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+# SET FUNCTIONS FOR TIME SERIES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Despite the eminent successes of deep neural networks, many architectures are often hard to transfer to irregularly-sampled and asynchronous time series that occur in many real-world datasets, such as healthcare applications. This paper proposes a novel framework for classifying irregularly sampled time series with unaligned measurements, focusing on high scalability and data efficiency. Our method SEFT (Set Functions for Time Series) is based on recent advances in differentiable set function learning, extremely parallelizable, and scales well to very large datasets and online monitoring scenarios. We extensively compare our method to competitors on multiple healthcare time series datasets and show that it performs competitively whilst significantly reducing runtime.
+
+# 1 INTRODUCTION
+
+With the increasing digitalization, measurements over extensive time periods are becoming ubiquitous. Nevertheless, in many application domains, in particular healthcare (Yadav et al., 2018), measurements might not necessarily be observed at a regular rate or could be misaligned. Moreover, the presence or absence of a measurement and its observation frequency may carry information of its own (Little & Rubin, 2014), such that imputing the missing values is not always desired.
+
+While some algorithms can be readily applied to datasets with varying length, these methods usually assume regular sampling of the data and/or require the measurements across modalities to be aligned/synchronized, preventing their application to the aforementioned settings. Existing approaches for unaligned measurements, by contrast, typically rely on imputation to obtain a regularlysampled version of a dataset for classification. Learning a suitable imputation scheme, however, requires understanding the underlying dynamics of a system; this task is significantly more complicated and not necessarily required when classification is the main goal. Furthermore, even though a decoupled imputation scheme followed by classification is generally more scalable, it may lose information (in terms of “missingness patterns”) that could be crucial for prediction tasks. In addition, the fact that decoupled schemes perform worse than methods that are trained end-to-end has been has been empirically demonstrated by Li & Marlin (2016). Approaches that jointly optimize both tasks also add a large computational overhead, thus suffering from poor scalability or high memory requirements.
+
+Our method is motivated by the understanding that, while RNNs and similar architectures are well suited for capturing and modelling the dynamics of a time series and thus excel at tasks such as forecasting, retaining the order of an input sequence can even be a disadvantage in classification scenarios (Vinyals et al., 2015). We show that by relaxing the condition that a sequence must be processed in order, we can naturally derive an architecture that directly accounts for (i) irregular sampling, and (ii) unsynchronized measurements. Our method SEFT: Set Functions for Time Series, extends recent advances in set function learning to irregular sampled time series classification tasks, yields state-of-the-art performance, is highly scalable and improves over current approaches by almost an order of magnitude in terms of runtime.
+
+With SEFT, we propose to rephrase the problem of classifying time series as classifying a set of observations. We show how set functions can be exploited to learn classifiers that are naturally applicable to unaligned and irregularly sampled time series, leading to state-of-the-art performance in irregularly-sampled time series classification tasks. Our approach can be interpreted as learning dataset-specific summary statistics of time series which are optimized to separate instances by class.
+
+Furthermore, our method is highly parallelizable and can be readily extended to an online monitoring setup with up to thousands of patients.
+
+# 2 RELATED WORK
+
+This paper focuses on classifying time series with irregular sampling and potentially unaligned measurements. We briefly discuss recent work in this field; all approaches can be broadly grouped into the following three categories.
+
+Irregular sampling as missing data While the problem of supervised classification in the presence of missing data is closely related to irregular sampling on time series, there are some core differences. Missing data is usually defined with respect to a number of features that could be observed, whereas time series themselves can have different lengths and a “typical” number of observed values might not exist. Generally, an irregularly-sampled time series can be converted into a missing data problem by discretizing the time axis into non-overlapping intervals, and declaring intervals in which no data was sampled as missing. This approach is followed by Marlin et al. (2012), where a Gaussian Mixture Model was used to do semi-supervised clustering on electronic health records. Similarly, Lipton et al. (2016) discretize the time series into intervals, aggregate multiple measurements within an interval, and add missingness indicators to the input of a Recurrent Neural Network. By contrast, Che et al. (2018) present several variants of the Gated Recurrent Unit (GRU) combined with imputation schemes. Most prominently, the GRU-model was extended to include a decay term (GRU-D), such that the last observed value is decayed to the empirical mean of the time series via a learnable decay term. While these approaches are applicable to irregularly-sampled data, they either rely on imputation schemes or empirical global estimates on the data distribution (our method, by contrast, requires neither), without directly exploiting the global structure of the time series.
+
+Frameworks supporting irregular sampling Some frameworks support missing data. For example, Lu et al. (2008) directly defined a kernel on irregularly-sampled time series, permitting subsequent classification and regression with kernel-based classifiers or regression schemes. Furthermore, Gaussian Processes (Williams & Rasmussen, 2006) constitute a common probabilistic model for time series; they directly permit modelling of continuous time data using mean and covariance functions. Along these lines, Li & Marlin (2015) derived a kernel on Gaussian Process Posteriors, allowing the comparison and classification of irregularly-sampled time series using kernel-based classifiers. Nevertheless, all of these approaches still rely on separate tuning/training of the imputation method and the classifier so that structures supporting the classification could be potentially missed in the imputation step. An emerging line of research employs Hawkes processes (Hawkes, 1971; Liniger, 2009), i.e. a specific class of self-exciting point processes, for time series modelling and forecasting (Mei & Eisner, 2017; Yang et al., 2017; Xiao et al., 2017). While Hawkes processes exhibit extraordinary performance in these domains, there is no standardised way of using them for classification. Previous work (Lukasik et al., 2016) trains multiple Hawkes processes (one for each label) and classifies a time series by assigning it the label that maximises the respective likelihood function. Since this approach does not scale to our datasets, we were unable to perform a fair comparison. We conjecture that further research will be required to make Hawkes processes applicable to general time series classification scenarios.
+
+End-to-end learning of imputation schemes Methods of this type are composed of two modules with separate responsibilities, namely an imputation scheme and a classifier, where both components are trained discriminatively and end-to-end using gradient-based training. Recently, Li & Marlin (2016) proposed the Gaussian Process Adapters (GP Adapters) framework, where the parameters of a Gaussian Process Kernel are trained alongside a classifier. The Gaussian Process gives rise to a fixed-size representation of the irregularly-sampled time series, making it possible to apply any differentiable classification architecture. This approach was further extended to multivariate time series by Futoma et al. (2017) using Multi-task Gaussian Processes (MGPs) (Bonilla et al., 2008), which allow correlations between the imputed channels. Moreover, Futoma et al. (2017) made the approach more compatible with time series of different lengths by applying a Long Short Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) classifier. Motivated by the limited scalability of approaches based on GP Adapters, Shukla & Marlin (2019) suggest an alternative imputation scheme, the interpolation prediction networks. It applies multiple semi-parametric interpolation schemes to obtain a regularly-sampled time series representation. The parameters of the interpolation network are trained with the classifier in an end-to-end setup.
+
+![](images/99a1ff8e5b7b0ed149a713759f89dce223919083906167cbd08a05c7333a4fb7.jpg)  
+Figure 1: Schematic overview of SEFT’s architecture. The first panel exemplifies a potential input, namely a multivariate time series, consisting of 3 modalities $m _ { 1 } , m _ { 2 } , m _ { 3 }$ . We treat the $j ^ { \mathrm { t h } }$ observation as a tuple $( t _ { j } , z _ { j } , m _ { j } )$ , comprising a time $t _ { j }$ , a value $z _ { j }$ , and a modality indicator $m _ { j }$ . All observations are summarized as a set of such tuples. Each set of tuples belonging to the same modality is then separately embedded $( f ^ { \prime } )$ and subsequently aggregated (agg). An attention mechanism (attn) as described in Section 3.3 is then applied to learn the importance of individual and consecutive observations. Respective query matrices for 2 attentions head are illustrated in purple and orange blocks. The results of each attention head are then concatenated and used as the input for final classification layers.
+
+# 3 PROPOSED METHOD
+
+Our paper focuses on the problem of time series classification of irregularly sampled and unaligned time series. We first define the required terms before describing our models
+
+# 3.1 NOTATION & REQUIREMENTS
+
+Definition 1 (Time series). We describe a time series of an instance i as a set $s _ { i }$ of $M : = \mathrm { l e n } ( S _ { i } )$ observations $s _ { j }$ such that $\mathcal { S } _ { i } : = \{ s _ { 1 } , . . . , s _ { M } \}$ . We assume each observation $s _ { j }$ to be represented as a tuple $( t _ { j } , z _ { j } , m _ { j } )$ , consisting of a time $t _ { j } \in \mathbb { R } ^ { + }$ , an observed value $z _ { j } \in \mathbb { R } ,$ , and a modality indicator $m _ { j } \in \{ 1 \cdot \dots D \}$ , where $D$ represents the dimensionality of the time series. We write $\Omega \subseteq \mathbb { R } ^ { + } \times \bar { \mathbb { R } } \times \mathbb { N } ^ { + }$ to denote the domain of observations. An entire time series can thus be represented as
+
+$$
+\begin{array} { r } { S _ { i } : = \left\{ \left( t _ { 1 } , z _ { 1 } , m _ { 1 } \right) , \ldots , \left( t _ { M } , z _ { M } , m _ { M } \right) \right\} , } \end{array}
+$$
+
+where for notational convenience we omitted the index $i$ .
+
+We leave this definition very general on purpose, allowing the length of each time series (comprising all channels, such as “heart rate”, “respiratory rate”, etc. of one instance) to differ, since our models are capable of handling this. Likewise, we neither enforce nor expect all time series to be synchronized, i.e. being sampled at the same time, but rather we permit unaligned or non-synchronized observations in the sense of not having to observe all modalities at each time point. Time series are collected in a dataset $\mathcal { D }$ .
+
+Definition 2 (Dataset). We consider a dataset $\mathcal { D }$ to contain $n$ time series. Elements of $\mathcal { D }$ are tuples, i.e. $\mathcal { D } : = \{ ( S _ { 1 } , y _ { 1 } ) , . . . , ( S _ { N } , y _ { N } ) \}$ , where $S _ { i }$ denotes the $i ^ { \mathrm { { t h } } }$ time series and $y _ { i } \in \{ 1 , \ldots , C \}$ its associated class label.
+
+Figure 1 gives a high-level overview of our method, including the individual steps required to perform classification. To get a more intuitive grasp of these definitions, we briefly illustrate our time series notation with an example. Let instance $i$ be an in-hospital patient, while the time series represent measurements of two channels of vital parameters during a hospital stay, namely heart rate (HR) and mean arterial blood pressure (MAP). We enumerate those channels as modalities 1 and 2. Counting from admission time, a HR of 60 and 65 beats per minute was measured after $0 . 5 \mathrm { h }$ and $3 . 0 \mathrm { h }$ , respectively, whereas MAP values of 80, 85, and $8 7 \mathrm { m m H g }$ were observed after $0 . 5 \mathrm { h }$ , $1 . 7 \mathrm { h }$ , and $2 . 5 \mathrm { h }$ . According to Definition 1, the time series is thus represented as $S _ { i } = \{ ( 0 . 5 , 6 0 , 1 ) , ( 3 , 6 5 , 1 ) , ( 0 . 5 , 8 0 , \bar { 2 } ) , ( 1 . 7 , 8 5 , 2 ) , ( 3 , 8 7 , 2 ) \}$ . In this example, observations are ordered by modality to increase readability; in practice, we are dealing with unordered sets.
+
+Definition 3 (Non-synchronized time series). We call a $D$ -dimensional time series nonsynchronized if there is at least one time point $t _ { j } ~ \in ~ \mathbb { R } ^ { + }$ at which at least one modality is not observed, i.e. if there exists $t _ { j } \in \mathbb { R } ^ { + }$ such that $| \{ ( t _ { k } , z _ { k } , m _ { k } ) \mid t _ { k } = t _ { j } \} | \neq D$ .
+
+Furthermore, we assume that no two measurements of the same modality $m _ { k }$ occur at the same time, i.e. $t _ { i } \neq t _ { j }$ for $i \neq j$ has to be satisfied for all measurements in $m _ { k }$ . This assumption is not required for technical reasons but for consistency. It also makes it possible to interpret the results later on.
+
+To summarize our generic setup, we do not require $M$ , the number of observations per time series, to be the same, i.e. $\mathrm { l e n } ( S _ { i } ) \neq \mathrm { l e n } ( S _ { j } )$ for $i \neq j$ is permitted, nor do we assume that the time points and modalities of the observations are the same across time series. This setting is common in biomedical time series, for example. Since typical machine learning algorithms are designed to operate on data of a fixed dimension, novel approaches to this non-trivial problem are required.
+
+# 3.2 OUR MODEL
+
+In the following, we describe an approach inspired by differentiable learning of functions that operate on sets (Zaheer et al., 2017; Wagstaff et al., 2019). We phrase the problem of classifying time series on irregular grids as learning a function $f$ on a set of arbitrarily many time series observations following Definition 1, i.e. ${ \cal S } = \{ ( t _ { 1 } , z _ { 1 } , m _ { 1 } ) , \ldots , ( t _ { M } , z _ { M } , m _ { M } ) \}$ , such that $f \colon S  \mathbb { R } ^ { C }$ , where $s$ represents a generic time series of arbitrary cardinality and $\mathbb { R } ^ { C }$ corresponds to the logits of the $C$ classes in the dataset. As we previously discussed, we interpret each time series as an unordered set of measurements, where all information is conserved because the observation time is included for each set element. Specifically, we define $f$ to be a set function, i.e. a function that operates on a set and thus has to be invariant to the ordering of the elements in the set. Multiple architectures are applicable to constructing set functions such as Transformers (Lee et al., 2019; Vaswani et al., 2017), or Deep Sets (Zaheer et al., 2017). Due to preliminary experiments, where Transformers suffered from lower generalization performance in our setting1, we base this work on the framework of Zaheer et al. (2017). Intuitively, this can be seen as computing multivariate dataset-specific summary statistics, which are optimized to maximize classification performance. Thus, we sum-decompose the set function $f$ into the form
+
+$$
+f ( S ) = g \left( \frac { 1 } { | S | } \sum _ { s _ { j } \in S } h ( s _ { j } ) \right)
+$$
+
+where $h \colon \Omega \to  { \mathbb { R } } ^ { d }$ and $g \colon  { \mathbb { R } ^ { d } } \to  { \mathbb { R } ^ { C } }$ are neural networks, $d \in \mathbb { N } ^ { + }$ determines the dimensionality of the latent representation, and $s _ { j }$ represents a single observation of the time series $s$ . We can view the averaged representations $1 / | \mathcal { \bar { S } } | \sum _ { s _ { j } \in \mathcal { S } } h ( s _ { j } )$ in general as a dataset-specific summary statistic learned to best distinguish the class labels. Equation 2 also implies the beneficial scalability properties of our approach: each embedding can be calculated independently of the others; hence, the constant computational cost of passing a single observation through the function $h$ is scaled by the number of observations, resulting in a runtime of $\mathcal { O } ( M )$ for a time series of length $M$ .
+
+Recently, Wagstaff et al. (2019) derived requirements for a practical universal function representation of sum-decomposable set functions, i.e the requirements necessary for a sum-decomposable function to represent an arbitrary set-function given that $h$ and $g$ are arbitrarily expressive. In particular, they show that a universal function representation can only be guaranteed provided that $d \geq \operatorname* { m a x } _ { i } \operatorname { l e n } ( S _ { i } )$ is satisfied. During hyperparameter search we thus independently sample the dimensionality of the aggregation space, and allow it to be in the order of the number of observations that are to be expected in the dataset. Further, we explored the utilization of max, sum, and mean as alternative aggregation functions inspired by Zaheer et al. (2017); Garnelo et al. (2018).
+
+Intuition Our method can be connected to Takens’s embedding theorem (Takens, 1981) for dynamical systems: we also observe a set of samples from some unknown (but deterministic) dynamical process; provided the dimensionality of our architecture is sufficiently large2, we are capable of reconstructing the system up to diffeomorphism. The crucial difference is that we do not have to construct a time-delay embedding but rather, we let the network learn an embedding that is suitable for classification.
+
+Time encoding In order to represent the time point of an observation on a normalized scale, we employ variant of positional encodings, as introduced by Vaswani et al. (2017). Preliminary results indicated that this encoding scheme reduces the sensitivity towards initialization and training hyperparameters of a model. Specifically, the time encoding converts the one-dimensional time axis into a multi-dimensional input by passing the time $t$ of each observation through multiple sine and cosine functions of varying frequencies. Given a dimensionality $\tau \in \mathbb { N } ^ { + }$ of the time encoding, we refer to the encoded position as $x \in \mathbb { R } ^ { \tau }$ , where
+
+$$
+\begin{array} { r } { x _ { 2 k } ( t ) : = \sin \bigg ( \frac { t } { \operatorname* { m a x } _ { - } \mathrm { t s } ^ { 2 k / \tau } } \bigg ) } \\ { x _ { 2 k + 1 } ( t ) : = \cos \bigg ( \frac { t } { \operatorname* { m a x } _ { - } \mathrm { t s } ^ { 2 k / \tau } } \bigg ) } \end{array}
+$$
+
+with $k \in \{ 0 , \ldots , \tau / 2 \}$ and max ts representing the maximal time scale that is expected in the data. Intuitively, we select the wavelengths using a geometric progression from $2 \pi$ to max ts $\cdot 2 \pi$ , and treat the number of steps and the maximum timescale max ts as hyperparameters of the model. For all experiments time encodings were used, such that an observation is represented as $s _ { j } =$ $( x ( t _ { j } ) , z _ { j } , m _ { j } )$ .
+
+Loss function If not mentioned otherwise, we choose $h$ and $g$ in Equation 2 to be multilayer perceptron deep neural networks, parametrized by weights $\theta$ and $\psi$ , respectively. We thus denote these neural networks by $h _ { \theta }$ and $g _ { \psi }$ ; their parameters are shared across all instances per dataset. In our training setup, we follow Zaheer et al. (2017) and apply the devised set function to the complete time series, i.e. to the set of all observations for each time series. Overall, we optimize a loss function that is defined as
+
+$$
+\mathcal { L } ( \boldsymbol { \theta } , \boldsymbol { \psi } ) : = \mathbb { E } _ { ( S , \boldsymbol { y } ) \in \mathcal { D } } \left[ \ell \left( \boldsymbol { y } ; \boldsymbol { g } _ { \boldsymbol { \psi } } \left( \frac { 1 } { | S | } \sum _ { s _ { j } \in S } h _ { \boldsymbol { \theta } } ( s _ { j } ) \right) \right) \right] ,
+$$
+
+where $\ell ( \cdot )$ represents a task-specific loss function. In out setup, we either utilize the binary crossentropy in combination with a sigmoid activation function in the last layer for binary classification or multi-label classification tasks and categorical cross-entropy in combination with a softmax activation function in the last layer for multi-class classification tasks.
+
+# 3.3 ATTENTION-BASED AGGREGATION
+
+So far, our method permits encoding sets of arbitrary sizes into a fixed-size representation. For increasingly large set sizes, however, many irrelevant observations could influence the result of the set function. The mean aggregation function is particularly susceptible to this because the influence of an observation to the embedding shrinks proportionally to the size of the set. We thus suggest to use a weighted mean in order to allow the model to decide which observations are relevant and which should be considered irrelevant. This is equivalent to computing an attention $a ( S , s _ { j } )$ over the set input elements, and subsequently, computing the sum over all elements in the set.
+
+Our approach is based on scaled dot-product attention with multiple heads $i \in \{ 1 , \ldots , m \}$ in order to be able to cover different aspects of the aggregated $\mathrm { s e t } ^ { 3 }$ . We define $a ( \cdot )$ , i.e. the attention weight function of an individual time series, to depend on the overall set of observations. This is achieved by computing an embedding of the set elements using a smaller set function $f ^ { \prime }$ , and projecting the concatenation of the set representation and the individual set elements into a $d$ -dimensional space. Specifically, we have $K _ { j , i } ~ = ~ [ f ^ { \prime } ( { \cal { S } } ) , s _ { j } ] ^ { T } { \cal { W } } _ { i }$ where $W _ { i } \ \in \ \mathbb { R } ^ { ( \mathrm { i m } ( f ^ { \prime } ) + | s _ { j } | ) \times d }$ and $K \in \mathbb { R } ^ { | s | \times d }$ . Furthermore, we define a matrix of query points $Q \in \mathbb { R } ^ { m \times d }$ , which allow the model to summarize different aspects of the dataset via
+
+$$
+e _ { j , i } = \frac { K _ { j , i } \cdot Q _ { i } } { \sqrt { d } } \qquad \mathrm { a n d } \qquad a _ { j , i } = \frac { \exp ( e _ { j , i } ) } { \sum _ { j } \exp ( e _ { j , i } ) }
+$$
+
+where $a _ { j , i }$ represents the amount of attention that head $i$ gives to set element $j$ . The head-specific row $Q _ { i }$ of the query matrix $Q$ allows a head to focus on individual aspects (such as the distribution of one or multiple modalities) of a time series. For each head, we multiply the set element embeddings computed via the set function $f$ with the attentions derived for the individual instances, i.e. $r _ { i } =$ $\textstyle \sum _ { j } a _ { j , i } f ( s _ { j } )$ . The computed representation is concatenated and passed to the aggregation network $h _ { \theta }$ as in a regular set function, i.e. $r * = [ r _ { 1 } \ldots r _ { m } ]$ . In our setup, we initialize $Q$ with zeros, such that at the beginning of training, the attention mechanism is equivalent to computing the unweighted mean over the set elements.
+
+Overall, this aggregation function is similar to Transformers (Vaswani et al., 2017), but differs from them in a few key aspects. Standard Transformer blocks would use the information from all set elements in order to compute the embedding of an individual set element, leading to a runtime and space complexity of $\mathcal { O } ( \bar { n } ^ { 2 } )$ . In contrast, our approach computes the embeddings of set elements independently, leading lower runtime and memory complexity of ${ \mathcal { O } } ( n )$ . Further, we observed that computing embeddings with information from other set elements (as the Transformer does) actually decreases generalization performance (see Table 1 for details).
+
+# 4 EXPERIMENTS
+
+We executed all experiments and implementations in a unified code base, which we also make available4 to the community. While some of the datasets used subsequently have access restrictions, anybody can gain access after satisfying the defined requirements. This ensures the reproducibility of our results. Please consult Appendix A.2 for further details.
+
+# 4.1 DATASETS
+
+In order to benchmark the proposed method we selected 4 datasets with irregularly-sampled and non-synchronized measurements.
+
+Healing MNIST The H-MNIST dataset was introduced by Krishnan et al. (2015) in order to simulate characteristics which typically occur in medical time series. In our setup, we use a variant of this dataset. Every instance of the dataset contains 10 frames, derived from a single instance of MNIST dataset, where the digit is rotated according to an angle uniformly sampled between $- 9 0 ^ { \circ }$ to $9 0 ^ { \circ }$ . Furthermore, 3 randomly-selected consecutive frames are augmented by a square artefact in the top left corner of the image in order to indicate seasonality in the time series. Finally, $60 \%$ of the data points are randomly discarded in order to yield a final high-dimensional irregularly-sampled time series with non-synchronized measurements. Using these settings each instance has on average 3, 136 observations.
+
+MIMIC-III Tasks MIMIC-III (Johnson et al., 2016) is a widely-used, freely-accessible dataset containing around 50, 000 distinct ICU stays. The median length of stay is $2 . 1 \mathrm { d }$ and a wide range of physiological measurements (e.g. arterial blood pressure, respiration rate, heart rate) are recorded with a resolution of $^ { 1 \mathrm { h } }$ . Furthermore, laboratory test results, collected at irregular time intervals are available. Recently, Harutyunyan et al. (2019) defined a set of machine learning tasks, labels, and benchmarks using a subset of the MIMIC-III dataset. We trained and evaluated our method and competing methods on the binary mortality prediction task (M3-Mortality) and on the multiclass problem of phenotype classification (M3-Phenotyping), while applying additional filtering described in Appendix A.1. The goal of the mortality prediction task is to predict whether a patient will die during his/her hospital stay using only data from the first 48 hours of the ICU stay. This dataset contains around 21, 000 stays of which approximately $10 \%$ result in death. The phenotype classification task consists of 40, 000 patients, each of which can suffer from a multitude of 25 acute care conditions.
+
+Physionet Mortality Prediction Challenge The 2012 Physionet challenge dataset (Goldberger et al., 2000), which we abbreviate P-Mortality, contains $1 2 , 0 0 0 \mathrm { I C U }$ stays each of which lasts at least $4 8 \mathrm { h }$ . For each stay, a set of general descriptors (such as gender, age, height, weight) were collected at admission time. Depending on the course of the stay and patient status, up to 37 time series variables were measured (e.g. blood pressure, lactate, respiration rate, temperature). While some modalities might be measured in regular time intervals (e.g. hourly or daily), some are only collected when required. Not all variables are available for each stay. The goal of the challenge was to predict if—and with which certainty —a patient will die during the hospital stay. The training set consists of 8, 000 stays while the testing set comprises 4, 000 ICU visits. Both datasets are similarly imbalanced, with a prevalence of around $14 \%$ . For simplicity, the general descriptors (such as age and weight), were included as time points with a single observation at the beginning of the stay. This treatment is similar to the approach by Harutyunyan et al. (2019) in the MIMIC-III benchmarking datasets. Please refer to Table A.1, Table A.2, and Table A.3 in the appendix for a more detailed enumeration of samples sizes and label distributions. The total number of samples may slightly deviate from the originally published splits, as time series of excessive length prevented fitting some methods in reasonable time, and were therefore excluded.
+
+# 4.2 COMPETITOR METHODS
+
+GRU-simple GRU-SIMPLE (Che et al., 2018) augments the input at time $t$ of a Gated-RecurrentUnit RNN with a measurement mask $m _ { t } ^ { d }$ and a $\delta _ { t }$ matrix, which contains the time since the last measurement of the corresponding modality $d$ , such that
+
+$$
+\delta _ { t } = \left\{ \begin{array} { l l } { s _ { t } - s _ { t - 1 } + \delta _ { t - 1 } ^ { d } } & { t > 1 , m _ { t - 1 } ^ { d } = 0 } \\ { s _ { t } - s _ { t - 1 } } & { t > 1 , m _ { t - 1 } ^ { d } = 1 } \\ { 0 } & { t = 0 } \end{array} \right.
+$$
+
+where $s _ { t }$ represents the time associated with time step $t$ .
+
+Phased-LSTM The PHASED-LSTM (Neil et al., 2016) introduced a biologically inspired time dependent gating mechanism which regulates access to the hidden and cell state of a Long short-term RNN cell (Hochreiter & Schmidhuber, 1997). While this allows the network to handle event-based sequences with irregularly spaced observations, the approach does not support unaligned measurements. In order to still provide the architecture with all relevant information, we augment the input in a similar fashion as described for the GRU-SIMPLE approach.
+
+GRU-D GRU-D or GRU-Decay (Che et al., 2018) contains modifications to the GRU RNN cell, allowing it to decay past observations to the mean imputation of a modality using a learnable decay rate. By additionally providing the measurement masks as an input the recurrent neural network the last feed in value. Learns how fast to decay back to a mean imputation of the missing data modality.
+
+Interpolation Prediction Networks IP-NETWORKS (Shukla & Marlin, 2019) apply multiple semiparametric interpolation schemes to irregularly-sampled time series to obtain regularly-sampled representations that cover long-term trends, transients, and also sampling information. The method combines a univariate interpolation step with a subsequent multivariate interpolation; the parameters of the interpolation network are trained with the classifier in an end-to-end fashion.
+
+Transformer In the TRANSFORMER architecture (Vaswani et al., 2017) the elements of a sequence are encoded simultaneously and information between sequence elements is captured using MultiHead-Attention blocks. In our case, an individual sequence element corresponds to all measurements available at a given time point, augmented with a measurement indicator. Transformers are normally used for sequence-to-sequence modelling tasks and in our setup were adapted to classification tasks by mean-aggregating the final representation. This representation is then fed into a one-layer MLP to predict logits for the individual classes.
+
+![](images/067d20b9dff90ce2160b9c55a84e97acb5c9306cbf172f64abc03a6a9df16919.jpg)  
+Figure 2: Visualizations of a single attention head on an instance of the P-Mortality dataset. We display a set of variables relevant for assessing patient stability and organ failure: Serum Potassium (K), Lactate, Systolic Arterial Blood Pressure (SysABP), and Urine output. Darker colors represent higher attention values.
