    {
+        "type": "text",
+        "text": "Note that the LSB sampler belongs to the family of local sampling strategies, thus inheriting their limitations. The locality assumption can be quite restrictive, for example when sampling from discrete distributions with deterministic dependencies among variables. In such situations, local sampling might fail to correctly sample from the target in a finite amount of time, as being required to cross regions with zero probability mass. This remains an open challenge to be investigated in future work. ",
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+    {
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+        "text": "It’s important to mention that this work is foundational and general. The proposed strategy reduces the amount of user intervention and improves the sampling efficiency. However, these results could have potential impacts on society. At first glance, one could argue that the automation of the sampling procedure could have a negative impact on society as reducing the amount of human labour required to run the sampling strategy. However, we think that the benefits are of a far greater number compared to the negative aspects. In fact, the procedure reduces the costs of domain knowledge, with the advantage of democratizing the sampling strategy and reducing suboptimal configurations of the algorithm resulting from possibly wrong human decisions. Furthermore, the proposed strategy introduces a small amount of additional computation, which is used to reduce the amount of queries to the target distribution, thus improving the query and the sampling efficiency. We think that this last aspect could contribute positively towards devising more energy-efficient algorithms and therefore being more environment friendly. ",
+        "bbox": [
+            166,
+            570,
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            746,
+            266,
+            761
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "[1] H. Mohasel Afshar, J. Domke, et al. Reflection, Refraction, and Hamiltonian Monte Carlo. In NeurIPS, pages 3007–3015, 2015. ",
+        "bbox": [
+            179,
+            770,
+            823,
+            797
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "[2] N. Alexeev, J. Isomurodov, V. Sukhov, G. Korotkevich, and A. Sergushichev. Markov Chain Monte Carlo for Active Module Identification Problem. BMC Bioinformatics, 21(6):1–20, 2020. ",
+        "bbox": [
+            179,
+            808,
+            825,
+            835
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "[3] G. Altekar, S. Dwarkadas, J. P. Huelsenbeck, and F. Ronquist. Parallel Metropolis Coupled Markov Chain Monte Carlo for Bayesian Phylogenetic Inference. Bioinformatics, 20(3):407– 415, 2004. ",
+        "bbox": [
+            178,
+            845,
+            826,
+            887
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "[4] C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan. An Introduction to MCMC for Machine Learning. Machine learning, 50(1):5–43, 2003.   \n[5] C. Andrieu and J. Thoms. A Tutorial on Adaptive MCMC. Statistics and Computing, 18(4):343– 373, 2008.   \n[6] V. Dinh, A. Bilge, C. Zhang, and Frederick A. Matsen I. V. Probabilistic Path Hamiltonian Monte Carlo. In ICML, pages 1009–1018, 2017.   \n[7] S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth. Hybrid Monte Carlo. Physics letters B, 195(2):216–222, 1987.   \n[8] W. Gong, Y. Li, and J. M. Hernández-Lobato. Meta-Learning for Stochastic Gradient MCMC. In ICLR, 2019.   \n[9] U. Grenander and M. I. Miller. Representations of Knowledge in Complex Systems. Journal of the Royal Statistical Society: Series B (Methodological), 56(4):549–581, 1994.   \n[10] R. Habib and D. Barber. Auxiliary Variational MCMC. In ICLR, 2018.   \n[11] C. Hans, A. Dobra, and M. West. Shotgun Stochastic Search for “Large P” Regression. Journal of the American Statistical Association, 102(478):507–516, 2007.   \n[12] D. M. Higdon. Auxiliary Variable Methods for Markov Chain Monte Carlo with Applications. Journal of the American statistical Association, 93(442):585–595, 1998.   \n[13] M. D. Hoffman and A. Gelman. The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo. J. Mach. Learn. Res., 15(1):1593–1623, 2014.   \n[14] D. Koller and N. Friedman. Probabilistic Graphical Models: Principles and Techniques. MIT press, 2009.   \n[15] D. Levy, Matt D. H., and J. Sohl-Dickstein. Generalizing Hamiltonian Monte Carlo with Neural Networks. In ICLR, 2018.   \n[16] D. J. C. MacKay. Information Theory, Inference and Learning Algorithms. Cambridge university press, 2003.   \n[17] R. Neal. Probabilistic Inference Using Markov Chain Monte Carlo Methods. Department of Computer Science, University of Toronto Toronto, Ontario, Canada, 1993.   \n[18] E. Nijkamp, M. Hill, T. Han, S. C. Zhu, and Y. N. Wu. On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models. In AAAI, pages 5272–5280, 2020.   \n[19] A. Nishimura, D. B. Dunson, and J. Lu. Discontinuous Hamiltonian Monte Carlo for Discrete Parameters and Discontinuous Likelihoods. Biometrika, 107(2):365–380, 2020.   \n[20] A. Pakman and L. Paninski. Auxiliary-Variable Exact Hamiltonian Monte Carlo Samplers for Binary Distributions. In NeurIPS, 2013.   \n[21] C. Pasarica and A. Gelman. Adaptively Scaling the Metropolis Algorithm Using Expected Squared Jumped Distance. Statistica Sinica, pages 343–364, 2010.   \n[22] Pompe, E. and Holmes, C. and Łatuszynski, K. and others. A Framework for Adaptive MCMC ´ Targeting Multimodal Distributions. Annals of Statistics, 48(5):2930–2952, 2020.   \n[23] C. Robert and G. Casella. Monte Carlo Statistical Methods. Springer Science & Business Media, 2013.   \n[24] J. Sill. Monotonic Networks. In NeurIPS, pages 661–667, 1997.   \n[25] J. Song, S. Zhao, and S. Ermon. A-NICE-MC: Adversarial Training for MCMC. In NeurIPS, 2017.   \n[26] M. K. Titsias and C. Yau. The Hamming Ball Sampler. Journal of the American Statistical Association, 112(520):1598–1611, 2017.   \n[27] T. Wang, Y. Wu, D. A. Moore, and S. J. Russell. Meta-Learning MCMC Proposals. In NeurIPS, pages 4150–4160, 2018.   \n[28] G. Zanella. Informed Proposals for Local MCMC in Discrete Spaces. Journal of the American Statistical Association, 115(530):852–865, 2020.   \n[29] Y. Zhang, Z. Ghahramani, A. J. Storkey, and C. Sutton. Continuous Relaxations for Discrete Hamiltonian Monte Carlo. NeurIPS, 25:3194–3202, 2012. ",
+        "bbox": [
+            165,
+            97,
+            828,
+            921
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "bbox": [
+            173,
+            90,
+            826,
+            200
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[30] M. Zhu. Sample Adaptive MCMC. In NeurIPS, volume 32, 2019. ",
+        "bbox": [
+            158,
+            208,
+            643,
+            223
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
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+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] We included a paragraph in the conclusions (Section 6) discussing the main limitations of our strategy.   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] We included a paragraph in the conclusions (Section 6) discussing both pros and cons of the proposed work from the societal perspective.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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parse/train/qGeqg4_hA2/qGeqg4_hA2.md

@@ -0,0 +1,355 @@
+# Post-processing for Individual Fairness
+
+Felix Petersen ∗ University of Konstanz felix.petersen@uni.kn
+
+Debarghya Mukherjee ∗ University of Michigan mdeb@umich.edu
+
+Yuekai Sun University of Michigan yuekai@umich.edu
+
+Mikhail Yurochkin IBM Research, MIT-IBM Watson AI Lab mikhail.yurochkin@ibm.com
+
+# Abstract
+
+Post-processing in algorithmic fairness is a versatile approach for correcting bias in ML systems that are already used in production. The main appeal of postprocessing is that it avoids expensive retraining. In this work, we propose general post-processing algorithms for individual fairness (IF). We consider a setting where the learner only has access to the predictions of the original model and a similarity graph between individuals guiding the desired fairness constraints. We cast the IF post-processing problem as a graph smoothing problem corresponding to graph Laplacian regularization that preserves the desired “treat similar individuals similarly” interpretation. Our theoretical results demonstrate the connection of the new objective function to a local relaxation of the original individual fairness. Empirically, our post-processing algorithms correct individual biases in large-scale NLP models such as BERT, while preserving accuracy.
+
+# 1 Introduction
+
+There are many instances of algorithmic bias in machine learning (ML) models [1]–[4], which has led to the development of methods for quantifying and correcting algorithmic bias. To quantify algorithmic bias, researchers have proposed numerous mathematical definitions of algorithmic fairness. Broadly speaking, these definitions fall into two categories: group fairness [5] and individual fairness [6]. The former formalizes the idea that ML system should treat certain groups of individuals similarly, e.g., requiring the average loan approval rate for applicants of different ethnicities be similar [7]. The latter asks for similar treatment of similar individuals, e.g., same outcome for applicants with resumes that differ only in names [8]. Researchers have also developed many ways of correcting algorithmic bias. These fairness interventions broadly fall into three categories: pre-processing the data, enforcing fairness during model training (also known as in-processing), and post-processing the outputs of a model.
+
+While both group and individual fairness (IF) definitions have their benefits and drawbacks [5], [6], [9], the existing suite of algorithmic fairness solutions mostly enforces group fairness. The few prior works on individual fairness are all in-processing methods [10]–[13]. Although in-processing is arguably the most-effective type of intervention, it has many practical limitations. For example, it requires training models from scratch. Nowadays, it is more common to fine-tune publicly available models (e.g., language models such as BERT [14] and GPT-3 [15]) than to train models afresh, as many practitioners do not have the necessary computational resources. Even with enough computational resources, training large deep learning models has a significant environmental impact [4], [16].
+
+Post-processing offers an easier path towards incorporating algorithmic fairness into deployed ML models, and has potential to reduce environmental harm from re-training with in-processing fairness techniques.
+
+In this paper, we propose a computationally efficient method for post-processing off-the-shelf models to be individually fair. We consider a setting where we are given the outputs of a (possibly unfair) ML model on a set of $n$ individuals, and side information about their similarity for the ML task at hand, which can either be obtained using a fair metric on the input space or from external (e.g., human) annotations. Our starting point is a post-processing version of the algorithm by Dwork et al. [6] (see (2.3)). Unfortunately, this method has two drawbacks: poor scalability and an unfavorable trade-off with accuracy. As we shall see, the sharp trade-off is due to the restrictions imposed on dissimilar individuals by Dwork et al. [6]’s global Lipschitz continuity condition. By relaxing these restrictions on dissimilar individuals, we obtain a better trade-off between accuracy and fairness, while preserving the intuition of treating similar individuals similarly. This leads us to consider a graph signal-processing approach to IF post-processing that only enforces similar outputs between similar individuals. The nodes in the underlying graph correspond to individuals, edges (possibly weighted) indicate similarity, and the signal on the graph is the output of the model on the corresponding node-individuals. To enforce IF, we use Laplacian regularization [17], which encourages the signal to
+
+be smooth on the graph. We illustrate this idea in Figure 1: a biased model decides whom to show a job ad for a Python programming job based on their CVs and chooses Charlie and Dave but excludes Alice. However, from the qualifications, we can see that Alice, Charlie and Dave are similar because they all have experience in Python, which is the job requirement, and thus should be treated similarly. We represent all five candidates as nodes in a graph, where the node signal (checkmark or cross) is the model’s decision for the corresponding candidate, and the edge weights are indicated by the thickness of the connecting line. Alice and Charlie have the same qualifications and are therefore connected with a large edge-weight. For the predictions to satisfy IF, the graph needs to be smooth, i.e., the similar / connected candidates should have similar node
+
+![](images/f23f92c1b7b23609fd732c54d7541ee19aa81b25fe90a8adab304cc5cb266631.jpg)  
+Figure 1: IF on a graph.
+
+signals, which can be accomplished by also offering the job to Alice. In contrast, directly enforcing IF constraints [6] requires a certain degree of output similarity on all pairs of candidates, and not just on those which are connected and thus similar. Our main contributions are summarized below.
+
+1. We cast post-processing for individual fairness as a graph smoothing problem and propose a coordinate descent algorithm to scale the approach to large data sets.   
+2. We demonstrate theoretically and verify empirically that graph smoothing enforces individual fairness constraints locally, i.e., it guarantees similar treatment of similar individuals.   
+3. We empirically compare the Laplacian smoothing method to the post-processing adaptation of the algorithm by Dwork et al. [6] enforcing global Lipschitz continuity. The Laplacian smoothing method is not only computationally more efficient but is also more effective in reducing algorithmic bias while preserving accuracy of the original model.   
+4. We demonstrate the efficacy of Laplacian smoothing on two large-scale text data sets by reducing biases in fine-tuned BERT models.
+
+# 2 Post-processing Problem Formulation
+
+Let $\mathcal { X }$ be the feature space, $\mathcal { V }$ be the set of possible labels/targets, and $h : \mathcal { X }  \mathcal { Y }$ be a (possibly unfair) ML model trained for the task. Our goal is to post-process the outputs of $h$ so that they are individually fair. Formally, the post-processor is provided with a set of inputs $\{ x _ { i } \} _ { i = 1 } ^ { n }$ and the outputs of $h$ on the inputs $\{ \widehat { y } _ { i } \triangleq h ( x _ { i } ) \} _ { i = 1 } ^ { n }$ , and its goal is to produce $\{ \widehat { f } _ { i } \} _ { i = 1 } ^ { n }$ that is both individually fair and similar to the $\widehat { y } _ { i }$ b ’s. Recall that individual fairness of $h$ is the Lipschitz continuity of $h$ with respect
+
+to a fair metric $d _ { \mathcal { X } }$ on the input space:
+
+$$
+d y ( h ( x ) , h ( x ^ { \prime } ) ) \leq L d _ { \mathcal { X } } ( x , x ^ { \prime } ) \mathrm { ~ f o r ~ a l l ~ } x , x ^ { \prime } \in \mathcal { X } ,
+$$
+
+where $L > 0$ is a Lipschitz constant. The fair metric encodes problem-specific intuition of which samples should be treated similarly by the ML model. It is analogous to the knowledge of protected attributes in group fairness needed to define corresponding fairness constraints. Recent literature proposes several practical methods for learning fair metric from data [18], [19]. We assume the postprocessor is either given access to the fair metric (it can evaluate the fair distance on any pair of points in $\mathcal { X }$ ), or receives feedback on which inputs should be treated similarly. We encode this information in an adjacency matrix $W \in \mathbb { R } ^ { n \times n }$ of a graph with individuals as nodes. If the post-processor is given the fair metric, then the entries of $W$ are
+
+$$
+W _ { i j } = { \left\{ \begin{array} { l l } { \exp ( - \theta d _ { \mathcal { X } } ( x _ { i } , x _ { j } ) ^ { 2 } ) } & { d _ { \mathcal { X } } ( x _ { i } , x _ { j } ) \leq \tau } \\ { 0 } & { { \mathrm { o t h e r w i s e } } , } \end{array} \right. }
+$$
+
+where $\theta > 0$ is a scale parameter and $\tau > 0$ is a threshold parameter. If the post-processor is given an annotator’s feedback, then $W$ is a binary matrix with $W _ { i j } = 1$ if $i$ and $j$ are considered to be treated similarly by the annotator and 0 otherwise. Extensions to multiple annotators are straightforward.
+
+We start with a simple post-processing adaptation of the algorithm by Dwork et al. [6] for enforcing individual fairness, that projects the (possibly unfair) outputs of $h$ onto a constraint set to enforce (2.1). In other words, the post-processor seeks the closest set of outputs to the $\widehat { y } _ { i }$ ’s that satisfies individual fairness:
+
+$$
+\begin{array} { r } { \{ \widehat { f } _ { i } \} _ { i = 1 } ^ { n } \in \left\{ \begin{array} { l l } { \arg \operatorname* { m i n } _ { f _ { 1 } , \ldots , f _ { n } } } & { \sum _ { i = 1 } ^ { n } \frac { 1 } { 2 } d y ( f _ { i } , \widehat { y } _ { i } ) ^ { 2 } } \\ { \mathrm { s u b j e c t ~ t o } } & { d y ( f _ { i } , f _ { j } ) \leq L d _ { \mathcal { X } } ( x _ { i } , x _ { j } ) } \end{array} \right\} . } \end{array}
+$$
+
+This objective function, though convex, scales poorly due to the order of $n ^ { 2 }$ constraints. Empirically, we observe that (2.3) leads to post-processed outputs that are dissimilar to the $\widehat { y } _ { i }$ ’s, leading to poor performance in practice. The goal of our method is to improve performance and scalability, while preserving the IF desiderata of treating similar individual similarly. Before presenting our method, we discuss other post-processing perspectives that differ in their applicability and input requirements.
+
+# 2.1 Alternative Post-processing Formulations
+
+We review three post-processing problem setups and the corresponding methods in the literature. First, one can fine-tune a model via an in-processing algorithm to reduce algorithmic biases. Yurochkin et al. [12] proposed an in-processing algorithm for IF and used it to train fair models for text classification using sentence BERT embeddings. This setting is the most demanding in terms of input and computational requirements: a user needs access to the original model parameters, fair metric function, and train a predictor, e.g., a moderately deep fully connected neural network, with a non-trivial fairness-promoting objective function.
+
+Second, it is possible to post-process by training additional models to correct the initial model’s behavior. For example, Kim et al. [20] propose a boosting-based method for group fairness postprocessing. This perspective can be adapted to individual fairness; however, it implicitly assumes that we can train weak-learners to boost. Lohia et al. [21], [22] propose to train a bias detector to post-process for group fairness and a special, group based, notion of individual fairness. Such methods are challenging to apply to text data or other non-tabular data types.
+
+The third perspective is the most generic: a user has access to original model outputs only, and a minimal additional feedback guiding fairness constraints. Wei et al. [23] consider such setting and propose a method to satisfy group fairness constraints; however, it is not applicable to individual fairness. Our problem formulation belongs to this post-processing setup. The main benefit of this approach is its broad applicability and ease of deployment.
+
+# 3 Graph Laplacian Individual Fairness
+
+To formulate our method, we cast IF post-processing as a graph smoothing problem. Using the fair metric or human annotations as discussed in Section 2, we obtain an $n \times n$ matrix $W$ that we treat as an adjacency matrix. As elaborated earlier, the goal of post-processing is to obtain a model $f$ that is individually fair and accurate. The accuracy is achieved by minimizing the distance between the outputs of $f$ and $h$ , a pre-trained model assumed to be accurate but possibly biased. Recall that we do not have access to the parameters of $h$ , but can evaluate its predictions. Our method enforces fairness using a graph Laplacian quadratic form [24] regularizer:
+
+$$
+\widehat { \mathbf { f } } = \arg \operatorname* { m i n } _ { \mathbf { f } } { g _ { \lambda } ( \mathbf { f } ) } = \arg \operatorname* { m i n } _ { \mathbf { f } } \ \| \mathbf { f } - \hat { \mathbf { y } } \| _ { 2 } ^ { 2 } + \lambda \mathbf { f } ^ { \top } \mathbb { L } _ { n } \mathbf { f } ,
+$$
+
+where $\hat { \mathbf { y } }$ is the output of the model $h$ , and $\widehat { \mathbf { f } }$ is the vector of the post-processed outputs, i.e., ${ \widehat { f } } _ { i } = f ( x _ { i } )$ for $i = 1 , \ldots , n$ . The matrix $\mathbb { L } _ { n } \in \mathbb { R } ^ { n \times n }$ is called graph Laplacian matrix and is a function of $W$ . There are multiple versions of $\mathbb { L } _ { n }$ popularized in the graph literature (see, e.g., [25] or [26]). To elucidwhere t, ual f for s, consider the unnormalized Laplaciais the degree matrix corresponding to $\mathbb { L } _ { u n , n } = D - W$ , $\begin{array} { r } { D _ { i i } = \sum _ { j = 1 } ^ { n } W _ { i j } } \end{array}$ $D _ { i j } = 0$ $i \neq j$ $W$ identity is:
+
+$$
+\begin{array} { r } { \mathbf { f } ^ { \top } \mathbb { L } _ { u n , n } \mathbf { f } = \frac { 1 } { 2 } \sum _ { i \neq j } W _ { i j } \left( f _ { i } - f _ { j } \right) ^ { 2 } . } \end{array}
+$$
+
+Hence, the Laplacian regularizer is small if the post-processed model outputs $\widehat { f } _ { i }$ and ${ \widehat { f } } _ { j }$ (i.e., treatment) are similar for large $W _ { i j }$ (i.e., for similar individuals $i$ and $j$ ). This promotes the philosophy of individual fairness: “treat similar individuals similarly”. This observation intuitively explains the motivation for minimizing the graph Laplacian quadratic form to achieve IF. In Section 4, we present a more formal discussion on the connections between the graph Laplacian regularization and IF.
+
+Our post-processing problem (3.1) is easy to solve: setting the gradient of $g _ { \lambda }$ to 0 implies that the optimal solution $\widehat { \mathbf { f } }$ is:
+
+$$
+\widehat { \mathbf { f } } = \left( I + \lambda \left( \frac { \mathbb { L } _ { n } + \mathbb { L } _ { n } ^ { \top } } { 2 } \right) \right) ^ { - 1 } \widehat { \mathbf { y } } .
+$$
+
+The Laplacian $\mathbb { L } _ { n }$ is a positive semi-definite matrix ensuring that (3.1) is strongly convex and that (3.3) is a global minimum. In comparison to the computationally expensive constraint optimization problem (2.3), this approach has a simple closed-form expression.
+
+Note that the symmetry of the unnormalized Laplacian $\mathbb { L } _ { u n , n }$ simplifies (3.3); however, there are also non-symmetric Laplacian variations. In this work, we also consider the normalized random walk Laplacian $\mathbb { L } _ { n r w , n } = ( I - \widetilde { D } ^ { - 1 } \widetilde { W } )$ , where $\widetilde { W } = D ^ { - 1 / 2 } W D ^ { - 1 / 2 }$ is the normalized adjacency matrix and De is its degree matrix. We discuss its properties in the context of IF in Section 4. Henceforth, we refer to our method as Graph Laplacian Individual Fairness (GLIF) when using the unnormalized Laplacian, and GLIF-NRW when using Normalized Random Walk Laplacian.
+
+# 3.1 Prior Work on Graph Laplacians
+
+Graph-based learning via a similarity matrix is prevalent in statistics and ML literature, specifically, in semi-supervised learning. The core idea is to gather information from similar unlabeled inputs to improve prediction accuracy (e.g., see [27], [28], [29] and references therein). Laplacian regularization is widely used in science engineering. We refer to Chapelle [17] for a survey.
+
+We note that [30], [31] also use graph Laplacian regularizers to enforce individual fairness. Our work builds on their work by elucidating the key role played by the graph Laplacian in enforcing individual fairness. In particular, we clarify the connection between the choice of the graph Laplacian and the exact notion of individual fairness the corresponding graph Laplacian regularizer enforces.
+
+# 3.2 Extensions of the Basic Method
+
+In this subsection, we present four extensions of our method: multi-dimensional outputs, coordinate descent for large-scale data, an inductive setting, and alternative output space discrepancy measures.
+
+# 3.2.1 Multi-dimensional Output
+
+We presented our objective function (3.1) and post-processing procedure (3.3) for the case of univariate outputs. This covers regression and binary classification. Our method readily extends to multi-dimensional output space, for example, in classification, $f _ { i } , \widehat { y } _ { i } \in \mathbb { R } ^ { K }$ can represent logits, i.e.,
+
+softmax inputs, of the $K$ classes. In this case, f and $\hat { \mathbf { y } }$ are $n \times K$ matrices, and the term f $\mathbf { : } ^ { \top } \mathbb { L } _ { n } \mathbf { f }$ is a $K \times K$ matrix. We use the trace of it as a regularizer. The optimization problem (3.1) then becomes:
+
+$$
+\widehat { \mathbf { f } } = \arg \operatorname* { m i n } _ { f } g _ { \lambda } ( \mathbf { f } ) = \arg \operatorname* { m i n } _ { f } \| \mathbf { f } - \widehat { \mathbf { y } } \| _ { F } ^ { 2 } + \lambda \operatorname { t r } \left( \mathbf { f } ^ { \top } \mathbb { L } _ { n } \mathbf { f } \right) ,
+$$
+
+where $\| \cdot \| _ { F }$ is the Frobenius norm. Similar to the univariate output case, this yields:
+
+$$
+\widehat { \mathbf { f } } = \left( I + \lambda \left( \frac { \mathbb { L } _ { n } + \mathbb { L } _ { n } ^ { \top } } { 2 } \right) \right) ^ { - 1 } \widehat { \mathbf { y } } .
+$$
+
+The solution is the same as (3.3); however, now it accounts for multi-dimensional outputs.
+
+# 3.2.2 Coordinate Descent for Large-Scale Data
+
+Although our method has a closed form solution, it is not immediately scalable, as we have to invert a $n \times n$ matrix to obtain the optimal solution. We propose a coordinate descent variant of our method that readily scales to any data size. The idea stems primarily from the gradient of equation (3.4), where we solve:
+
+$$
+\mathbf { f } - \widehat { \mathbf { y } } + \lambda \frac { \mathbb { L } _ { n } + \mathbb { L } _ { n } ^ { \top } } { 2 } \mathbf { f } = 0 .
+$$
+
+Fixing $\{ f _ { j } \} _ { j \neq i }$ , we can solve (3.6) for $f _ { i }$
+
+$$
+f _ { i } \gets \frac { \hat { y } _ { i } - \frac { \lambda } { 2 } \sum _ { j \neq i } ( \mathbb { L } _ { n , i j } + \mathbb { L } _ { n , j i } ) f _ { j } } { 1 + \lambda \mathbb { L } _ { n , i i } } .
+$$
+
+This gives rise to the coordinate descent algorithm. We perform asynchronous updates over randomly selected coordinate batches until convergence. We refer the reader to Wright [32] and the references therein for the convergence properties of (asynchronous) coordinate descent.
+
+# 3.2.3 Extension to the Inductive Setting
+
+This coordinate descent update is key to extending our approach to the inductive setting. To handle new unseen points, we assume we have a set of test points on which we have already post-processed the outputs of the ML model. To post-process new unseen points, we simply fix the outputs of the other test points and perform a single coordinate descent step with respect to the output of the new point. Similar strategies are often employed to extend transductive graph-based algorithms to the inductive setting [17].
+
+# 3.2.4 Alternative Discrepancy Measures on the Output Space
+
+So far, we have considered the squared Euclidean distance as a measure of discrepancy between outputs. This is a natural choice for post-processing models with continuous-valued outputs. For models that output a probability distribution over the possible classes, we consider alternative discrepancy measures on the output space. It is possible to replace the squared Euclidean distance with a Bregman divergence with very little change to the algorithm in the case of the unnormalized Laplacian. Below, we work through the details for the KL divergence as a demonstration of the idea. A result for the general Bregman divergence can be found in Appendix B.3 (see Theorem B.4).
+
+Suppose the output of the pre-trained model $h$ is $\hat { y } _ { i } \in \Delta ^ { K }$ , where $\begin{array} { r } { \hat { y } _ { i } = \{ e ^ { o _ { i , j } } / \sum _ { k = 1 } ^ { K } e ^ { o _ { i , k } } \} _ { j = 1 } ^ { K } } \end{array}$ a $K$ - dimensional probability vector corresponding to a $K$ class classification problem $\left. { o _ { i , j } } \right\}$ is the output of tand del and is the p $\hat { y } _ { i }$ is obtained by passbability simplex in it thr. Let h softmax) denote the $\Delta ^ { K } = \{ x \in \mathbb { R } ^ { K } : x _ { i } \geq 0 , \sum _ { i = 1 } ^ { K } x _ { i } = 1 \}$ $\mathbb { R } ^ { K }$ $P _ { v }$ multinomial distribution with success probabilities $v$ for any $v \in \Delta ^ { k }$ . Define $\hat { \eta } _ { i } \in \mathbb { R } ^ { K - 1 }$ (resp. $\eta _ { i }$ ) as the natural parameter corresponding to $\hat { y } _ { i }$ (resp. $f _ { i }$ ), i.e., $\hat { \eta } _ { i , j } = \log \left( \hat { y } _ { i , j } / \hat { y } _ { i , K } \right) = o _ { i , j } - o _ { i , K }$ for $1 \leq j \leq K - 1$ . The (unnormalized) Laplacian smoothing problem with the KL divergence is
+
+$$
+\left( \widetilde { y } _ { 1 } , \dots , \widetilde { y } _ { n } \right) = \arg \operatorname* { m i n } _ { y _ { 1 } , \dots , y _ { n } \in \Delta ^ { K } } \left[ \sum _ { i } \left\{ \mathsf { K L } \left( P _ { y _ { i } } | | P _ { \widetilde { y } _ { i } } \right) + \frac { \lambda } { 2 } \sum _ { j = 1 , j \neq i } ^ { n } W _ { i j } \mathsf { K L } \left( P _ { y _ { i } } | | P _ { y _ { j } } \right) \right\} \right] .
+$$
+
+A coordinate descent approach for solving the above equation is:
+
+$$
+\begin{array} { r } { \tilde { y } _ { i } = \arg \operatorname* { m i n } _ { y \in \Delta ^ { k } } \left\{ \mathsf { K L } \left( P _ { y } | | P _ { \hat { y } _ { i } } \right) + \frac { \lambda } { 2 } \sum _ { j = 1 , j \neq i } ^ { n } W _ { i j } \mathsf { K L } \left( P _ { y } | | P _ { \tilde { y } _ { j } } \right) \right\} . } \end{array}
+$$
+
+The following theorem establishes that (3.5) solves the above problem in the logit space, or equivalently in the space of the corresponding natural parameters (see Appendix $\mathbf { B }$ for the proof):
+
+Theorem 3.1. Consider the following optimization problem on the space of natural parameters:
+
+$$
+\begin{array} { r } { \widetilde { \eta } _ { i } = \arg \operatorname* { m i n } _ { \eta } \left[ \| \eta - \widehat { \eta } _ { i } \| ^ { 2 } + \frac { \lambda } { 2 } \sum _ { j = 1 , j \neq i } ^ { n } W _ { i j } \| \eta - \widetilde { \eta } _ { j } \| ^ { 2 } \right] . } \end{array}
+$$
+
+Then, the minimizer $\tilde { \eta } _ { i }$ of equation (3.10) is the natural parameter corresponding to the minimizer $\tilde { y } _ { i }$ of (3.8).