+
+# 4.3 EXPERIMENTAL SETUP
+
+To permit a fair comparison between the methods, we executed hyperparameter searches for each model on each dataset, composed of uniformly sampling 20 parameters according to Appendix A.3. Training was stopped after 20 epochs without improvement of the validation loss, the hyperparameters with the best overall validation performance were selected for quantifying the performance on the test set. The train, validation, and test splits were the same for all models and all evaluations. Final performance on the test set was calculated by 3 independent runs of the models; evaluation took place after the model was restored to the state with the best validation loss. In all subsequent benchmarks, we use the standard deviation of the test performance of these runs as generalization performance estimates.
+
+# 4.4 RESULTS
+
+The results are shown in Table 1. Overall, our proposed method exhibits the lowest per-epoch runtime on most datasets, while either yielding competitive or state-the-art performance. Further, the trade-off between runtime and performance of the proposed method is very good on all datasets (see Figure A.1 and Figure A.2 in the appendix for a visualization of this argument). In order to elucidate the contribution of individual model components, we also provide an ablation study in Table A.4. Here we see that the attention mechanism contributes more to the model performance, while the positional encoding seems to be beneficial for datasets with highly-varying time series lengths, in particular M3-Phenotyping.
+
+Opening the black box In the medical domain, it is of particular interest to understand the decisions a model makes based on the input it is provided with. The formulation of our model and its per observation perspective on time series gives it the unique property of being able to quantify to which extent an individual observation contributed to the output of the model. We exemplify this in Figure 2 with a patient time series that was combined with our models attention values, displayed for a set of clinically relevant variables. After reviewing these records with our medical expert, we find that our model is able to pick up regions with drastic changes in individual modalities. Moreover, it is able to inspect other modalities at the same associated time (for instance, at hour 20). This is behaviour similar to what one would expect from an alerted clinician reviewing the logged medical records. Interestingly, we observe that the model attends to known trends (that are consisting with domain knowledge about patient deterioration ultimately resulting in death) such as increase in lactate or hemodynamic instability, as indicated by drops in blood pressure. Furthermore, the model appears to be alerted by persisting low urine output. After several hours, this can be indicative of kidney failure.
+
+Table 1: Performance comparison of methods on benchmarking datasets. Performance metrics have been rescaled to 100 for readability reasons. “AUC” denotes the area under the Receiver Operating Characteristic (ROC) curve; “PR AUC” denotes the area under the precision recall curve. “MICRO” refers to evaluating the metric globally by treating each entry of the label indicator matrix as a label. For “MACRO”, the metric is computed for each class and then averaged, whereas in “WEIGHTED” the class-wise metrics are weighted by class imbalance. Values denoted with “OOM” were not obtainable due to restrictions in GPU memory.   
+
+<table><tr><td>DATASET</td><td>MODEL</td><td>MICRO AUC</td><td>MACRO AUC</td><td>WEIGHTED AUC</td><td>RUNTIME</td></tr><tr><td rowspan="6">H-MNIST</td><td>GRU-SIMPLE</td><td>99.09 ± 0.05</td><td>99.01 ± 0.05</td><td>99.03 ± 0.05</td><td>11.43 ± 0.47</td></tr><tr><td>PHASED-LSTM</td><td>98.63 ± 0.13</td><td>98.50 ± 0.15</td><td>98.52 ± 0.14</td><td>33.93 ±1.11</td></tr><tr><td>GRU-D</td><td>99.42 ± 0.01</td><td>99.37 ± 0.02</td><td>99.38 ±0.02</td><td>11.81 ± 0.44</td></tr><tr><td>IP-NETS</td><td>99.06 ± 0.05</td><td>98.96 ± 0.03</td><td>98.98 ± 0.03</td><td>127.76 ± 0.95</td></tr><tr><td>TRANSFORMER</td><td>99.59 ± 0.05</td><td>99.55 ± 0.05</td><td>99.56 ± 0.05</td><td>21.62 ± 0.90</td></tr><tr><td>SEFT*</td><td>99.76 ± 0.01</td><td>99.75 ± 0.01</td><td>99.75 ± 0.01</td><td>4.05 ± 0.35</td></tr><tr><td rowspan="7">M3-Phenotyping</td><td>GRU-SIMPLE</td><td>79.89 ± 0.14</td><td>73.91 ± 0.19</td><td>72.55 ± 0.16</td><td>112.58 ± 2.03</td></tr><tr><td>PHASED-LSTM</td><td>80.00±0.06</td><td>73.91 ± 0.09</td><td>72.65 ± 0.08</td><td>400.41 ± 14.14</td></tr><tr><td>GRU-D</td><td>82.16 ± 0.04</td><td>77.14 ± 0.03</td><td>76.08 ± 0.01</td><td>288.70± 16.66</td></tr><tr><td>IP-NETS</td><td>-OOM—</td><td>-OOM-</td><td>-OOM—</td><td>-0OM-</td></tr><tr><td>TRANSFORMER</td><td>—O0M-</td><td>-00M—</td><td>—00M-</td><td>—00M-</td></tr><tr><td>SEFT</td><td>81.22 ±0.12</td><td>75.95 ± 0.09</td><td>74.90 ± 0.11</td><td>56.27 ± 2.14</td></tr><tr><td>SEFT-ATTN</td><td>82.00 ± 0.06</td><td>76.95 ± 0.09</td><td>75.88 ± 0.09</td><td>52.32 ± 0.74</td></tr><tr><td></td><td></td><td>ACCURACY</td><td>PR AUC</td><td>AUC</td><td>RUNTIME</td></tr><tr><td rowspan="7">M3-Mortality</td><td>GRU-SIMPLE</td><td>88.24±0.38</td><td>36.36 ±1.31</td><td>79.36 ± 0.26</td><td>22.80 ±0.56</td></tr><tr><td>PHASED-LSTM</td><td>88.32 ±0.31</td><td>35.30 ±1.38</td><td>80.16 ±0.22</td><td>25.54 ± 0.26</td></tr><tr><td>GRU-D</td><td>89.56 ± 0.38</td><td>46.76 ± 0.65</td><td>83.73 ± 0.21</td><td>31.85 ± 0.86</td></tr><tr><td>IP-NETS</td><td>89.73 ± 0.16</td><td>45.88 ± 0.87</td><td>83.30 ± 0.56</td><td>101.12 ± 4.52</td></tr><tr><td>TRANSFORMER</td><td>89.14 ± 0.15</td><td>42.32 ± 0.41</td><td>82.60 ± 0.55</td><td>4.79 ± 0.02</td></tr><tr><td>SEFT</td><td>88.65 ± 0.49</td><td>36.18 ± 5.07</td><td>79.15 ± 3.00</td><td>3.72 ± 0.11</td></tr><tr><td>SEFT-ATTN</td><td>89.48 ± 0.16</td><td>45.25 ± 0.96</td><td>83.79 ± 0.59</td><td>16.64 ± 0.20</td></tr><tr><td rowspan="7">P-Mortality</td><td>GRU-SIMPLE</td><td>85.66 ± 0.14</td><td>39.43 ± 0.71</td><td>79.79 ±0.16</td><td>5.16 ± 0.06</td></tr><tr><td>PHASED-LSTM</td><td>85.57 ± 0.11</td><td>39.55 ± 0.62</td><td>78.71 ±0.76</td><td>18.59 ± 1.15</td></tr><tr><td>GRU-D</td><td>87.19 ± 0.30</td><td>54.95 ± 0.54</td><td>86.58 ± 0.32</td><td>14.08 ± 0.38</td></tr><tr><td>IP-NETS</td><td>87.23 ± 0.18</td><td>54.87 ± 0.41</td><td>86.42 ±0.18</td><td>7.21 ± 0.46</td></tr><tr><td>TRANSFORMER</td><td>86.47 ± 0.08</td><td>48.72 ± 0.61</td><td>83.49 ± 0.46</td><td>2.69 ± 0.43</td></tr><tr><td>SEFT</td><td>87.11 ± 0.32</td><td>52.07 ± 0.41</td><td>84.12 ±0.32</td><td>3.07 ± 0.03</td></tr><tr><td>SEFT-ATTN</td><td>87.62 ± 0.16</td><td>54.05 ± 0.27</td><td>85.50 ± 0.13</td><td>7.54 ± 0.08</td></tr></table>
+
+\*: Due to the high dimensionality of H-MNIST and associated memory issues, the set elements were constructed by concatenating the observation time with all values associated with the time point and measurement indicators. Furthermore, as this dataset features only 10 time steps and missingness is induced randomly, we refrained from applying the attention-based aggregation.
+
+# 5 CONCLUSION
+
+In this work, we presented a novel approach for classifying time series with irregularly-sampled and unaligned, that is non-synchronized, observations. Our approach yields state-of-the-art to strongly competitive performance on numerous simulated and real-world datasets, while reducing runtime by almost half. Moreover, we demonstrated that combining the perspective of individual observations with an attention mechanism permits increasing the interpretability of the model. This is particularly relevant for the medical and healthcare applications.
+
+For future work, we reserve a more extensive exploration of the learned latent representation to evaluate its utility for clustering of time series or visualization of their similarity.
+
+# REFERENCES
+
+Edwin V. Bonilla, Kian M Chai, and Christopher Williams. Multi-task gaussian process prediction. In Advances in Neural Information Processing Systems (NeurIPS), pp. 153–160, 2008.
+
+Zhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, and Yan Liu. Recurrent neural networks for multivariate time series with missing values. Scientific reports, 8(1):6085, 2018.
+
+Joseph Futoma, Sanjay Hariharan, and Katherine Heller. Learning to detect sepsis with a Multitask Gaussian Process RNN classifier. In International Conference on Machine Learning (ICML), pp. 1174–1182, 2017.
+
+Marta Garnelo, Dan Rosenbaum, Christopher Maddison, Tiago Ramalho, David Saxton, Murray Shanahan, Yee Whye Teh, Danilo Rezende, and SM Ali Eslami. Conditional neural processes. In International Conference on Machine Learning, pp. 1690–1699, 2018.
+
+Ary L. Goldberger, Luis A.N. Amaral, Leon Glass, Jeffrey M. Hausdorff, Plamen Ivanov, Roger G. Mark, Joseph E. Mietus, George B. Moody, Chung-Kang Peng, and H. Eugene Stanley. Physiobank, physiotoolkit, and physionet: components of a new research resource for complex physiologic signals. Circulation, 101(23):e215–e220, 2000.
+
+Hrayr Harutyunyan, Hrant Khachatrian, David C. Kale, Greg Ver Steeg, and Aram Galstyan. Multitask learning and benchmarking with clinical time series data. Scientific Data, 6(1):96, 2019. ISSN 2052-4463.
+
+Alan G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1):83–90, 1971.
+
+Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural Computation, 9(8): 1735–1780, 1997.
+
+Alistair E. W. Johnson, Tom J. Pollard, Lu Shen, Li-wei H. Lehman, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Anthony Celi, and Roger G. Mark. MIMIC-III, a freely accessible critical care database. Scientific Data, 3, 2016.
+
+Rahul G. Krishnan, Uri Shalit, and David Sontag. Deep Kalman Filters. arXiv e-prints, art. arXiv:1511.05121, Nov 2015.
+
+Juho Lee, Yoonho Lee, Jungtaek Kim, Adam Kosiorek, Seungjin Choi, and Yee Whye Teh. Set transformer: A framework for attention-based permutation-invariant neural networks. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.), Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pp. 3744–3753, Long Beach, California, USA, 09–15 Jun 2019. PMLR.
+
+Steven Cheng-Xian Li and Benjamin M Marlin. Classification of sparse and irregularly sampled time series with mixtures of expected gaussian kernels and random features. In UAI, pp. 484– 493, 2015.
+
+Steven Cheng-Xian Li and Benjamin M. Marlin. A scalable end-to-end Gaussian process adapter for irregularly sampled time series classification. In Advances In Neural Information Processing Systems 29, pp. 1804–1812, 2016.
+
+Larry S. Liebovitch and Tibor Toth. A fast algorithm to determine fractal dimensions by box counting. Physics Letters A, 141(8):386–390, 1989.
+
+Thomas Josef Liniger. Multivariate Hawkes processes. PhD thesis, ETH Zurich, 2009.
+
+Zachary C Lipton, David Kale, and Randall Wetzel. Directly modeling missing data in sequences with rnns: Improved classification of clinical time series. In Machine Learning for Healthcare Conference, pp. 253–270, 2016.
+
+Roderick JA Little and Donald B Rubin. Statistical analysis with missing data, volume 333. John Wiley & Sons, 2014.
+
+Zhengdong Lu, Todd K Leen, Yonghong Huang, and Deniz Erdogmus. A reproducing kernel hilbert space framework for pairwise time series distances. In Proceedings of the 25th International Conference on Machine learning, pp. 624–631, 2008.
+
+Michal Lukasik, P. K. Srijith, Duy Vu, Kalina Bontcheva, Arkaitz Zubiaga, and Trevor Cohn. Hawkes processes for continuous time sequence classification: an application to rumour stance classification in Twitter. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pp. 393–398, Berlin, Germany, August 2016. Association for Computational Linguistics. doi: 10.18653/v1/P16-2064. URL https: //www.aclweb.org/anthology/P16-2064.
+
+Benjamin M. Marlin, David C. Kale, Robinder G. Khemani, and Randall C. Wetzel. Unsupervised pattern discovery in electronic health care data using probabilistic clustering models. In Proceedings of the 2nd ACM SIGHIT International Health Informatics Symposium, pp. 389–398. ACM, 2012.
+
+Hongyuan Mei and Jason M. Eisner. The neural Hawkes process: A neurally self-modulating multivariate point process. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems 30, pp. 6754–6764. Curran Associates, Inc., 2017.
+
+Daniel Neil, Michael Pfeiffer, and Shih-Chii Liu. Phased lstm: Accelerating recurrent network training for long or event-based sequences. In Advances in Neural Information Processing Systems 29, pp. 3882–3890, 2016.
+
+Satya Narayan Shukla and Benjamin Marlin. Interpolation-prediction networks for irregularly sampled time series. In International Conference on Learning Representations, 2019.
+
+Floris Takens. Detecting strange attractors in turbulence. In David Rand and Lai-Sang Young (eds.), Dynamical Systems and Turbulence, pp. 366–381, Heidelberg, Germany, 1981. Springer.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems (NeurIPS), pp. 5998–6008, 2017.
+
+Oriol Vinyals, Samy Bengio, and Manjunath Kudlur. Order matters: Sequence to sequence for sets. arXiv preprint arXiv:1511.06391, 2015.
+
+Edward Wagstaff, Fabian B. Fuchs, Martin Engelcke, Ingmar Posner, and Michael Osborne. On the Limitations of Representing Functions on Sets. 2019.
+
+Christopher K. Williams and Carl Edward Rasmussen. Gaussian processes for machine learning. MIT Press, 2(3):4, 2006.
+
+Shuai Xiao, Junchi Yan, Mehrdad Farajtabar, Le Song, Xiaokang Yang, and Hongyuan Zha. Joint modeling of event sequence and time series with attentional twin recurrent neural networks. arXiv e-prints, art. arXiv:1703.08524, 2017.
+
+Pranjul Yadav, Michael Steinbach, Vipin Kumar, and Gyorgy Simon. Mining electronic health records (EHRs): a survey. ACM Computing Surveys, 50(6):85, 2018.
+
+Yingxiang Yang, Jalal Etesami, Niao He, and Negar Kiyavash. Online learning for multivariate hawkes processes. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems 30, pp. 4937–4946. Curran Associates, Inc., 2017.
+
+Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Ruslan R. Salakhutdinov, and Alexander J. Smola. Deep sets. In Advances in Neural Information Processing Systems (NeurIPS), pp. 3391–3401, 2017.
+
+# A APPENDIX
+
+Table A.1: M3-Mortality prevalence of labels for the binary classification task   
+
+<table><tr><td></td><td>Training Prevalence</td><td>Testing Prevalence</td></tr><tr><td>In-hospital deaths</td><td>0.135</td><td>0.116</td></tr></table>
+
+Table A.2: P-Mortality prevalence of labels for the binary classification task   
+
+<table><tr><td></td><td></td><td>Training PrevalenceTesting Prevalence</td></tr><tr><td>In-hospital deaths S</td><td>0.140</td><td>0.146</td></tr></table>
+
+# A.1 DATA FILTERING
+
+Due to memory requirements of some of the competitor methods, it was necassary to excluded time series with extremly high number of measurements. For the M3-Phenotyping patients with more than 2000 distinct time points were discarded from training. For M3-Mortality patients with more than 1000 time points were discarded as they contained dramatically different measuring frequencies compared to the rest of the dataset.
+
+# A.2 IMPLEMENTATIONAL DETAILS
+
+All experiments were run using tensorflow $0 . 1 5 . 0 \Sigma \mathsf { c } 0$ and training was performed on NVIDIA Geforce GTX 1080 GPUs. In order to allow a fair comparison between methods, the input processing pipeline cached model specific representations and transformations of the data. To further increase efficiency of the RNNs, sequences were binned in to buckets of jointly trained instances depending on their sequence length. The buckets were determined according to the (0.25, 0.5, 0.75) quantiles of the length distributions of the datasets.
+
+# A.3 TRAINING, MODEL ARCHITECTURES AND HYPERPARAMETER SEARCH
+
+General All models were trained using the Adam optimizer, while randomly sampling the learning rate from (0.001, 0.0005, 0.00025, 0.0001). Further, the batch size of all methods was sampled from the values (32, 64, 128, 256).
+
+Recurrent neural networks For the RNN based methods (GRU-SIMPLE, PHASEDLSTM, GRU-D and IP-NETS), the number of units was sampled in from the values (16, 32, 64, 128, 256, 512). Further, recurrent dropout and input dropout were sampled from the values (0.0, 0.1, 0.2, 0.3). Solely, for the PHASED-LSTM method, we did not apply dropout to the recurrent state and the inputs, as the learnt frequencies were hypothesized to fulfill a similar function as dropout (Neil et al., 2016).
+
+SEFT We vary the number of layers, dropout in between the layers and the number of nodes per layer for both the encoding network $h _ { \theta }$ and the aggregation network $g _ { \psi }$ from the same ranges. The number of layers is randomly sampled between 1 and 5, the number of nodes in a layer are uniformly sampled from the range (16, 32, 64, 128, 256, 512) and the dropout fraction is sampled from the values $( 0 . 0 , 0 . 1 , 0 . 2 , 0 . 3 )$ . The width of the embedding space prior to aggregation is sampled from the values (32, 64, 128, 256, 512, 1024, 2048). The aggregation function selected to be one of mean, sum and max. The number of dimensions used for the positional embedding $\tau$ is selected uniformly from $( 4 , 8 , 1 6 )$ and max ts us selected from the values (10, 100, 1000).
+
+SEFT-Attn The parameters for the encoding and aggregation networks are sampled in a similar fashion as for SEFT. In contrast we set the aggregation function to be sum as described in the text.
+
+Table A.3: M3-Phenotyping prevalence of labels for the multi label classification task   
+
+<table><tr><td>Phenotype</td><td>Training</td><td>Validation</td><td>Testing</td></tr><tr><td>Acute and unspecified renal failure</td><td>0.216</td><td>0.207</td><td>0.211</td></tr><tr><td>Acute cerebrovascular disease</td><td>0.0746</td><td>0.0753</td><td>0.0662</td></tr><tr><td>Acute myocardial infarction</td><td>0.103</td><td>0.103</td><td>0.108</td></tr><tr><td>Cardiac dysrhythmias</td><td>0.322</td><td>0.317</td><td>0.323</td></tr><tr><td>Chronic kidney disease</td><td>0.135</td><td>0.131</td><td>0.132</td></tr><tr><td>Chronic obstructive pulmonary disease and bronchiectasis</td><td>0.132</td><td>0.128</td><td>0.126</td></tr><tr><td>Complications of surgical procedures or</td><td>0.207</td><td>0.201</td><td>0.213</td></tr><tr><td>medical care Conduction disorders</td><td>0.0726</td><td>0.07</td><td>0.0704</td></tr><tr><td>Congestive heart failure； nonhyperten- sive</td><td>0.268</td><td>0.264</td><td>0.268</td></tr><tr><td>Coronary atherosclerosis and other heart disease</td><td>0.323</td><td>0.317</td><td>0.331</td></tr><tr><td>Diabetes mellitus with complications</td><td>0.0955</td><td>0.0945</td><td>0.094</td></tr><tr><td>Diabetes mellitus without complication</td><td>0.194</td><td>0.187</td><td>0.192</td></tr><tr><td>Disorders of lipid metabolism</td><td>0.291</td><td>0.287</td><td>0.289</td></tr><tr><td>Essential hypertension</td><td>0.421</td><td>0.41</td><td>0.424</td></tr><tr><td>Fluid and electrolyte disorders</td><td>0.267</td><td>0.276</td><td>0.265</td></tr><tr><td>Gastrointestinal hemorrhage</td><td>0.0715</td><td>0.0747</td><td>0.0788</td></tr><tr><td>Hypertension with complications and</td><td>0.133</td><td>0.131</td><td>0.13</td></tr><tr><td>secondary hypertension Other liver diseases</td><td>0.0884</td><td>0.0904</td><td>0.0883</td></tr><tr><td>Other lower respiratory disease</td><td>0.0514</td><td>0.0484</td><td>0.0565</td></tr><tr><td> Other upper respiratory disease</td><td>0.0408</td><td>0.0371</td><td>0.0429</td></tr><tr><td>Pleurisy; pneumothorax; pulmonary</td><td>0.0858</td><td>0.09</td><td>0.0905</td></tr><tr><td>collapse Pneumonia (except that caused by tu- berculosis or sexually transmitted dis-</td><td>0.14</td><td>0.135</td><td>0.135</td></tr><tr><td>ease) Respiratory failure; insufficiency; arrest</td><td>0.18</td><td>0.184</td><td>0.177</td></tr><tr><td>(adult) Septicemia (except in labor)</td><td>0.142</td><td>0.145</td><td>0.138</td></tr><tr><td>Shock</td><td>0.0783</td><td>0.0745</td><td>0.0811</td></tr><tr><td>Total samples</td><td>29 208</td><td>6359</td><td>6266</td></tr></table>
+
+Further we use a constant architecture for the attention network $f ^ { \prime }$ with 2 layers, 64 nodes per layer, 4 heads and a dimensionality of the dot product space $d$ of 128. We solely sample the amount of attention dropout uniformly from the values $( 0 . 0 , 0 . 1 , 0 . 2 5 , 0 . 5 )$ .
+
+Transformer We utilize the same model architecture as defined in Vaswani et al. (2017), where we use a one hidden layer MLP as a feed-forward network, with dimensionality of the hidden layer selected to be twice the model dimensionality. The parameters for the Transformer network were sampled according to the following criteria. The dimensionality of the model was sampled uniformly from the values (64, 128, 256, 512, 1024), the number of attention heads per layer from the values $( 2 , 4 , 8 )$ and the number of layers from the range $[ 1 , 6 ] \in \mathbb { N }$ . Further, we sampled the amount of dropout of the residual connections and the amount of attention dropout uniformly from the values $( 0 . 0 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 5 )$ , and the maximal timescale for the time embedding from the values $( 1 0 , 1 0 0 , 1 0 0 0 )$ (similar to the SEFT approach).
+
+Table A.4: Ablation study of individual components of SEFT. “AUC” denotes the area under the Receiver Operating Characteristic (ROC) curve; “PR AUC” denotes the area under the precision recall curve; “RUNTIME” denotes the runtime of one training epoch in seconds.   
+
+<table><tr><td>DATASET</td><td>MODEL</td><td>MICRO AUC</td><td>MACRO AUC</td><td>WEIGHTED AUC</td><td>RUNTIME</td></tr><tr><td>H-MNIST</td><td>SEFT</td><td>99.76 ± 0.01</td><td>99.75 ± 0.01</td><td>99.75 ± 0.01</td><td>4.05 ± 0.35</td></tr><tr><td rowspan="3">M3-Phenotyping</td><td>SEFT (NO ATTENTION)</td><td>81.22 ± 0.12</td><td>75.95 ± 0.09</td><td>74.90 ± 0.11</td><td>56.27 ± 2.14</td></tr><tr><td>SEFT-ATTN (NO TIME ENC.)</td><td>80.46±0.86</td><td>74.70 ± 1.12</td><td>73.48 ± 1.18</td><td>50.17 ± 0.84</td></tr><tr><td>SEFT-ATTN</td><td>82.00 ± 0.06</td><td>76.95 ± 0.09</td><td>75.88 ± 0.09</td><td>52.32 ± 0.74</td></tr><tr><td></td><td></td><td>ACCURACY</td><td>PR AUC</td><td>AUC</td><td>RUNTIME</td></tr><tr><td rowspan="3">M3-Mortality</td><td>SEFT (NO ATTENTION)</td><td>88.65 ± 0.49</td><td>36.18 ± 5.07</td><td>79.15 ± 3.00</td><td>3.72 ± 0.11</td></tr><tr><td>SEFT-ATTN (NO TIME ENC.)</td><td>89.31±0.08</td><td>44.12 ± 0.06</td><td>83.72 ± 0.34</td><td>17.60 ± 0.43</td></tr><tr><td>SEFT-ATTN</td><td>89.48 ± 0.16</td><td>45.25 ± 0.96</td><td>83.79 ± 0.59</td><td>16.64± 0.20</td></tr><tr><td rowspan="3">P-Mortality</td><td>SEFT (NO ATTENTION)</td><td>87.11 ± 0.32</td><td>52.07 ±0.41</td><td>84.12 ± 0.32</td><td>3.07 ± 0.03</td></tr><tr><td>SEFT-ATTN (NO TIME ENC.)</td><td>87.03 ±0.06</td><td>51.86 ± 1.04</td><td>84.91 ± 0.29</td><td>7.04 ± 0.04</td></tr><tr><td>SEFT-ATTN</td><td>87.62 ± 0.16</td><td>54.05± 0.27</td><td>85.50 ± 0.13</td><td>7.54 ± 0.08</td></tr></table>
+
+![](images/5a022c75092c637eed0c7fba11b58915df9fb99274cb37712336626f29f9b3bb.jpg)  
+Figure A.1: A visualisation of the runtime of all methods and their AUC for datasets with a binary classification scenario.
+
+![](images/7bfd5f3b99f47c4b2eeb4f901e15232fff222b513a48bd633e440baaacd0c2d9.jpg)  
+Figure A.2: A visualisation of the runtime of all methods and their AUC for datasets with a multilabel classification scenario. Please note that the model definition for SEFT changes between the left and the right column; please see Table 1 for more details.