+
+# 4 Local IF and Graph Laplacian Regularization
+
+In this section, we provide theoretical insights into why the graph Laplacian regularizer enforces individual fairness. As pointed out in Section 2, enforcing IF globally is expensive and often reduces a significant amount of accuracy of the final classifier. Here, we establish that solving (3.1) is tantamount to enforcing a localized version of individual fairness, namely Local Individual Fairness, which is defined below:
+
+Definition 4.1 (Local Individual Fairness). An ML model $h$ is said to be locally individually fair if it
+
+$$
+\mathbb { E } _ { x \sim P } \left[ \operatorname* { l i m } _ { x ^ { \prime } : d _ { \mathcal { X } } ( x , x ^ { \prime } ) \downarrow 0 } \frac { d _ { \mathcal { Y } } ( h ( x ) , h ( x ^ { \prime } ) ) } { d _ { \mathcal { X } } ( x , x ^ { \prime } ) } \right] \leq L < \infty .
+$$
+
+For practical purposes, this means that $h$ is locally individually fair with constants $\epsilon$ and $L$ if it satisfies
+
+$$
+d y ( h ( x ) , h ( x ^ { \prime } ) ) \leq L d _ { \mathcal { X } } ( x , x ^ { \prime } ) \mathrm { ~ f o r ~ a l l ~ } x , x ^ { \prime } \in \mathcal { X } \mathrm { ~ w h e r e ~ } d _ { \mathcal { X } } ( x , x ^ { \prime } ) \leq \epsilon
+$$
+
+in analogy to equation (2.1). Equation (4.2) is a relaxation of traditional IF, where we only care about the Lipschitz-constraint for all pairs of points with small fair distances, i.e., where it is less than some user-defined threshold $\epsilon$ .
+
+Example 4.2. For our theoretical analysis, we need to specify a functional form of the fair metric. A popular choice is a Mahalanobis fair metric proposed by $I I 9 J$ , which is defined as:
+
+$$
+d _ { \mathcal { X } } ^ { 2 } ( \boldsymbol { x } , \boldsymbol { x } ^ { \prime } ) = ( \boldsymbol { x } - \boldsymbol { x } ^ { \prime } ) ^ { \top } \Sigma ( \boldsymbol { x } - \boldsymbol { x } ^ { \prime } ) ,
+$$
+
+where $\Sigma$ is a dispersion matrix that puts lower weight in the directions of sensitive attributes and higher weight in the directions of relevant attributes. [19] also proposed several algorithms to learn such a fair metric from the data. If we further assume $d y ( y _ { 1 } , y _ { 2 } ) = | y _ { 1 } - y _ { 2 } |$ , then a simple application of Lagrange’s mean value theorem yields:
+
+$$
+\operatorname* { l i m } _ { x ^ { \prime } : d _ { \mathcal { X } } ( x , x ^ { \prime } ) \downarrow 0 } \frac { | h ( x ) - h ( x ^ { \prime } ) | } { d _ { \mathcal { X } } ( x , x ^ { \prime } ) } \leq \| \Sigma ^ { - 1 / 2 } \nabla h ( x ) \| .
+$$
+
+This immediately implies:
+
+$$
+\mathbb { E } _ { x \sim P } \left[ \operatorname* { l i m } _ { { x ^ { \prime } : d _ { \mathcal { X } } ( x , x ^ { \prime } ) } \downarrow 0 } \frac { d _ { \mathcal { Y } } ( h ( x ) , h ( x ^ { \prime } ) ) } { d _ { \mathcal { X } } ( x , x ^ { \prime } ) } \right] \leq \mathbb { E } [ \| { \Sigma } ^ { - 1 / 2 } \nabla h ( x ) \| ] ,
+$$
+
+i.e., h satisfies local individual fairness constraint as long as $\mathbb { E } [ | | \Sigma ^ { - 1 / 2 } \nabla h ( x ) | | ] < \infty$ . On the other hand, the global $I F$ constraint necessitates $\begin{array} { r } { \operatorname* { s u p } _ { x \in \mathcal { X } } \| \Sigma ^ { - 1 / 2 } \nabla h ( x ) \| < \infty , } \end{array}$ , i.e., h is Lipschitz continuous with respect to the Mahalanobis distance.
+
+The main advantage of this local notion of IF over its global counterpart is that the local definition concentrates on the input pairs with smaller fair distance and ignores those with larger distance. For example, in Figure 1, the edge-weights among Alice, Charlie, and Dave are much larger than among any other pairs (which have a weight of 0); therefore, our local notion enforces fairness constraint on the corresponding similar pairs, while ignoring (or being less stringent on) others. This prevents over-smoothing and consequently preserves accuracy while enforcing fairness as is evident from our real data experiments in Section 5.
+
+We now present our main theorem, which establishes that, under certain assumptions on the underlying hypothesis class and the distribution of inputs, the graph Laplacian regularizers (both unnormalized and normalized random walk) enforce the local IF constraint (as defined in Definition 4.1) in the limit. For our theory, we work with $d _ { \mathcal { X } }$ as the Mahalanobis distance introduced in Example 4.2 in equation (2.2) along with $\theta = 1 / ( 2 \sigma ^ { 2 } )$ $\sigma$ is a bandwidth parameter which goes to 0 at an appropriate rate as $n \to \infty$ ) and $\tau = \infty$ . All our results will be thorough for any finite $\tau$ but with more tedious technical analysis. Therefore, our weight matrix $W$ becomes:
+
+$$
+W _ { i j } = \frac { | \Sigma | ^ { 1 / 2 } } { ( 2 \pi ) ^ { d / 2 } \sigma ^ { d } } \mathrm { e x p } \left( - \frac { 1 } { 2 \sigma ^ { 2 } } ( x _ { i } - x _ { j } ) ^ { \top } \Sigma \left( x _ { i } - x _ { j } \right) \right) .
+$$
+
+The constant $| \Sigma | ^ { 1 / 2 } / ( ( 2 \pi ) ^ { d / 2 } \sigma ^ { d } )$ is for the normalization purpose and can be absorbed into the penalty parameter $\lambda$ . We start by listing our assumptions:
+
+Assumption 4.3 (Assumption on the domain). The domain of the inputs $\mathcal { X }$ is a compact subset of $\mathbb { R } ^ { d }$ where $d$ is the underlying dimension.
+
+Assumption 4.4 (Assumption on the hypothesis). All functions $f \in { \mathcal { F } }$ of the hypothesis class satisfy the following:
+
+1. The $i ^ { t h }$ derivative $f ^ { ( i ) }$ is uniformly bounded over the domain $\mathcal { X }$ of inputs for $i \in \{ 0 , 1 , 2 \}$ .
+
+2. $f ^ { ( 1 ) } ( x ) = 0$ for all $x \in \partial X$ , where $\partial X$ denotes the boundary of $\mathcal { X }$ .
+
+Assumption 4.5 (Assumption on the density of inputs). The density $p$ of the input random variable x on the domain $\mathcal { X }$ satisfies the following:
+
+1. There exists $p _ { \operatorname* { m a x } } < \infty$ and $p _ { \operatorname* { m i n } } > 0$ such that, for all $x \in \mathcal { X }$ , we have $p _ { \operatorname* { m i n } } \le p ( x ) \le p _ { \operatorname* { m a x } }$ .   
+2. The derivatives $\{ p ^ { ( i ) } \} _ { i = 0 , 1 , 2 }$ of the density $p$ are uniformly bounded on the domain $\mathcal { X }$ .
+
+Discussion on the assumptions Most of our assumptions (e.g., compactness of the domain, bounded derivatives of $f$ or $p$ ) are for technical simplicity and are fairly common for the asymptotic analysis of graph regularization (see, e.g., Hein et al. [25], [33] and references therein). It is possible to relax some of the assumptions: for example, if the domain $\mathcal { X }$ of inputs is unbounded, then the target function $f$ and the density $p$ should decay at certain rate so that observations far away will not be able to affect the convergence (e.g., sub-exponential tails). Part (2.) of Assumption 4.4 can be relaxed if we assume $p ( x )$ is 0 at boundary. However, we do not pursue these extensions further in this manuscript, as they are purely technical and do not add anything of significance to the main intuition of the result.
+
+Theorem 4.6. Under Assumptions 4.3 - 4.5, we have:
+
+1. If the sequence of bandwidths $\sigma \equiv \sigma _ { n } \downarrow 0$ such that $n \sigma _ { n } ^ { 2 } \to \infty$ and $\mathbb { L } _ { u n , n }$ is unnormalized Laplacian matrix, then
+
+$$
+\frac { 2 } { n ^ { 2 } \sigma ^ { 2 } } \mathbf { f } ^ { \top } \mathbb { L } _ { u n , n } \mathbf { f } \xrightarrow { P } \mathbb { E } _ { x \sim p } \left[ \nabla f ( x ) ^ { \top } \Sigma ^ { - 1 } \nabla f ( x ) p ( x ) \right] .
+$$
+
+2. If the sequence of bandwidths $\sigma \equiv \sigma _ { n } \downarrow 0$ such that $( n \sigma ^ { d + 4 } ) / ( \log { ( 1 / \sigma ) } ) \to \infty$ and $\mathbb { L } _ { n r w , n }$ is the normalized random walk Laplacian matrix, then:
+
+$$
+\frac { 1 } { n \sigma ^ { 2 } } \mathbf { f } ^ { \top } \mathbb { L } _ { n r w , n } \mathbf { f } \stackrel { P } { \longrightarrow } \mathbb { E } _ { x \sim p } \left[ \nabla f ( x ) ^ { \top } { \Sigma } ^ { - 1 } \nabla f ( x ) \right] .
+$$
+
+where $\mathbf { f } = \{ f ( x _ { i } ) \} _ { i = 1 } ^ { n }$ . Consequently, both Laplacian regularizers asymptotically enforce local IF.
+
+The proof of the above theorem can be found in Appendix B. When we use a normalized random walk graph Laplacian matrix $\mathbb { L } _ { n r w , n }$ as regularizer, the regularizer does (asymptotically) penalize $\mathbb { E } \left[ \nabla f ( x ) ^ { \top } \Sigma ^ { - 1 } \nabla f ( x ) \right] = \mathbb { E } \left[ \| \Sigma ^ { - 1 / 2 } \nabla f ( x ) \| ^ { 2 } \right]$ , which, by Example 4.2, is equivalent to enforcing the local IF constraint. Similarly, the un-normalized Laplacian matrix $\mathbb { L } _ { u n , n }$ , also enforces the same under Assumption 4.5 as:
+
+$$
+\mathbb { E } \left[ \Vert \Sigma ^ { - 1 / 2 } \nabla f ( x ) \Vert ^ { 2 } \right] \leq \frac { 1 } { p _ { \operatorname* { m i n } } } \mathbb { E } \left[ \nabla f ( x ) ^ { \top } \Sigma ^ { - 1 } \nabla f ( x ) p ( x ) \right] , \mathrm { ~ w h e r e ~ } p _ { \operatorname* { m i n } } = \operatorname* { i n f } _ { x \in \mathcal { X } } p ( x ) .
+$$
+
+Although both the Laplacian matrices enforce local IF, the primary difference between them is that the limit of the unnormalized Laplacian involves the density $p ( x )$ , i.e., it upweights the high-density region (consequently stringent imposition of fairness constraint), whereas it down-weights the underrepresented/low-density region. On the other hand, the limit corresponding to the normalized random walk Laplacian matrix does not depend on $p ( x )$ and enforces fairness constraint with equal intensity on the entire input space. We used both regularizers in our experiments, comparing and contrasting their performance on several practical ML problems.
+
+# 5 Experiments
+
+The goals of our experiments are threefold:
+
+1. Exploring the trade-offs between post-processing for local IF with GLIF and post-processing with (global) IF constraints using our adaptation of the algorithm by Dwork et al. [6] described in (2.3).   
+2. Studying practical implications of theoretical differences between GLIF and GLIF-NRW, i.e., different graph Laplacians, presented in Section 4.   
+3. Evaluating the effectiveness of GLIF in its main application, i.e., computationally light debiasing of large deep learning models such as BERT.
+
+The implementation of this work is available at github.com/Felix-Petersen/fairness-post-processing.
+
+# 5.1 Comparing GLIF and Global IF-constraints
+
+For our first experiment, we consider the sentiment prediction task [34], where our goal is to classify words as having a positive or negative sentiment. The baseline model is a neural network trained with GloVe word embeddings [35]. Following Yurochkin et al. [11], we evaluate the model on a set of names and observe that it assigns varying sentiments to names. An individually fair model should assign similar sentiment scores to all names. Further, we observe that there is a gap between average sentiments of names typical for Caucasian and African-American ethnic groups [36], which is violating group fairness. Yurochkin et al. [11] propose a fair metric learning procedure for this task using a side data set of names, and an in-processing technique for achieving individual fairness. We use their method to obtain the fair metric and compare post-processing of the baseline model with GLIF, GLIF-NRW and the global IF-constraints method. The test set comprises 663 words from the original task and 94 names. For post-processing, no problem specific knowledge is used. The resulting post-processed predictions for the original test set are used to evaluate accuracy, and the predictions on the names are used for evaluating fairness metrics. Even for this small problem, the global IF-constraints method, i.e., a CVXPY [37] implementation of (2.3), takes 7 minutes to run. Due to the poor scalability of the global IF-constraints method, we can use it only for the study of this smaller data set and can not consider it for the large language model experiments in Section 5.2. For GLIF(-NRW), we implement the closed-form solution (3.3) that takes less than a tenth of a second to run. See Appendix A for additional experimental details and a runtime analysis.
+
+We evaluate the fairness-accuracy trade-offs for a range of threshold parameters $\tau$ (for GLIF and GLIF-NRW) and for a range of Lipschitz-constants $L$ (for IF-constraints) in Figure 2. Figure 2 (left) shows the standard deviation of the post-processed outputs on all names as a function of test accuracy on the original sentiment task. Lower standard deviations imply that all names received similar predictions, which is the goal of individual fairness. Figure 2 (center) visualizes group fairness and accuracy, i.e., difference in average name sentiment scores for the two ethnic groups. In this problem, individual fairness is a stronger notion of fairness: achieving similar predictions for all names implies similar group averages, but not vice a versa. Therefore, for this task, post-processing for individual fairness also corrects group disparities.