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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": "Despite the eminent successes of deep neural networks, many architectures are often hard to transfer to irregularly-sampled and asynchronous time series that occur in many real-world datasets, such as healthcare applications. This paper proposes a novel framework for classifying irregularly sampled time series with unaligned measurements, focusing on high scalability and data efficiency. Our method SEFT (Set Functions for Time Series) is based on recent advances in differentiable set function learning, extremely parallelizable, and scales well to very large datasets and online monitoring scenarios. We extensively compare our method to competitors on multiple healthcare time series datasets and show that it performs competitively whilst significantly reducing runtime. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "With the increasing digitalization, measurements over extensive time periods are becoming ubiquitous. Nevertheless, in many application domains, in particular healthcare (Yadav et al., 2018), measurements might not necessarily be observed at a regular rate or could be misaligned. Moreover, the presence or absence of a measurement and its observation frequency may carry information of its own (Little & Rubin, 2014), such that imputing the missing values is not always desired. ",
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+    {
+        "type": "text",
+        "text": "While some algorithms can be readily applied to datasets with varying length, these methods usually assume regular sampling of the data and/or require the measurements across modalities to be aligned/synchronized, preventing their application to the aforementioned settings. Existing approaches for unaligned measurements, by contrast, typically rely on imputation to obtain a regularlysampled version of a dataset for classification. Learning a suitable imputation scheme, however, requires understanding the underlying dynamics of a system; this task is significantly more complicated and not necessarily required when classification is the main goal. Furthermore, even though a decoupled imputation scheme followed by classification is generally more scalable, it may lose information (in terms of “missingness patterns”) that could be crucial for prediction tasks. In addition, the fact that decoupled schemes perform worse than methods that are trained end-to-end has been has been empirically demonstrated by Li & Marlin (2016). Approaches that jointly optimize both tasks also add a large computational overhead, thus suffering from poor scalability or high memory requirements. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Our method is motivated by the understanding that, while RNNs and similar architectures are well suited for capturing and modelling the dynamics of a time series and thus excel at tasks such as forecasting, retaining the order of an input sequence can even be a disadvantage in classification scenarios (Vinyals et al., 2015). We show that by relaxing the condition that a sequence must be processed in order, we can naturally derive an architecture that directly accounts for (i) irregular sampling, and (ii) unsynchronized measurements. Our method SEFT: Set Functions for Time Series, extends recent advances in set function learning to irregular sampled time series classification tasks, yields state-of-the-art performance, is highly scalable and improves over current approaches by almost an order of magnitude in terms of runtime. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "With SEFT, we propose to rephrase the problem of classifying time series as classifying a set of observations. We show how set functions can be exploited to learn classifiers that are naturally applicable to unaligned and irregularly sampled time series, leading to state-of-the-art performance in irregularly-sampled time series classification tasks. Our approach can be interpreted as learning dataset-specific summary statistics of time series which are optimized to separate instances by class. ",
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+    {
+        "type": "text",
+        "text": "Furthermore, our method is highly parallelizable and can be readily extended to an online monitoring setup with up to thousands of patients. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This paper focuses on classifying time series with irregular sampling and potentially unaligned measurements. We briefly discuss recent work in this field; all approaches can be broadly grouped into the following three categories. ",
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+    {
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+        "text": "Irregular sampling as missing data While the problem of supervised classification in the presence of missing data is closely related to irregular sampling on time series, there are some core differences. Missing data is usually defined with respect to a number of features that could be observed, whereas time series themselves can have different lengths and a “typical” number of observed values might not exist. Generally, an irregularly-sampled time series can be converted into a missing data problem by discretizing the time axis into non-overlapping intervals, and declaring intervals in which no data was sampled as missing. This approach is followed by Marlin et al. (2012), where a Gaussian Mixture Model was used to do semi-supervised clustering on electronic health records. Similarly, Lipton et al. (2016) discretize the time series into intervals, aggregate multiple measurements within an interval, and add missingness indicators to the input of a Recurrent Neural Network. By contrast, Che et al. (2018) present several variants of the Gated Recurrent Unit (GRU) combined with imputation schemes. Most prominently, the GRU-model was extended to include a decay term (GRU-D), such that the last observed value is decayed to the empirical mean of the time series via a learnable decay term. While these approaches are applicable to irregularly-sampled data, they either rely on imputation schemes or empirical global estimates on the data distribution (our method, by contrast, requires neither), without directly exploiting the global structure of the time series. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Frameworks supporting irregular sampling Some frameworks support missing data. For example, Lu et al. (2008) directly defined a kernel on irregularly-sampled time series, permitting subsequent classification and regression with kernel-based classifiers or regression schemes. Furthermore, Gaussian Processes (Williams & Rasmussen, 2006) constitute a common probabilistic model for time series; they directly permit modelling of continuous time data using mean and covariance functions. Along these lines, Li & Marlin (2015) derived a kernel on Gaussian Process Posteriors, allowing the comparison and classification of irregularly-sampled time series using kernel-based classifiers. Nevertheless, all of these approaches still rely on separate tuning/training of the imputation method and the classifier so that structures supporting the classification could be potentially missed in the imputation step. An emerging line of research employs Hawkes processes (Hawkes, 1971; Liniger, 2009), i.e. a specific class of self-exciting point processes, for time series modelling and forecasting (Mei & Eisner, 2017; Yang et al., 2017; Xiao et al., 2017). While Hawkes processes exhibit extraordinary performance in these domains, there is no standardised way of using them for classification. Previous work (Lukasik et al., 2016) trains multiple Hawkes processes (one for each label) and classifies a time series by assigning it the label that maximises the respective likelihood function. Since this approach does not scale to our datasets, we were unable to perform a fair comparison. We conjecture that further research will be required to make Hawkes processes applicable to general time series classification scenarios. ",
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+    {
+        "type": "text",
+        "text": "End-to-end learning of imputation schemes Methods of this type are composed of two modules with separate responsibilities, namely an imputation scheme and a classifier, where both components are trained discriminatively and end-to-end using gradient-based training. Recently, Li & Marlin (2016) proposed the Gaussian Process Adapters (GP Adapters) framework, where the parameters of a Gaussian Process Kernel are trained alongside a classifier. The Gaussian Process gives rise to a fixed-size representation of the irregularly-sampled time series, making it possible to apply any differentiable classification architecture. This approach was further extended to multivariate time series by Futoma et al. (2017) using Multi-task Gaussian Processes (MGPs) (Bonilla et al., 2008), which allow correlations between the imputed channels. Moreover, Futoma et al. (2017) made the approach more compatible with time series of different lengths by applying a Long Short Term Memory (LSTM) (Hochreiter & Schmidhuber, 1997) classifier. Motivated by the limited scalability of approaches based on GP Adapters, Shukla & Marlin (2019) suggest an alternative imputation scheme, the interpolation prediction networks. It applies multiple semi-parametric interpolation schemes to obtain a regularly-sampled time series representation. The parameters of the interpolation network are trained with the classifier in an end-to-end setup. ",
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+        "img_path": "images/99a1ff8e5b7b0ed149a713759f89dce223919083906167cbd08a05c7333a4fb7.jpg",
+        "image_caption": [
+            "Figure 1: Schematic overview of SEFT’s architecture. The first panel exemplifies a potential input, namely a multivariate time series, consisting of 3 modalities $m _ { 1 } , m _ { 2 } , m _ { 3 }$ . We treat the $j ^ { \\mathrm { t h } }$ observation as a tuple $( t _ { j } , z _ { j } , m _ { j } )$ , comprising a time $t _ { j }$ , a value $z _ { j }$ , and a modality indicator $m _ { j }$ . All observations are summarized as a set of such tuples. Each set of tuples belonging to the same modality is then separately embedded $( f ^ { \\prime } )$ and subsequently aggregated (agg). An attention mechanism (attn) as described in Section 3.3 is then applied to learn the importance of individual and consecutive observations. Respective query matrices for 2 attentions head are illustrated in purple and orange blocks. The results of each attention head are then concatenated and used as the input for final classification layers. "
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+    {
+        "type": "text",
+        "text": "3 PROPOSED METHOD ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our paper focuses on the problem of time series classification of irregularly sampled and unaligned time series. We first define the required terms before describing our models ",
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+        "type": "text",
+        "text": "3.1 NOTATION & REQUIREMENTS ",
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+        "text": "Definition 1 (Time series). We describe a time series of an instance i as a set $s _ { i }$ of $M : = \\mathrm { l e n } ( S _ { i } )$ observations $s _ { j }$ such that $\\mathcal { S } _ { i } : = \\{ s _ { 1 } , . . . , s _ { M } \\}$ . We assume each observation $s _ { j }$ to be represented as a tuple $( t _ { j } , z _ { j } , m _ { j } )$ , consisting of a time $t _ { j } \\in \\mathbb { R } ^ { + }$ , an observed value $z _ { j } \\in \\mathbb { R } ,$ , and a modality indicator $m _ { j } \\in \\{ 1 \\cdot \\dots D \\}$ , where $D$ represents the dimensionality of the time series. We write $\\Omega \\subseteq \\mathbb { R } ^ { + } \\times \\bar { \\mathbb { R } } \\times \\mathbb { N } ^ { + }$ to denote the domain of observations. An entire time series can thus be represented as ",
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+    {
+        "type": "equation",
+        "img_path": "images/ed33a79f565b1399c87a809152ace45da37790bf0f7a51f7ccf6fbc884217766.jpg",
+        "text": "$$\n\\begin{array} { r } { S _ { i } : = \\left\\{ \\left( t _ { 1 } , z _ { 1 } , m _ { 1 } \\right) , \\ldots , \\left( t _ { M } , z _ { M } , m _ { M } \\right) \\right\\} , } \\end{array}\n$$",
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+    {
+        "type": "text",
+        "text": "where for notational convenience we omitted the index $i$ . ",
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+        "type": "text",
+        "text": "We leave this definition very general on purpose, allowing the length of each time series (comprising all channels, such as “heart rate”, “respiratory rate”, etc. of one instance) to differ, since our models are capable of handling this. Likewise, we neither enforce nor expect all time series to be synchronized, i.e. being sampled at the same time, but rather we permit unaligned or non-synchronized observations in the sense of not having to observe all modalities at each time point. Time series are collected in a dataset $\\mathcal { D }$ . ",
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+        "type": "text",
+        "text": "Definition 2 (Dataset). We consider a dataset $\\mathcal { D }$ to contain $n$ time series. Elements of $\\mathcal { D }$ are tuples, i.e. $\\mathcal { D } : = \\{ ( S _ { 1 } , y _ { 1 } ) , . . . , ( S _ { N } , y _ { N } ) \\}$ , where $S _ { i }$ denotes the $i ^ { \\mathrm { { t h } } }$ time series and $y _ { i } \\in \\{ 1 , \\ldots , C \\}$ its associated class label. ",
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+        "type": "text",
+        "text": "Figure 1 gives a high-level overview of our method, including the individual steps required to perform classification. To get a more intuitive grasp of these definitions, we briefly illustrate our time series notation with an example. Let instance $i$ be an in-hospital patient, while the time series represent measurements of two channels of vital parameters during a hospital stay, namely heart rate (HR) and mean arterial blood pressure (MAP). We enumerate those channels as modalities 1 and 2. Counting from admission time, a HR of 60 and 65 beats per minute was measured after $0 . 5 \\mathrm { h }$ and $3 . 0 \\mathrm { h }$ , respectively, whereas MAP values of 80, 85, and $8 7 \\mathrm { m m H g }$ were observed after $0 . 5 \\mathrm { h }$ , $1 . 7 \\mathrm { h }$ , and $2 . 5 \\mathrm { h }$ . According to Definition 1, the time series is thus represented as $S _ { i } = \\{ ( 0 . 5 , 6 0 , 1 ) , ( 3 , 6 5 , 1 ) , ( 0 . 5 , 8 0 , \\bar { 2 } ) , ( 1 . 7 , 8 5 , 2 ) , ( 3 , 8 7 , 2 ) \\}$ . In this example, observations are ordered by modality to increase readability; in practice, we are dealing with unordered sets. ",
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+        "type": "text",
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+        "type": "text",
+        "text": "Definition 3 (Non-synchronized time series). We call a $D$ -dimensional time series nonsynchronized if there is at least one time point $t _ { j } ~ \\in ~ \\mathbb { R } ^ { + }$ at which at least one modality is not observed, i.e. if there exists $t _ { j } \\in \\mathbb { R } ^ { + }$ such that $| \\{ ( t _ { k } , z _ { k } , m _ { k } ) \\mid t _ { k } = t _ { j } \\} | \\neq D$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Furthermore, we assume that no two measurements of the same modality $m _ { k }$ occur at the same time, i.e. $t _ { i } \\neq t _ { j }$ for $i \\neq j$ has to be satisfied for all measurements in $m _ { k }$ . This assumption is not required for technical reasons but for consistency. It also makes it possible to interpret the results later on. ",
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+    {
+        "type": "text",
+        "text": "To summarize our generic setup, we do not require $M$ , the number of observations per time series, to be the same, i.e. $\\mathrm { l e n } ( S _ { i } ) \\neq \\mathrm { l e n } ( S _ { j } )$ for $i \\neq j$ is permitted, nor do we assume that the time points and modalities of the observations are the same across time series. This setting is common in biomedical time series, for example. Since typical machine learning algorithms are designed to operate on data of a fixed dimension, novel approaches to this non-trivial problem are required. ",
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+    {
+        "type": "text",
+        "text": "3.2 OUR MODEL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the following, we describe an approach inspired by differentiable learning of functions that operate on sets (Zaheer et al., 2017; Wagstaff et al., 2019). We phrase the problem of classifying time series on irregular grids as learning a function $f$ on a set of arbitrarily many time series observations following Definition 1, i.e. ${ \\cal S } = \\{ ( t _ { 1 } , z _ { 1 } , m _ { 1 } ) , \\ldots , ( t _ { M } , z _ { M } , m _ { M } ) \\}$ , such that $f \\colon S  \\mathbb { R } ^ { C }$ , where $s$ represents a generic time series of arbitrary cardinality and $\\mathbb { R } ^ { C }$ corresponds to the logits of the $C$ classes in the dataset. As we previously discussed, we interpret each time series as an unordered set of measurements, where all information is conserved because the observation time is included for each set element. Specifically, we define $f$ to be a set function, i.e. a function that operates on a set and thus has to be invariant to the ordering of the elements in the set. Multiple architectures are applicable to constructing set functions such as Transformers (Lee et al., 2019; Vaswani et al., 2017), or Deep Sets (Zaheer et al., 2017). Due to preliminary experiments, where Transformers suffered from lower generalization performance in our setting1, we base this work on the framework of Zaheer et al. (2017). Intuitively, this can be seen as computing multivariate dataset-specific summary statistics, which are optimized to maximize classification performance. Thus, we sum-decompose the set function $f$ into the form ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f82cc0c73281cf55da43abf5ec01d62bf0f2c08b3a2c2bd009221f00947e7d3b.jpg",
+        "text": "$$\nf ( S ) = g \\left( \\frac { 1 } { | S | } \\sum _ { s _ { j } \\in S } h ( s _ { j } ) \\right)\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $h \\colon \\Omega \\to  { \\mathbb { R } } ^ { d }$ and $g \\colon  { \\mathbb { R } ^ { d } } \\to  { \\mathbb { R } ^ { C } }$ are neural networks, $d \\in \\mathbb { N } ^ { + }$ determines the dimensionality of the latent representation, and $s _ { j }$ represents a single observation of the time series $s$ . We can view the averaged representations $1 / | \\mathcal { \\bar { S } } | \\sum _ { s _ { j } \\in \\mathcal { S } } h ( s _ { j } )$ in general as a dataset-specific summary statistic learned to best distinguish the class labels. Equation 2 also implies the beneficial scalability properties of our approach: each embedding can be calculated independently of the others; hence, the constant computational cost of passing a single observation through the function $h$ is scaled by the number of observations, resulting in a runtime of $\\mathcal { O } ( M )$ for a time series of length $M$ . ",
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+    {
+        "type": "text",
+        "text": "Recently, Wagstaff et al. (2019) derived requirements for a practical universal function representation of sum-decomposable set functions, i.e the requirements necessary for a sum-decomposable function to represent an arbitrary set-function given that $h$ and $g$ are arbitrarily expressive. In particular, they show that a universal function representation can only be guaranteed provided that $d \\geq \\operatorname* { m a x } _ { i } \\operatorname { l e n } ( S _ { i } )$ is satisfied. During hyperparameter search we thus independently sample the dimensionality of the aggregation space, and allow it to be in the order of the number of observations that are to be expected in the dataset. Further, we explored the utilization of max, sum, and mean as alternative aggregation functions inspired by Zaheer et al. (2017); Garnelo et al. (2018). ",
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+        "type": "text",
+        "text": "Intuition Our method can be connected to Takens’s embedding theorem (Takens, 1981) for dynamical systems: we also observe a set of samples from some unknown (but deterministic) dynamical process; provided the dimensionality of our architecture is sufficiently large2, we are capable of reconstructing the system up to diffeomorphism. The crucial difference is that we do not have to construct a time-delay embedding but rather, we let the network learn an embedding that is suitable for classification. ",
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+    {
+        "type": "text",
+        "text": "Time encoding In order to represent the time point of an observation on a normalized scale, we employ variant of positional encodings, as introduced by Vaswani et al. (2017). Preliminary results indicated that this encoding scheme reduces the sensitivity towards initialization and training hyperparameters of a model. Specifically, the time encoding converts the one-dimensional time axis into a multi-dimensional input by passing the time $t$ of each observation through multiple sine and cosine functions of varying frequencies. Given a dimensionality $\\tau \\in \\mathbb { N } ^ { + }$ of the time encoding, we refer to the encoded position as $x \\in \\mathbb { R } ^ { \\tau }$ , where ",
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+        "type": "equation",
+        "img_path": "images/4512c7e93124eb17e9be9c70f88a7a08e3595e37f0664d45b5c8c91fb2af8485.jpg",
+        "text": "$$\n\\begin{array} { r } { x _ { 2 k } ( t ) : = \\sin \\bigg ( \\frac { t } { \\operatorname* { m a x } _ { - } \\mathrm { t s } ^ { 2 k / \\tau } } \\bigg ) } \\\\ { x _ { 2 k + 1 } ( t ) : = \\cos \\bigg ( \\frac { t } { \\operatorname* { m a x } _ { - } \\mathrm { t s } ^ { 2 k / \\tau } } \\bigg ) } \\end{array}\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "with $k \\in \\{ 0 , \\ldots , \\tau / 2 \\}$ and max ts representing the maximal time scale that is expected in the data. Intuitively, we select the wavelengths using a geometric progression from $2 \\pi$ to max ts $\\cdot 2 \\pi$ , and treat the number of steps and the maximum timescale max ts as hyperparameters of the model. For all experiments time encodings were used, such that an observation is represented as $s _ { j } =$ $( x ( t _ { j } ) , z _ { j } , m _ { j } )$ . ",
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+        "type": "text",
+        "text": "Loss function If not mentioned otherwise, we choose $h$ and $g$ in Equation 2 to be multilayer perceptron deep neural networks, parametrized by weights $\\theta$ and $\\psi$ , respectively. We thus denote these neural networks by $h _ { \\theta }$ and $g _ { \\psi }$ ; their parameters are shared across all instances per dataset. In our training setup, we follow Zaheer et al. (2017) and apply the devised set function to the complete time series, i.e. to the set of all observations for each time series. Overall, we optimize a loss function that is defined as ",
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+        "img_path": "images/2ea5e40376d66506d9769650fd7474d8f7d24656a94290677e4128db18f672c9.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { \\theta } , \\boldsymbol { \\psi } ) : = \\mathbb { E } _ { ( S , \\boldsymbol { y } ) \\in \\mathcal { D } } \\left[ \\ell \\left( \\boldsymbol { y } ; \\boldsymbol { g } _ { \\boldsymbol { \\psi } } \\left( \\frac { 1 } { | S | } \\sum _ { s _ { j } \\in S } h _ { \\boldsymbol { \\theta } } ( s _ { j } ) \\right) \\right) \\right] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\ell ( \\cdot )$ represents a task-specific loss function. In out setup, we either utilize the binary crossentropy in combination with a sigmoid activation function in the last layer for binary classification or multi-label classification tasks and categorical cross-entropy in combination with a softmax activation function in the last layer for multi-class classification tasks. ",
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+    {
+        "type": "text",
+        "text": "3.3 ATTENTION-BASED AGGREGATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "So far, our method permits encoding sets of arbitrary sizes into a fixed-size representation. For increasingly large set sizes, however, many irrelevant observations could influence the result of the set function. The mean aggregation function is particularly susceptible to this because the influence of an observation to the embedding shrinks proportionally to the size of the set. We thus suggest to use a weighted mean in order to allow the model to decide which observations are relevant and which should be considered irrelevant. This is equivalent to computing an attention $a ( S , s _ { j } )$ over the set input elements, and subsequently, computing the sum over all elements in the set. ",
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+        "text": "Our approach is based on scaled dot-product attention with multiple heads $i \\in \\{ 1 , \\ldots , m \\}$ in order to be able to cover different aspects of the aggregated $\\mathrm { s e t } ^ { 3 }$ . We define $a ( \\cdot )$ , i.e. the attention weight function of an individual time series, to depend on the overall set of observations. This is achieved by computing an embedding of the set elements using a smaller set function $f ^ { \\prime }$ , and projecting the concatenation of the set representation and the individual set elements into a $d$ -dimensional space. Specifically, we have $K _ { j , i } ~ = ~ [ f ^ { \\prime } ( { \\cal { S } } ) , s _ { j } ] ^ { T } { \\cal { W } } _ { i }$ where $W _ { i } \\ \\in \\ \\mathbb { R } ^ { ( \\mathrm { i m } ( f ^ { \\prime } ) + | s _ { j } | ) \\times d }$ and $K \\in \\mathbb { R } ^ { | s | \\times d }$ . Furthermore, we define a matrix of query points $Q \\in \\mathbb { R } ^ { m \\times d }$ , which allow the model to summarize different aspects of the dataset via ",
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+        "text": "",
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+        "type": "equation",
+        "img_path": "images/4684aafff794954dde4f53801cb7dd959c4e0bcd438470be5ec5e6220d8c8dc2.jpg",
+        "text": "$$\ne _ { j , i } = \\frac { K _ { j , i } \\cdot Q _ { i } } { \\sqrt { d } } \\qquad \\mathrm { a n d } \\qquad a _ { j , i } = \\frac { \\exp ( e _ { j , i } ) } { \\sum _ { j } \\exp ( e _ { j , i } ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $a _ { j , i }$ represents the amount of attention that head $i$ gives to set element $j$ . The head-specific row $Q _ { i }$ of the query matrix $Q$ allows a head to focus on individual aspects (such as the distribution of one or multiple modalities) of a time series. For each head, we multiply the set element embeddings computed via the set function $f$ with the attentions derived for the individual instances, i.e. $r _ { i } =$ $\\textstyle \\sum _ { j } a _ { j , i } f ( s _ { j } )$ . The computed representation is concatenated and passed to the aggregation network $h _ { \\theta }$ as in a regular set function, i.e. $r * = [ r _ { 1 } \\ldots r _ { m } ]$ . In our setup, we initialize $Q$ with zeros, such that at the beginning of training, the attention mechanism is equivalent to computing the unweighted mean over the set elements. ",
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+    {
+        "type": "text",
+        "text": "Overall, this aggregation function is similar to Transformers (Vaswani et al., 2017), but differs from them in a few key aspects. Standard Transformer blocks would use the information from all set elements in order to compute the embedding of an individual set element, leading to a runtime and space complexity of $\\mathcal { O } ( \\bar { n } ^ { 2 } )$ . In contrast, our approach computes the embeddings of set elements independently, leading lower runtime and memory complexity of ${ \\mathcal { O } } ( n )$ . Further, we observed that computing embeddings with information from other set elements (as the Transformer does) actually decreases generalization performance (see Table 1 for details). ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We executed all experiments and implementations in a unified code base, which we also make available4 to the community. While some of the datasets used subsequently have access restrictions, anybody can gain access after satisfying the defined requirements. This ensures the reproducibility of our results. Please consult Appendix A.2 for further details. ",
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+    {
+        "type": "text",
+        "text": "4.1 DATASETS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In order to benchmark the proposed method we selected 4 datasets with irregularly-sampled and non-synchronized measurements. ",
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+    {
+        "type": "text",
+        "text": "Healing MNIST The H-MNIST dataset was introduced by Krishnan et al. (2015) in order to simulate characteristics which typically occur in medical time series. In our setup, we use a variant of this dataset. Every instance of the dataset contains 10 frames, derived from a single instance of MNIST dataset, where the digit is rotated according to an angle uniformly sampled between $- 9 0 ^ { \\circ }$ to $9 0 ^ { \\circ }$ . Furthermore, 3 randomly-selected consecutive frames are augmented by a square artefact in the top left corner of the image in order to indicate seasonality in the time series. Finally, $60 \\%$ of the data points are randomly discarded in order to yield a final high-dimensional irregularly-sampled time series with non-synchronized measurements. Using these settings each instance has on average 3, 136 observations. ",
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+    {
+        "type": "text",
+        "text": "MIMIC-III Tasks MIMIC-III (Johnson et al., 2016) is a widely-used, freely-accessible dataset containing around 50, 000 distinct ICU stays. The median length of stay is $2 . 1 \\mathrm { d }$ and a wide range of physiological measurements (e.g. arterial blood pressure, respiration rate, heart rate) are recorded with a resolution of $^ { 1 \\mathrm { h } }$ . Furthermore, laboratory test results, collected at irregular time intervals are available. Recently, Harutyunyan et al. (2019) defined a set of machine learning tasks, labels, and benchmarks using a subset of the MIMIC-III dataset. We trained and evaluated our method and competing methods on the binary mortality prediction task (M3-Mortality) and on the multiclass problem of phenotype classification (M3-Phenotyping), while applying additional filtering described in Appendix A.1. The goal of the mortality prediction task is to predict whether a patient will die during his/her hospital stay using only data from the first 48 hours of the ICU stay. This dataset contains around 21, 000 stays of which approximately $10 \\%$ result in death. The phenotype classification task consists of 40, 000 patients, each of which can suffer from a multitude of 25 acute care conditions. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "Physionet Mortality Prediction Challenge The 2012 Physionet challenge dataset (Goldberger et al., 2000), which we abbreviate P-Mortality, contains $1 2 , 0 0 0 \\mathrm { I C U }$ stays each of which lasts at least $4 8 \\mathrm { h }$ . For each stay, a set of general descriptors (such as gender, age, height, weight) were collected at admission time. Depending on the course of the stay and patient status, up to 37 time series variables were measured (e.g. blood pressure, lactate, respiration rate, temperature). While some modalities might be measured in regular time intervals (e.g. hourly or daily), some are only collected when required. Not all variables are available for each stay. The goal of the challenge was to predict if—and with which certainty —a patient will die during the hospital stay. The training set consists of 8, 000 stays while the testing set comprises 4, 000 ICU visits. Both datasets are similarly imbalanced, with a prevalence of around $14 \\%$ . For simplicity, the general descriptors (such as age and weight), were included as time points with a single observation at the beginning of the stay. This treatment is similar to the approach by Harutyunyan et al. (2019) in the MIMIC-III benchmarking datasets. Please refer to Table A.1, Table A.2, and Table A.3 in the appendix for a more detailed enumeration of samples sizes and label distributions. The total number of samples may slightly deviate from the originally published splits, as time series of excessive length prevented fitting some methods in reasonable time, and were therefore excluded. ",
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+    {
+        "type": "text",
+        "text": "4.2 COMPETITOR METHODS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "GRU-simple GRU-SIMPLE (Che et al., 2018) augments the input at time $t$ of a Gated-RecurrentUnit RNN with a measurement mask $m _ { t } ^ { d }$ and a $\\delta _ { t }$ matrix, which contains the time since the last measurement of the corresponding modality $d$ , such that ",
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+        "img_path": "images/d10488a6636f310961d7ab3702ca72956853d82f1a28e5fbe47f3bdf292cafac.jpg",
+        "text": "$$\n\\delta _ { t } = \\left\\{ \\begin{array} { l l } { s _ { t } - s _ { t - 1 } + \\delta _ { t - 1 } ^ { d } } & { t > 1 , m _ { t - 1 } ^ { d } = 0 } \\\\ { s _ { t } - s _ { t - 1 } } & { t > 1 , m _ { t - 1 } ^ { d } = 1 } \\\\ { 0 } & { t = 0 } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $s _ { t }$ represents the time associated with time step $t$ . ",
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+    {
+        "type": "text",
+        "text": "Phased-LSTM The PHASED-LSTM (Neil et al., 2016) introduced a biologically inspired time dependent gating mechanism which regulates access to the hidden and cell state of a Long short-term RNN cell (Hochreiter & Schmidhuber, 1997). While this allows the network to handle event-based sequences with irregularly spaced observations, the approach does not support unaligned measurements. In order to still provide the architecture with all relevant information, we augment the input in a similar fashion as described for the GRU-SIMPLE approach. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "GRU-D GRU-D or GRU-Decay (Che et al., 2018) contains modifications to the GRU RNN cell, allowing it to decay past observations to the mean imputation of a modality using a learnable decay rate. By additionally providing the measurement masks as an input the recurrent neural network the last feed in value. Learns how fast to decay back to a mean imputation of the missing data modality. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Interpolation Prediction Networks IP-NETWORKS (Shukla & Marlin, 2019) apply multiple semiparametric interpolation schemes to irregularly-sampled time series to obtain regularly-sampled representations that cover long-term trends, transients, and also sampling information. The method combines a univariate interpolation step with a subsequent multivariate interpolation; the parameters of the interpolation network are trained with the classifier in an end-to-end fashion. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Transformer In the TRANSFORMER architecture (Vaswani et al., 2017) the elements of a sequence are encoded simultaneously and information between sequence elements is captured using MultiHead-Attention blocks. In our case, an individual sequence element corresponds to all measurements available at a given time point, augmented with a measurement indicator. Transformers are normally used for sequence-to-sequence modelling tasks and in our setup were adapted to classification tasks by mean-aggregating the final representation. This representation is then fed into a one-layer MLP to predict logits for the individual classes. ",