+
+![](images/918fbdc2656ad0da27cbad9688ad1474168f32e745c722f2bcebde4c7b9eda00.jpg)  
+Figure 2: Sentiment experiment. Left: Trade-off between standard deviations of logits of names (measuring individual fairness) and accuracy. Center: Trade-off between race gap (measuring group fairness) and accuracy. Right: Frequencies of violations of the global IF constraints after applying GLIF, constraints corresponding to names, and GLIF’s global IF constraint violations for names.
+
+In both settings, GLIF and GLIF-NRW achieve substantially better fairness metrics for the same levels of test accuracy in comparison to the IF-constraints method. To understand the reason for this, we study which global IF constraints are violated after applying the GLIF method in Figure 2 (right). Corresponding to the unique pairs of words in our test set, there are $n ( n - 1 ) / 2$ unique constraints in (2.3), and the global IF-constraints method satisfies all of them by design. Each constraint (i.e., each pair of words) corresponds to a fair distance, which is small for (under the fair metric) similar words and large for dissimilar words. We bin the constraints by fair distance and present the proportion of global IF constraints violated after applying the GLIF method for each bin in the histogram in Figure 2 (right). Here, we set the Lipschitz-constant $L$ in (2.3) to $L = 2 . 2 5$ corresponding to a $8 9 . 4 \%$ accuracy of the IF-constraints method and show global IF constraint violations of GLIF corresponding to $9 5 \%$ accuracy in blue. This means that we use strong global IF constraints and use a setting of the GLIF method which maintains most of the accuracy, which would not be possible using the IF-constraints method. GLIF does not violate any constraints corresponding to small fair distances, i.e., it satisfies IF on similar individuals, while violating many large fair distance constraints. This can be seen as basically all constraint violations (blue) are at large fair distances of greater or equal 6. This demonstrates the effect of enforcing local individual fairness from our theoretical analysis in Section 4. At the same time, we display frequency of constraints that correspond to pairs of names in orange, where we can see that almost all constraints corresponding to names occur at small fair distances of smaller or equal to 6. This is expected in this task because we consider all names similar, so fair distances between them should be small. We can see that the distributions of constraint violations after applying GLIF (blue, right) and names (orange, left) are almost disjoint. We mark all global IF constraint violations after applying GLIF that correspond to names in green, and observe that there are none. Summarizing, GLIF ignores unnecessary (in the context of this problem) constraints allowing it to achieve higher accuracy, while satisfying the more relevant local IF constraints leading to improved fairness.
+
+Regarding the practical differences between GLIF and GLIF-NRW, in Figure 2 (left) GLIF has smaller standard deviations on the name outputs, but in in Figure 2 (center) GLIF-NRW achieves lower race gap. In Theorem 4.6, we showed that GLIF penalizes fairness violations in high density data regions stronger. As a result, GLIF may favor enforcing similar outputs in the high density region causing lower standard deviation, while leaving outputs nearly unchanged in the lower density region, resulting in larger race gaps. GLIF-NRW weights all data density regions equally, i.e., it is less likely to miss a small subset of names, but is less stringent in the high density regions.
+
+# 5.2 Post-processing for Debiasing Large Language Models
+
+Large language models have achieved impressive results on many tasks; however, there is also significant evidence demonstrating that they are prone to biases [4], [38], [39]. Debiasing these models remains largely an open problem: most in-processing algorithms are not applicable or computationally prohibitive due to large and highly complex model architectures, and challenges in handling text inputs. Even if an appropriate in-processing algorithm arises, significant environmental impact due to re-training is unavoidable [4], [16]. In our experiments, we evaluate effectiveness of GLIF as a simple post-processing technique to debias BERT-based models for text classification. Another possible solution is to fine-tune BERT with an in-processing technique as was done by Yurochkin et al. [12]. The two approaches are not directly comparable: fine-tuning with SenSeI [12] requires knowledge of the model parameters, alleviates only part of the computational burden, and has more stringent requirements on the fair metric, while post-processing with GLIF is transductive, i.e., it requires access to unlabeled test data (see extended discussion in Section 2.1).
+
+![](images/d8797f1f8a189470f5f4dac9da7fcb31f67a3d412338cd47d6ced12d932c6f3f.jpg)  
+Figure 3: Accuracy-Consistency trade-offs for Bios (left) and Toxicity (right).
+
+Table 1: Results for the Bios task.   
+
+<table><tr><td>Method</td><td>Test Acc.</td><td>Pred. Consist.</td></tr><tr><td>Baseline</td><td>0.846 ± 0.003</td><td>0.942 ± 0.002</td></tr><tr><td>GLIF</td><td>0.830± 0.004</td><td>0.986 ± 0.002</td></tr><tr><td>GLIF-NRW</td><td>0.834 ± 0.003</td><td>0.988 ± 0.002</td></tr><tr><td>SenSEI</td><td>0.843 ± 0.003</td><td>0.977 ± 0.001</td></tr></table>
+
+Table 2: Results for the Toxicity task.   
+
+<table><tr><td>Method</td><td>Test Acc.</td><td>Pred. Consist.</td></tr><tr><td>Baseline</td><td>0.809 ± 0.004</td><td>0.614 ± 0.013</td></tr><tr><td>GLIF</td><td>0.803 ±0.003</td><td>0.835 ± 0.012</td></tr><tr><td>GLIF-NRW</td><td>0.803 ± 0.003</td><td>0.844 ± 0.013</td></tr><tr><td>SenSEI</td><td>0.791 ± 0.005</td><td>0.773 ± 0.043</td></tr></table>
+
+We replicate the experiments of Yurochkin et al. [12] on Bios [40] and Toxicity1 data sets. They use the approach of Mukherjee et al. [19] for fair metric learning which we reproduce. We refer to the Appendix B.1 of [12] for details. In both tasks, following [12], we quantify performance with balanced accuracy due to class imbalance, and measure individual fairness via prediction consistency, i.e., the fraction of test points where the prediction remains unchanged when performing task-specific input modifications. For implementation details, see Appendix A. In Appendix A.4, we analyze the runtime and distinguish between the closed-form and coordinate descent variants of GLIF.
+
+In Bios, the goal is to predict the occupation of a person based on their textual biography. Such models can be useful for recruiting purposes. However, due to historical gender bias in some occupations, the baseline BERT model learns to associate gender pronouns and names with the corresponding occupations. Individual fairness is measured with prediction consistency with respect to gender pronouns and names alterations. A prediction is considered consistent if it is the same after swapping the gender pronouns and names. We present the fairness-accuracy trade-off in Figure 3 (left) for a range of threshold parameters $\tau$ , and compare performance based on hyperparameter values selected with a validation data in Table 1. Both GLIF and GLIF-NRW noticeably improve individual fairness measured with prediction consistency, while retaining most of the accuracy.
+
+In Toxicity, the task is to identify toxic comments—an important tool for facilitating inclusive discussions online. The baseline BERT model learns to associate certain identity words with toxicity (e.g., “gay”) because they are often abused in online conversations. The prediction consistency is measured with respect to changes to identity words in the inputs. There are 50 identity words, e.g., “gay”, “muslim”, “asian”, etc. and a prediction is considered consistent if it is the same for all 50 identities. We present the trade-off plots in Figure 3 (right) and compare performance in Table 2 (right). Our methods reduce individual biases in BERT predictions. We note that in both Toxicity and Bios experiments, we observe no practical differences between GLIF and GLIF-NRW.
+
+# 6 Summary and Discussion
+
+We studied post-processing methods for enforcing individual fairness. The methods provably enforce a local form of IF and scale readily to large data sets. We hope this broadens the appeal of IF by (i) alleviating the computational costs of operationalizing IF and (ii) allowing practitioners to use off-the-shelf models for standard ML tasks. We also note that it is possible to use our objective for in-processing.
+
+We conclude with two warnings: First, enforcing any algorithmic fairness definition does not guarantee complete fairness from the perspective of the user. The problem-specific meaning of fairness is often hard to encode exactly with a mathematical fairness definition. Second, while local individual fairness is a reasonable choice in many applications, this choice should be understood and verified by the practitioner depending on the situation.
+
+# Acknowledgments and Disclosure of Funding
+
+This note is based upon work supported by the National Science Foundation (NSF) under grants no. 1916271, 2027737, and 2113373 and supported by the German Research Foundation (DFG) under Germany’s Excellence Strategy EXC–2117–390829875. Any opinions, findings, and conclusions or recommendations expressed in this note are those of the authors and do not necessarily reflect the views of the NSF nor the DFG.
+
+# References
+
+[1] S. Barocas and A. D. Selbst, “Big data’s disparate impact,” Calif. L. Rev., vol. 104, p. 671, 2016.   
+[2] J. Larson, J. Angwin, L. Kirchner, and S. Mattu, “How we examined racial discrimination in auto insurance prices,” ProPublica, 2017.   
+[3] J. Buolamwini and T. Gebru, “Gender shades: Intersectional accuracy disparities in commercial gender classification,” in Conference on fairness, accountability and transparency, PMLR, 2018, pp. 77–91.   
+[4] E. M. Bender, T. Gebru, A. McMillan-Major, and S. Shmitchell, “On the dangers of stochastic parrots: Can language models be too big?” In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, 2021, pp. 610–623.   
+[5] A. Chouldechova and A. Roth, “A snapshot of the frontiers of fairness in machine learning,” Communications of the ACM, vol. 63, no. 5, pp. 82–89, 2020.   
+[6] C. Dwork, M. Hardt, T. Pitassi, O. Reingold, and R. Zemel, “Fairness through awareness,” in Proceedings of the 3rd innovations in theoretical computer science conference, 2012, pp. 214–226.   
+[7] D. C. Hsia, “Credit scoring and the equal credit opportunity act,” Hastings LJ, vol. 30, p. 371, 1978.   
+[8] M. Bertrand and S. Mullainathan, “Are emily and greg more employable than lakisha and jamal? a field experiment on labor market discrimination,” American Economic Review, vol. 94, no. 4, pp. 991–1013, 2004.   
+[9] W. Fleisher, “What’s fair about individual fairness?” Available at SSRN 3819799, 2021.   
+[10] C. Jung, M. Kearns, S. Neel, A. Roth, L. Stapleton, and Z. S. Wu, “An algorithmic framework for fairness elicitation,” arXiv preprint arXiv:1905.10660, 2019.   
+[11] M. Yurochkin, A. Bower, and Y. Sun, “Training individually fair ML models with sensitive subspace robustness,” in International Conference on Learning Representations (ICLR), 2020.   
+[12] M. Yurochkin and Y. Sun, “Sensei: Sensitive set invariance for enforcing individual fairness,” in International Conference on Learning Representations (ICLR), 2021.   
+[13] A. Vargo, F. Zhang, M. Yurochkin, and Y. Sun, “Individually fair gradient boosting,” in International Conference on Learning Representations (ICLR), 2021.   
+[14] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “BERT: Pre-training of deep bidirectional transformers for language understanding. arxiv,” in Proc. Conference of the North American Chapter of the Association for Computational Linguistics (ACL), 2018.   
+[15] T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al., “Language models are few-shot learners,” arXiv preprint arXiv:2005.14165, 2020.   
+[16] E. Strubell, A. Ganesh, and A. McCallum, “Energy and policy considerations for deep learning in nlp,” in 57th Annual Meeting of the Association for Computational Linguistics, 2019.   
+[17] O. Chapelle, B. Schölkopf, and A. Zien, Eds., Semi-Supervised Learning. Cambridge, Mass: MIT Press, 2006.   
+[18] C. Ilvento, “Metric Learning for Individual Fairness,” in 1st Symposium on Foundations of Responsible Computing (FORC 2020), Leibniz International Proceedings in Informatics (LIPIcs), vol. 156, Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2020.   
+[19] D. Mukherjee, M. Yurochkin, M. Banerjee, and Y. Sun, “Two simple ways to learn individual fairness metrics from data,” in International Conference on Machine Learning (ICML), 2020.   
+[20] M. P. Kim, A. Ghorbani, and J. Zou, “Multiaccuracy: Black-box post-processing for fairness in classification,” in Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society, 2019.   
+[21] P. K. Lohia, K. N. Ramamurthy, M. Bhide, D. Saha, K. R. Varshney, and R. Puri, “Bias mitigation post-processing for individual and group fairness,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2019, pp. 2847–2851.   
+[22] P. Lohia, “Priority-based post-processing bias mitigation for individual and group fairness,” arXiv preprint arXiv:2102.00417, 2021.   
+[23] D. Wei, K. N. Ramamurthy, and F. d. P. Calmon, “Optimized score transformation for fair classification,” in International Conference on Artificial Intelligence and Statistics (AISTATS), 2020.   
+[24] D. Spielman, “Spectral graph theory,” Combinatorial scientific computing, no. 18, 2012.   
+[25] M. Hein, J.-Y. Audibert, and U. v. Luxburg, “Graph laplacians and their convergence on random neighborhood graphs.,” Journal of Machine Learning Research, vol. 8, no. 6, 2007.   
+[26] R. R. Coifman and S. Lafon, “Diffusion maps,” Applied and computational harmonic analysis, vol. 21, no. 1, 2006.   
+[27] D. Zhou and B. Schölkopf, “A regularization framework for learning from graph data,” in ICML 2004 Workshop on Statistical Relational Learning and Its Connections to Other Fields (SRL 2004), 2004, pp. 132–137.   
+[28] M. Belkin, I. Matveeva, and P. Niyogi, “Regularization and semi-supervised learning on large graphs,” in International Conference on Computational Learning Theory, Springer, 2004, pp. 624–638.   
+[29] A. J. Smola and R. Kondor, “Kernels and regularization on graphs,” in Learning theory and kernel machines, Springer, 2003.   
+[30] P. Lahoti, K. P. Gummadi, and G. Weikum, “Ifair: Learning individually fair data representations for algorithmic decision making,” in 2019 ieee 35th international conference on data engineering (icde), IEEE, 2019, pp. 1334–1345.   
+[31] J. Kang, J. He, R. Maciejewski, and H. Tong, “Inform: Individual fairness on graph mining,” in Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 2020, pp. 379–389.   
+[32] S. J. Wright, “Coordinate descent algorithms,” Mathematical Programming, 2015.   
+[33] M. Hein, J.-Y. Audibert, and U. Von Luxburg, “From graphs to manifolds–weak and strong pointwise consistency of graph laplacians,” in International Conference on Computational Learning Theory, Springer, 2005.   
+[34] M. Hu and B. Liu, “Mining and summarizing customer reviews,” in Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining, 2004.   
+[35] J. Pennington, R. Socher, and C. D. Manning, “Glove: Global vectors for word representation,” in Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP), 2014, pp. 1532–1543.   
+[36] A. Caliskan, J. J. Bryson, and A. Narayanan, “Semantics derived automatically from language corpora contain human-like biases,” Science, 2017.   
+[37] S. Diamond and S. Boyd, “Cvxpy: A python-embedded modeling language for convex optimization,” The Journal of Machine Learning Research, vol. 17, no. 1, pp. 2909–2913, 2016.   
+[38] K. Kurita, N. Vyas, A. Pareek, A. W. Black, and Y. Tsvetkov, “Measuring bias in contextualized word representations,” in First Workshop on Gender Bias in Natural Language Processing, 2019.   