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+    {
+        "type": "image",
+        "img_path": "images/067d20b9dff90ce2160b9c55a84e97acb5c9306cbf172f64abc03a6a9df16919.jpg",
+        "image_caption": [
+            "Figure 2: Visualizations of a single attention head on an instance of the P-Mortality dataset. We display a set of variables relevant for assessing patient stability and organ failure: Serum Potassium (K), Lactate, Systolic Arterial Blood Pressure (SysABP), and Urine output. Darker colors represent higher attention values. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.3 EXPERIMENTAL SETUP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To permit a fair comparison between the methods, we executed hyperparameter searches for each model on each dataset, composed of uniformly sampling 20 parameters according to Appendix A.3. Training was stopped after 20 epochs without improvement of the validation loss, the hyperparameters with the best overall validation performance were selected for quantifying the performance on the test set. The train, validation, and test splits were the same for all models and all evaluations. Final performance on the test set was calculated by 3 independent runs of the models; evaluation took place after the model was restored to the state with the best validation loss. In all subsequent benchmarks, we use the standard deviation of the test performance of these runs as generalization performance estimates. ",
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+    {
+        "type": "text",
+        "text": "4.4 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The results are shown in Table 1. Overall, our proposed method exhibits the lowest per-epoch runtime on most datasets, while either yielding competitive or state-the-art performance. Further, the trade-off between runtime and performance of the proposed method is very good on all datasets (see Figure A.1 and Figure A.2 in the appendix for a visualization of this argument). In order to elucidate the contribution of individual model components, we also provide an ablation study in Table A.4. Here we see that the attention mechanism contributes more to the model performance, while the positional encoding seems to be beneficial for datasets with highly-varying time series lengths, in particular M3-Phenotyping. ",
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+        "type": "text",
+        "text": "Opening the black box In the medical domain, it is of particular interest to understand the decisions a model makes based on the input it is provided with. The formulation of our model and its per observation perspective on time series gives it the unique property of being able to quantify to which extent an individual observation contributed to the output of the model. We exemplify this in Figure 2 with a patient time series that was combined with our models attention values, displayed for a set of clinically relevant variables. After reviewing these records with our medical expert, we find that our model is able to pick up regions with drastic changes in individual modalities. Moreover, it is able to inspect other modalities at the same associated time (for instance, at hour 20). This is behaviour similar to what one would expect from an alerted clinician reviewing the logged medical records. Interestingly, we observe that the model attends to known trends (that are consisting with domain knowledge about patient deterioration ultimately resulting in death) such as increase in lactate or hemodynamic instability, as indicated by drops in blood pressure. Furthermore, the model appears to be alerted by persisting low urine output. After several hours, this can be indicative of kidney failure. ",
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+    {
+        "type": "table",
+        "img_path": "images/b84fc3abe980cf67bb31e56de1ba78ee2a2df84fa7f4f25ca5c16efa0e4e893a.jpg",
+        "table_caption": [
+            "Table 1: Performance comparison of methods on benchmarking datasets. Performance metrics have been rescaled to 100 for readability reasons. “AUC” denotes the area under the Receiver Operating Characteristic (ROC) curve; “PR AUC” denotes the area under the precision recall curve. “MICRO” refers to evaluating the metric globally by treating each entry of the label indicator matrix as a label. For “MACRO”, the metric is computed for each class and then averaged, whereas in “WEIGHTED” the class-wise metrics are weighted by class imbalance. Values denoted with “OOM” were not obtainable due to restrictions in GPU memory. "
+        ],
+        "table_footnote": [
+            "\\*: Due to the high dimensionality of H-MNIST and associated memory issues, the set elements were constructed by concatenating the observation time with all values associated with the time point and measurement indicators. Furthermore, as this dataset features only 10 time steps and missingness is induced randomly, we refrained from applying the attention-based aggregation. "
+        ],
+        "table_body": "<table><tr><td>DATASET</td><td>MODEL</td><td>MICRO AUC</td><td>MACRO AUC</td><td>WEIGHTED AUC</td><td>RUNTIME</td></tr><tr><td rowspan=\"6\">H-MNIST</td><td>GRU-SIMPLE</td><td>99.09 ± 0.05</td><td>99.01 ± 0.05</td><td>99.03 ± 0.05</td><td>11.43 ± 0.47</td></tr><tr><td>PHASED-LSTM</td><td>98.63 ± 0.13</td><td>98.50 ± 0.15</td><td>98.52 ± 0.14</td><td>33.93 ±1.11</td></tr><tr><td>GRU-D</td><td>99.42 ± 0.01</td><td>99.37 ± 0.02</td><td>99.38 ±0.02</td><td>11.81 ± 0.44</td></tr><tr><td>IP-NETS</td><td>99.06 ± 0.05</td><td>98.96 ± 0.03</td><td>98.98 ± 0.03</td><td>127.76 ± 0.95</td></tr><tr><td>TRANSFORMER</td><td>99.59 ± 0.05</td><td>99.55 ± 0.05</td><td>99.56 ± 0.05</td><td>21.62 ± 0.90</td></tr><tr><td>SEFT*</td><td>99.76 ± 0.01</td><td>99.75 ± 0.01</td><td>99.75 ± 0.01</td><td>4.05 ± 0.35</td></tr><tr><td rowspan=\"7\">M3-Phenotyping</td><td>GRU-SIMPLE</td><td>79.89 ± 0.14</td><td>73.91 ± 0.19</td><td>72.55 ± 0.16</td><td>112.58 ± 2.03</td></tr><tr><td>PHASED-LSTM</td><td>80.00±0.06</td><td>73.91 ± 0.09</td><td>72.65 ± 0.08</td><td>400.41 ± 14.14</td></tr><tr><td>GRU-D</td><td>82.16 ± 0.04</td><td>77.14 ± 0.03</td><td>76.08 ± 0.01</td><td>288.70± 16.66</td></tr><tr><td>IP-NETS</td><td>-OOM—</td><td>-OOM-</td><td>-OOM—</td><td>-0OM-</td></tr><tr><td>TRANSFORMER</td><td>—O0M-</td><td>-00M—</td><td>—00M-</td><td>—00M-</td></tr><tr><td>SEFT</td><td>81.22 ±0.12</td><td>75.95 ± 0.09</td><td>74.90 ± 0.11</td><td>56.27 ± 2.14</td></tr><tr><td>SEFT-ATTN</td><td>82.00 ± 0.06</td><td>76.95 ± 0.09</td><td>75.88 ± 0.09</td><td>52.32 ± 0.74</td></tr><tr><td></td><td></td><td>ACCURACY</td><td>PR AUC</td><td>AUC</td><td>RUNTIME</td></tr><tr><td rowspan=\"7\">M3-Mortality</td><td>GRU-SIMPLE</td><td>88.24±0.38</td><td>36.36 ±1.31</td><td>79.36 ± 0.26</td><td>22.80 ±0.56</td></tr><tr><td>PHASED-LSTM</td><td>88.32 ±0.31</td><td>35.30 ±1.38</td><td>80.16 ±0.22</td><td>25.54 ± 0.26</td></tr><tr><td>GRU-D</td><td>89.56 ± 0.38</td><td>46.76 ± 0.65</td><td>83.73 ± 0.21</td><td>31.85 ± 0.86</td></tr><tr><td>IP-NETS</td><td>89.73 ± 0.16</td><td>45.88 ± 0.87</td><td>83.30 ± 0.56</td><td>101.12 ± 4.52</td></tr><tr><td>TRANSFORMER</td><td>89.14 ± 0.15</td><td>42.32 ± 0.41</td><td>82.60 ± 0.55</td><td>4.79 ± 0.02</td></tr><tr><td>SEFT</td><td>88.65 ± 0.49</td><td>36.18 ± 5.07</td><td>79.15 ± 3.00</td><td>3.72 ± 0.11</td></tr><tr><td>SEFT-ATTN</td><td>89.48 ± 0.16</td><td>45.25 ± 0.96</td><td>83.79 ± 0.59</td><td>16.64 ± 0.20</td></tr><tr><td rowspan=\"7\">P-Mortality</td><td>GRU-SIMPLE</td><td>85.66 ± 0.14</td><td>39.43 ± 0.71</td><td>79.79 ±0.16</td><td>5.16 ± 0.06</td></tr><tr><td>PHASED-LSTM</td><td>85.57 ± 0.11</td><td>39.55 ± 0.62</td><td>78.71 ±0.76</td><td>18.59 ± 1.15</td></tr><tr><td>GRU-D</td><td>87.19 ± 0.30</td><td>54.95 ± 0.54</td><td>86.58 ± 0.32</td><td>14.08 ± 0.38</td></tr><tr><td>IP-NETS</td><td>87.23 ± 0.18</td><td>54.87 ± 0.41</td><td>86.42 ±0.18</td><td>7.21 ± 0.46</td></tr><tr><td>TRANSFORMER</td><td>86.47 ± 0.08</td><td>48.72 ± 0.61</td><td>83.49 ± 0.46</td><td>2.69 ± 0.43</td></tr><tr><td>SEFT</td><td>87.11 ± 0.32</td><td>52.07 ± 0.41</td><td>84.12 ±0.32</td><td>3.07 ± 0.03</td></tr><tr><td>SEFT-ATTN</td><td>87.62 ± 0.16</td><td>54.05 ± 0.27</td><td>85.50 ± 0.13</td><td>7.54 ± 0.08</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we presented a novel approach for classifying time series with irregularly-sampled and unaligned, that is non-synchronized, observations. Our approach yields state-of-the-art to strongly competitive performance on numerous simulated and real-world datasets, while reducing runtime by almost half. Moreover, we demonstrated that combining the perspective of individual observations with an attention mechanism permits increasing the interpretability of the model. This is particularly relevant for the medical and healthcare applications. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            217
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "For future work, we reserve a more extensive exploration of the learned latent representation to evaluate its utility for clustering of time series or visualization of their similarity. ",
+        "bbox": [
+            173,
+            224,
+            823,
+            252
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            287,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Edwin V. Bonilla, Kian M Chai, and Christopher Williams. Multi-task gaussian process prediction. In Advances in Neural Information Processing Systems (NeurIPS), pp. 153–160, 2008. ",
+        "bbox": [
+            173,
+            126,
+            821,
+            155
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhengping Che, Sanjay Purushotham, Kyunghyun Cho, David Sontag, and Yan Liu. Recurrent neural networks for multivariate time series with missing values. Scientific reports, 8(1):6085, 2018. ",
+        "bbox": [
+            173,
+            162,
+            823,
+            205
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Joseph Futoma, Sanjay Hariharan, and Katherine Heller. Learning to detect sepsis with a Multitask Gaussian Process RNN classifier. In International Conference on Machine Learning (ICML), pp. 1174–1182, 2017. ",
+        "bbox": [
+            173,
+            214,
+            823,
+            257
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Marta Garnelo, Dan Rosenbaum, Christopher Maddison, Tiago Ramalho, David Saxton, Murray Shanahan, Yee Whye Teh, Danilo Rezende, and SM Ali Eslami. Conditional neural processes. In International Conference on Machine Learning, pp. 1690–1699, 2018. ",
+        "bbox": [
+            174,
+            266,
+            825,
+            310
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ary L. Goldberger, Luis A.N. Amaral, Leon Glass, Jeffrey M. Hausdorff, Plamen Ivanov, Roger G. Mark, Joseph E. Mietus, George B. Moody, Chung-Kang Peng, and H. Eugene Stanley. Physiobank, physiotoolkit, and physionet: components of a new research resource for complex physiologic signals. Circulation, 101(23):e215–e220, 2000. ",
+        "bbox": [
+            174,
+            318,
+            825,
+            375
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Hrayr Harutyunyan, Hrant Khachatrian, David C. Kale, Greg Ver Steeg, and Aram Galstyan. Multitask learning and benchmarking with clinical time series data. Scientific Data, 6(1):96, 2019. ISSN 2052-4463. ",
+        "bbox": [
+            173,
+            383,
+            825,
+            426
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Alan G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1):83–90, 1971. ",
+        "bbox": [
+            174,
+            435,
+            823,
+            464
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural Computation, 9(8): 1735–1780, 1997. ",
+        "bbox": [
+            173,
+            473,
+            823,
+            501
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Alistair E. W. Johnson, Tom J. Pollard, Lu Shen, Li-wei H. Lehman, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Anthony Celi, and Roger G. Mark. MIMIC-III, a freely accessible critical care database. Scientific Data, 3, 2016. ",
+        "bbox": [
+            173,
+            510,
+            825,
+            553
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Rahul G. Krishnan, Uri Shalit, and David Sontag. Deep Kalman Filters. arXiv e-prints, art. arXiv:1511.05121, Nov 2015. ",
+        "bbox": [
+            173,
+            561,
+            823,
+            590
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Juho Lee, Yoonho Lee, Jungtaek Kim, Adam Kosiorek, Seungjin Choi, and Yee Whye Teh. Set transformer: A framework for attention-based permutation-invariant neural networks. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.), Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pp. 3744–3753, Long Beach, California, USA, 09–15 Jun 2019. PMLR. ",
+        "bbox": [
+            173,
+            599,
+            825,
+            670
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Steven Cheng-Xian Li and Benjamin M Marlin. Classification of sparse and irregularly sampled time series with mixtures of expected gaussian kernels and random features. In UAI, pp. 484– 493, 2015. ",
+        "bbox": [
+            171,
+            679,
+            823,
+            722
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Steven Cheng-Xian Li and Benjamin M. Marlin. A scalable end-to-end Gaussian process adapter for irregularly sampled time series classification. In Advances In Neural Information Processing Systems 29, pp. 1804–1812, 2016. ",
+        "bbox": [
+            173,
+            731,
+            823,
+            773
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Larry S. Liebovitch and Tibor Toth. A fast algorithm to determine fractal dimensions by box counting. Physics Letters A, 141(8):386��390, 1989. ",
+        "bbox": [
+            173,
+            781,
+            821,
+            810
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Thomas Josef Liniger. Multivariate Hawkes processes. PhD thesis, ETH Zurich, 2009. ",
+        "bbox": [
+            173,
+            819,
+            743,
+            835
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zachary C Lipton, David Kale, and Randall Wetzel. Directly modeling missing data in sequences with rnns: Improved classification of clinical time series. In Machine Learning for Healthcare Conference, pp. 253–270, 2016. ",
+        "bbox": [
+            174,
+            843,
+            825,
+            886
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Roderick JA Little and Donald B Rubin. Statistical analysis with missing data, volume 333. John Wiley & Sons, 2014. ",
+        "bbox": [
+            173,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhengdong Lu, Todd K Leen, Yonghong Huang, and Deniz Erdogmus. A reproducing kernel hilbert space framework for pairwise time series distances. In Proceedings of the 25th International Conference on Machine learning, pp. 624–631, 2008. ",
+        "bbox": [
+            176,
+            103,
+            821,
+            146
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Michal Lukasik, P. K. Srijith, Duy Vu, Kalina Bontcheva, Arkaitz Zubiaga, and Trevor Cohn. Hawkes processes for continuous time sequence classification: an application to rumour stance classification in Twitter. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pp. 393–398, Berlin, Germany, August 2016. Association for Computational Linguistics. doi: 10.18653/v1/P16-2064. URL https: //www.aclweb.org/anthology/P16-2064. ",
+        "bbox": [
+            173,
+            155,
+            825,
+            239
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Benjamin M. Marlin, David C. Kale, Robinder G. Khemani, and Randall C. Wetzel. Unsupervised pattern discovery in electronic health care data using probabilistic clustering models. In Proceedings of the 2nd ACM SIGHIT International Health Informatics Symposium, pp. 389–398. ACM, 2012. ",
+        "bbox": [
+            173,
+            247,
+            825,
+            304
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hongyuan Mei and Jason M. Eisner. The neural Hawkes process: A neurally self-modulating multivariate point process. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems 30, pp. 6754–6764. Curran Associates, Inc., 2017. ",
+        "bbox": [
+            174,
+            313,
+            825,
+            369
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Daniel Neil, Michael Pfeiffer, and Shih-Chii Liu. Phased lstm: Accelerating recurrent network training for long or event-based sequences. In Advances in Neural Information Processing Systems 29, pp. 3882–3890, 2016. ",
+        "bbox": [
+            173,
+            378,
+            821,
+            421
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Satya Narayan Shukla and Benjamin Marlin. Interpolation-prediction networks for irregularly sampled time series. In International Conference on Learning Representations, 2019. ",
+        "bbox": [
+            171,
+            430,
+            821,
+            459
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Floris Takens. Detecting strange attractors in turbulence. In David Rand and Lai-Sang Young (eds.), Dynamical Systems and Turbulence, pp. 366–381, Heidelberg, Germany, 1981. Springer. ",
+        "bbox": [
+            171,
+            468,
+            823,
+            497
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems (NeurIPS), pp. 5998–6008, 2017. ",
+        "bbox": [
+            173,
+            506,
+            825,
+            549
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Oriol Vinyals, Samy Bengio, and Manjunath Kudlur. Order matters: Sequence to sequence for sets. arXiv preprint arXiv:1511.06391, 2015. ",
+        "bbox": [
+            174,
+            558,
+            821,
+            587
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Edward Wagstaff, Fabian B. Fuchs, Martin Engelcke, Ingmar Posner, and Michael Osborne. On the Limitations of Representing Functions on Sets. 2019. ",
+        "bbox": [
+            174,
+            594,
+            823,
+            625
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Christopher K. Williams and Carl Edward Rasmussen. Gaussian processes for machine learning. MIT Press, 2(3):4, 2006. ",
+        "bbox": [
+            173,
+            632,
+            821,
+            661
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Shuai Xiao, Junchi Yan, Mehrdad Farajtabar, Le Song, Xiaokang Yang, and Hongyuan Zha. Joint modeling of event sequence and time series with attentional twin recurrent neural networks. arXiv e-prints, art. arXiv:1703.08524, 2017. ",
+        "bbox": [
+            173,
+            670,
+            825,
+            713
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Pranjul Yadav, Michael Steinbach, Vipin Kumar, and Gyorgy Simon. Mining electronic health records (EHRs): a survey. ACM Computing Surveys, 50(6):85, 2018. ",
+        "bbox": [
+            173,
+            722,
+            823,
+            751
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yingxiang Yang, Jalal Etesami, Niao He, and Negar Kiyavash. Online learning for multivariate hawkes processes. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems 30, pp. 4937–4946. Curran Associates, Inc., 2017. ",
+        "bbox": [
+            174,
+            760,
+            825,
+            816
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Ruslan R. Salakhutdinov, and Alexander J. Smola. Deep sets. In Advances in Neural Information Processing Systems (NeurIPS), pp. 3391–3401, 2017. ",
+        "bbox": [
+            173,
+            825,
+            821,
+            867
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/41d94d646bb62c047a22e75e4067770c1a126a1b33e29ae4d298141419bb6293.jpg",
+        "table_caption": [
+            "Table A.1: M3-Mortality prevalence of labels for the binary classification task "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Training Prevalence</td><td>Testing Prevalence</td></tr><tr><td>In-hospital deaths</td><td>0.135</td><td>0.116</td></tr></table>",
+        "bbox": [
+            282,
+            167,
+            714,
+            213
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/a3ff0fc5c57723666eb9ca5ac506c4f1185c094809720ab8f5d850daa4e52dc5.jpg",
+        "table_caption": [
+            "Table A.2: P-Mortality prevalence of labels for the binary classification task "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td>Training PrevalenceTesting Prevalence</td></tr><tr><td>In-hospital deaths S</td><td>0.140</td><td>0.146</td></tr></table>",
+        "bbox": [
+            282,
+            270,
+            714,
+            314
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 DATA FILTERING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            347,
+            334,
+            361
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Due to memory requirements of some of the competitor methods, it was necassary to excluded time series with extremly high number of measurements. For the M3-Phenotyping patients with more than 2000 distinct time points were discarded from training. For M3-Mortality patients with more than 1000 time points were discarded as they contained dramatically different measuring frequencies compared to the rest of the dataset. ",
+        "bbox": [
+            174,
+            372,
+            825,
+            443
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 IMPLEMENTATIONAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            460,
+            426,
+            474
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "All experiments were run using tensorflow $0 . 1 5 . 0 \\Sigma \\mathsf { c } 0$ and training was performed on NVIDIA Geforce GTX 1080 GPUs. In order to allow a fair comparison between methods, the input processing pipeline cached model specific representations and transformations of the data. To further increase efficiency of the RNNs, sequences were binned in to buckets of jointly trained instances depending on their sequence length. The buckets were determined according to the (0.25, 0.5, 0.75) quantiles of the length distributions of the datasets. ",
+        "bbox": [
+            174,
+            486,
+            825,
+            569
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 TRAINING, MODEL ARCHITECTURES AND HYPERPARAMETER SEARCH ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            585,
+            709,
+            601
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "General All models were trained using the Adam optimizer, while randomly sampling the learning rate from (0.001, 0.0005, 0.00025, 0.0001). Further, the batch size of all methods was sampled from the values (32, 64, 128, 256). ",
+        "bbox": [
+            174,
+            612,
+            825,
+            655
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Recurrent neural networks For the RNN based methods (GRU-SIMPLE, PHASEDLSTM, GRU-D and IP-NETS), the number of units was sampled in from the values (16, 32, 64, 128, 256, 512). Further, recurrent dropout and input dropout were sampled from the values (0.0, 0.1, 0.2, 0.3). Solely, for the PHASED-LSTM method, we did not apply dropout to the recurrent state and the inputs, as the learnt frequencies were hypothesized to fulfill a similar function as dropout (Neil et al., 2016). ",
+        "bbox": [
+            174,
+            670,
+            825,
+            753
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "SEFT We vary the number of layers, dropout in between the layers and the number of nodes per layer for both the encoding network $h _ { \\theta }$ and the aggregation network $g _ { \\psi }$ from the same ranges. The number of layers is randomly sampled between 1 and 5, the number of nodes in a layer are uniformly sampled from the range (16, 32, 64, 128, 256, 512) and the dropout fraction is sampled from the values $( 0 . 0 , 0 . 1 , 0 . 2 , 0 . 3 )$ . The width of the embedding space prior to aggregation is sampled from the values (32, 64, 128, 256, 512, 1024, 2048). The aggregation function selected to be one of mean, sum and max. The number of dimensions used for the positional embedding $\\tau$ is selected uniformly from $( 4 , 8 , 1 6 )$ and max ts us selected from the values (10, 100, 1000). ",
+        "bbox": [
+            174,
+            768,
+            825,
+            881
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "SEFT-Attn The parameters for the encoding and aggregation networks are sampled in a similar fashion as for SEFT. In contrast we set the aggregation function to be sum as described in the text. ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/7efe28a597f61c6d36f67f864536723cc541ff02f85916fb90c892d4d4c20e19.jpg",
+        "table_caption": [
+            "Table A.3: M3-Phenotyping prevalence of labels for the multi label classification task "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Phenotype</td><td>Training</td><td>Validation</td><td>Testing</td></tr><tr><td>Acute and unspecified renal failure</td><td>0.216</td><td>0.207</td><td>0.211</td></tr><tr><td>Acute cerebrovascular disease</td><td>0.0746</td><td>0.0753</td><td>0.0662</td></tr><tr><td>Acute myocardial infarction</td><td>0.103</td><td>0.103</td><td>0.108</td></tr><tr><td>Cardiac dysrhythmias</td><td>0.322</td><td>0.317</td><td>0.323</td></tr><tr><td>Chronic kidney disease</td><td>0.135</td><td>0.131</td><td>0.132</td></tr><tr><td>Chronic obstructive pulmonary disease and bronchiectasis</td><td>0.132</td><td>0.128</td><td>0.126</td></tr><tr><td>Complications of surgical procedures or</td><td>0.207</td><td>0.201</td><td>0.213</td></tr><tr><td>medical care Conduction disorders</td><td>0.0726</td><td>0.07</td><td>0.0704</td></tr><tr><td>Congestive heart failure； nonhyperten- sive</td><td>0.268</td><td>0.264</td><td>0.268</td></tr><tr><td>Coronary atherosclerosis and other heart disease</td><td>0.323</td><td>0.317</td><td>0.331</td></tr><tr><td>Diabetes mellitus with complications</td><td>0.0955</td><td>0.0945</td><td>0.094</td></tr><tr><td>Diabetes mellitus without complication</td><td>0.194</td><td>0.187</td><td>0.192</td></tr><tr><td>Disorders of lipid metabolism</td><td>0.291</td><td>0.287</td><td>0.289</td></tr><tr><td>Essential hypertension</td><td>0.421</td><td>0.41</td><td>0.424</td></tr><tr><td>Fluid and electrolyte disorders</td><td>0.267</td><td>0.276</td><td>0.265</td></tr><tr><td>Gastrointestinal hemorrhage</td><td>0.0715</td><td>0.0747</td><td>0.0788</td></tr><tr><td>Hypertension with complications and</td><td>0.133</td><td>0.131</td><td>0.13</td></tr><tr><td>secondary hypertension Other liver diseases</td><td>0.0884</td><td>0.0904</td><td>0.0883</td></tr><tr><td>Other lower respiratory disease</td><td>0.0514</td><td>0.0484</td><td>0.0565</td></tr><tr><td> Other upper respiratory disease</td><td>0.0408</td><td>0.0371</td><td>0.0429</td></tr><tr><td>Pleurisy; pneumothorax; pulmonary</td><td>0.0858</td><td>0.09</td><td>0.0905</td></tr><tr><td>collapse Pneumonia (except that caused by tu- berculosis or sexually transmitted dis-</td><td>0.14</td><td>0.135</td><td>0.135</td></tr><tr><td>ease) Respiratory failure; insufficiency; arrest</td><td>0.18</td><td>0.184</td><td>0.177</td></tr><tr><td>(adult) Septicemia (except in labor)</td><td>0.142</td><td>0.145</td><td>0.138</td></tr><tr><td>Shock</td><td>0.0783</td><td>0.0745</td><td>0.0811</td></tr><tr><td>Total samples</td><td>29 208</td><td>6359</td><td>6266</td></tr></table>",
+        "bbox": [
+            243,
+            128,
+            758,
+            657
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Further we use a constant architecture for the attention network $f ^ { \\prime }$ with 2 layers, 64 nodes per layer, 4 heads and a dimensionality of the dot product space $d$ of 128. We solely sample the amount of attention dropout uniformly from the values $( 0 . 0 , 0 . 1 , 0 . 2 5 , 0 . 5 )$ . ",
+        "bbox": [
+            174,
+            685,
+            825,
+            728
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Transformer We utilize the same model architecture as defined in Vaswani et al. (2017), where we use a one hidden layer MLP as a feed-forward network, with dimensionality of the hidden layer selected to be twice the model dimensionality. The parameters for the Transformer network were sampled according to the following criteria. The dimensionality of the model was sampled uniformly from the values (64, 128, 256, 512, 1024), the number of attention heads per layer from the values $( 2 , 4 , 8 )$ and the number of layers from the range $[ 1 , 6 ] \\in \\mathbb { N }$ . Further, we sampled the amount of dropout of the residual connections and the amount of attention dropout uniformly from the values $( 0 . 0 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 5 )$ , and the maximal timescale for the time embedding from the values $( 1 0 , 1 0 0 , 1 0 0 0 )$ (similar to the SEFT approach). ",
+        "bbox": [
+            173,
+            742,
+            825,
+            868
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/cd413b9baeddcf30bbff41cb3d041c0bf8ed063d22b7b0f35a4f09834f97f9ff.jpg",
+        "table_caption": [
+            "Table A.4: Ablation study of individual components of SEFT. “AUC” denotes the area under the Receiver Operating Characteristic (ROC) curve; “PR AUC” denotes the area under the precision recall curve; “RUNTIME” denotes the runtime of one training epoch in seconds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>DATASET</td><td>MODEL</td><td>MICRO AUC</td><td>MACRO AUC</td><td>WEIGHTED AUC</td><td>RUNTIME</td></tr><tr><td>H-MNIST</td><td>SEFT</td><td>99.76 ± 0.01</td><td>99.75 ± 0.01</td><td>99.75 ± 0.01</td><td>4.05 ± 0.35</td></tr><tr><td rowspan=\"3\">M3-Phenotyping</td><td>SEFT (NO ATTENTION)</td><td>81.22 ± 0.12</td><td>75.95 ± 0.09</td><td>74.90 ± 0.11</td><td>56.27 ± 2.14</td></tr><tr><td>SEFT-ATTN (NO TIME ENC.)</td><td>80.46±0.86</td><td>74.70 ± 1.12</td><td>73.48 ± 1.18</td><td>50.17 ± 0.84</td></tr><tr><td>SEFT-ATTN</td><td>82.00 ± 0.06</td><td>76.95 ± 0.09</td><td>75.88 ± 0.09</td><td>52.32 ± 0.74</td></tr><tr><td></td><td></td><td>ACCURACY</td><td>PR AUC</td><td>AUC</td><td>RUNTIME</td></tr><tr><td rowspan=\"3\">M3-Mortality</td><td>SEFT (NO ATTENTION)</td><td>88.65 ± 0.49</td><td>36.18 ± 5.07</td><td>79.15 ± 3.00</td><td>3.72 ± 0.11</td></tr><tr><td>SEFT-ATTN (NO TIME ENC.)</td><td>89.31±0.08</td><td>44.12 ± 0.06</td><td>83.72 ± 0.34</td><td>17.60 ± 0.43</td></tr><tr><td>SEFT-ATTN</td><td>89.48 ± 0.16</td><td>45.25 ± 0.96</td><td>83.79 ± 0.59</td><td>16.64± 0.20</td></tr><tr><td rowspan=\"3\">P-Mortality</td><td>SEFT (NO ATTENTION)</td><td>87.11 ± 0.32</td><td>52.07 ±0.41</td><td>84.12 ± 0.32</td><td>3.07 ± 0.03</td></tr><tr><td>SEFT-ATTN (NO TIME ENC.)</td><td>87.03 ±0.06</td><td>51.86 ± 1.04</td><td>84.91 ± 0.29</td><td>7.04 ± 0.04</td></tr><tr><td>SEFT-ATTN</td><td>87.62 ± 0.16</td><td>54.05± 0.27</td><td>85.50 ± 0.13</td><td>7.54 ± 0.08</td></tr></table>",
+        "bbox": [
+            174,
+            227,
+            816,
+            417
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/5a022c75092c637eed0c7fba11b58915df9fb99274cb37712336626f29f9b3bb.jpg",
+        "image_caption": [
+            "Figure A.1: A visualisation of the runtime of all methods and their AUC for datasets with a binary classification scenario. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            568,
+            782,
+            808
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/7bfd5f3b99f47c4b2eeb4f901e15232fff222b513a48bd633e440baaacd0c2d9.jpg",
+        "image_caption": [
+            "Figure A.2: A visualisation of the runtime of all methods and their AUC for datasets with a multilabel classification scenario. Please note that the model definition for SEFT changes between the left and the right column; please see Table 1 for more details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            364,
+            792,
+            602
+        ],
+        "page_idx": 15
+    }
+]

parse/train/byCQ9Uu4PD/byCQ9Uu4PD.md

@@ -0,0 +1,308 @@
+# Personalized Federated Learning with Gaussian Processes
+
+Idan Achituve Bar-Ilan University, Israel idan.achituve@biu.ac.il
+
+Aviv Shamsian Bar-Ilan University, Israel aviv.shamsian@biu.ac.il
+
+Aviv Navon Bar-Ilan University, Israel aviv.navon@biu.ac.il
+
+Gal Chechik   
+Bar-Ilan University, Israel NVIDIA, Isreal   
+gal.chechik@biu.ac.il
+
+Ethan Fetaya Bar-Ilan University, Israel ethan.fetaya@biu.ac.il
+
+# Abstract
+
+Federated learning aims to learn a global model that performs well on client devices with limited cross-client communication. Personalized federated learning (PFL) further extends this setup to handle data heterogeneity between clients by learning personalized models. A key challenge in this setting is to learn effectively across clients even though each client has unique data that is often limited in size. Here we present pFedGP, a solution to PFL that is based on Gaussian processes (GPs) with deep kernel learning. GPs are highly expressive models that work well in the low data regime due to their Bayesian nature. However, applying GPs to PFL raises multiple challenges. Mainly, GPs performance depends heavily on access to a good kernel function, and learning a kernel requires a large training set. Therefore, we propose learning a shared kernel function across all clients, parameterized by a neural network, with a personal GP classifier for each client. We further extend pFedGP to include inducing points using two novel methods, the first helps to improve generalization in the low data regime and the second reduces the computational cost. We derive a PAC-Bayes generalization bound on novel clients and empirically show that it gives non-vacuous guarantees. Extensive experiments on standard PFL benchmarks with CIFAR-10, CIFAR-100, and CINIC-10, and on a new setup of learning under input noise show that pFedGP achieves well-calibrated predictions while significantly outperforming baseline methods, reaching up to $21 \%$ in accuracy gain.