+[39] M. Nadeem, A. Bethke, and S. Reddy, “Stereoset: Measuring stereotypical bias in pretrained language models,” arXiv preprint arXiv:2004.09456, 2020.   
+[40] M. De-Arteaga, A. Romanov, H. Wallach, J. Chayes, C. Borgs, A. Chouldechova, S. Geyik, K. Kenthapadi, and A. T. Kalai, “Bias in bios: A case study of semantic representation bias in a high-stakes setting,” in Proceedings of the Conference on Fairness, Accountability, and Transparency (ACM FAccT), 2019.   
+[41] L. Dixon, J. Li, J. Sorensen, N. Thain, and L. Vasserman, “Measuring and mitigating unintended bias in text classification,” in Proceedings of the 2018 AAAI/ACM Conference on AI, Ethics, and Society, 2018.   
+[42] B. Sen, A gentle introduction to empirical process theory and applications, 2018.

parse/train/qGeqg4_hA2/qGeqg4_hA2_content_list.json

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+        "text": "Post-processing for Individual Fairness ",
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+    {
+        "type": "text",
+        "text": "Felix Petersen ∗ University of Konstanz felix.petersen@uni.kn ",
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+        "text": "Debarghya Mukherjee ∗ University of Michigan mdeb@umich.edu ",
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+    {
+        "type": "text",
+        "text": "Yuekai Sun University of Michigan yuekai@umich.edu ",
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+        "type": "text",
+        "text": "Mikhail Yurochkin IBM Research, MIT-IBM Watson AI Lab mikhail.yurochkin@ibm.com ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "Post-processing in algorithmic fairness is a versatile approach for correcting bias in ML systems that are already used in production. The main appeal of postprocessing is that it avoids expensive retraining. In this work, we propose general post-processing algorithms for individual fairness (IF). We consider a setting where the learner only has access to the predictions of the original model and a similarity graph between individuals guiding the desired fairness constraints. We cast the IF post-processing problem as a graph smoothing problem corresponding to graph Laplacian regularization that preserves the desired “treat similar individuals similarly” interpretation. Our theoretical results demonstrate the connection of the new objective function to a local relaxation of the original individual fairness. Empirically, our post-processing algorithms correct individual biases in large-scale NLP models such as BERT, while preserving accuracy. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "There are many instances of algorithmic bias in machine learning (ML) models [1]–[4], which has led to the development of methods for quantifying and correcting algorithmic bias. To quantify algorithmic bias, researchers have proposed numerous mathematical definitions of algorithmic fairness. Broadly speaking, these definitions fall into two categories: group fairness [5] and individual fairness [6]. The former formalizes the idea that ML system should treat certain groups of individuals similarly, e.g., requiring the average loan approval rate for applicants of different ethnicities be similar [7]. The latter asks for similar treatment of similar individuals, e.g., same outcome for applicants with resumes that differ only in names [8]. Researchers have also developed many ways of correcting algorithmic bias. These fairness interventions broadly fall into three categories: pre-processing the data, enforcing fairness during model training (also known as in-processing), and post-processing the outputs of a model. ",
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+        "text": "While both group and individual fairness (IF) definitions have their benefits and drawbacks [5], [6], [9], the existing suite of algorithmic fairness solutions mostly enforces group fairness. The few prior works on individual fairness are all in-processing methods [10]–[13]. Although in-processing is arguably the most-effective type of intervention, it has many practical limitations. For example, it requires training models from scratch. Nowadays, it is more common to fine-tune publicly available models (e.g., language models such as BERT [14] and GPT-3 [15]) than to train models afresh, as many practitioners do not have the necessary computational resources. Even with enough computational resources, training large deep learning models has a significant environmental impact [4], [16]. ",
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+        "text": "Post-processing offers an easier path towards incorporating algorithmic fairness into deployed ML models, and has potential to reduce environmental harm from re-training with in-processing fairness techniques. ",
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+        "text": "In this paper, we propose a computationally efficient method for post-processing off-the-shelf models to be individually fair. We consider a setting where we are given the outputs of a (possibly unfair) ML model on a set of $n$ individuals, and side information about their similarity for the ML task at hand, which can either be obtained using a fair metric on the input space or from external (e.g., human) annotations. Our starting point is a post-processing version of the algorithm by Dwork et al. [6] (see (2.3)). Unfortunately, this method has two drawbacks: poor scalability and an unfavorable trade-off with accuracy. As we shall see, the sharp trade-off is due to the restrictions imposed on dissimilar individuals by Dwork et al. [6]’s global Lipschitz continuity condition. By relaxing these restrictions on dissimilar individuals, we obtain a better trade-off between accuracy and fairness, while preserving the intuition of treating similar individuals similarly. This leads us to consider a graph signal-processing approach to IF post-processing that only enforces similar outputs between similar individuals. The nodes in the underlying graph correspond to individuals, edges (possibly weighted) indicate similarity, and the signal on the graph is the output of the model on the corresponding node-individuals. To enforce IF, we use Laplacian regularization [17], which encourages the signal to ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "be smooth on the graph. We illustrate this idea in Figure 1: a biased model decides whom to show a job ad for a Python programming job based on their CVs and chooses Charlie and Dave but excludes Alice. However, from the qualifications, we can see that Alice, Charlie and Dave are similar because they all have experience in Python, which is the job requirement, and thus should be treated similarly. We represent all five candidates as nodes in a graph, where the node signal (checkmark or cross) is the model’s decision for the corresponding candidate, and the edge weights are indicated by the thickness of the connecting line. Alice and Charlie have the same qualifications and are therefore connected with a large edge-weight. For the predictions to satisfy IF, the graph needs to be smooth, i.e., the similar / connected candidates should have similar node ",
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+    {
+        "type": "image",
+        "img_path": "images/f23f92c1b7b23609fd732c54d7541ee19aa81b25fe90a8adab304cc5cb266631.jpg",
+        "image_caption": [
+            "Figure 1: IF on a graph. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "signals, which can be accomplished by also offering the job to Alice. In contrast, directly enforcing IF constraints [6] requires a certain degree of output similarity on all pairs of candidates, and not just on those which are connected and thus similar. Our main contributions are summarized below. ",
+        "bbox": [
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+        "text": "1. We cast post-processing for individual fairness as a graph smoothing problem and propose a coordinate descent algorithm to scale the approach to large data sets.   \n2. We demonstrate theoretically and verify empirically that graph smoothing enforces individual fairness constraints locally, i.e., it guarantees similar treatment of similar individuals.   \n3. We empirically compare the Laplacian smoothing method to the post-processing adaptation of the algorithm by Dwork et al. [6] enforcing global Lipschitz continuity. The Laplacian smoothing method is not only computationally more efficient but is also more effective in reducing algorithmic bias while preserving accuracy of the original model.   \n4. We demonstrate the efficacy of Laplacian smoothing on two large-scale text data sets by reducing biases in fine-tuned BERT models. ",
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+        "type": "text",
+        "text": "2 Post-processing Problem Formulation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Let $\\mathcal { X }$ be the feature space, $\\mathcal { V }$ be the set of possible labels/targets, and $h : \\mathcal { X }  \\mathcal { Y }$ be a (possibly unfair) ML model trained for the task. Our goal is to post-process the outputs of $h$ so that they are individually fair. Formally, the post-processor is provided with a set of inputs $\\{ x _ { i } \\} _ { i = 1 } ^ { n }$ and the outputs of $h$ on the inputs $\\{ \\widehat { y } _ { i } \\triangleq h ( x _ { i } ) \\} _ { i = 1 } ^ { n }$ , and its goal is to produce $\\{ \\widehat { f } _ { i } \\} _ { i = 1 } ^ { n }$ that is both individually fair and similar to the $\\widehat { y } _ { i }$ b ’s. Recall that individual fairness of $h$ is the Lipschitz continuity of $h$ with respect ",
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+    {
+        "type": "text",
+        "text": "to a fair metric $d _ { \\mathcal { X } }$ on the input space: ",
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+    {
+        "type": "equation",
+        "img_path": "images/a4811805f8214c0932bc029d8f17d5db72f39e62820dbc31d74e2fc330e5fab3.jpg",
+        "text": "$$\nd y ( h ( x ) , h ( x ^ { \\prime } ) ) \\leq L d _ { \\mathcal { X } } ( x , x ^ { \\prime } ) \\mathrm { ~ f o r ~ a l l ~ } x , x ^ { \\prime } \\in \\mathcal { X } ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $L > 0$ is a Lipschitz constant. The fair metric encodes problem-specific intuition of which samples should be treated similarly by the ML model. It is analogous to the knowledge of protected attributes in group fairness needed to define corresponding fairness constraints. Recent literature proposes several practical methods for learning fair metric from data [18], [19]. We assume the postprocessor is either given access to the fair metric (it can evaluate the fair distance on any pair of points in $\\mathcal { X }$ ), or receives feedback on which inputs should be treated similarly. We encode this information in an adjacency matrix $W \\in \\mathbb { R } ^ { n \\times n }$ of a graph with individuals as nodes. If the post-processor is given the fair metric, then the entries of $W$ are ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/c9eb76df7d7fb0ce1db8cc3b55eb155865a61794ab44781ad87248725c0e4c83.jpg",
+        "text": "$$\nW _ { i j } = { \\left\\{ \\begin{array} { l l } { \\exp ( - \\theta d _ { \\mathcal { X } } ( x _ { i } , x _ { j } ) ^ { 2 } ) } & { d _ { \\mathcal { X } } ( x _ { i } , x _ { j } ) \\leq \\tau } \\\\ { 0 } & { { \\mathrm { o t h e r w i s e } } , } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\theta > 0$ is a scale parameter and $\\tau > 0$ is a threshold parameter. If the post-processor is given an annotator’s feedback, then $W$ is a binary matrix with $W _ { i j } = 1$ if $i$ and $j$ are considered to be treated similarly by the annotator and 0 otherwise. Extensions to multiple annotators are straightforward. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We start with a simple post-processing adaptation of the algorithm by Dwork et al. [6] for enforcing individual fairness, that projects the (possibly unfair) outputs of $h$ onto a constraint set to enforce (2.1). In other words, the post-processor seeks the closest set of outputs to the $\\widehat { y } _ { i }$ ’s that satisfies individual fairness: ",
+        "bbox": [
+            174,
+            340,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/37cb1b16f13da1f475472566017c32fa51357ba261c45329114db958659e71c0.jpg",
+        "text": "$$\n\\begin{array} { r } { \\{ \\widehat { f } _ { i } \\} _ { i = 1 } ^ { n } \\in \\left\\{ \\begin{array} { l l } { \\arg \\operatorname* { m i n } _ { f _ { 1 } , \\ldots , f _ { n } } } & { \\sum _ { i = 1 } ^ { n } \\frac { 1 } { 2 } d y ( f _ { i } , \\widehat { y } _ { i } ) ^ { 2 } } \\\\ { \\mathrm { s u b j e c t ~ t o } } & { d y ( f _ { i } , f _ { j } ) \\leq L d _ { \\mathcal { X } } ( x _ { i } , x _ { j } ) } \\end{array} \\right\\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            396,
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+            439
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This objective function, though convex, scales poorly due to the order of $n ^ { 2 }$ constraints. Empirically, we observe that (2.3) leads to post-processed outputs that are dissimilar to the $\\widehat { y } _ { i }$ ’s, leading to poor performance in practice. The goal of our method is to improve performance and scalability, while preserving the IF desiderata of treating similar individual similarly. Before presenting our method, we discuss other post-processing perspectives that differ in their applicability and input requirements. ",
+        "bbox": [
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+            445,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Alternative Post-processing Formulations ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            531,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We review three post-processing problem setups and the corresponding methods in the literature. First, one can fine-tune a model via an in-processing algorithm to reduce algorithmic biases. Yurochkin et al. [12] proposed an in-processing algorithm for IF and used it to train fair models for text classification using sentence BERT embeddings. This setting is the most demanding in terms of input and computational requirements: a user needs access to the original model parameters, fair metric function, and train a predictor, e.g., a moderately deep fully connected neural network, with a non-trivial fairness-promoting objective function. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Second, it is possible to post-process by training additional models to correct the initial model’s behavior. For example, Kim et al. [20] propose a boosting-based method for group fairness postprocessing. This perspective can be adapted to individual fairness; however, it implicitly assumes that we can train weak-learners to boost. Lohia et al. [21], [22] propose to train a bias detector to post-process for group fairness and a special, group based, notion of individual fairness. Such methods are challenging to apply to text data or other non-tabular data types. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The third perspective is the most generic: a user has access to original model outputs only, and a minimal additional feedback guiding fairness constraints. Wei et al. [23] consider such setting and propose a method to satisfy group fairness constraints; however, it is not applicable to individual fairness. Our problem formulation belongs to this post-processing setup. The main benefit of this approach is its broad applicability and ease of deployment. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Graph Laplacian Individual Fairness ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            838,
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+            856
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To formulate our method, we cast IF post-processing as a graph smoothing problem. Using the fair metric or human annotations as discussed in Section 2, we obtain an $n \\times n$ matrix $W$ that we treat as an adjacency matrix. As elaborated earlier, the goal of post-processing is to obtain a model $f$ that is individually fair and accurate. The accuracy is achieved by minimizing the distance between the outputs of $f$ and $h$ , a pre-trained model assumed to be accurate but possibly biased. Recall that we do not have access to the parameters of $h$ , but can evaluate its predictions. Our method enforces fairness using a graph Laplacian quadratic form [24] regularizer: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            825,
+            147
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/45ba0bd0856ffe15c83f6456d6053f2e94cd7915945e175ec3e19e4b71e1a379.jpg",