+
+# 1 Introduction
+
+In recent years, there is a growing interest in applying learning in decentralized systems under the setup of federated learning (FL) [37, 51, 66]. In FL, a server node stores a global model and connects to multiple end-devices (“clients"), which have private data that cannot be shared. The goal is to learn the global model in a communication-efficient manner. However, learning a single shared model across all clients may perform poorly when the data distribution varies significantly across clients. Personalized Federated Learning (PFL) [67] addresses this challenge by jointly learning a personalized model for each client. While significant progress had been made in recent years, leading approaches still struggle in realistic scenarios. First, when the amount of data per client is limited, even though this is one of the original motivations behind federated learning [4, 51, 72]. Second, when the input distribution shifts between clients, which is often the case, as clients use different devices and sensors. Last, when we require well-calibrated predictions, which is an important demand from medical and other safety-critical applications.
+
+![](images/8d507a18b380127d903c3606092df2f26e1b1b418f81b6c10c35d88528776c5e.jpg)  
+Figure 1: pFedGP - learning a shared deep kernel function with client-specific GP models. Each client stores private data, possibly from a different distribution. The data is first mapped to an embedding space with a shared neural network across all clients. Then, using common kernels a GP is applied to the data of the client for model learning and inference. We illustrate the per-client kernel matrix $k _ { \theta } ( \mathbf { x } _ { i } , \mathbf { x } _ { j } )$ . Bold cells indicate a stronger covariance.
+
+Here, we show how Gaussian Processes (GPs) with deep kernel learning (DKL) [80] is an effective alternative for handling these challenges. GPs have good predictive performance in a wide range of dataset sizes [2, 81], they are robust to input noise [75], can adapt to shifts in the data distribution [48], and provide well-calibrated predictions [69]. While regression tasks are more natural for GPs, here we focus on classification tasks for consistency with common benchmarks and learning procedures in the field; however, our approach is also applicable to regression tasks.
+
+Consider a naive approach that fits a separate GP classifier to each client based on its personal data. Its performance heavily depends on the quality of the kernel, and standard kernels tend to work poorly in domains such as images. A popular solution to this problem is to use deep kernel learning (DKL) [80], where a kernel is applied to features outputted by a neural network (NN). Unfortunately, GPs with DKL can strongly overfit, often even worse than standard NNs [56], and thus negate the main benefit of using a GP. We solve this issue by jointly learning a shared kernel function across clients. As the kernel captures similarities between inputs, a single kernel should work well across clients, while using a separate GP per client will give the required flexibility for personalization.
+
+We adapt a GP classifier recently proposed in [2] which uses the Pólya-Gamma augmentation [57] in a tree-structure model to the federated setting. We term our method pFedGP. We extend pFedGP by tailoring two inducing points (IPs) methods [58, 70]. The first helps generalization in the low data regime and, unlike common inducing point methods, does not reduce the computational costs. The second does focus on reducing the computational cost to make our approach scalable and work in low-resource clients. We also adjust previous PAC-Bayes generalization bounds for GPs [60, 64] to include the Pólya-Gamma augmentation scheme. These bounds are suitable for cases where the kernel is not learned, such as when new clients arrive after the shared NN was already learned.
+
+Therefore, this paper makes the following contributions: (i) introduce pFedGP as a natural solution to PFL; (ii) develop two IP methods to enhance GP classifiers that use the Pólya-Gamma augmentation scheme and integrate them with pFedGP; (iii) derive a PAC-Bayes generalization bound on novel clients and show empirically that it gives meaningful guarantees; (iv) achieve state-of-the-art results in a wide array of experiments, improving accuracy by up to $2 1 \%$ 1 .
+
+# 2 Related work
+
+Federated learning. In FL, clients collaboratively solve a learning task while preserving data privacy and maintaining communication efficiency [1, 34, 42, 51, 54, 82, 85]. FedAvg [51] is an early but effective FL approach that updates models locally and averages them into a global model. Several optimization methods have been proposed for improving convergence in FL [41, 45, 71, 77]. Other approaches focus on preserving client privacy [3, 20, 52, 90], improving robustness to statistical diversity [26, 27, 31, 35, 87, 88], and reducing communication cost [13, 61]. These methods aim to learn a global model across clients, limiting their ability to deal with heterogeneous datasets.
+
+Personalized federated learning. To overcome client heterogeneity, PFL aims to introduce some personalization for each client in the federation [39, 73]. Recent methods include adapting multitask learning [18, 67], meta-learning approaches [6, 21, 22, 33, 43, 89], and model mixing, where clients learn a mixture of the global and local models [4, 17, 27, 44]. Other approaches utilize different regularization schemes to enforce soft parameter sharing [32, 72]. Personalization in FL has also been explored through clustering approaches in which similar clients within the federation have a greater effect on one another [49, 86]. Recently, [65] proposed learning a central hypernetworks that acts on client representation vectors for generating personalized models.
+
+Bayesian FL. Some studies put forward a Bayesian treatment to the FL setup. [8, 12] used variational inference with Bayesian NNs. [76, 84] proposed a matching algorithm between local models based on the Beta-Bernoulli process to construct a global model. [14] extended Bayesian optimization to FL setting via Thompson sampling. To scale the GP model they used random Fourier features. We use inducing points instead. [83] proposed a federated learning framework that uses a global GP model for regression tasks and without DKL. Unlike this study, we focus on classification tasks with a personal GP classifier per client and advocate sharing information between clients through the kernel. [74] used GPs in a client selection strategy. In [36] an approach based on stein variational gradient descent was suggested. This method does not scale beyond small-sized networks. [47] proposed a multivariate Gaussian product mechanism to aggregate local models. As we will show, this method is less suited when the data heterogeneity between clients is large.
+
+Gaussian process classification. Unlike regression, in classification approximations must be used since the likelihood is not a Gaussian [59]. Classic approaches include the Laplace approximation [79], expectation-propagation [53], and least squares [62]. Recently, several methods were proposed based on the Pólya-Gamma augmentation [57] for modeling multinomial distributions [46], GP classification [23, 24, 78], few-shot learning [69], and incremental learning [2]. Here we build on the last approach. Classification with GPs is commonly done with variational inference techniques [30], here we wish to exploit the conjugacy of the model to take Gibbs samples from the posterior. This approach yields well calibrated [69] and more accurate models [2].
+
+# 3 Gaussian processes background
+
+We first provide a brief introduction to the main components of our model. Detailed explanations are deferred to the Appendix. Scalars are denoted with lower-case letters (e.g., $x$ ), vectors with bold lowercase letters (e.g., x), and matrices with bold capital letters (e.g., X). In general, $\mathbf { y } = [ y _ { 1 } , . . . , y _ { N } ] ^ { T }$ is the vector of labels, and $\mathbf { X } \in \mathbb { R } ^ { N \times d }$ is the design matrix with $N$ data points whose $i ^ { t h }$ row is $\mathbf { x } _ { i }$ .
+
+Gaussian processes. GPs map input points to target output values via a random latent function $f$ . $f$ is assumed to follow a Gaussian process prior $\mathbf { \bar { \boldsymbol { f } } } \sim \mathcal { G P } ( m ( \mathbf { x } ) , \ k ( \mathbf { x } , \mathbf { x } ^ { \prime } ) )$ , where the evaluation vector of $f$ on $\mathbf { X }$ , $\mathbf { f } = [ f ( \mathbf { x } _ { 1 } ) , . . . , f ( \mathbf { \bar { x } } _ { N } ) ] ^ { T }$ , has a Gaussian distribution f $\sim \mathcal { N } ( \mu , \kappa )$ with means $\pmb { \mu } _ { i } = m ( \mathbf { x } _ { i } )$ and covariance $K _ { i j } = k ( \mathbf { x } _ { i } , \mathbf { x } _ { j } )$ . The mean $m ( \mathbf { x } )$ is often set to be the constant zero function, and the kernel $k ( \mathbf { x } , \mathbf { x } ^ { \prime } )$ is a positive semi-definite function. The target values are assumed to be independent when conditioned on f . For Gaussian process regression the likelihood is Gaussian, $p ( y | f ) = \operatorname { \bar { \mathcal { N } } } ( f , \sigma ^ { 2 } )$ . Therefore, the posterior $p ( f | y , \mathbf { X } )$ is also Gaussian, and both the marginal and the predictive distributions have known analytic expressions. This is one of the main motivations behind using GPs, as most other Bayesian models have intractable inference.
+
+Unfortunately, for Gaussian process classification (GPC) the likelihood, $p ( y | f )$ , is not a Gaussian and the posterior does not admit a closed-form expression. One approach for applying GPs to binary classification tasks is the Pólya-Gamma augmentation [57]. Using this approach, we can augment the GP model with random variables $\omega$ from a Pólya-Gamma distribution, one for each example. As a result, $p ( \mathbf { f } | \mathbf { y } , \mathbf { X } , \omega )$ is a Gaussian density and $p ( \omega | \mathbf { y } , \mathbf { X } , \mathbf { f } )$ is a Pólya-Gamma density. This allows to use Gibbs sampling to efficiently sample from the posterior $p ( \mathbf { f } , \omega | \mathbf { y } , \mathbf { X } )$ for inference and prediction. A key advantage of the Pólya-Gamma augmentation is that it benefits from fast mixing and has the ability of even a single value of $\omega$ to capture much of the volume of the marginal distribution over function values [46]. Full equations and further details on the Pólya-Gamma augmentation scheme are given in Appendix A.1.
+
+Deep kernel learning (DKL). The quality of the GP model heavily depends on the kernel function $k ( \mathbf { x } _ { i } , \mathbf { x } _ { j } )$ . For many data modalities, such as images, common kernels are not a good measure of semantic similarity. Therefore, in [10, 80] standard kernels are used over features outputted by a neural network In regression, it is p $g _ { \boldsymbol { \theta } }$ . For example, the RBF kernel ible to directly backpropagate th $\begin{array} { r } { k _ { \theta } ( \mathbf { x } _ { i } , \mathbf { x } _ { j } ) = \exp \left( - \frac { | | g _ { \theta } ( \mathbf { x } _ { i } ) - g _ { \theta } ( \mathbf { x } _ { j } ) | | ^ { 2 } } { 2 \ell ^ { 2 } } \right) } \end{array}$ closed-form. In our case, we use Fisher’s identity [19] to obtain stochastic gradients [69].
+
+Inducing points. GPs require storing and inverting a kernel matrix on the entire training set which often limits its usage. A common solution to this problem is to use inducing point methods [58, 70]. The key idea is to replace the exact kernel with an approximation for fast computation. Usually, $M \ll N$ pseudo-inputs are learned such that the main computational bottleneck is in inverting $M \times M$ matrices.
+
+GP-Tree. We build on GP-Tree [2], a recent GP classifier that was shown to scale well with dataset size and the number of classes. GP-Tree turns the multi-class classification problem into a sequence of binary decisions along the tree nodes. Each node in the tree fits a binary GP classifier based on the Pólya-Gamma augmentation scheme and the data associated with that node. The leaf nodes correspond to the classes in the dataset. The tree is constructed by first computing a prototype for each class and then recursively performing divisive hierarchical clustering on these prototypes to two clusters at each node. Further details are given in Appendix A.2.
+
+# 4 pFedGP: federated learning with Gaussian processes
+
+Now we describe our approach for applying personalized federated learning (PFL) with Gaussian processes. First, we extend GP-Tree to the FL setup and show how to use Gibbs sampling to learn the NN parameters. Then, we present two alternatives for this method that use inducing points. The first is for extremely limited-size datasets, while the second allows controlling the computational resources. We name our method pFedGP. An illustration of our method is given in Figure 1.
+
+# 4.1 A full GP model
+
+The training procedure follows the standard protocol in this field [4, 44, 51]. We assume the existence of a server that holds the shared parameters $\theta$ (a NN). Let $C$ denote the set of clients. For each client $c \in C$ we denote by $D _ { c }$ its local dataset of size $N _ { c }$ . At each training iteration (round) the model is sent to $S$ clients to perform kernel learning $( | S | \leq | C | )$ . Each client $c \in S$ updates its copy of the global model and then sends the updated model to the server. The server then averages over the updates to obtain a new global model.
+
+At each client $c$ , we perform kernel learning in the following manner. We first compute the feature representation of the data samples associated with the client using the shared network. Then, we build the hierarchical classification tree as discussed in Section $_ { 3 \& }$ Appendix A.2. In [2] the tree was built only once after a pre-training stage and the model parameters were learned using a variational inference approach. Here, we re-build the tree at each round using the most recent features and we use a Gibbs sampling procedure, as it allows this flexibility in building the tree and performs better when not prohibitive by computational limitations. Learning the network parameters $\theta$ with the Gibbs sampling approach can be done with two common objectives, the marginal likelihood, and the predictive distribution.
+
+We denote by $\mathbf { X } _ { v }$ the data associated with the tree node $v$ , i.e., the data points which have $v$ on the path from the root node to their class leaf node. We denote by $\mathbf { y } _ { v }$ the binary label of these points, i.e., does their path go left or right at this node. And we denote by $\omega _ { v }$ the Pólya-Gamma random variables associated with node $v$ . The marginal likelihood term for the full hierarchical classification tree is the sum of the separate marginal likelihood terms of all the nodes $v$ in the tree:
+
+$$
+\mathcal { L } _ { c } ^ { M L } ( \theta ; D _ { c } ) = \sum _ { v } \log \ p _ { \theta } ( \mathbf { y } _ { v } | \mathbf { X } _ { v } ) = \sum _ { v } \log \ \int p _ { \theta } ( \mathbf { y } _ { v } | \omega _ { v } , \mathbf { X } _ { v } ) p ( \omega _ { v } ) d \omega _ { v } .
+$$
+
+Similar to [69] we use a gradient estimator based on Fisher’s identity [19]:
+
+$$
+\nabla _ { \theta } \mathcal { L } _ { c } ^ { M L } ( \theta ; D _ { c } ) = \sum _ { v } \int p _ { \theta } ( \omega _ { v } | \mathbf { y } _ { v } , \mathbf { X } _ { v } ) \nabla _ { \theta } l o g \ p _ { \theta } ( \mathbf { y } _ { v } | \omega _ { v } , \mathbf { X } _ { v } ) d \omega _ { v } \approx \sum _ { v } \frac { 1 } { L } \sum _ { l = 1 } ^ { L } \nabla _ { \theta } l o g \ p _ { \theta } ( \mathbf { y } _ { v } | \omega _ { v } ^ { ( l ) } , \mathbf { X } _ { v } ) .
+$$
+
+Here, (1)v , ..., ω(L)v ar e samples from the posterior at node $v$ . Due to the Pólya-Gamma augmentation $p _ { \theta } \big ( \mathbf { y } _ { v } | \omega _ { v } ^ { ( l ) } , \mathbf { X } _ { v } \big )$ is proportional to a Gaussian density. The exact expression is give in Appendix A.2.
+
+To use the predictive distribution as an objective, in each training iteration, after building the tree model, at each node we randomly draw a portion from the (node) training data and use it to predict the class label for the remaining part. We denote with $\mathbf { X } _ { v }$ and $\mathbf { y } _ { v }$ the training portion, $\mathbf { x } _ { v } ^ { * }$ and $y _ { v } ^ { * }$ the input and the label of the point we are predicting, and $P ^ { y ^ { * } }$ the path from the root node to the $y ^ { * }$ leaf node (i.e., the original class). Here we also take advantage of the independence between nodes to maximize the predictive distribution per node individually. The predictive distribution for a single data point:
+
+$$
+\mathbf { \Psi } _ { c } ^ { P D } ( \theta ; \mathbf { x } ^ { * } , y ^ { * } ) = \sum _ { v \in P ^ { y ^ { * } } } \log \ p \varrho ( y _ { v } ^ { * } | \mathbf { x } _ { v } ^ { * } , \mathbf { y } _ { v } , \mathbf { X } _ { v } ) = \sum _ { v \in P ^ { y ^ { * } } } \log \ \int p \varrho ( y _ { v } ^ { * } | \omega _ { v } , \mathbf { x } _ { v } ^ { * } , \mathbf { y } _ { v } , \mathbf { X } _ { v } ) p ( \omega _ { v } | \mathbf { y } _ { v } , \mathbf { X } _ { v } ) d \omega _ { v } .
+$$
+
+We use an approximate-gradient estimator based on posterior samples of $\omega$ :
+
+$$
+\nabla _ { \theta } \mathcal { L } _ { c } ^ { P D } ( \theta ; \mathbf { x } ^ { * } , y ^ { * } ) \approx \sum _ { v \in P ^ { y ^ { * } } } \frac { 1 } { L } \sum _ { l = 1 } ^ { L } \nabla _ { \theta } l o g p _ { \theta } ( y _ { v } ^ { * } | \boldsymbol { \omega } _ { v } ^ { ( l ) } , \mathbf { x } _ { v } ^ { * } , \mathbf { y } _ { v } , \mathbf { X } _ { v } ) .
+$$
+
+Where $\begin{array} { r } { p _ { \theta } ( y _ { v } ^ { * } | \omega _ { v } ^ { ( l ) } , \mathbf { x } _ { v } ^ { * } , \mathbf { y } _ { v } , \mathbf { X } _ { v } ) = \int p ( y _ { v } ^ { * } | f ^ { * } ) p _ { \theta } ( f ^ { * } | \omega _ { v } ^ { ( l ) } , \mathbf { x } _ { v } ^ { * } , \mathbf { y } _ { v } , \mathbf { X } _ { v } ) d f ^ { * } \ d \mathbf { x } } \end{array}$ does not have an analytical expression, but $\begin{array} { r } { p _ { \theta } \big ( f ^ { * } | \omega _ { v } ^ { ( l ) } , \mathbf { x } _ { v } ^ { * } , \mathbf { y } _ { v } , \mathbf { X } _ { v } \big ) = \int p _ { \theta } \big ( f ^ { * } | \mathbf { f } , \mathbf { x } _ { v } ^ { * } , \mathbf { X } _ { v } \big ) p _ { \theta } \big ( \mathbf { f } | \omega _ { v } ^ { ( l ) } , \mathbf { y } _ { v } , \mathbf { X } _ { v } \big ) d f ^ { * } } \end{array}$ is Gaussian with known parameters. We then compute the predictive distribution by performing Gauss-Hermite integration over $f ^ { * }$ . See exact expression in Appendix A.2.
+
+# 4.2 Augmenting the model with inducing points: sample efficiency
+
+The GP model described in Section 4.1 works well in most situations. However, when the number of data points per client is small, performance naturally degrades. To increase information sharing between clients and improve the per-client performance, we suggest augmenting the model with global inducing points shared across clients. When sending the model from the server to a client, we also send the inducing inputs and their labels. To streamline optimization and reduce the communication burden, we define the inducing inputs in the feature space of the last embedding layer of the shared NN. Therefore, usually, their size will be negligible compared to the network size.
+
+We denote by $\bar { \bf X }$ the learned inducing inputs and by $\bar { \mathbf { y } }$ their fixed class labels. They are set evenly across classes. During training, we regard only the set of inducing inputs-labels $( { \bar { \mathbf { X } } } , { \bar { \mathbf { y } } } )$ as the available (training) data and use them for posterior inference. More formally, we first compute $\begin{array} { r } { p _ { \theta } ( \mathbf { f } | \boldsymbol { \bar { \omega } } , \bar { \mathbf { y } } , \bar { \mathbf { X } } , \mathbf { X } ) \overset {  } { = } \int p _ { \theta } ( \mathbf { f } | \bar { \mathbf { f } } , \bar { \mathbf { X } } , \mathbf { X } ) p _ { \theta } ( \bar { \mathbf { f } } | \boldsymbol { \bar { \omega } } , \bar { \mathbf { y } } , \bar { \mathbf { X } } ) d \bar { \mathbf { f } } } \end{array}$ using its analytical expression for the actual training data and then compute the probability of $\mathbf { y }$ using Gauss-Hermite integration. Then we use Eq. 3 & 4 for learning the network parameters and the inducing locations. At test time, to make full use of the training data, we combine the inducing inputs with the training data and use both to obtain the GP formulas and to make predictions. We note that with just using the inducing inputs at test time the model performs remarkably well, despite having almost no personalization component. See Appendix E.8 for a further discussion.
+
+One potential issue with using IPs in this manner is that it distorts the true class distribution. As a result, the classifier may be more likely to predict a low-probability class during test time. We address this issue by adjusting the output distribution. In general, let $p ( y , \mathbf x )$ and $q ( y , \mathbf x )$ be two distributions that differ only in the class probabilities, i.e. $p ( \mathbf { x } | y ) = q ( \mathbf { x } | y )$ , the predictive distribution follows:
+
+$$
+\frac { q ( y ^ { * } | \mathbf { x } ^ { * } ) } { p ( y ^ { * } | \mathbf { x } ^ { * } ) } \propto \frac { q ( \mathbf { x } ^ { * } | y ^ { * } ) q ( y ^ { * } ) } { p ( \mathbf { x } ^ { * } | y ^ { * } ) p ( y ^ { * } ) } \Longrightarrow q ( y ^ { * } | \mathbf { x } ^ { * } ) \propto \frac { q ( y ^ { * } ) } { p ( y ^ { * } ) } p ( y ^ { * } | \mathbf { x } ^ { * } ) .
+$$
+
+We use this to correct the GP predictions to the original class ratios at each tree node. We found in our experiments that this correction generally improves the classifier performance for class imbalanced data. As an example for this phenomena, consider a binary classification problem having 90 examples from the first class and 10 examples from the second class (therefore, $q ( y = 0 ) = 0 . 9$ , and $q ( y = 1 ) = 0 . 1 \AA$ . Assume we defined 50 inducing inputs per class, so now during test time the model sees 140 samples from the first class and 60 samples from the second class which corresponds to probabilities $p ( y = 0 ) = 0 . 7$ and $p ( y = 1 ) = 0 . 3$ .
+
+# 4.3 Augmenting the model with inducing points: computational efficiency
+
+Learning the full GP model described in Section 4.1 requires inverting a matrix of size $N _ { c }$ in the worst case (at the root node), which has $\mathcal { O } ( N _ { c } ^ { 3 } )$ run-time complexity and $\hat { \mathcal { O } } ( N _ { c } ^ { 2 } )$ memory complexity.
+
+Algorithm 1 pFedGP. $C$ clients indexed by c; $E$ - number of local epochs; $| S |$ - number of sampled clients; $M$ - number of inducing inputs per class
+
+# Server executes:
+
+Initialize shared network $\theta  \theta _ { 0 }$ Initialize $M$ inducing inputs per class for all classes in the system # in pFedGP-IP variants only for each round $t  1 , 2 , \dots \mathbf { d }$ o: Sample $S$ clients uniformly at random for each client $c \in S$ in parallel: $\theta _ { t + 1 } ^ { c }$ ${ } _ { 1 } , M _ { t + 1 } ^ { c } \gets$ ClientUpdate $( \theta _ { t } , M _ { t } )$ # obtain updates from client c Update $\theta _ { t + 1 } , M _ { t + 1 }$ using FedAvg [51] update rule. ClientUpdate $( \theta , M )$ : for each local epoch $e \gets 1 , . . . , E$ do: if $e = 1$ : Build GP tree classifier using the personal dataset $D _ { c }$ Update $\theta , M$ using gradient-based optimization methods on $D _ { c }$ with $\mathcal { L } ^ { M L }$ or $\mathcal { L } ^ { P D }$ return $\theta , M$
+
+Therefore, we propose an additional procedure based on inducing points to allow reduced complexity in low resource environments and scalability to larger dataset sizes.
+
+This variant is based on the fully independent training conditional (FITC) method [70]. The key idea is to cast all the dependence on the inducing points and assume independence between the latent function values given the inducing points. Here for brevity, we omit the subscripts denoting the client and the tree node. However, all quantities and data points are those that belong to a specific client and tree node. Let $\bar { \mathbf { X } } \in \mathbb { R } ^ { M \times d }$ denote the pseudo-inputs (defined in the embedding space of the last layer of the NN), and $\bar { \mathbf { f } } \in \mathbb { R } ^ { M }$ the corresponding latent function values. Here as well, the inducing inputs are defined globally at the server level and they are set evenly across classes. We assume the following GP prior $p ( \mathbf { f } , \bar { \mathbf { f } } ) = \mathcal { N } \Big ( \mathbf { 0 } , \big [ \mathbf { K } _ { M N } \quad \mathbf { K } _ { N M } \big ] \Big )$ , where ${ \bf K } _ { M M }$ is the kernel between the inducing inputs, ${ \bf K } _ { N N }$ is a diagonal matrix between the actual training data, ${ \bf K } _ { N M }$ is the kernel between the data and the inducing inputs, and we placed a zero mean prior. The likelihood of the dataset when factoring the inducing variables and the Pólya-Gamma variables (one per training sample), and the posterior over f, both have known analytical expressions. We can then obtain the posterior and marginal distributions by marginalizing over $\bar { \bf f }$ . Here we will present the posterior and marginal distributions:
+
+$$
+\begin{array} { r l } & { p ( \mathbf { f } | \mathbf { X } , \mathbf { y } , \omega , \bar { \mathbf { X } } ) = \mathcal { N } ( \mathbf { f } | \mathbf { K } _ { N M } \mathbf { B } ^ { - 1 } \mathbf { K } _ { M N } \mathbf { A } ^ { - 1 } \Omega ^ { - 1 } \boldsymbol { \kappa } , \mathbf { K } _ { N N } - \mathbf { K } _ { N M } ( \mathbf { K } _ { M M } ^ { - 1 } - \mathbf { B } ^ { - 1 } ) \mathbf { K } _ { M N } ) , } \\ & { \quad p ( \mathbf { y } | \mathbf { X } , \omega , \bar { \mathbf { X } } ) \propto \mathcal { N } ( \Omega ^ { - 1 } \boldsymbol { \kappa } | \mathbf { 0 } , \ \Lambda + \mathbf { K } _ { N M } \mathbf { K } _ { M M } ^ { - 1 } \mathbf { K } _ { M N } ) . } \end{array}
+$$
+
+Where $\pmb { \Omega } = d i a g ( \omega )$ , $\kappa _ { j } = y _ { j } - 1 / 2$ , $\pmb { \Lambda } = \pmb { \Omega } ^ { - 1 } + d i a g ( \mathbf { K } _ { N N } - \mathbf { K } _ { N M } \mathbf { K } _ { M M } ^ { - 1 } \mathbf { K } _ { M N } )$ , and $\mathbf { B } =$ $\mathbf { K } _ { M M } + \mathbf { K } _ { M N } \mathbf { A } ^ { - 1 } \mathbf { K } _ { N M }$ . Importantly, we only need to invert $M \times M$ or diagonal matrices. See full derivation in Appendix B. During test time, we use $\bar { \bf f }$ to get the posterior of $f ^ { * }$ to compute the predictive distribution.
+
+Now we can use either the marginal or the predictive distribution to learn the shared NN parameters and the inducing locations. The complexity of applying this procedure is reduced to $\mathcal { O } ( M ^ { \frac { . } { 2 } } N _ { c } + M ^ { 3 } )$ in run-time, and $\mathcal { O } ( M N _ { c } + M ^ { 2 } )$ in memory. While the (conditional) independence assumption between the latent function values may be restrictive, we found this method to be comparable with the full GP alternative in our experiments. Potentially, this can be attributed to the effect of sharing the inducing inputs among clients and the information that $\omega$ stores on $\mathbf { f }$ .
+
+# 5 Generalization bound
+
+It is reasonable to expect that after we learned the system new clients will arrive. In such cases, we would like to use pFedGP without re-training the kernel function. Under this scenario, we can derive generalization bounds concerning only the GP classifier without taking into account the fixed
+
+neural network using PAC-Bayes bound [50]. Having meaningful guarantees can be very important in safety-critical applications. The PAC-Bayes bound for GPC [64] (with the Gibbs risk):
+
+Theorem 1. Given i.i.d. samples $D _ { c } = \{ ( \mathbf { x } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N _ { c } }$ of size $N _ { c }$ drawn from any data distribution over $\mathcal { X } \times \{ - 1 , 1 \}$ , a posterior $Q$ , and a $G P$ prior , the following bound holds, where the probability is over random data samples:
+
+$$
+{ \cal P } r _ { D _ { c } } \{ { \cal R } ( Q ) > { \cal R } _ { D _ { c } } ( Q ) + K L _ { b e r } ^ { - 1 } ( { \cal R } _ { D _ { c } } ( Q ) , \epsilon ( \delta , n , P , Q ) ) \} \le \delta .
+$$
+
+Here, we have,
+
+$$
+\begin{array} { l } { { \displaystyle \mathrm { \normalfont { \gtrsim } } ( Q ) = \mathbb { E } _ { ( \mathbf { x ^ { * } } , y ^ { * } ) } [ P r _ { f ^ { * } \sim Q ( f ^ { * } \mid \mathbf { x ^ { * } } , D _ { c } ) } \{ s i g n f ^ { * } \neq y ^ { * } \} ] , \quad R _ { D _ { c } } ( Q ) = \displaystyle \frac { 1 } { N _ { c } } \sum _ { i = 1 } ^ { N _ { c } } P r _ { f _ { i } \sim Q ( f _ { i } \mid D _ { c } ) } \{ s i g n f _ { i } \neq y _ { i } \} } } \\ { { \displaystyle ( \delta , N _ { c } , P , Q ) = \frac { 1 } { N _ { c } } \Big ( K L [ Q \mid | \mathbf { \nabla } P | + \log \frac { N _ { c } + 1 } { \delta } \Big ) , \quad K L _ { b e r } ^ { - 1 } ( q , \epsilon ) = m a x _ { p \in [ 0 , 1 ] } K L _ { b e r } [ q \mid | \mathbf { \nabla } p ] \le \epsilon } } \end{array}
+$$
+
+An important observation in [64] is that the KL-divergence between the posterior and prior Gaussian processes is equivalent to the KL-divergence between the posterior and prior distribution of their values on the $N _ { c }$ training samples. While [64] assumed $Q$ to be Gaussian, this observation still holds even without this assumption. However, when $Q$ is no longer Gaussian, as is the case here, $K L [ Q ( \mathbf { f } ) \mid \mid P ( \mathbf { f } ) ]$ no longer has a closed-form expression. We can show that for the Pólya-Gamma augmentation:
+
+$$
+\begin{array} { l } { { \displaystyle K L [ Q ( { \bf f } ) \mathbf { \nabla } | | \mathbf { \nabla } P ( { \bf f } ) ] = \mathbb { E } _ { Q ( \omega ) } \{ K L [ Q ( { \bf f } | \omega ) | | P ( { \bf f } ) ] \} - M I [ { \bf f } ; \omega ] } \ ~ } \\ { { \displaystyle ~ \quad = \mathbb { E } _ { Q ( \omega ) } \{ K L [ Q ( { \bf f } | \omega ) | | P ( { \bf f } ) ] \} + \mathbb { E } _ { Q ( { \bf f } , \omega ) } \left[ \log \frac { Q ( \omega ) } { Q ( \omega | { \bf f } ) } \right] } } \end{array}
+$$
+
+where MI denotes the mutual information. Since $Q ( \mathbf { f } \vert \boldsymbol { \omega } )$ and $P ( \mathbf { f } )$ are Gaussian, the $K L [ Q ( \mathbf { f } | \omega ) | | P ( \mathbf { f } ) ]$ term has a close form expression so we only need to perform MonteCarlo approximation on the expectation on $\omega$ on the first element. In the second expectation, $Q ( \omega )$ does not have a known expression. To estimate it, given $\{ ( \omega _ { i } , \mathbf { f } _ { i } ) \} _ { i = 1 } ^ { N }$ samples, we use $\begin{array} { r } { Q ( \omega _ { i } ) \approx \frac { 1 } { N - 1 } \sum _ { j \neq i } Q ( \omega _ { i } | \mathbf { f } _ { j } ) . } \end{array}$ . Note that if the summation for $j$ includes $\mathbf { f } _ { i }$ , it might result in a biased estimator. Further details on estimating $K L [ Q ( \mathbf { f } ) \mid \mid P ( \mathbf { f } ) ]$ are in Appendix C.
+
+To assess the quality of the bound, we partitioned the CIFAR-10 dataset to 100 clients. We trained a shared network using our full-GP variant on 90 clients and then recorded the generalization and test error on the remaining 10 clients four times, each with a different training set size. Figure 2 shows the estimation of the generalization error bound $\tilde { \delta } = 0 . 0 1 \mathrm { \Omega }$ vs the actual error on the novel clients with the Gibbs classifier. First, we observe that indeed the bound is greater than the actual test error for all points and that it is not vacuous. There is a strong correlation between the actual error and the bound. Secondly, unlike worst-case bounds (e.g. VC-dimension), this bound depends on the actual data and not only the number of data points.
+
+![](images/9edfb836139c4fc8a311d92a715076ede5cc00bc3b202b4cc8243a02879bdfef.jpg)  
+Figure 2: Test error vs an estimated upper bound over 10 clients with varying degrees of a training set data size. Each dot represents a combination of client and data size. In parenthesis - the average difference between the empirical and the test error.
+
+# 6 Experiments
+
+We evaluated pFedGP against baseline methods in various learning setups. We present the result for the following model variants: (i) pFedGP, the full GP model (Section 4.1); (ii) pFedGP-IP-data, the model with IPs described in Section 4.2; and (iii) pFedGP-IP-compute, the model with IPs described in Section 4.3. For pFedGP and pFedGP-IP-compute, the results obtained by maximizing the predictive and marginal likelihood were similar, with a slight advantage to the former. Therefore, we present here the results only for the predictive alternative and defer the results of the marginal alternative to the Appendix. Additional experiments, ablation study, and further analyses are provided in Appendix E. Unless stated otherwise, we report the average and the standard error of the mean (SEM) over three random seeds of the federated accuracy, defined as the average accuracy across all clients and samples.
+
+Table 1: Test accuracy $\pm$ SEM) over 50, 100, 500 clients on CIFAR-10, CIFAR-100, and CINIC-10. The # samples/client indicates the average number of training samples per client.   
+
+<table><tr><td>#clients</td><td colspan="3">CIFAR-10</td><td colspan="3">CIFAR-100</td><td colspan="3">CINIC-10</td></tr><tr><td># samples/client</td><td>50 800</td><td>100 400</td><td>500 80</td><td>50 800</td><td>100 400</td><td>500 80</td><td>50 1800</td><td>100 900</td><td>500 180</td></tr><tr><td>Local</td><td>86.2 ±0.2</td><td>82.9 ± 0.4</td><td>74.8 ± 0.5</td><td>52.1± 0.2</td><td>45.6 ± 0.3</td><td>30.9 ±0.2</td><td>61.1 ± 0.3</td><td>56.9 ± 0.7</td><td>46.4 ± 0.1</td></tr><tr><td>FedAvg [51]</td><td>56.4 ± 0.5</td><td>59.7 ± 0.5</td><td>54.0 ±0.5</td><td>23.6±0.2</td><td>24.0±0.2</td><td>20.4±0.0</td><td>45.6± 0.4</td><td>44.7 ± 0.5</td><td>45.7± 0.5</td></tr><tr><td>FOLA [47]</td><td>55.9± 3.3</td><td>52.1 ± 3.1</td><td>45.9±0.3</td><td>25.5 ± 1.5</td><td>22.4 ± 1.3</td><td>18.7 ± 0.1</td><td>45.2 ± 0.3</td><td>43.4±0.3</td><td>38.3±0.2</td></tr><tr><td>FedPer [4]</td><td>83.8±0.8</td><td>81.5 ± 0.5</td><td>76.8 ±1.2</td><td>48.3±0.6</td><td>43.6±0.2</td><td>25.6±0.3</td><td>70.6±0.2</td><td>68.4± 0.5</td><td>62.2± .05</td></tr><tr><td>LG-FedAvg [44]</td><td>87.9±0.3</td><td>83.6±0.7</td><td>64.7 ± 0.7</td><td>43.6±0.2</td><td>37.5 ± 0.9</td><td>20.3±0.5</td><td>59.5 ± 1.1</td><td>59.9 ± 2.1</td><td>52.5±0.8</td></tr><tr><td>pFedMe[72]</td><td>86.4±0.8</td><td>85.0±0.3</td><td>80.3±0.5</td><td>49.8 ± 0.5</td><td>47.7 ± 0.4</td><td>32.5±0.8</td><td>69.9 ± 0.5</td><td>68.9 ±0.7</td><td>58.8 ±0.1</td></tr><tr><td>FedU[18]</td><td>80.6±0.3</td><td>78.1 ± 0.5</td><td>65.6±0.4</td><td>41.1 ± 0.2</td><td>36.0±0.2</td><td>15.9 ± 0.4</td><td>59.3 ± 0.2</td><td>55.4±0.6</td><td>41.6± 0.5</td></tr><tr><td>pFedHN [65]</td><td>90.2 ±0.6</td><td>87.4±0.2</td><td>83.2 ±0.8</td><td>60.0 ±1.0</td><td>52.3±0.5</td><td>34.1 ±0.1</td><td>70.4 ±0.4</td><td>69.4± 0.5</td><td>64.2±.05</td></tr><tr><td>Ours</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>pFedGP-IP-data</td><td>88.6±0.2 87.4±0.2</td><td></td><td>86.9±0.7</td><td>60.2±0.3</td><td>58.5±0.3</td><td>55.7±0.4</td><td>69.8±0.2</td><td>68.3 ± 0.6 67.6± 0.3</td><td></td></tr><tr><td>pFedGP-IP-compute</td><td>89.9±0.6 88.8±0.1</td><td></td><td>86.8 ±0.4</td><td>61.2 ± 0.4</td><td>59.8±0.3</td><td>49.2 ± 0.3</td><td>72.0 ±0.3</td><td>71.5± 0.5</td><td>68.2±0.2</td></tr><tr><td>pFedGP</td><td>89.2±0.3</td><td>88.8 ±0.2</td><td>87.6± 0.4</td><td>63.3± 0.1</td><td>61.3 ± 0.2</td><td>50.6±0.2</td><td>71.8 ± 0.3</td><td>71.3 ± 0.4</td><td>68.1 ±0.3</td></tr></table>
+
+Datasets. All methods were evaluated on CIFAR-10, CIFAR-100 [38], and CINIC-10 [15] datasets.   
+CINIC-10 is more diverse since it combines images from CIFAR-10 and ImageNet [16].
+
+Compared methods. We compared our method against the following baselines: (1) Local, pFedGP full model on each client with a private network and no collaboration with other clients; (2) FedAvg [51], a standard FL model with no personalization component; (3) FOLA [47], a Bayesian method that used a multivariate Gaussian product mechanism to aggregate local models; (4) FedPer [4], a PFL approach that learns a personal classifier for each client on top of a shared feature extractor; (5) LG-FedAvg [44], a PFL method that uses local feature extractor per client and global output layers; (6) pFedMe [72], a PFL method which adds a Moreau-envelopes loss term; (7) FedU [18], a recent multi-task learning approach for PFL that learns a model per client; (8) pFedHN [65], a recent PFL approach that uses a hypernetwork to generate client-specific networks.
+
+Training protocol. We follow the training strategy proposed in [65]. We limit the training process to 1000 communication rounds, in each we sample five clients uniformly at random for model updates. The training procedure is different in the FOLA and pFedHN baselines, so we used an equivalent communication cost. In LG-FedAvg, we made an extra 200 communication rounds after a pre-training stage with the FedAvg model for 1000 communication rounds. In the local model, we performed 100 epochs of training for each client. In all experiments, we used a LeNet-based network [40] having two convolution layers followed by two fully connected layers and an additional linear layer. We tuned the hyperparameters of all methods using a pre-allocated held-out validation set. Full experimental details are given in Appendix D.
+
+# 6.1 Standard PFL setting
+
+We first evaluated all methods in a standard PFL setting [65, 72]. We varied the total number of clients in the system from 50 to 500 and we set the number of classes per client to two/ten/four for CIFAR-10/CIFAR-100/CINIC-10 respectively. Since the total number of samples in the system is fixed, the number of samples per client changed accordingly. For each client, the same classes appeared in the training and test set.
+
+The results are presented in Table 1. They show that: (1) The performance of the local baseline is significantly impaired when the number of samples per client decreases, emphasizing the importance of federated learning in the presence of limited local data. (2) FedAvg and FOLA, which do not use personalized FL, perform poorly in this heterogeneous setup. (3) pFedGP outperforms or is on par with previous state-of-the-art approaches when local data is sufficient (e.g., 50 clients on all datasets). When the data per client becomes limited, pFedGP achieves significant improvements over competing methods; note the $9 \%$ and $2 1 \%$ difference in CIFAR-100 over 100 and 500 clients, respectively. (4) pFedGP-IP-compute often achieves comparable results to pFedGP and is often superior to pFedGP-IP-data. We believe that it can be attributed to the fact that in pFedGP-IP-compute the training data take an active part in the GP inference formulas (Eq. 6), while in pFedGP-IP-data the data impact in a weak manner only through the loss function. (5) pFedGP-IP-data is especially helpful when few samples per class are available, e.g., CIFAR-100 with 500 clients. That last point is further illustrated by decoupling the effect of the number of clients from that of the training set size. To illustrate that, in Appendix E.2 we fixed the number of clients and varied the number of training samples per class. From this experiment, we deduced that both factors (individually) contribute to pFedGP success.
+
+![](images/93e79ff70f1ee350f1b37b3442cc3d9fab6ebd89fce89f22f647595b6d4fd26a.jpg)  
+Figure 3: Reliability diagrams on CIFAR-100 with 50 clients. Diagonal indicates perfect calibration. Each plot also shows the expected & maximum calibration error (ECE & MCE) and the Brier Score (BRI). Lower is better.
+
+A desired property from PFL classifiers is the ability to provide uncertainty estimation. For example, in decision support systems, such as in healthcare applications, the decision-maker should have an accurate estimation of the classifier confidence in the prediction. Here, we quantify the uncertainty through calibration. Figure 3 compares all methods both visually and using common metrics [7, 25, 55] on the CIFAR-100 dataset with 50 clients. Expected calibration error (ECE) measures the weighted average between the classifier confidence and accuracy. Maximum calibration error (MCE) takes the maximum instead of the average. And, Brier score (BRI) [7] measures the average squared error between the labels and the prediction prob
+
+Table 2: Test accuracy ( $\pm$ SEM) over 100 clients on noisy CIFAR-100. We also provide the relative accuracy decrease $( \% )$ w.r.t. the performance on the original CIFAR-100 data (see Table 1).   
+
+<table><tr><td>Method</td><td>Accuracy</td><td>Decrease (%)</td></tr><tr><td>FedPer [4]</td><td>28.1 ± 0.9</td><td>-35.6</td></tr><tr><td>LG-FedAvg [44]</td><td>26.9 ± 0.9</td><td>-28.3</td></tr><tr><td>pFedme [72]</td><td>33.2 ± 0.6</td><td>-30.4</td></tr><tr><td>FedU[18]</td><td>35.0 ± 0.2</td><td>-2.8</td></tr><tr><td>pFedHN[65]</td><td>38.9 ± 0.5</td><td>-25.7</td></tr><tr><td>Ours</td><td></td><td></td></tr><tr><td>pFedGP-IP-data</td><td>45.0± 0.3</td><td>-23.1</td></tr><tr><td>pFedGP-IP-compute</td><td>47.1 ± .05</td><td>-21.2</td></tr><tr><td>pFedGP</td><td>49.5 ± 0.1</td><td>-19.2</td></tr></table>
+
+abilities. The figure shows that pFedGP classifiers are best calibrated across all metrics in almost all cases. We note that with temperature scaling, the calibration of the baseline methods can be improved [25]; however, choosing the right temperature requires optimization over a separate validation set, which our model does not need. Additional calibration results, including temperature scaling, are presented in Appendix E.9.
+
+# 6.2 PFL with input noise
+
+In real-world federated systems, the clients may employ different measurement devices for data collection (cameras, sensors, etc.), resulting in different input noise characteristics per client. Here, we investigate pFedGP performance in this type of personalization. To simulate that, we partitioned CIFAR-10/100 to 100 clients similar to the protocol described in Section 6.1, we defined 57 unique distributions of image corruption noise [29], and we assigned a noise model to each client. Then for each example in each client, we sampled a corruption noise according to the noise model allocated to that client. Here we show the results for the noisy CIFAR-100 dataset in Table 2. Further details on the perturbations performed and result for the noisy CIFAR-10 are given in the Appendix2. We observe a significant gap in favor of the pFedGP variants compared to baseline methods. Note that using global inducing points is slightly less beneficial in this case since they are defined globally and therefore are not tied to a specific noise type as the real client data is.
+
+# 6.3 Generalization to out-of-distribution (OOD) novel clients
+
+FL are dynamic systems. For example, novel clients may enter the system after the model was trained, possibly with a data distribution shift. Adapting to a new OOD client is both challenging and important for real-world FL systems. To evaluate pFedGP in this scenario, we followed the learning protocol proposed in [65]. We partitioned the CIFAR-10 dataset into two groups. The data in the first group was distributed between 90 clients for model training. The remaining data from the second group was distributed between an additional 10 clients that were excluded during training. Within each group, we set the class probabilities in each client by sam
+
+![](images/8e392c0adfd99111f0d066660fe0fb5834a68cb130b237f21eca7eb8685cd8b0.jpg)  
+Figure 4: Generalization to novel clients on CIFAR-10.
+
+pling from a Dirichlet distribution with the same $\alpha$ parameter. For the training group, we set $\alpha = 0 . 1$ , trained the shared model using these clients, and froze it. Then, we evaluated the models on the second group by varying $\alpha \in \{ . 1 , . 2 5 , . 5 , . 7 5 , 1 \}$ , on the remaining 10 clients. As $\alpha$ moves away from 0.1 the distribution shift between the two groups increases, resulting in more challenging OOD clients. Figure 4 reports the generalization gap as a function of the Dirichlet parameter $\alpha$ . The generalization gap is computed by taking the difference between the average test accuracy of the (ten) novel clients and the average test accuracy of the (ninety) clients used for training. From the figure, here as well, pFedGP achieves the best generalization performance for all values of $\alpha$ . Moreover, unlike baseline methods, pFedGP does not require any parameter tuning. Several baselines were excluded from the figure since they had a large generalization gap.
+
+# 7 Conclusion
+
+In this study, we proposed pFedGP, a novel method for PFL. pFedGP learns a kernel function, parameterized by a NN, that is shared between all clients using a personal GP classifier on each client. We proposed three variants for pFedGP, a full model approach that generally shows the best performance and two extensions to it. The first is most beneficial when the number of examples per class are small while the second allows controlling the computational requirements of the model. We also derived PAC-Bayes generalization bound on novel clients and empirically showed that it gives non-vacuous guarantees. pFedGP provides well-calibrated predictions, generalizes well to OOD novel clients, and consistently outperforms competing methods.
+
+Broader impact: Our method shares the standard communication procedure of FL approaches, where no private data is directly communicated across the different nodes in the system. This protocol does not explicitly guarantee that no private information can be inferred at this time. As we show, pFedGP is particularly useful for clients with little data, and for clients that have strongly different distribution. This has great potential to improve client personalization in real-world systems, and do better at handling less common data. The latter is of great interest for decision support systems in sensitive domains such as health care or legal.
+
+# Acknowledgements
+
+This study was funded by a grant to GC from the Israel Science Foundation (ISF 737/2018), and by an equipment grant to GC and Bar-Ilan University from the Israel Science Foundation (ISF 2332/18). IA was funded by a grant from the Israeli innovation authority, through the AVATAR consortium.
+
+References   
+[1] Sawsan Abdulrahman, Hanine Tout, Hakima Ould-Slimane, Azzam Mourad, Chamseddine Talhi, and Mohsen Guizani. A survey on federated learning: The journey from centralized to distributed on-site learning and beyond. IEEE Internet of Things Journal, 8(7):5476–5497, 2021.   
+[2] Idan Achituve, Aviv Navon, Yochai Yemini, Gal Chechik, and Ethan Fetaya. GP-Tree: A Gaussian process classifier for few-shot incremental learning. In Proceedings of the 38th International Conference on Machine Learning, pages 54–65. PMLR, 2021.   
+[3] Naman Agarwal, Ananda Theertha Suresh, Felix Xinnan X Yu, Sanjiv Kumar, and Brendan McMahan. cpSGD: Communication-efficient and differentially-private distributed SGD. In Advances in Neural Information Processing Systems, pages 7564–7575, 2018.   
+[4] Manoj Ghuhan Arivazhagan, Vinay Aggarwal, Aaditya Kumar Singh, and Sunav Choudhary. Federated learning with personalization layers. arXiv preprint arXiv:1912.00818, 2019.   
+[5] David Arthur and Sergei Vassilvitskii. k-means $^ { + + }$ the advantages of careful seeding. In Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, pages 1027–1035, 2007.   
+[6] Harkirat Singh Behl, Atılım Güne¸s Baydin, and Philip HS Torr. Alpha MAML: Adaptive model-agnostic meta-learning. arXiv preprint arXiv:1905.07435, 2019.   
+[7] Glenn W Brier. Verification of forecasts expressed in terms of probability. Monthly weather review, 78(1):1–3, 1950.   
+[8] Thang D Bui, Cuong V Nguyen, Siddharth Swaroop, and Richard E Turner. Partitioned variational inference: A unified framework encompassing federated and continual learning. arXiv preprint arXiv:1811.11206, 2018.   
+[9] Dongqi Cai, Qipeng Wang, Yuanqiang Liu, Yunxin Liu, Shangguang Wang, and Mengwei Xu. Towards ubiquitous learning: A first measurement of on-device training performance. In Proceedings of the 5th International Workshop on Embedded and Mobile Deep Learning, pages 31–36, 2021.   
+[10] Roberto Calandra, Jan Peters, Carl Edward Rasmussen, and Marc Peter Deisenroth. Manifold Gaussian processes for regression. In 2016 International Joint Conference on Neural Networks (IJCNN), pages 3338–3345. IEEE, 2016.   
+[11] Shuxiao Chen, Qinqing Zheng, Qi Long, and Weijie J Su. A theorem of the alternative for personalized federated learning. arXiv preprint arXiv:2103.01901, 2021.   
+[12] Luca Corinzia, Ami Beuret, and Joachim M Buhmann. Variational federated multi-task learning. arXiv preprint arXiv:1906.06268, 2019.   
+[13] Xinyan Dai, Xiao Yan, Kaiwen Zhou, Han Yang, Kelvin KW Ng, James Cheng, and Yu Fan. Hyper-sphere quantization: Communication-efficient SGD for federated learning. arXiv preprint arXiv:1911.04655, 2019.   
+[14] Zhongxiang Dai, Bryan Kian Hsiang Low, and Patrick Jaillet. Federated Bayesian optimization via Thompson sampling. Advances in Neural Information Processing Systems, 33, 2020.   
+[15] Luke N Darlow, Elliot J Crowley, Antreas Antoniou, and Amos J Storkey. CINIC-10 is not Imagenet or CIFAR-10. arXiv preprint arXiv:1810.03505, 2018.   
+[16] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A largescale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.   
+[17] Yuyang Deng, Mohammad Mahdi Kamani, and Mehrdad Mahdavi. Adaptive personalized federated learning. arXiv preprint arXiv:2003.13461, 2020.   
+[18] Canh T Dinh, Tung T Vu, Nguyen H Tran, Minh N Dao, and Hongyu Zhang. FedU: A unified framework for federated multi-task learning with Laplacian regularization. arXiv preprint arXiv:2102.07148, 2021.   