+        "text": "$$\n\\widehat { \\mathbf { f } } = \\arg \\operatorname* { m i n } _ { \\mathbf { f } } { g _ { \\lambda } ( \\mathbf { f } ) } = \\arg \\operatorname* { m i n } _ { \\mathbf { f } } \\ \\| \\mathbf { f } - \\hat { \\mathbf { y } } \\| _ { 2 } ^ { 2 } + \\lambda \\mathbf { f } ^ { \\top } \\mathbb { L } _ { n } \\mathbf { f } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            155,
+            676,
+            181
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { \\mathbf { y } }$ is the output of the model $h$ , and $\\widehat { \\mathbf { f } }$ is the vector of the post-processed outputs, i.e., ${ \\widehat { f } } _ { i } = f ( x _ { i } )$ for $i = 1 , \\ldots , n$ . The matrix $\\mathbb { L } _ { n } \\in \\mathbb { R } ^ { n \\times n }$ is called graph Laplacian matrix and is a function of $W$ . There are multiple versions of $\\mathbb { L } _ { n }$ popularized in the graph literature (see, e.g., [25] or [26]). To elucidwhere t, ual f for s, consider the unnormalized Laplaciais the degree matrix corresponding to $\\mathbb { L } _ { u n , n } = D - W$ , $\\begin{array} { r } { D _ { i i } = \\sum _ { j = 1 } ^ { n } W _ { i j } } \\end{array}$ $D _ { i j } = 0$ $i \\neq j$ $W$ identity is: ",
+        "bbox": [
+            173,
+            190,
+            826,
+            279
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cfffaa922892d5295494c0391dbbc1ad6e519175c2569ffd61bcd3f82b58bed.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { f } ^ { \\top } \\mathbb { L } _ { u n , n } \\mathbf { f } = \\frac { 1 } { 2 } \\sum _ { i \\neq j } W _ { i j } \\left( f _ { i } - f _ { j } \\right) ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            280,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Hence, the Laplacian regularizer is small if the post-processed model outputs $\\widehat { f } _ { i }$ and ${ \\widehat { f } } _ { j }$ (i.e., treatment) are similar for large $W _ { i j }$ (i.e., for similar individuals $i$ and $j$ ). This promotes the philosophy of individual fairness: “treat similar individuals similarly”. This observation intuitively explains the motivation for minimizing the graph Laplacian quadratic form to achieve IF. In Section 4, we present a more formal discussion on the connections between the graph Laplacian regularization and IF. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our post-processing problem (3.1) is easy to solve: setting the gradient of $g _ { \\lambda }$ to 0 implies that the optimal solution $\\widehat { \\mathbf { f } }$ is: ",
+        "bbox": [
+            174,
+            385,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf1f3c26af9044b24d9bef75a67a86584b276607bddb9a41ea94f8267d796373.jpg",
+        "text": "$$\n\\widehat { \\mathbf { f } } = \\left( I + \\lambda \\left( \\frac { \\mathbb { L } _ { n } + \\mathbb { L } _ { n } ^ { \\top } } { 2 } \\right) \\right) ^ { - 1 } \\widehat { \\mathbf { y } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            411,
+            611,
+            449
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The Laplacian $\\mathbb { L } _ { n }$ is a positive semi-definite matrix ensuring that (3.1) is strongly convex and that (3.3) is a global minimum. In comparison to the computationally expensive constraint optimization problem (2.3), this approach has a simple closed-form expression. ",
+        "bbox": [
+            174,
+            454,
+            823,
+            497
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that the symmetry of the unnormalized Laplacian $\\mathbb { L } _ { u n , n }$ simplifies (3.3); however, there are also non-symmetric Laplacian variations. In this work, we also consider the normalized random walk Laplacian $\\mathbb { L } _ { n r w , n } = ( I - \\widetilde { D } ^ { - 1 } \\widetilde { W } )$ , where $\\widetilde { W } = D ^ { - 1 / 2 } W D ^ { - 1 / 2 }$ is the normalized adjacency matrix and De is its degree matrix. We discuss its properties in the context of IF in Section 4. Henceforth, we refer to our method as Graph Laplacian Individual Fairness (GLIF) when using the unnormalized Laplacian, and GLIF-NRW when using Normalized Random Walk Laplacian. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Prior Work on Graph Laplacians ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            611,
+            444,
+            626
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Graph-based learning via a similarity matrix is prevalent in statistics and ML literature, specifically, in semi-supervised learning. The core idea is to gather information from similar unlabeled inputs to improve prediction accuracy (e.g., see [27], [28], [29] and references therein). Laplacian regularization is widely used in science engineering. We refer to Chapelle [17] for a survey. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We note that [30], [31] also use graph Laplacian regularizers to enforce individual fairness. Our work builds on their work by elucidating the key role played by the graph Laplacian in enforcing individual fairness. In particular, we clarify the connection between the choice of the graph Laplacian and the exact notion of individual fairness the corresponding graph Laplacian regularizer enforces. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Extensions of the Basic Method ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            773,
+            431,
+            787
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we present four extensions of our method: multi-dimensional outputs, coordinate descent for large-scale data, an inductive setting, and alternative output space discrepancy measures. ",
+        "bbox": [
+            174,
+            799,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2.1 Multi-dimensional Output ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            844,
+            408,
+            859
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We presented our objective function (3.1) and post-processing procedure (3.3) for the case of univariate outputs. This covers regression and binary classification. Our method readily extends to multi-dimensional output space, for example, in classification, $f _ { i } , \\widehat { y } _ { i } \\in \\mathbb { R } ^ { K }$ can represent logits, i.e., ",
+        "bbox": [
+            174,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "softmax inputs, of the $K$ classes. In this case, f and $\\hat { \\mathbf { y } }$ are $n \\times K$ matrices, and the term f $\\mathbf { : } ^ { \\top } \\mathbb { L } _ { n } \\mathbf { f }$ is a $K \\times K$ matrix. We use the trace of it as a regularizer. The optimization problem (3.1) then becomes: ",
+        "bbox": [
+            171,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d2c3e75b1fc3ac083b1de4bcc4e16cd1836e2dbde0518597f1f42cf4be2b6f9.jpg",
+        "text": "$$\n\\widehat { \\mathbf { f } } = \\arg \\operatorname* { m i n } _ { f } g _ { \\lambda } ( \\mathbf { f } ) = \\arg \\operatorname* { m i n } _ { f } \\| \\mathbf { f } - \\widehat { \\mathbf { y } } \\| _ { F } ^ { 2 } + \\lambda \\operatorname { t r } \\left( \\mathbf { f } ^ { \\top } \\mathbb { L } _ { n } \\mathbf { f } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            123,
+            692,
+            150
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\| \\cdot \\| _ { F }$ is the Frobenius norm. Similar to the univariate output case, this yields: ",
+        "bbox": [
+            174,
+            155,
+            728,
+            170
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/676116820a88f7eec0d6242c6ce4fc54e491844342c82fdd770cf7612a29ea8a.jpg",
+        "text": "$$\n\\widehat { \\mathbf { f } } = \\left( I + \\lambda \\left( \\frac { \\mathbb { L } _ { n } + \\mathbb { L } _ { n } ^ { \\top } } { 2 } \\right) \\right) ^ { - 1 } \\widehat { \\mathbf { y } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            172,
+            611,
+            212
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The solution is the same as (3.3); however, now it accounts for multi-dimensional outputs. ",
+        "bbox": [
+            173,
+            214,
+            761,
+            229
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.2 Coordinate Descent for Large-Scale Data ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            242,
+            513,
+            257
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Although our method has a closed form solution, it is not immediately scalable, as we have to invert a $n \\times n$ matrix to obtain the optimal solution. We propose a coordinate descent variant of our method that readily scales to any data size. The idea stems primarily from the gradient of equation (3.4), where we solve: ",
+        "bbox": [
+            173,
+            266,
+            825,
+            320
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/83ff825ff15b7fd79b2390905d08830423451ccf025d16ec8f9023eed1c64aeb.jpg",
+        "text": "$$\n\\mathbf { f } - \\widehat { \\mathbf { y } } + \\lambda \\frac { \\mathbb { L } _ { n } + \\mathbb { L } _ { n } ^ { \\top } } { 2 } \\mathbf { f } = 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            318,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Fixing $\\{ f _ { j } \\} _ { j \\neq i }$ , we can solve (3.6) for $f _ { i }$ ",
+        "bbox": [
+            176,
+            351,
+            447,
+            366
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5955c091ff740a9916da61ad64bdbb54c5e09bedc4ce986449eef6aadac0fd19.jpg",
+        "text": "$$\nf _ { i } \\gets \\frac { \\hat { y } _ { i } - \\frac { \\lambda } { 2 } \\sum _ { j \\neq i } ( \\mathbb { L } _ { n , i j } + \\mathbb { L } _ { n , j i } ) f _ { j } } { 1 + \\lambda \\mathbb { L } _ { n , i i } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            369,
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+            409
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This gives rise to the coordinate descent algorithm. We perform asynchronous updates over randomly selected coordinate batches until convergence. We refer the reader to Wright [32] and the references therein for the convergence properties of (asynchronous) coordinate descent. ",
+        "bbox": [
+            174,
+            410,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.3 Extension to the Inductive Setting ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            467,
+            460,
+            481
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This coordinate descent update is key to extending our approach to the inductive setting. To handle new unseen points, we assume we have a set of test points on which we have already post-processed the outputs of the ML model. To post-process new unseen points, we simply fix the outputs of the other test points and perform a single coordinate descent step with respect to the output of the new point. Similar strategies are often employed to extend transductive graph-based algorithms to the inductive setting [17]. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.4 Alternative Discrepancy Measures on the Output Space ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            588,
+            611,
+            603
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "So far, we have considered the squared Euclidean distance as a measure of discrepancy between outputs. This is a natural choice for post-processing models with continuous-valued outputs. For models that output a probability distribution over the possible classes, we consider alternative discrepancy measures on the output space. It is possible to replace the squared Euclidean distance with a Bregman divergence with very little change to the algorithm in the case of the unnormalized Laplacian. Below, we work through the details for the KL divergence as a demonstration of the idea. A result for the general Bregman divergence can be found in Appendix B.3 (see Theorem B.4). ",
+        "bbox": [
+            173,
+            611,
+            826,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Suppose the output of the pre-trained model $h$ is $\\hat { y } _ { i } \\in \\Delta ^ { K }$ , where $\\begin{array} { r } { \\hat { y } _ { i } = \\{ e ^ { o _ { i , j } } / \\sum _ { k = 1 } ^ { K } e ^ { o _ { i , k } } \\} _ { j = 1 } ^ { K } } \\end{array}$ a $K$ - dimensional probability vector corresponding to a $K$ class classification problem $\\left. { o _ { i , j } } \\right\\}$ is the output of tand del and is the p $\\hat { y } _ { i }$ is obtained by passbability simplex in it thr. Let h softmax) denote the $\\Delta ^ { K } = \\{ x \\in \\mathbb { R } ^ { K } : x _ { i } \\geq 0 , \\sum _ { i = 1 } ^ { K } x _ { i } = 1 \\}$ $\\mathbb { R } ^ { K }$ $P _ { v }$ multinomial distribution with success probabilities $v$ for any $v \\in \\Delta ^ { k }$ . Define $\\hat { \\eta } _ { i } \\in \\mathbb { R } ^ { K - 1 }$ (resp. $\\eta _ { i }$ ) as the natural parameter corresponding to $\\hat { y } _ { i }$ (resp. $f _ { i }$ ), i.e., $\\hat { \\eta } _ { i , j } = \\log \\left( \\hat { y } _ { i , j } / \\hat { y } _ { i , K } \\right) = o _ { i , j } - o _ { i , K }$ for $1 \\leq j \\leq K - 1$ . The (unnormalized) Laplacian smoothing problem with the KL divergence is ",
+        "bbox": [
+            173,
+            715,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9dd0506a9937275452912317c54ac78019bd3935d9985875cde83d895333bedc.jpg",
+        "text": "$$\n\\left( \\widetilde { y } _ { 1 } , \\dots , \\widetilde { y } _ { n } \\right) = \\arg \\operatorname* { m i n } _ { y _ { 1 } , \\dots , y _ { n } \\in \\Delta ^ { K } } \\left[ \\sum _ { i } \\left\\{ \\mathsf { K L } \\left( P _ { y _ { i } } | | P _ { \\widetilde { y } _ { i } } \\right) + \\frac { \\lambda } { 2 } \\sum _ { j = 1 , j \\neq i } ^ { n } W _ { i j } \\mathsf { K L } \\left( P _ { y _ { i } } | | P _ { y _ { j } } \\right) \\right\\} \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            824,
+            785,
+            868
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A coordinate descent approach for solving the above equation is: ",
+        "bbox": [
+            174,
+            871,
+            599,
+            886
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f18aa002a91c448315e3a6e6dcf6582a3e3d0be7968e1e7e69415ec500fc988d.jpg",
+        "text": "$$\n\\begin{array} { r } { \\tilde { y } _ { i } = \\arg \\operatorname* { m i n } _ { y \\in \\Delta ^ { k } } \\left\\{ \\mathsf { K L } \\left( P _ { y } | | P _ { \\hat { y } _ { i } } \\right) + \\frac { \\lambda } { 2 } \\sum _ { j = 1 , j \\neq i } ^ { n } W _ { i j } \\mathsf { K L } \\left( P _ { y } | | P _ { \\tilde { y } _ { j } } \\right) \\right\\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            888,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The following theorem establishes that (3.5) solves the above problem in the logit space, or equivalently in the space of the corresponding natural parameters (see Appendix $\\mathbf { B }$ for the proof): ",
+        "bbox": [
+            173,
+            90,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1. Consider the following optimization problem on the space of natural parameters: ",
+        "bbox": [
+            171,
+            123,
+            803,
+            138
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec3fd3b5e15cbbf2a02400539ce77d86a3b1eb29be2aebb244a78e27332fc30b.jpg",
+        "text": "$$\n\\begin{array} { r } { \\widetilde { \\eta } _ { i } = \\arg \\operatorname* { m i n } _ { \\eta } \\left[ \\| \\eta - \\widehat { \\eta } _ { i } \\| ^ { 2 } + \\frac { \\lambda } { 2 } \\sum _ { j = 1 , j \\neq i } ^ { n } W _ { i j } \\| \\eta - \\widetilde { \\eta } _ { j } \\| ^ { 2 } \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            143,
+            692,
+            171
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Then, the minimizer $\\tilde { \\eta } _ { i }$ of equation (3.10) is the natural parameter corresponding to the minimizer $\\tilde { y } _ { i }$ of (3.8). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Local IF and Graph Laplacian Regularization ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            223,
+            591,
+            242
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide theoretical insights into why the graph Laplacian regularizer enforces individual fairness. As pointed out in Section 2, enforcing IF globally is expensive and often reduces a significant amount of accuracy of the final classifier. Here, we establish that solving (3.1) is tantamount to enforcing a localized version of individual fairness, namely Local Individual Fairness, which is defined below: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Definition 4.1 (Local Individual Fairness). An ML model $h$ is said to be locally individually fair if it ",
+        "bbox": [
+            169,
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+    {
+        "type": "equation",
+        "img_path": "images/41742888f38d5111cad39e02d426cf79090d47d4475a5d7f98c808b5a5b54c19.jpg",
+        "text": "$$\n\\mathbb { E } _ { x \\sim P } \\left[ \\operatorname* { l i m } _ { x ^ { \\prime } : d _ { \\mathcal { X } } ( x , x ^ { \\prime } ) \\downarrow 0 } \\frac { d _ { \\mathcal { Y } } ( h ( x ) , h ( x ^ { \\prime } ) ) } { d _ { \\mathcal { X } } ( x , x ^ { \\prime } ) } \\right] \\leq L < \\infty .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For practical purposes, this means that $h$ is locally individually fair with constants $\\epsilon$ and $L$ if it satisfies ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dd63b0be464b4931f26357ade74c6c476be9d179a457b30a464228c65a3df717.jpg",
+        "text": "$$\nd y ( h ( x ) , h ( x ^ { \\prime } ) ) \\leq L d _ { \\mathcal { X } } ( x , x ^ { \\prime } ) \\mathrm { ~ f o r ~ a l l ~ } x , x ^ { \\prime } \\in \\mathcal { X } \\mathrm { ~ w h e r e ~ } d _ { \\mathcal { X } } ( x , x ^ { \\prime } ) \\leq \\epsilon\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "in analogy to equation (2.1). Equation (4.2) is a relaxation of traditional IF, where we only care about the Lipschitz-constraint for all pairs of points with small fair distances, i.e., where it is less than some user-defined threshold $\\epsilon$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Example 4.2. For our theoretical analysis, we need to specify a functional form of the fair metric. A popular choice is a Mahalanobis fair metric proposed by $I I 9 J$ , which is defined as: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0861429dcc4b63fc5165914d35c3d59d296c32b06ccdb232e46b3d7b81ebe133.jpg",
+        "text": "$$\nd _ { \\mathcal { X } } ^ { 2 } ( \\boldsymbol { x } , \\boldsymbol { x } ^ { \\prime } ) = ( \\boldsymbol { x } - \\boldsymbol { x } ^ { \\prime } ) ^ { \\top } \\Sigma ( \\boldsymbol { x } - \\boldsymbol { x } ^ { \\prime } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\Sigma$ is a dispersion matrix that puts lower weight in the directions of sensitive attributes and higher weight in the directions of relevant attributes. [19] also proposed several algorithms to learn such a fair metric from the data. If we further assume $d y ( y _ { 1 } , y _ { 2 } ) = | y _ { 1 } - y _ { 2 } |$ , then a simple application of Lagrange’s mean value theorem yields: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/7b0c5559893ff3d76f6eafe4f4b6c6bc3fe268870e454e6dacaeae9980ed1f22.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { x ^ { \\prime } : d _ { \\mathcal { X } } ( x , x ^ { \\prime } ) \\downarrow 0 } \\frac { | h ( x ) - h ( x ^ { \\prime } ) | } { d _ { \\mathcal { X } } ( x , x ^ { \\prime } ) } \\leq \\| \\Sigma ^ { - 1 / 2 } \\nabla h ( x ) \\| .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This immediately implies: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5ce2efb5402acfdd60734388c8ecc732d47faa036222a91e9d36a65191a967bf.jpg",
+        "text": "$$\n\\mathbb { E } _ { x \\sim P } \\left[ \\operatorname* { l i m } _ { { x ^ { \\prime } : d _ { \\mathcal { X } } ( x , x ^ { \\prime } ) } \\downarrow 0 } \\frac { d _ { \\mathcal { Y } } ( h ( x ) , h ( x ^ { \\prime } ) ) } { d _ { \\mathcal { X } } ( x , x ^ { \\prime } ) } \\right] \\leq \\mathbb { E } [ \\| { \\Sigma } ^ { - 1 / 2 } \\nabla h ( x ) \\| ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "i.e., h satisfies local individual fairness constraint as long as $\\mathbb { E } [ | | \\Sigma ^ { - 1 / 2 } \\nabla h ( x ) | | ] < \\infty$ . On the other hand, the global $I F$ constraint necessitates $\\begin{array} { r } { \\operatorname* { s u p } _ { x \\in \\mathcal { X } } \\| \\Sigma ^ { - 1 / 2 } \\nabla h ( x ) \\| < \\infty , } \\end{array}$ , i.e., h is Lipschitz continuous with respect to the Mahalanobis distance. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The main advantage of this local notion of IF over its global counterpart is that the local definition concentrates on the input pairs with smaller fair distance and ignores those with larger distance. For example, in Figure 1, the edge-weights among Alice, Charlie, and Dave are much larger than among any other pairs (which have a weight of 0); therefore, our local notion enforces fairness constraint on the corresponding similar pairs, while ignoring (or being less stringent on) others. This prevents over-smoothing and consequently preserves accuracy while enforcing fairness as is evident from our real data experiments in Section 5. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We now present our main theorem, which establishes that, under certain assumptions on the underlying hypothesis class and the distribution of inputs, the graph Laplacian regularizers (both unnormalized and normalized random walk) enforce the local IF constraint (as defined in Definition 4.1) in the limit. For our theory, we work with $d _ { \\mathcal { X } }$ as the Mahalanobis distance introduced in Example 4.2 in equation (2.2) along with $\\theta = 1 / ( 2 \\sigma ^ { 2 } )$ $\\sigma$ is a bandwidth parameter which goes to 0 at an appropriate rate as $n \\to \\infty$ ) and $\\tau = \\infty$ . All our results will be thorough for any finite $\\tau$ but with more tedious technical analysis. Therefore, our weight matrix $W$ becomes: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f3af875483e730a88c2510ee0184d31983dba7063014ac0c4bd8ee4ee0a67514.jpg",
+        "text": "$$\nW _ { i j } = \\frac { | \\Sigma | ^ { 1 / 2 } } { ( 2 \\pi ) ^ { d / 2 } \\sigma ^ { d } } \\mathrm { e x p } \\left( - \\frac { 1 } { 2 \\sigma ^ { 2 } } ( x _ { i } - x _ { j } ) ^ { \\top } \\Sigma \\left( x _ { i } - x _ { j } \\right) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The constant $| \\Sigma | ^ { 1 / 2 } / ( ( 2 \\pi ) ^ { d / 2 } \\sigma ^ { d } )$ is for the normalization purpose and can be absorbed into the penalty parameter $\\lambda$ . We start by listing our assumptions: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.3 (Assumption on the domain). The domain of the inputs $\\mathcal { X }$ is a compact subset of $\\mathbb { R } ^ { d }$ where $d$ is the underlying dimension. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.4 (Assumption on the hypothesis). All functions $f \\in { \\mathcal { F } }$ of the hypothesis class satisfy the following: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1. The $i ^ { t h }$ derivative $f ^ { ( i ) }$ is uniformly bounded over the domain $\\mathcal { X }$ of inputs for $i \\in \\{ 0 , 1 , 2 \\}$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "2. $f ^ { ( 1 ) } ( x ) = 0$ for all $x \\in \\partial X$ , where $\\partial X$ denotes the boundary of $\\mathcal { X }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.5 (Assumption on the density of inputs). The density $p$ of the input random variable x on the domain $\\mathcal { X }$ satisfies the following: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1. There exists $p _ { \\operatorname* { m a x } } < \\infty$ and $p _ { \\operatorname* { m i n } } > 0$ such that, for all $x \\in \\mathcal { X }$ , we have $p _ { \\operatorname* { m i n } } \\le p ( x ) \\le p _ { \\operatorname* { m a x } }$ .   \n2. The derivatives $\\{ p ^ { ( i ) } \\} _ { i = 0 , 1 , 2 }$ of the density $p$ are uniformly bounded on the domain $\\mathcal { X }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Discussion on the assumptions Most of our assumptions (e.g., compactness of the domain, bounded derivatives of $f$ or $p$ ) are for technical simplicity and are fairly common for the asymptotic analysis of graph regularization (see, e.g., Hein et al. [25], [33] and references therein). It is possible to relax some of the assumptions: for example, if the domain $\\mathcal { X }$ of inputs is unbounded, then the target function $f$ and the density $p$ should decay at certain rate so that observations far away will not be able to affect the convergence (e.g., sub-exponential tails). Part (2.) of Assumption 4.4 can be relaxed if we assume $p ( x )$ is 0 at boundary. However, we do not pursue these extensions further in this manuscript, as they are purely technical and do not add anything of significance to the main intuition of the result. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.6. Under Assumptions 4.3 - 4.5, we have: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1. If the sequence of bandwidths $\\sigma \\equiv \\sigma _ { n } \\downarrow 0$ such that $n \\sigma _ { n } ^ { 2 } \\to \\infty$ and $\\mathbb { L } _ { u n , n }$ is unnormalized Laplacian matrix, then ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/35452ebefc4c3e5c55e3350d27c72744a583cfce03568ad5fb9ef985006b895b.jpg",
+        "text": "$$\n\\frac { 2 } { n ^ { 2 } \\sigma ^ { 2 } } \\mathbf { f } ^ { \\top } \\mathbb { L } _ { u n , n } \\mathbf { f } \\xrightarrow { P } \\mathbb { E } _ { x \\sim p } \\left[ \\nabla f ( x ) ^ { \\top } \\Sigma ^ { - 1 } \\nabla f ( x ) p ( x ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "2. If the sequence of bandwidths $\\sigma \\equiv \\sigma _ { n } \\downarrow 0$ such that $( n \\sigma ^ { d + 4 } ) / ( \\log { ( 1 / \\sigma ) } ) \\to \\infty$ and $\\mathbb { L } _ { n r w , n }$ is the normalized random walk Laplacian matrix, then: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c1c42276b6b29eaba683336661ba5adfe275a6223b5df15a8b5f117976fd34d6.jpg",
+        "text": "$$\n\\frac { 1 } { n \\sigma ^ { 2 } } \\mathbf { f } ^ { \\top } \\mathbb { L } _ { n r w , n } \\mathbf { f } \\stackrel { P } { \\longrightarrow } \\mathbb { E } _ { x \\sim p } \\left[ \\nabla f ( x ) ^ { \\top } { \\Sigma } ^ { - 1 } \\nabla f ( x ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { f } = \\{ f ( x _ { i } ) \\} _ { i = 1 } ^ { n }$ . Consequently, both Laplacian regularizers asymptotically enforce local IF. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The proof of the above theorem can be found in Appendix B. When we use a normalized random walk graph Laplacian matrix $\\mathbb { L } _ { n r w , n }$ as regularizer, the regularizer does (asymptotically) penalize $\\mathbb { E } \\left[ \\nabla f ( x ) ^ { \\top } \\Sigma ^ { - 1 } \\nabla f ( x ) \\right] = \\mathbb { E } \\left[ \\| \\Sigma ^ { - 1 / 2 } \\nabla f ( x ) \\| ^ { 2 } \\right]$ , which, by Example 4.2, is equivalent to enforcing the local IF constraint. Similarly, the un-normalized Laplacian matrix $\\mathbb { L } _ { u n , n }$ , also enforces the same under Assumption 4.5 as: ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "equation",
+        "img_path": "images/e0f800bb3be1d0d6984b0eac9d91672cc99824b2dce72e2f3542aa7dfa4e4da9.jpg",
+        "text": "$$\n\\mathbb { E } \\left[ \\Vert \\Sigma ^ { - 1 / 2 } \\nabla f ( x ) \\Vert ^ { 2 } \\right] \\leq \\frac { 1 } { p _ { \\operatorname* { m i n } } } \\mathbb { E } \\left[ \\nabla f ( x ) ^ { \\top } \\Sigma ^ { - 1 } \\nabla f ( x ) p ( x ) \\right] , \\mathrm { ~ w h e r e ~ } p _ { \\operatorname* { m i n } } = \\operatorname* { i n f } _ { x \\in \\mathcal { X } } p ( x ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Although both the Laplacian matrices enforce local IF, the primary difference between them is that the limit of the unnormalized Laplacian involves the density $p ( x )$ , i.e., it upweights the high-density region (consequently stringent imposition of fairness constraint), whereas it down-weights the underrepresented/low-density region. On the other hand, the limit corresponding to the normalized random walk Laplacian matrix does not depend on $p ( x )$ and enforces fairness constraint with equal intensity on the entire input space. We used both regularizers in our experiments, comparing and contrasting their performance on several practical ML problems. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The goals of our experiments are threefold: ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "1. Exploring the trade-offs between post-processing for local IF with GLIF and post-processing with (global) IF constraints using our adaptation of the algorithm by Dwork et al. [6] described in (2.3).   \n2. Studying practical implications of theoretical differences between GLIF and GLIF-NRW, i.e., different graph Laplacians, presented in Section 4.   \n3. Evaluating the effectiveness of GLIF in its main application, i.e., computationally light debiasing of large deep learning models such as BERT. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The implementation of this work is available at github.com/Felix-Petersen/fairness-post-processing. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.1 Comparing GLIF and Global IF-constraints ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For our first experiment, we consider the sentiment prediction task [34], where our goal is to classify words as having a positive or negative sentiment. The baseline model is a neural network trained with GloVe word embeddings [35]. Following Yurochkin et al. [11], we evaluate the model on a set of names and observe that it assigns varying sentiments to names. An individually fair model should assign similar sentiment scores to all names. Further, we observe that there is a gap between average sentiments of names typical for Caucasian and African-American ethnic groups [36], which is violating group fairness. Yurochkin et al. [11] propose a fair metric learning procedure for this task using a side data set of names, and an in-processing technique for achieving individual fairness. We use their method to obtain the fair metric and compare post-processing of the baseline model with GLIF, GLIF-NRW and the global IF-constraints method. The test set comprises 663 words from the original task and 94 names. For post-processing, no problem specific knowledge is used. The resulting post-processed predictions for the original test set are used to evaluate accuracy, and the predictions on the names are used for evaluating fairness metrics. Even for this small problem, the global IF-constraints method, i.e., a CVXPY [37] implementation of (2.3), takes 7 minutes to run. Due to the poor scalability of the global IF-constraints method, we can use it only for the study of this smaller data set and can not consider it for the large language model experiments in Section 5.2. For GLIF(-NRW), we implement the closed-form solution (3.3) that takes less than a tenth of a second to run. See Appendix A for additional experimental details and a runtime analysis. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the fairness-accuracy trade-offs for a range of threshold parameters $\\tau$ (for GLIF and GLIF-NRW) and for a range of Lipschitz-constants $L$ (for IF-constraints) in Figure 2. Figure 2 (left) shows the standard deviation of the post-processed outputs on all names as a function of test accuracy on the original sentiment task. Lower standard deviations imply that all names received similar predictions, which is the goal of individual fairness. Figure 2 (center) visualizes group fairness and accuracy, i.e., difference in average name sentiment scores for the two ethnic groups. In this problem, individual fairness is a stronger notion of fairness: achieving similar predictions for all names implies similar group averages, but not vice a versa. Therefore, for this task, post-processing for individual fairness also corrects group disparities. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/918fbdc2656ad0da27cbad9688ad1474168f32e745c722f2bcebde4c7b9eda00.jpg",
+        "image_caption": [