+[19] Randal Douc, Eric Moulines, and David Stoffer. Nonlinear time series: Theory, methods and applications with R examples. CRC press, 2014.   
+[20] John C Duchi, Michael I Jordan, and Martin J Wainwright. Privacy aware learning. Journal of the ACM (JACM), 61(6):1–57, 2014.   
+[21] Alireza Fallah, Aryan Mokhtari, and A. Ozdaglar. Personalized federated learning: A metalearning approach. Advances in Neural Information Processing Systems, 2020.   
+[22] Alireza Fallah, Aryan Mokhtari, and Asuman Ozdaglar. On the convergence theory of gradientbased model-agnostic meta-learning algorithms. In International Conference on Artificial Intelligence and Statistics, pages 1082–1092. PMLR, 2020.   
+[23] Théo Galy-Fajou, Florian Wenzel, Christian Donner, and Manfred Opper. Multi-class Gaussian process classification made conjugate: Efficient inference via data augmentation. In Uncertainty in Artificial Intelligence, pages 755–765. PMLR, 2020.   
+[24] Théo Galy-Fajou, Florian Wenzel, and Manfred Opper. Automated augmented conjugate inference for non-conjugate gaussian process models. In International Conference on Artificial Intelligence and Statistics, pages 3025–3035. PMLR, 2020.   
+[25] Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In International Conference on Machine Learning, pages 1321–1330. PMLR, 2017.   
+[26] Farzin Haddadpour and Mehrdad Mahdavi. On the convergence of local descent methods in federated learning. arXiv preprint arXiv:1910.14425, 2019.   
+[27] Filip Hanzely and Peter Richtárik. Federated learning of a mixture of global and local models. arXiv preprint arXiv:2002.05516, 2020.   
+[28] 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, pages 770–778, 2016.   
+[29] Dan Hendrycks and Thomas Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. In International Conference on Learning Representations, 2019.   
+[30] James Hensman, Alexander Matthews, and Zoubin Ghahramani. Scalable variational Gaussian process classification. In Artificial Intelligence and Statistics, pages 351–360. PMLR, 2015.   
+[31] T. H. Hsu, Hang Qi, and M. Brown. Measuring the effects of non-identical data distribution for federated visual classification. ArXiv, abs/1909.06335, 2019.   
+[32] Yutao Huang, Lingyang Chu, Zirui Zhou, Lanjun Wang, Jiangchuan Liu, Jian Pei, and Yong Zhang. Personalized cross-silo federated learning on non-IID data. In Proceedings of the AAAI Conference on Artificial Intelligence, 2021.   
+[33] Yihan Jiang, Jakub Konecnˇ y, Keith Rush, and Sreeram Kannan. Improving federated learning \` personalization via model agnostic meta learning. arXiv preprint arXiv:1909.12488, 2019.   
+[34] Peter Kairouz, H Brendan McMahan, Brendan Avent, Aurélien Bellet, Mehdi Bennis, Arjun Nitin Bhagoji, Keith Bonawitz, Zachary Charles, Graham Cormode, Rachel Cummings, et al. Advances and open problems in federated learning. arXiv preprint arXiv:1912.04977, 2019.   
+[35] Sai Praneeth Karimireddy, Satyen Kale, Mehryar Mohri, Sashank Reddi, Sebastian Stich, and Ananda Theertha Suresh. SCAFFOLD: Stochastic controlled averaging for federated learning. In International Conference on Machine Learning, pages 5132–5143. PMLR, 2020.   
+[36] Rahif Kassab and Osvaldo Simeone. Federated generalized Bayesian learning via distributed Stein variational gradient descent. arXiv preprint arXiv:2009.06419, 2020.   
+[37] Jakub Konecnˇ y, H Brendan McMahan, Daniel Ramage, and Peter Richtárik. Federated optimiza- \` tion: Distributed machine learning for on-device intelligence. arXiv preprint arXiv:1610.02527, 2016.   
+[38] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.   
+[39] V. Kulkarni, Milind Kulkarni, and A. Pant. Survey of personalization techniques for federated learning. 2020 Fourth World Conference on Smart Trends in Systems, Security and Sustainability (WorldS4), pages 794–797, 2020.   
+[40] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   
+[41] Tian Li, Anit Kumar Sahu, Maziar Sanjabi, Manzil Zaheer, Ameet Talwalkar, and Virginia Smith. On the convergence of federated optimization in heterogeneous networks. arXiv preprint arXiv:1812.06127, 2018.   
+[42] Tian Li, Anit Kumar Sahu, Ameet Talwalkar, and Virginia Smith. Federated learning: Challenges, methods, and future directions. IEEE Signal Processing Magazine, 37(3):50–60, 2020.   
+[43] Zhenguo Li, Fengwei Zhou, Fei Chen, and Hang Li. Meta-SGD: Learning to learn quickly for few-shot learning. arXiv preprint arXiv:1707.09835, 2017.   
+[44] Paul Pu Liang, Terrance Liu, Liu Ziyin, Nicholas B Allen, Randy P Auerbach, David Brent, Ruslan Salakhutdinov, and Louis-Philippe Morency. Think locally, act globally: Federated learning with local and global representations. arXiv preprint arXiv:2001.01523, 2020.   
+[45] Tao Lin, Sebastian U Stich, Kumar Kshitij Patel, and Martin Jaggi. Don’t use large mini-batches, use local SGD. In International Conference on Learning Representations, 2019.   
+[46] Scott W Linderman, Matthew J Johnson, and Ryan P Adams. Dependent multinomial models made easy: stick breaking with the Pólya-Gamma augmentation. In Proceedings of the 28th International Conference on Neural Information Processing Systems, pages 3456–3464, 2015.   
+[47] Liangxi Liu and Feng Zheng. A Bayesian federated learning framework with multivariate gaussian product. arXiv preprint arXiv:2102.01936, 2021.   
+[48] Wesley Maddox, Shuai Tang, Pablo Moreno, Andrew Gordon Wilson, and Andreas Damianou. Fast adaptation with linearized neural networks. In International Conference on Artificial Intelligence and Statistics, pages 2737–2745. PMLR, 2021.   
+[49] Y. Mansour, M. Mohri, J. Ro, and A. T. Suresh. Three approaches for personalization with applications to federated learning. ArXiv, abs/2002.10619, 2020.   
+[50] David A McAllester. PAC-Bayesian stochastic model selection. Machine Learning, 51(1):5–21, 2003.   
+[51] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas. Communication-efficient learning of deep networks from decentralized data. In Artificial Intelligence and Statistics, pages 1273–1282. PMLR, 2017.   
+[52] H Brendan McMahan, Daniel Ramage, Kunal Talwar, and Li Zhang. Learning differentially private recurrent language models. In International Conference on Learning Representations, 2018.   
+[53] Thomas Peter Minka. A family of algorithms for approximate Bayesian inference. PhD thesis, Massachusetts Institute of Technology, 2001.   
+[54] Viraaji Mothukuri, Reza M Parizi, Seyedamin Pouriyeh, Yan Huang, Ali Dehghantanha, and Gautam Srivastava. A survey on security and privacy of federated learning. Future Generation Computer Systems, 115:619–640, 2021.   
+[55] Mahdi Pakdaman Naeini, Gregory Cooper, and Milos Hauskrecht. Obtaining well calibrated probabilities using Bayesian binning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 29, 2015.   
+[56] Sebastian W. Ober, Carl E. Rasmussen, and Mark van der Wilk. The promises and pitfalls of deep kernel learning. CoRR, 2021.   
+[57] Nicholas G. Polson, James G. Scott, and Jesse Windle. Bayesian inference for logistic models using Pólya–Gamma latent variables. Journal of the American Statistical Association, pages 1339–1349, 2013.   
+[58] Joaquin Quinonero-Candela and Carl Edward Rasmussen. A unifying view of sparse approximate Gaussian process regression. The Journal of Machine Learning Research, 6:1939–1959, 2005.   
+[59] Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian Processes for Machine Learning. The MIT Press, 2006.   
+[60] David Reeb, Andreas Doerr, Sebastian Gerwinn, and Barbara Rakitsch. Learning Gaussian processes by minimizing PAC-Bayesian generalization bounds. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 3341–3351, 2018.   
+[61] Amirhossein Reisizadeh, Aryan Mokhtari, Hamed Hassani, Ali Jadbabaie, and Ramtin Pedarsani. FedPAQ: A communication-efficient federated learning method with periodic averaging and quantization. In International Conference on Artificial Intelligence and Statistics, pages 2021– 2031. PMLR, 2020.   
+[62] Ryan Rifkin and Aldebaro Klautau. In defense of one-vs-all classification. The Journal of Machine Learning Research, 5:101–141, 2004.   
+[63] Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4510–4520, 2018.   
+[64] Matthias Seeger. PAC-Bayesian generalisation error bounds for Gaussian process classification. Journal of machine learning research, 3(Oct):233–269, 2002.   
+[65] Aviv Shamsian, Aviv Navon, Ethan Fetaya, and Gal Chechik. Personalized federated learning using hypernetworks. In International Conference on Machine Learning. PMLR, 2021.   
+[66] Reza Shokri and Vitaly Shmatikov. Privacy-preserving deep learning. In Proceedings of the 22nd ACM SIGSAC conference on computer and communications security, pages 1310–1321, 2015.   
+[67] Virginia Smith, Chao-Kai Chiang, Maziar Sanjabi, and Ameet Talwalkar. Federated multi-task learning. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 4427–4437, 2017.   
+[68] Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 4080–4090, 2017.   
+[69] Jake Snell and Richard Zemel. Bayesian few-shot classification with one-vs-each Pólya-Gamma augmented Gaussian processes. In International Conference on Learning Representations, 2021.   
+[70] Edward Snelson and Zoubin Ghahramani. Sparse Gaussian processes using pseudo-inputs. In Advances in Neural Information Processing Systems, pages 1257–1264. MIT Press, 2006.   
+[71] Sebastian U Stich. Local SGD converges fast and communicates little. In International Conference on Learning Representations, 2018.   
+[72] Canh T Dinh, Nguyen Tran, and Tuan Dung Nguyen. Personalized federated learning with Moreau envelopes. Advances in Neural Information Processing Systems, 33, 2020.   
+[73] Alysa Ziying Tan, Han Yu, Lizhen Cui, and Qiang Yang. Towards personalized federated learning. arXiv preprint arXiv:2103.00710, 2021.   
+[74] Minxue Tang, Xuefei Ning, Yitu Wang, Yu Wang, and Yiran Chen. Fedgp: Correlation-based active client selection for heterogeneous federated learning. arXiv preprint arXiv:2103.13822, 2021.   
+[75] Carlos Villacampa-Calvo, Bryan Zaldívar, Eduardo C Garrido-Merchán, and Daniel HernándezLobato. Multi-class Gaussian process classification with noisy inputs. Journal of Machine Learning Research, 22(36):1–52, 2021.   
+[76] Hongyi Wang, Mikhail Yurochkin, Yuekai Sun, Dimitris Papailiopoulos, and Yasaman Khazaeni. Federated learning with matched averaging. In International Conference on Learning Representations, 2019.   
+[77] Jianyu Wang and Gauri Joshi. Cooperative SGD: A unified framework for the design and analysis of communication-efficient SGD algorithms. In ICML Workshop on Coding Theory for Machine Learning, 2019.   
+[78] Florian Wenzel, Théo Galy-Fajou, Christian Donner, Marius Kloft, and Manfred Opper. Efficient Gaussian process classification using Pólya-Gamma data augmentation. In The AAAI Conference on Artificial Intelligence, pages 5417–5424. AAAI Press, 2019.   
+[79] Christopher KI Williams and David Barber. Bayesian classification with Gaussian processes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(12):1342–1351, 1998.   
+[80] Andrew Gordon Wilson, Zhiting Hu, Ruslan Salakhutdinov, and Eric P. Xing. Deep kernel learning. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2016.   
+[81] Andrew Gordon Wilson, Zhiting Hu, Ruslan Salakhutdinov, and Eric P Xing. Stochastic variational deep kernel learning. In Proceedings of the 30th International Conference on Neural Information Processing Systems, pages 2594–2602, 2016.   
+[82] Qiang Yang, Yang Liu, Tianjian Chen, and Yongxin Tong. Federated machine learning: Concept and applications. ACM Transactions on Intelligent Systems and Technology (TIST), 10(2):1–19, 2019.   
+[83] Feng Yin, Zhidi Lin, Qinglei Kong, Yue Xu, Deshi Li, Sergios Theodoridis, and Shuguang Robert Cui. FedLoc: Federated learning framework for data-driven cooperative localization and location data processing. IEEE Open Journal of Signal Processing, 1:187–215, 2020.   
+[84] Mikhail Yurochkin, Mayank Agarwal, Soumya Ghosh, Kristjan Greenewald, Nghia Hoang, and Yasaman Khazaeni. Bayesian nonparametric federated learning of neural networks. In International Conference on Machine Learning, pages 7252–7261. PMLR, 2019.   
+[85] Chen Zhang, Yu Xie, Hang Bai, Bin Yu, Weihong Li, and Yuan Gao. A survey on federated learning. Knowledge-Based Systems, 216:106775, 2021.   
+[86] Michael Zhang, Karan Sapra, Sanja Fidler, Serena Yeung, and Jose M Alvarez. Personalized federated learning with first order model optimization. arXiv preprint arXiv:2012.08565, 2020.   
+[87] Yue Zhao, Meng Li, Liangzhen Lai, Naveen Suda, Damon Civin, and Vikas Chandra. Federated learning with non-IID data. arXiv preprint arXiv:1806.00582, 2018.   
+[88] Fan Zhou and Guojing Cong. On the convergence properties of a K-step averaging stochastic gradient descent algorithm for nonconvex optimization. In Proceedings of the 27th International Joint Conference on Artificial Intelligence, pages 3219–3227, 2018.   
+[89] Pan Zhou, Xiaotong Yuan, Huan Xu, Shuicheng Yan, and Jiashi Feng. Efficient meta learning via minibatch proximal update. Advances in Neural Information Processing Systems, 32:1534–1544, 2019.   
+[90] Wennan Zhu, Peter Kairouz, Brendan McMahan, Haicheng Sun, and Wei Li. Federated heavy hitters discovery with differential privacy. In International Conference on Artificial Intelligence and Statistics, pages 3837–3847. PMLR, 2020.

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+        "text": "Personalized Federated Learning with Gaussian Processes ",
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+        "type": "text",
+        "text": "Idan Achituve Bar-Ilan University, Israel idan.achituve@biu.ac.il ",
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+        "text": "Aviv Shamsian Bar-Ilan University, Israel aviv.shamsian@biu.ac.il ",
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+        "text": "Aviv Navon Bar-Ilan University, Israel aviv.navon@biu.ac.il ",
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+        "text": "Gal Chechik   \nBar-Ilan University, Israel NVIDIA, Isreal   \ngal.chechik@biu.ac.il ",
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+        "text": "Ethan Fetaya Bar-Ilan University, Israel ethan.fetaya@biu.ac.il ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "Federated learning aims to learn a global model that performs well on client devices with limited cross-client communication. Personalized federated learning (PFL) further extends this setup to handle data heterogeneity between clients by learning personalized models. A key challenge in this setting is to learn effectively across clients even though each client has unique data that is often limited in size. Here we present pFedGP, a solution to PFL that is based on Gaussian processes (GPs) with deep kernel learning. GPs are highly expressive models that work well in the low data regime due to their Bayesian nature. However, applying GPs to PFL raises multiple challenges. Mainly, GPs performance depends heavily on access to a good kernel function, and learning a kernel requires a large training set. Therefore, we propose learning a shared kernel function across all clients, parameterized by a neural network, with a personal GP classifier for each client. We further extend pFedGP to include inducing points using two novel methods, the first helps to improve generalization in the low data regime and the second reduces the computational cost. We derive a PAC-Bayes generalization bound on novel clients and empirically show that it gives non-vacuous guarantees. Extensive experiments on standard PFL benchmarks with CIFAR-10, CIFAR-100, and CINIC-10, and on a new setup of learning under input noise show that pFedGP achieves well-calibrated predictions while significantly outperforming baseline methods, reaching up to $21 \\%$ in accuracy gain. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "In recent years, there is a growing interest in applying learning in decentralized systems under the setup of federated learning (FL) [37, 51, 66]. In FL, a server node stores a global model and connects to multiple end-devices (“clients\"), which have private data that cannot be shared. The goal is to learn the global model in a communication-efficient manner. However, learning a single shared model across all clients may perform poorly when the data distribution varies significantly across clients. Personalized Federated Learning (PFL) [67] addresses this challenge by jointly learning a personalized model for each client. While significant progress had been made in recent years, leading approaches still struggle in realistic scenarios. First, when the amount of data per client is limited, even though this is one of the original motivations behind federated learning [4, 51, 72]. Second, when the input distribution shifts between clients, which is often the case, as clients use different devices and sensors. Last, when we require well-calibrated predictions, which is an important demand from medical and other safety-critical applications. ",
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+        "image_caption": [
+            "Figure 1: pFedGP - learning a shared deep kernel function with client-specific GP models. Each client stores private data, possibly from a different distribution. The data is first mapped to an embedding space with a shared neural network across all clients. Then, using common kernels a GP is applied to the data of the client for model learning and inference. We illustrate the per-client kernel matrix $k _ { \\theta } ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } )$ . Bold cells indicate a stronger covariance. "
+        ],
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+    {
+        "type": "text",
+        "text": "Here, we show how Gaussian Processes (GPs) with deep kernel learning (DKL) [80] is an effective alternative for handling these challenges. GPs have good predictive performance in a wide range of dataset sizes [2, 81], they are robust to input noise [75], can adapt to shifts in the data distribution [48], and provide well-calibrated predictions [69]. While regression tasks are more natural for GPs, here we focus on classification tasks for consistency with common benchmarks and learning procedures in the field; however, our approach is also applicable to regression tasks. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Consider a naive approach that fits a separate GP classifier to each client based on its personal data. Its performance heavily depends on the quality of the kernel, and standard kernels tend to work poorly in domains such as images. A popular solution to this problem is to use deep kernel learning (DKL) [80], where a kernel is applied to features outputted by a neural network (NN). Unfortunately, GPs with DKL can strongly overfit, often even worse than standard NNs [56], and thus negate the main benefit of using a GP. We solve this issue by jointly learning a shared kernel function across clients. As the kernel captures similarities between inputs, a single kernel should work well across clients, while using a separate GP per client will give the required flexibility for personalization. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We adapt a GP classifier recently proposed in [2] which uses the Pólya-Gamma augmentation [57] in a tree-structure model to the federated setting. We term our method pFedGP. We extend pFedGP by tailoring two inducing points (IPs) methods [58, 70]. The first helps generalization in the low data regime and, unlike common inducing point methods, does not reduce the computational costs. The second does focus on reducing the computational cost to make our approach scalable and work in low-resource clients. We also adjust previous PAC-Bayes generalization bounds for GPs [60, 64] to include the Pólya-Gamma augmentation scheme. These bounds are suitable for cases where the kernel is not learned, such as when new clients arrive after the shared NN was already learned. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Therefore, this paper makes the following contributions: (i) introduce pFedGP as a natural solution to PFL; (ii) develop two IP methods to enhance GP classifiers that use the Pólya-Gamma augmentation scheme and integrate them with pFedGP; (iii) derive a PAC-Bayes generalization bound on novel clients and show empirically that it gives meaningful guarantees; (iv) achieve state-of-the-art results in a wide array of experiments, improving accuracy by up to $2 1 \\%$ 1 . ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Federated learning. In FL, clients collaboratively solve a learning task while preserving data privacy and maintaining communication efficiency [1, 34, 42, 51, 54, 82, 85]. FedAvg [51] is an early but effective FL approach that updates models locally and averages them into a global model. Several optimization methods have been proposed for improving convergence in FL [41, 45, 71, 77]. Other approaches focus on preserving client privacy [3, 20, 52, 90], improving robustness to statistical diversity [26, 27, 31, 35, 87, 88], and reducing communication cost [13, 61]. These methods aim to learn a global model across clients, limiting their ability to deal with heterogeneous datasets. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Personalized federated learning. To overcome client heterogeneity, PFL aims to introduce some personalization for each client in the federation [39, 73]. Recent methods include adapting multitask learning [18, 67], meta-learning approaches [6, 21, 22, 33, 43, 89], and model mixing, where clients learn a mixture of the global and local models [4, 17, 27, 44]. Other approaches utilize different regularization schemes to enforce soft parameter sharing [32, 72]. Personalization in FL has also been explored through clustering approaches in which similar clients within the federation have a greater effect on one another [49, 86]. Recently, [65] proposed learning a central hypernetworks that acts on client representation vectors for generating personalized models. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Bayesian FL. Some studies put forward a Bayesian treatment to the FL setup. [8, 12] used variational inference with Bayesian NNs. [76, 84] proposed a matching algorithm between local models based on the Beta-Bernoulli process to construct a global model. [14] extended Bayesian optimization to FL setting via Thompson sampling. To scale the GP model they used random Fourier features. We use inducing points instead. [83] proposed a federated learning framework that uses a global GP model for regression tasks and without DKL. Unlike this study, we focus on classification tasks with a personal GP classifier per client and advocate sharing information between clients through the kernel. [74] used GPs in a client selection strategy. In [36] an approach based on stein variational gradient descent was suggested. This method does not scale beyond small-sized networks. [47] proposed a multivariate Gaussian product mechanism to aggregate local models. As we will show, this method is less suited when the data heterogeneity between clients is large. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Gaussian process classification. Unlike regression, in classification approximations must be used since the likelihood is not a Gaussian [59]. Classic approaches include the Laplace approximation [79], expectation-propagation [53], and least squares [62]. Recently, several methods were proposed based on the Pólya-Gamma augmentation [57] for modeling multinomial distributions [46], GP classification [23, 24, 78], few-shot learning [69], and incremental learning [2]. Here we build on the last approach. Classification with GPs is commonly done with variational inference techniques [30], here we wish to exploit the conjugacy of the model to take Gibbs samples from the posterior. This approach yields well calibrated [69] and more accurate models [2]. ",
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+        "type": "text",
+        "text": "3 Gaussian processes background ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first provide a brief introduction to the main components of our model. Detailed explanations are deferred to the Appendix. Scalars are denoted with lower-case letters (e.g., $x$ ), vectors with bold lowercase letters (e.g., x), and matrices with bold capital letters (e.g., X). In general, $\\mathbf { y } = [ y _ { 1 } , . . . , y _ { N } ] ^ { T }$ is the vector of labels, and $\\mathbf { X } \\in \\mathbb { R } ^ { N \\times d }$ is the design matrix with $N$ data points whose $i ^ { t h }$ row is $\\mathbf { x } _ { i }$ . ",
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+        "type": "text",
+        "text": "Gaussian processes. GPs map input points to target output values via a random latent function $f$ . $f$ is assumed to follow a Gaussian process prior $\\mathbf { \\bar { \\boldsymbol { f } } } \\sim \\mathcal { G P } ( m ( \\mathbf { x } ) , \\ k ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } ) )$ , where the evaluation vector of $f$ on $\\mathbf { X }$ , $\\mathbf { f } = [ f ( \\mathbf { x } _ { 1 } ) , . . . , f ( \\mathbf { \\bar { x } } _ { N } ) ] ^ { T }$ , has a Gaussian distribution f $\\sim \\mathcal { N } ( \\mu , \\kappa )$ with means $\\pmb { \\mu } _ { i } = m ( \\mathbf { x } _ { i } )$ and covariance $K _ { i j } = k ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } )$ . The mean $m ( \\mathbf { x } )$ is often set to be the constant zero function, and the kernel $k ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } )$ is a positive semi-definite function. The target values are assumed to be independent when conditioned on f . For Gaussian process regression the likelihood is Gaussian, $p ( y | f ) = \\operatorname { \\bar { \\mathcal { N } } } ( f , \\sigma ^ { 2 } )$ . Therefore, the posterior $p ( f | y , \\mathbf { X } )$ is also Gaussian, and both the marginal and the predictive distributions have known analytic expressions. This is one of the main motivations behind using GPs, as most other Bayesian models have intractable inference. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Unfortunately, for Gaussian process classification (GPC) the likelihood, $p ( y | f )$ , is not a Gaussian and the posterior does not admit a closed-form expression. One approach for applying GPs to binary classification tasks is the Pólya-Gamma augmentation [57]. Using this approach, we can augment the GP model with random variables $\\omega$ from a Pólya-Gamma distribution, one for each example. As a result, $p ( \\mathbf { f } | \\mathbf { y } , \\mathbf { X } , \\omega )$ is a Gaussian density and $p ( \\omega | \\mathbf { y } , \\mathbf { X } , \\mathbf { f } )$ is a Pólya-Gamma density. This allows to use Gibbs sampling to efficiently sample from the posterior $p ( \\mathbf { f } , \\omega | \\mathbf { y } , \\mathbf { X } )$ for inference and prediction. A key advantage of the Pólya-Gamma augmentation is that it benefits from fast mixing and has the ability of even a single value of $\\omega$ to capture much of the volume of the marginal distribution over function values [46]. Full equations and further details on the Pólya-Gamma augmentation scheme are given in Appendix A.1. ",
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+    {
+        "type": "text",
+        "text": "Deep kernel learning (DKL). The quality of the GP model heavily depends on the kernel function $k ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } )$ . For many data modalities, such as images, common kernels are not a good measure of semantic similarity. Therefore, in [10, 80] standard kernels are used over features outputted by a neural network In regression, it is p $g _ { \\boldsymbol { \\theta } }$ . For example, the RBF kernel ible to directly backpropagate th $\\begin{array} { r } { k _ { \\theta } ( \\mathbf { x } _ { i } , \\mathbf { x } _ { j } ) = \\exp \\left( - \\frac { | | g _ { \\theta } ( \\mathbf { x } _ { i } ) - g _ { \\theta } ( \\mathbf { x } _ { j } ) | | ^ { 2 } } { 2 \\ell ^ { 2 } } \\right) } \\end{array}$ closed-form. In our case, we use Fisher’s identity [19] to obtain stochastic gradients [69]. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Inducing points. GPs require storing and inverting a kernel matrix on the entire training set which often limits its usage. A common solution to this problem is to use inducing point methods [58, 70]. The key idea is to replace the exact kernel with an approximation for fast computation. Usually, $M \\ll N$ pseudo-inputs are learned such that the main computational bottleneck is in inverting $M \\times M$ matrices. ",
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+    {
+        "type": "text",