+            "Figure 2: Sentiment experiment. Left: Trade-off between standard deviations of logits of names (measuring individual fairness) and accuracy. Center: Trade-off between race gap (measuring group fairness) and accuracy. Right: Frequencies of violations of the global IF constraints after applying GLIF, constraints corresponding to names, and GLIF’s global IF constraint violations for names. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In both settings, GLIF and GLIF-NRW achieve substantially better fairness metrics for the same levels of test accuracy in comparison to the IF-constraints method. To understand the reason for this, we study which global IF constraints are violated after applying the GLIF method in Figure 2 (right). Corresponding to the unique pairs of words in our test set, there are $n ( n - 1 ) / 2$ unique constraints in (2.3), and the global IF-constraints method satisfies all of them by design. Each constraint (i.e., each pair of words) corresponds to a fair distance, which is small for (under the fair metric) similar words and large for dissimilar words. We bin the constraints by fair distance and present the proportion of global IF constraints violated after applying the GLIF method for each bin in the histogram in Figure 2 (right). Here, we set the Lipschitz-constant $L$ in (2.3) to $L = 2 . 2 5$ corresponding to a $8 9 . 4 \\%$ accuracy of the IF-constraints method and show global IF constraint violations of GLIF corresponding to $9 5 \\%$ accuracy in blue. This means that we use strong global IF constraints and use a setting of the GLIF method which maintains most of the accuracy, which would not be possible using the IF-constraints method. GLIF does not violate any constraints corresponding to small fair distances, i.e., it satisfies IF on similar individuals, while violating many large fair distance constraints. This can be seen as basically all constraint violations (blue) are at large fair distances of greater or equal 6. This demonstrates the effect of enforcing local individual fairness from our theoretical analysis in Section 4. At the same time, we display frequency of constraints that correspond to pairs of names in orange, where we can see that almost all constraints corresponding to names occur at small fair distances of smaller or equal to 6. This is expected in this task because we consider all names similar, so fair distances between them should be small. We can see that the distributions of constraint violations after applying GLIF (blue, right) and names (orange, left) are almost disjoint. We mark all global IF constraint violations after applying GLIF that correspond to names in green, and observe that there are none. Summarizing, GLIF ignores unnecessary (in the context of this problem) constraints allowing it to achieve higher accuracy, while satisfying the more relevant local IF constraints leading to improved fairness. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Regarding the practical differences between GLIF and GLIF-NRW, in Figure 2 (left) GLIF has smaller standard deviations on the name outputs, but in in Figure 2 (center) GLIF-NRW achieves lower race gap. In Theorem 4.6, we showed that GLIF penalizes fairness violations in high density data regions stronger. As a result, GLIF may favor enforcing similar outputs in the high density region causing lower standard deviation, while leaving outputs nearly unchanged in the lower density region, resulting in larger race gaps. GLIF-NRW weights all data density regions equally, i.e., it is less likely to miss a small subset of names, but is less stringent in the high density regions. ",
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+        "text": "5.2 Post-processing for Debiasing Large Language Models ",
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+        "text": "Large language models have achieved impressive results on many tasks; however, there is also significant evidence demonstrating that they are prone to biases [4], [38], [39]. Debiasing these models remains largely an open problem: most in-processing algorithms are not applicable or computationally prohibitive due to large and highly complex model architectures, and challenges in handling text inputs. Even if an appropriate in-processing algorithm arises, significant environmental impact due to re-training is unavoidable [4], [16]. In our experiments, we evaluate effectiveness of GLIF as a simple post-processing technique to debias BERT-based models for text classification. Another possible solution is to fine-tune BERT with an in-processing technique as was done by Yurochkin et al. [12]. The two approaches are not directly comparable: fine-tuning with SenSeI [12] requires knowledge of the model parameters, alleviates only part of the computational burden, and has more stringent requirements on the fair metric, while post-processing with GLIF is transductive, i.e., it requires access to unlabeled test data (see extended discussion in Section 2.1). ",
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+        "image_caption": [
+            "Figure 3: Accuracy-Consistency trade-offs for Bios (left) and Toxicity (right). "
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+            "Table 1: Results for the Bios task. "
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+        "table_body": "<table><tr><td>Method</td><td>Test Acc.</td><td>Pred. Consist.</td></tr><tr><td>Baseline</td><td>0.846 ± 0.003</td><td>0.942 ± 0.002</td></tr><tr><td>GLIF</td><td>0.830± 0.004</td><td>0.986 ± 0.002</td></tr><tr><td>GLIF-NRW</td><td>0.834 ± 0.003</td><td>0.988 ± 0.002</td></tr><tr><td>SenSEI</td><td>0.843 ± 0.003</td><td>0.977 ± 0.001</td></tr></table>",
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+            "Table 2: Results for the Toxicity task. "
+        ],
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+        "table_body": "<table><tr><td>Method</td><td>Test Acc.</td><td>Pred. Consist.</td></tr><tr><td>Baseline</td><td>0.809 ± 0.004</td><td>0.614 ± 0.013</td></tr><tr><td>GLIF</td><td>0.803 ±0.003</td><td>0.835 ± 0.012</td></tr><tr><td>GLIF-NRW</td><td>0.803 ± 0.003</td><td>0.844 ± 0.013</td></tr><tr><td>SenSEI</td><td>0.791 ± 0.005</td><td>0.773 ± 0.043</td></tr></table>",
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+        "text": "We replicate the experiments of Yurochkin et al. [12] on Bios [40] and Toxicity1 data sets. They use the approach of Mukherjee et al. [19] for fair metric learning which we reproduce. We refer to the Appendix B.1 of [12] for details. In both tasks, following [12], we quantify performance with balanced accuracy due to class imbalance, and measure individual fairness via prediction consistency, i.e., the fraction of test points where the prediction remains unchanged when performing task-specific input modifications. For implementation details, see Appendix A. In Appendix A.4, we analyze the runtime and distinguish between the closed-form and coordinate descent variants of GLIF. ",
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+        "type": "text",
+        "text": "In Bios, the goal is to predict the occupation of a person based on their textual biography. Such models can be useful for recruiting purposes. However, due to historical gender bias in some occupations, the baseline BERT model learns to associate gender pronouns and names with the corresponding occupations. Individual fairness is measured with prediction consistency with respect to gender pronouns and names alterations. A prediction is considered consistent if it is the same after swapping the gender pronouns and names. We present the fairness-accuracy trade-off in Figure 3 (left) for a range of threshold parameters $\\tau$ , and compare performance based on hyperparameter values selected with a validation data in Table 1. Both GLIF and GLIF-NRW noticeably improve individual fairness measured with prediction consistency, while retaining most of the accuracy. ",
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+        "type": "text",
+        "text": "In Toxicity, the task is to identify toxic comments—an important tool for facilitating inclusive discussions online. The baseline BERT model learns to associate certain identity words with toxicity (e.g., “gay”) because they are often abused in online conversations. The prediction consistency is measured with respect to changes to identity words in the inputs. There are 50 identity words, e.g., “gay”, “muslim”, “asian”, etc. and a prediction is considered consistent if it is the same for all 50 identities. We present the trade-off plots in Figure 3 (right) and compare performance in Table 2 (right). Our methods reduce individual biases in BERT predictions. We note that in both Toxicity and Bios experiments, we observe no practical differences between GLIF and GLIF-NRW. ",
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+    {
+        "type": "text",
+        "text": "6 Summary and Discussion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We studied post-processing methods for enforcing individual fairness. The methods provably enforce a local form of IF and scale readily to large data sets. We hope this broadens the appeal of IF by (i) alleviating the computational costs of operationalizing IF and (ii) allowing practitioners to use off-the-shelf models for standard ML tasks. We also note that it is possible to use our objective for in-processing. ",
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+        "text": "We conclude with two warnings: First, enforcing any algorithmic fairness definition does not guarantee complete fairness from the perspective of the user. The problem-specific meaning of fairness is often hard to encode exactly with a mathematical fairness definition. Second, while local individual fairness is a reasonable choice in many applications, this choice should be understood and verified by the practitioner depending on the situation. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+        "text": "This note is based upon work supported by the National Science Foundation (NSF) under grants no. 1916271, 2027737, and 2113373 and supported by the German Research Foundation (DFG) under Germany’s Excellence Strategy EXC–2117–390829875. Any opinions, findings, and conclusions or recommendations expressed in this note are those of the authors and do not necessarily reflect the views of the NSF nor the DFG. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "text": "[1] S. Barocas and A. D. Selbst, “Big data’s disparate impact,” Calif. L. Rev., vol. 104, p. 671, 2016.   \n[2] J. Larson, J. Angwin, L. Kirchner, and S. Mattu, “How we examined racial discrimination in auto insurance prices,” ProPublica, 2017.   \n[3] J. Buolamwini and T. Gebru, “Gender shades: Intersectional accuracy disparities in commercial gender classification,” in Conference on fairness, accountability and transparency, PMLR, 2018, pp. 77–91.   \n[4] E. M. Bender, T. Gebru, A. McMillan-Major, and S. Shmitchell, “On the dangers of stochastic parrots: Can language models be too big?” In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, 2021, pp. 610–623.   \n[5] A. Chouldechova and A. Roth, “A snapshot of the frontiers of fairness in machine learning,” Communications of the ACM, vol. 63, no. 5, pp. 82–89, 2020.   \n[6] C. Dwork, M. Hardt, T. Pitassi, O. Reingold, and R. Zemel, “Fairness through awareness,” in Proceedings of the 3rd innovations in theoretical computer science conference, 2012, pp. 214–226.   \n[7] D. C. Hsia, “Credit scoring and the equal credit opportunity act,” Hastings LJ, vol. 30, p. 371, 1978.   \n[8] M. Bertrand and S. Mullainathan, “Are emily and greg more employable than lakisha and jamal? a field experiment on labor market discrimination,” American Economic Review, vol. 94, no. 4, pp. 991–1013, 2004.   \n[9] W. Fleisher, “What’s fair about individual fairness?” Available at SSRN 3819799, 2021.   \n[10] C. Jung, M. Kearns, S. Neel, A. Roth, L. Stapleton, and Z. S. Wu, “An algorithmic framework for fairness elicitation,” arXiv preprint arXiv:1905.10660, 2019.   \n[11] M. Yurochkin, A. Bower, and Y. Sun, “Training individually fair ML models with sensitive subspace robustness,” in International Conference on Learning Representations (ICLR), 2020.   \n[12] M. Yurochkin and Y. Sun, “Sensei: Sensitive set invariance for enforcing individual fairness,” in International Conference on Learning Representations (ICLR), 2021.   \n[13] A. Vargo, F. Zhang, M. Yurochkin, and Y. Sun, “Individually fair gradient boosting,” in International Conference on Learning Representations (ICLR), 2021.   \n[14] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “BERT: Pre-training of deep bidirectional transformers for language understanding. arxiv,” in Proc. Conference of the North American Chapter of the Association for Computational Linguistics (ACL), 2018.   \n[15] T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al., “Language models are few-shot learners,” arXiv preprint arXiv:2005.14165, 2020.   \n[16] E. Strubell, A. Ganesh, and A. McCallum, “Energy and policy considerations for deep learning in nlp,” in 57th Annual Meeting of the Association for Computational Linguistics, 2019.   \n[17] O. Chapelle, B. Schölkopf, and A. Zien, Eds., Semi-Supervised Learning. Cambridge, Mass: MIT Press, 2006.   \n[18] C. Ilvento, “Metric Learning for Individual Fairness,” in 1st Symposium on Foundations of Responsible Computing (FORC 2020), Leibniz International Proceedings in Informatics (LIPIcs), vol. 156, Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2020.   \n[19] D. Mukherjee, M. Yurochkin, M. Banerjee, and Y. Sun, “Two simple ways to learn individual fairness metrics from data,” in International Conference on Machine Learning (ICML), 2020.   \n[20] M. P. Kim, A. Ghorbani, and J. Zou, “Multiaccuracy: Black-box post-processing for fairness in classification,” in Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society, 2019.   \n[21] P. K. Lohia, K. N. Ramamurthy, M. Bhide, D. Saha, K. R. Varshney, and R. Puri, “Bias mitigation post-processing for individual and group fairness,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2019, pp. 2847–2851.   \n[22] P. Lohia, “Priority-based post-processing bias mitigation for individual and group fairness,” arXiv preprint arXiv:2102.00417, 2021.   \n[23] D. Wei, K. N. Ramamurthy, and F. d. P. Calmon, “Optimized score transformation for fair classification,” in International Conference on Artificial Intelligence and Statistics (AISTATS), 2020.   \n[24] D. Spielman, “Spectral graph theory,” Combinatorial scientific computing, no. 18, 2012.   \n[25] M. Hein, J.-Y. Audibert, and U. v. Luxburg, “Graph laplacians and their convergence on random neighborhood graphs.,” Journal of Machine Learning Research, vol. 8, no. 6, 2007.   \n[26] R. R. Coifman and S. Lafon, “Diffusion maps,” Applied and computational harmonic analysis, vol. 21, no. 1, 2006.   \n[27] D. Zhou and B. Schölkopf, “A regularization framework for learning from graph data,” in ICML 2004 Workshop on Statistical Relational Learning and Its Connections to Other Fields (SRL 2004), 2004, pp. 132–137.   \n[28] M. Belkin, I. Matveeva, and P. Niyogi, “Regularization and semi-supervised learning on large graphs,” in International Conference on Computational Learning Theory, Springer, 2004, pp. 624–638.   \n[29] A. J. Smola and R. Kondor, “Kernels and regularization on graphs,” in Learning theory and kernel machines, Springer, 2003.   \n[30] P. Lahoti, K. P. Gummadi, and G. Weikum, “Ifair: Learning individually fair data representations for algorithmic decision making,” in 2019 ieee 35th international conference on data engineering (icde), IEEE, 2019, pp. 1334–1345.   \n[31] J. Kang, J. He, R. Maciejewski, and H. Tong, “Inform: Individual fairness on graph mining,” in Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 2020, pp. 379–389.   \n[32] S. J. Wright, “Coordinate descent algorithms,” Mathematical Programming, 2015.   \n[33] M. Hein, J.-Y. Audibert, and U. Von Luxburg, “From graphs to manifolds–weak and strong pointwise consistency of graph laplacians,” in International Conference on Computational Learning Theory, Springer, 2005.   \n[34] M. Hu and B. Liu, “Mining and summarizing customer reviews,” in Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining, 2004.   \n[35] J. Pennington, R. Socher, and C. D. Manning, “Glove: Global vectors for word representation,” in Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP), 2014, pp. 1532–1543.   \n[36] A. Caliskan, J. J. Bryson, and A. Narayanan, “Semantics derived automatically from language corpora contain human-like biases,” Science, 2017.   \n[37] S. Diamond and S. Boyd, “Cvxpy: A python-embedded modeling language for convex optimization,” The Journal of Machine Learning Research, vol. 17, no. 1, pp. 2909–2913, 2016.   \n[38] K. Kurita, N. Vyas, A. Pareek, A. W. Black, and Y. Tsvetkov, “Measuring bias in contextualized word representations,” in First Workshop on Gender Bias in Natural Language Processing, 2019.   \n[39] M. Nadeem, A. Bethke, and S. Reddy, “Stereoset: Measuring stereotypical bias in pretrained language models,” arXiv preprint arXiv:2004.09456, 2020.   \n[40] M. De-Arteaga, A. Romanov, H. Wallach, J. Chayes, C. Borgs, A. Chouldechova, S. Geyik, K. Kenthapadi, and A. T. Kalai, “Bias in bios: A case study of semantic representation bias in a high-stakes setting,” in Proceedings of the Conference on Fairness, Accountability, and Transparency (ACM FAccT), 2019.   \n[41] L. Dixon, J. Li, J. Sorensen, N. Thain, and L. Vasserman, “Measuring and mitigating unintended bias in text classification,” in Proceedings of the 2018 AAAI/ACM Conference on AI, Ethics, and Society, 2018.   \n[42] B. Sen, A gentle introduction to empirical process theory and applications, 2018. ",
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