+        "text": "GP-Tree. We build on GP-Tree [2], a recent GP classifier that was shown to scale well with dataset size and the number of classes. GP-Tree turns the multi-class classification problem into a sequence of binary decisions along the tree nodes. Each node in the tree fits a binary GP classifier based on the Pólya-Gamma augmentation scheme and the data associated with that node. The leaf nodes correspond to the classes in the dataset. The tree is constructed by first computing a prototype for each class and then recursively performing divisive hierarchical clustering on these prototypes to two clusters at each node. Further details are given in Appendix A.2. ",
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+    {
+        "type": "text",
+        "text": "4 pFedGP: federated learning with Gaussian processes ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Now we describe our approach for applying personalized federated learning (PFL) with Gaussian processes. First, we extend GP-Tree to the FL setup and show how to use Gibbs sampling to learn the NN parameters. Then, we present two alternatives for this method that use inducing points. The first is for extremely limited-size datasets, while the second allows controlling the computational resources. We name our method pFedGP. An illustration of our method is given in Figure 1. ",
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+    {
+        "type": "text",
+        "text": "4.1 A full GP model ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The training procedure follows the standard protocol in this field [4, 44, 51]. We assume the existence of a server that holds the shared parameters $\\theta$ (a NN). Let $C$ denote the set of clients. For each client $c \\in C$ we denote by $D _ { c }$ its local dataset of size $N _ { c }$ . At each training iteration (round) the model is sent to $S$ clients to perform kernel learning $( | S | \\leq | C | )$ . Each client $c \\in S$ updates its copy of the global model and then sends the updated model to the server. The server then averages over the updates to obtain a new global model. ",
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+        "type": "text",
+        "text": "At each client $c$ , we perform kernel learning in the following manner. We first compute the feature representation of the data samples associated with the client using the shared network. Then, we build the hierarchical classification tree as discussed in Section $_ { 3 \\& }$ Appendix A.2. In [2] the tree was built only once after a pre-training stage and the model parameters were learned using a variational inference approach. Here, we re-build the tree at each round using the most recent features and we use a Gibbs sampling procedure, as it allows this flexibility in building the tree and performs better when not prohibitive by computational limitations. Learning the network parameters $\\theta$ with the Gibbs sampling approach can be done with two common objectives, the marginal likelihood, and the predictive distribution. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We denote by $\\mathbf { X } _ { v }$ the data associated with the tree node $v$ , i.e., the data points which have $v$ on the path from the root node to their class leaf node. We denote by $\\mathbf { y } _ { v }$ the binary label of these points, i.e., does their path go left or right at this node. And we denote by $\\omega _ { v }$ the Pólya-Gamma random variables associated with node $v$ . The marginal likelihood term for the full hierarchical classification tree is the sum of the separate marginal likelihood terms of all the nodes $v$ in the tree: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/3c2b2a654258aabc71355fbcc8a0aa981aa1fd8f37c7426abfd7cda309b2ba41.jpg",
+        "text": "$$\n\\mathcal { L } _ { c } ^ { M L } ( \\theta ; D _ { c } ) = \\sum _ { v } \\log \\ p _ { \\theta } ( \\mathbf { y } _ { v } | \\mathbf { X } _ { v } ) = \\sum _ { v } \\log \\ \\int p _ { \\theta } ( \\mathbf { y } _ { v } | \\omega _ { v } , \\mathbf { X } _ { v } ) p ( \\omega _ { v } ) d \\omega _ { v } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Similar to [69] we use a gradient estimator based on Fisher’s identity [19]: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eb696607128f26ececfd47d7d3c9167d75f82637f7725d960f8da3066ced8a83.jpg",
+        "text": "$$\n\\nabla _ { \\theta } \\mathcal { L } _ { c } ^ { M L } ( \\theta ; D _ { c } ) = \\sum _ { v } \\int p _ { \\theta } ( \\omega _ { v } | \\mathbf { y } _ { v } , \\mathbf { X } _ { v } ) \\nabla _ { \\theta } l o g \\ p _ { \\theta } ( \\mathbf { y } _ { v } | \\omega _ { v } , \\mathbf { X } _ { v } ) d \\omega _ { v } \\approx \\sum _ { v } \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } \\nabla _ { \\theta } l o g \\ p _ { \\theta } ( \\mathbf { y } _ { v } | \\omega _ { v } ^ { ( l ) } , \\mathbf { X } _ { v } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, (1)v , ..., ω(L)v ar e samples from the posterior at node $v$ . Due to the Pólya-Gamma augmentation $p _ { \\theta } \\big ( \\mathbf { y } _ { v } | \\omega _ { v } ^ { ( l ) } , \\mathbf { X } _ { v } \\big )$ is proportional to a Gaussian density. The exact expression is give in Appendix A.2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To use the predictive distribution as an objective, in each training iteration, after building the tree model, at each node we randomly draw a portion from the (node) training data and use it to predict the class label for the remaining part. We denote with $\\mathbf { X } _ { v }$ and $\\mathbf { y } _ { v }$ the training portion, $\\mathbf { x } _ { v } ^ { * }$ and $y _ { v } ^ { * }$ the input and the label of the point we are predicting, and $P ^ { y ^ { * } }$ the path from the root node to the $y ^ { * }$ leaf node (i.e., the original class). Here we also take advantage of the independence between nodes to maximize the predictive distribution per node individually. The predictive distribution for a single data point: ",
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+    {
+        "type": "equation",
+        "img_path": "images/5a80fae03b0b2559f3597f3f9a0d95687af4e373ac581cde56ec717a8f848ed0.jpg",
+        "text": "$$\n\\mathbf { \\Psi } _ { c } ^ { P D } ( \\theta ; \\mathbf { x } ^ { * } , y ^ { * } ) = \\sum _ { v \\in P ^ { y ^ { * } } } \\log \\ p \\varrho ( y _ { v } ^ { * } | \\mathbf { x } _ { v } ^ { * } , \\mathbf { y } _ { v } , \\mathbf { X } _ { v } ) = \\sum _ { v \\in P ^ { y ^ { * } } } \\log \\ \\int p \\varrho ( y _ { v } ^ { * } | \\omega _ { v } , \\mathbf { x } _ { v } ^ { * } , \\mathbf { y } _ { v } , \\mathbf { X } _ { v } ) p ( \\omega _ { v } | \\mathbf { y } _ { v } , \\mathbf { X } _ { v } ) d \\omega _ { v } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We use an approximate-gradient estimator based on posterior samples of $\\omega$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/04920a550765f61da726617b5a1c2a659f40a6f34ff17745c7ae5e267c9b4777.jpg",
+        "text": "$$\n\\nabla _ { \\theta } \\mathcal { L } _ { c } ^ { P D } ( \\theta ; \\mathbf { x } ^ { * } , y ^ { * } ) \\approx \\sum _ { v \\in P ^ { y ^ { * } } } \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } \\nabla _ { \\theta } l o g p _ { \\theta } ( y _ { v } ^ { * } | \\boldsymbol { \\omega } _ { v } ^ { ( l ) } , \\mathbf { x } _ { v } ^ { * } , \\mathbf { y } _ { v } , \\mathbf { X } _ { v } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Where $\\begin{array} { r } { p _ { \\theta } ( y _ { v } ^ { * } | \\omega _ { v } ^ { ( l ) } , \\mathbf { x } _ { v } ^ { * } , \\mathbf { y } _ { v } , \\mathbf { X } _ { v } ) = \\int p ( y _ { v } ^ { * } | f ^ { * } ) p _ { \\theta } ( f ^ { * } | \\omega _ { v } ^ { ( l ) } , \\mathbf { x } _ { v } ^ { * } , \\mathbf { y } _ { v } , \\mathbf { X } _ { v } ) d f ^ { * } \\ d \\mathbf { x } } \\end{array}$ does not have an analytical expression, but $\\begin{array} { r } { p _ { \\theta } \\big ( f ^ { * } | \\omega _ { v } ^ { ( l ) } , \\mathbf { x } _ { v } ^ { * } , \\mathbf { y } _ { v } , \\mathbf { X } _ { v } \\big ) = \\int p _ { \\theta } \\big ( f ^ { * } | \\mathbf { f } , \\mathbf { x } _ { v } ^ { * } , \\mathbf { X } _ { v } \\big ) p _ { \\theta } \\big ( \\mathbf { f } | \\omega _ { v } ^ { ( l ) } , \\mathbf { y } _ { v } , \\mathbf { X } _ { v } \\big ) d f ^ { * } } \\end{array}$ is Gaussian with known parameters. We then compute the predictive distribution by performing Gauss-Hermite integration over $f ^ { * }$ . See exact expression in Appendix A.2. ",
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+    {
+        "type": "text",
+        "text": "4.2 Augmenting the model with inducing points: sample efficiency ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The GP model described in Section 4.1 works well in most situations. However, when the number of data points per client is small, performance naturally degrades. To increase information sharing between clients and improve the per-client performance, we suggest augmenting the model with global inducing points shared across clients. When sending the model from the server to a client, we also send the inducing inputs and their labels. To streamline optimization and reduce the communication burden, we define the inducing inputs in the feature space of the last embedding layer of the shared NN. Therefore, usually, their size will be negligible compared to the network size. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We denote by $\\bar { \\bf X }$ the learned inducing inputs and by $\\bar { \\mathbf { y } }$ their fixed class labels. They are set evenly across classes. During training, we regard only the set of inducing inputs-labels $( { \\bar { \\mathbf { X } } } , { \\bar { \\mathbf { y } } } )$ as the available (training) data and use them for posterior inference. More formally, we first compute $\\begin{array} { r } { p _ { \\theta } ( \\mathbf { f } | \\boldsymbol { \\bar { \\omega } } , \\bar { \\mathbf { y } } , \\bar { \\mathbf { X } } , \\mathbf { X } ) \\overset {  } { = } \\int p _ { \\theta } ( \\mathbf { f } | \\bar { \\mathbf { f } } , \\bar { \\mathbf { X } } , \\mathbf { X } ) p _ { \\theta } ( \\bar { \\mathbf { f } } | \\boldsymbol { \\bar { \\omega } } , \\bar { \\mathbf { y } } , \\bar { \\mathbf { X } } ) d \\bar { \\mathbf { f } } } \\end{array}$ using its analytical expression for the actual training data and then compute the probability of $\\mathbf { y }$ using Gauss-Hermite integration. Then we use Eq. 3 & 4 for learning the network parameters and the inducing locations. At test time, to make full use of the training data, we combine the inducing inputs with the training data and use both to obtain the GP formulas and to make predictions. We note that with just using the inducing inputs at test time the model performs remarkably well, despite having almost no personalization component. See Appendix E.8 for a further discussion. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One potential issue with using IPs in this manner is that it distorts the true class distribution. As a result, the classifier may be more likely to predict a low-probability class during test time. We address this issue by adjusting the output distribution. In general, let $p ( y , \\mathbf x )$ and $q ( y , \\mathbf x )$ be two distributions that differ only in the class probabilities, i.e. $p ( \\mathbf { x } | y ) = q ( \\mathbf { x } | y )$ , the predictive distribution follows: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/55c9a4fec6d60a747922b220edb2a65a20067410432a3676bd4a984821a39542.jpg",
+        "text": "$$\n\\frac { q ( y ^ { * } | \\mathbf { x } ^ { * } ) } { p ( y ^ { * } | \\mathbf { x } ^ { * } ) } \\propto \\frac { q ( \\mathbf { x } ^ { * } | y ^ { * } ) q ( y ^ { * } ) } { p ( \\mathbf { x } ^ { * } | y ^ { * } ) p ( y ^ { * } ) } \\Longrightarrow q ( y ^ { * } | \\mathbf { x } ^ { * } ) \\propto \\frac { q ( y ^ { * } ) } { p ( y ^ { * } ) } p ( y ^ { * } | \\mathbf { x } ^ { * } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use this to correct the GP predictions to the original class ratios at each tree node. We found in our experiments that this correction generally improves the classifier performance for class imbalanced data. As an example for this phenomena, consider a binary classification problem having 90 examples from the first class and 10 examples from the second class (therefore, $q ( y = 0 ) = 0 . 9$ , and $q ( y = 1 ) = 0 . 1 \\AA$ . Assume we defined 50 inducing inputs per class, so now during test time the model sees 140 samples from the first class and 60 samples from the second class which corresponds to probabilities $p ( y = 0 ) = 0 . 7$ and $p ( y = 1 ) = 0 . 3$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.3 Augmenting the model with inducing points: computational efficiency ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Learning the full GP model described in Section 4.1 requires inverting a matrix of size $N _ { c }$ in the worst case (at the root node), which has $\\mathcal { O } ( N _ { c } ^ { 3 } )$ run-time complexity and $\\hat { \\mathcal { O } } ( N _ { c } ^ { 2 } )$ memory complexity. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 pFedGP. $C$ clients indexed by c; $E$ - number of local epochs; $| S |$ - number of sampled clients; $M$ - number of inducing inputs per class ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Server executes: ",
+        "text_level": 1,
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+        "text": "Initialize shared network $\\theta  \\theta _ { 0 }$ Initialize $M$ inducing inputs per class for all classes in the system # in pFedGP-IP variants only for each round $t  1 , 2 , \\dots \\mathbf { d }$ o: Sample $S$ clients uniformly at random for each client $c \\in S$ in parallel: $\\theta _ { t + 1 } ^ { c }$ ${ } _ { 1 } , M _ { t + 1 } ^ { c } \\gets$ ClientUpdate $( \\theta _ { t } , M _ { t } )$ # obtain updates from client c Update $\\theta _ { t + 1 } , M _ { t + 1 }$ using FedAvg [51] update rule. ClientUpdate $( \\theta , M )$ : for each local epoch $e \\gets 1 , . . . , E$ do: if $e = 1$ : Build GP tree classifier using the personal dataset $D _ { c }$ Update $\\theta , M$ using gradient-based optimization methods on $D _ { c }$ with $\\mathcal { L } ^ { M L }$ or $\\mathcal { L } ^ { P D }$ return $\\theta , M$ ",
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+        "type": "text",
+        "text": "Therefore, we propose an additional procedure based on inducing points to allow reduced complexity in low resource environments and scalability to larger dataset sizes. ",
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+        "text": "This variant is based on the fully independent training conditional (FITC) method [70]. The key idea is to cast all the dependence on the inducing points and assume independence between the latent function values given the inducing points. Here for brevity, we omit the subscripts denoting the client and the tree node. However, all quantities and data points are those that belong to a specific client and tree node. Let $\\bar { \\mathbf { X } } \\in \\mathbb { R } ^ { M \\times d }$ denote the pseudo-inputs (defined in the embedding space of the last layer of the NN), and $\\bar { \\mathbf { f } } \\in \\mathbb { R } ^ { M }$ the corresponding latent function values. Here as well, the inducing inputs are defined globally at the server level and they are set evenly across classes. We assume the following GP prior $p ( \\mathbf { f } , \\bar { \\mathbf { f } } ) = \\mathcal { N } \\Big ( \\mathbf { 0 } , \\big [ \\mathbf { K } _ { M N } \\quad \\mathbf { K } _ { N M } \\big ] \\Big )$ , where ${ \\bf K } _ { M M }$ is the kernel between the inducing inputs, ${ \\bf K } _ { N N }$ is a diagonal matrix between the actual training data, ${ \\bf K } _ { N M }$ is the kernel between the data and the inducing inputs, and we placed a zero mean prior. The likelihood of the dataset when factoring the inducing variables and the Pólya-Gamma variables (one per training sample), and the posterior over f, both have known analytical expressions. We can then obtain the posterior and marginal distributions by marginalizing over $\\bar { \\bf f }$ . Here we will present the posterior and marginal distributions: ",
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+        "type": "equation",
+        "img_path": "images/f15ddc4e9fb9dc1f1744e8dc666a30f3679dfe51543ca44adf54d2ac0d23ff4d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { p ( \\mathbf { f } | \\mathbf { X } , \\mathbf { y } , \\omega , \\bar { \\mathbf { X } } ) = \\mathcal { N } ( \\mathbf { f } | \\mathbf { K } _ { N M } \\mathbf { B } ^ { - 1 } \\mathbf { K } _ { M N } \\mathbf { A } ^ { - 1 } \\Omega ^ { - 1 } \\boldsymbol { \\kappa } , \\mathbf { K } _ { N N } - \\mathbf { K } _ { N M } ( \\mathbf { K } _ { M M } ^ { - 1 } - \\mathbf { B } ^ { - 1 } ) \\mathbf { K } _ { M N } ) , } \\\\ & { \\quad p ( \\mathbf { y } | \\mathbf { X } , \\omega , \\bar { \\mathbf { X } } ) \\propto \\mathcal { N } ( \\Omega ^ { - 1 } \\boldsymbol { \\kappa } | \\mathbf { 0 } , \\ \\Lambda + \\mathbf { K } _ { N M } \\mathbf { K } _ { M M } ^ { - 1 } \\mathbf { K } _ { M N } ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Where $\\pmb { \\Omega } = d i a g ( \\omega )$ , $\\kappa _ { j } = y _ { j } - 1 / 2$ , $\\pmb { \\Lambda } = \\pmb { \\Omega } ^ { - 1 } + d i a g ( \\mathbf { K } _ { N N } - \\mathbf { K } _ { N M } \\mathbf { K } _ { M M } ^ { - 1 } \\mathbf { K } _ { M N } )$ , and $\\mathbf { B } =$ $\\mathbf { K } _ { M M } + \\mathbf { K } _ { M N } \\mathbf { A } ^ { - 1 } \\mathbf { K } _ { N M }$ . Importantly, we only need to invert $M \\times M$ or diagonal matrices. See full derivation in Appendix B. During test time, we use $\\bar { \\bf f }$ to get the posterior of $f ^ { * }$ to compute the predictive distribution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Now we can use either the marginal or the predictive distribution to learn the shared NN parameters and the inducing locations. The complexity of applying this procedure is reduced to $\\mathcal { O } ( M ^ { \\frac { . } { 2 } } N _ { c } + M ^ { 3 } )$ in run-time, and $\\mathcal { O } ( M N _ { c } + M ^ { 2 } )$ in memory. While the (conditional) independence assumption between the latent function values may be restrictive, we found this method to be comparable with the full GP alternative in our experiments. Potentially, this can be attributed to the effect of sharing the inducing inputs among clients and the information that $\\omega$ stores on $\\mathbf { f }$ . ",
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+        "type": "text",
+        "text": "5 Generalization bound ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "It is reasonable to expect that after we learned the system new clients will arrive. In such cases, we would like to use pFedGP without re-training the kernel function. Under this scenario, we can derive generalization bounds concerning only the GP classifier without taking into account the fixed ",
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+        "type": "text",
+        "text": "neural network using PAC-Bayes bound [50]. Having meaningful guarantees can be very important in safety-critical applications. The PAC-Bayes bound for GPC [64] (with the Gibbs risk): ",
+        "bbox": [
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+        "type": "text",
+        "text": "Theorem 1. Given i.i.d. samples $D _ { c } = \\{ ( \\mathbf { x } _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N _ { c } }$ of size $N _ { c }$ drawn from any data distribution over $\\mathcal { X } \\times \\{ - 1 , 1 \\}$ , a posterior $Q$ , and a $G P$ prior , the following bound holds, where the probability is over random data samples: ",
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+        "img_path": "images/fbe22bf9a7cda8e1ea2b2d2a5105b4c2a44de3b632a47df34fd33bdb737a95c6.jpg",
+        "text": "$$\n{ \\cal P } r _ { D _ { c } } \\{ { \\cal R } ( Q ) > { \\cal R } _ { D _ { c } } ( Q ) + K L _ { b e r } ^ { - 1 } ( { \\cal R } _ { D _ { c } } ( Q ) , \\epsilon ( \\delta , n , P , Q ) ) \\} \\le \\delta .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here, we have, ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/107fe7a459e13dff93bd72a4f74aad0d47abe79f73429b882f4af6f7a7fd67ee.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\mathrm { \\normalfont { \\gtrsim } } ( Q ) = \\mathbb { E } _ { ( \\mathbf { x ^ { * } } , y ^ { * } ) } [ P r _ { f ^ { * } \\sim Q ( f ^ { * } \\mid \\mathbf { x ^ { * } } , D _ { c } ) } \\{ s i g n f ^ { * } \\neq y ^ { * } \\} ] , \\quad R _ { D _ { c } } ( Q ) = \\displaystyle \\frac { 1 } { N _ { c } } \\sum _ { i = 1 } ^ { N _ { c } } P r _ { f _ { i } \\sim Q ( f _ { i } \\mid D _ { c } ) } \\{ s i g n f _ { i } \\neq y _ { i } \\} } } \\\\ { { \\displaystyle ( \\delta , N _ { c } , P , Q ) = \\frac { 1 } { N _ { c } } \\Big ( K L [ Q \\mid | \\mathbf { \\nabla } P | + \\log \\frac { N _ { c } + 1 } { \\delta } \\Big ) , \\quad K L _ { b e r } ^ { - 1 } ( q , \\epsilon ) = m a x _ { p \\in [ 0 , 1 ] } K L _ { b e r } [ q \\mid | \\mathbf { \\nabla } p ] \\le \\epsilon } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "An important observation in [64] is that the KL-divergence between the posterior and prior Gaussian processes is equivalent to the KL-divergence between the posterior and prior distribution of their values on the $N _ { c }$ training samples. While [64] assumed $Q$ to be Gaussian, this observation still holds even without this assumption. However, when $Q$ is no longer Gaussian, as is the case here, $K L [ Q ( \\mathbf { f } ) \\mid \\mid P ( \\mathbf { f } ) ]$ no longer has a closed-form expression. We can show that for the Pólya-Gamma augmentation: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/1ccc413ab84e93cc84fe9aad206a4037b5d6ddf75b6db3d36bc781cd2b3b8804.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle K L [ Q ( { \\bf f } ) \\mathbf { \\nabla } | | \\mathbf { \\nabla } P ( { \\bf f } ) ] = \\mathbb { E } _ { Q ( \\omega ) } \\{ K L [ Q ( { \\bf f } | \\omega ) | | P ( { \\bf f } ) ] \\} - M I [ { \\bf f } ; \\omega ] } \\ ~ } \\\\ { { \\displaystyle ~ \\quad = \\mathbb { E } _ { Q ( \\omega ) } \\{ K L [ Q ( { \\bf f } | \\omega ) | | P ( { \\bf f } ) ] \\} + \\mathbb { E } _ { Q ( { \\bf f } , \\omega ) } \\left[ \\log \\frac { Q ( \\omega ) } { Q ( \\omega | { \\bf f } ) } \\right] } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where MI denotes the mutual information. Since $Q ( \\mathbf { f } \\vert \\boldsymbol { \\omega } )$ and $P ( \\mathbf { f } )$ are Gaussian, the $K L [ Q ( \\mathbf { f } | \\omega ) | | P ( \\mathbf { f } ) ]$ term has a close form expression so we only need to perform MonteCarlo approximation on the expectation on $\\omega$ on the first element. In the second expectation, $Q ( \\omega )$ does not have a known expression. To estimate it, given $\\{ ( \\omega _ { i } , \\mathbf { f } _ { i } ) \\} _ { i = 1 } ^ { N }$ samples, we use $\\begin{array} { r } { Q ( \\omega _ { i } ) \\approx \\frac { 1 } { N - 1 } \\sum _ { j \\neq i } Q ( \\omega _ { i } | \\mathbf { f } _ { j } ) . } \\end{array}$ . Note that if the summation for $j$ includes $\\mathbf { f } _ { i }$ , it might result in a biased estimator. Further details on estimating $K L [ Q ( \\mathbf { f } ) \\mid \\mid P ( \\mathbf { f } ) ]$ are in Appendix C. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To assess the quality of the bound, we partitioned the CIFAR-10 dataset to 100 clients. We trained a shared network using our full-GP variant on 90 clients and then recorded the generalization and test error on the remaining 10 clients four times, each with a different training set size. Figure 2 shows the estimation of the generalization error bound $\\tilde { \\delta } = 0 . 0 1 \\mathrm { \\Omega }$ vs the actual error on the novel clients with the Gibbs classifier. First, we observe that indeed the bound is greater than the actual test error for all points and that it is not vacuous. There is a strong correlation between the actual error and the bound. Secondly, unlike worst-case bounds (e.g. VC-dimension), this bound depends on the actual data and not only the number of data points. ",
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+    {
+        "type": "image",
+        "img_path": "images/9edfb836139c4fc8a311d92a715076ede5cc00bc3b202b4cc8243a02879bdfef.jpg",
+        "image_caption": [
+            "Figure 2: Test error vs an estimated upper bound over 10 clients with varying degrees of a training set data size. Each dot represents a combination of client and data size. In parenthesis - the average difference between the empirical and the test error. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluated pFedGP against baseline methods in various learning setups. We present the result for the following model variants: (i) pFedGP, the full GP model (Section 4.1); (ii) pFedGP-IP-data, the model with IPs described in Section 4.2; and (iii) pFedGP-IP-compute, the model with IPs described in Section 4.3. For pFedGP and pFedGP-IP-compute, the results obtained by maximizing the predictive and marginal likelihood were similar, with a slight advantage to the former. Therefore, we present here the results only for the predictive alternative and defer the results of the marginal alternative to the Appendix. Additional experiments, ablation study, and further analyses are provided in Appendix E. Unless stated otherwise, we report the average and the standard error of the mean (SEM) over three random seeds of the federated accuracy, defined as the average accuracy across all clients and samples. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/470e6af135062db9c3644dc52431a7d4229e7e39c4c1413f1aad34e1682b9858.jpg",
+        "table_caption": [
+            "Table 1: Test accuracy $\\pm$ SEM) over 50, 100, 500 clients on CIFAR-10, CIFAR-100, and CINIC-10. The # samples/client indicates the average number of training samples per client. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>#clients</td><td colspan=\"3\">CIFAR-10</td><td colspan=\"3\">CIFAR-100</td><td colspan=\"3\">CINIC-10</td></tr><tr><td># samples/client</td><td>50 800</td><td>100 400</td><td>500 80</td><td>50 800</td><td>100 400</td><td>500 80</td><td>50 1800</td><td>100 900</td><td>500 180</td></tr><tr><td>Local</td><td>86.2 ±0.2</td><td>82.9 ± 0.4</td><td>74.8 ± 0.5</td><td>52.1± 0.2</td><td>45.6 ± 0.3</td><td>30.9 ±0.2</td><td>61.1 ± 0.3</td><td>56.9 ± 0.7</td><td>46.4 ± 0.1</td></tr><tr><td>FedAvg [51]</td><td>56.4 ± 0.5</td><td>59.7 ± 0.5</td><td>54.0 ±0.5</td><td>23.6±0.2</td><td>24.0±0.2</td><td>20.4±0.0</td><td>45.6± 0.4</td><td>44.7 ± 0.5</td><td>45.7± 0.5</td></tr><tr><td>FOLA [47]</td><td>55.9± 3.3</td><td>52.1 ± 3.1</td><td>45.9±0.3</td><td>25.5 ± 1.5</td><td>22.4 ± 1.3</td><td>18.7 ± 0.1</td><td>45.2 ± 0.3</td><td>43.4±0.3</td><td>38.3±0.2</td></tr><tr><td>FedPer [4]</td><td>83.8±0.8</td><td>81.5 ± 0.5</td><td>76.8 ±1.2</td><td>48.3±0.6</td><td>43.6±0.2</td><td>25.6±0.3</td><td>70.6±0.2</td><td>68.4± 0.5</td><td>62.2± .05</td></tr><tr><td>LG-FedAvg [44]</td><td>87.9±0.3</td><td>83.6±0.7</td><td>64.7 ± 0.7</td><td>43.6±0.2</td><td>37.5 ± 0.9</td><td>20.3±0.5</td><td>59.5 ± 1.1</td><td>59.9 ± 2.1</td><td>52.5±0.8</td></tr><tr><td>pFedMe[72]</td><td>86.4±0.8</td><td>85.0±0.3</td><td>80.3±0.5</td><td>49.8 ± 0.5</td><td>47.7 ± 0.4</td><td>32.5±0.8</td><td>69.9 ± 0.5</td><td>68.9 ±0.7</td><td>58.8 ±0.1</td></tr><tr><td>FedU[18]</td><td>80.6±0.3</td><td>78.1 ± 0.5</td><td>65.6±0.4</td><td>41.1 ± 0.2</td><td>36.0±0.2</td><td>15.9 ± 0.4</td><td>59.3 ± 0.2</td><td>55.4±0.6</td><td>41.6± 0.5</td></tr><tr><td>pFedHN [65]</td><td>90.2 ±0.6</td><td>87.4±0.2</td><td>83.2 ±0.8</td><td>60.0 ±1.0</td><td>52.3±0.5</td><td>34.1 ±0.1</td><td>70.4 ±0.4</td><td>69.4± 0.5</td><td>64.2±.05</td></tr><tr><td>Ours</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>pFedGP-IP-data</td><td>88.6±0.2 87.4±0.2</td><td></td><td>86.9±0.7</td><td>60.2±0.3</td><td>58.5±0.3</td><td>55.7±0.4</td><td>69.8±0.2</td><td>68.3 ± 0.6 67.6± 0.3</td><td></td></tr><tr><td>pFedGP-IP-compute</td><td>89.9±0.6 88.8±0.1</td><td></td><td>86.8 ±0.4</td><td>61.2 ± 0.4</td><td>59.8±0.3</td><td>49.2 ± 0.3</td><td>72.0 ±0.3</td><td>71.5± 0.5</td><td>68.2±0.2</td></tr><tr><td>pFedGP</td><td>89.2±0.3</td><td>88.8 ±0.2</td><td>87.6± 0.4</td><td>63.3± 0.1</td><td>61.3 ± 0.2</td><td>50.6±0.2</td><td>71.8 ± 0.3</td><td>71.3 ± 0.4</td><td>68.1 ±0.3</td></tr></table>",
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+        "type": "text",
+        "text": "Datasets. All methods were evaluated on CIFAR-10, CIFAR-100 [38], and CINIC-10 [15] datasets.   \nCINIC-10 is more diverse since it combines images from CIFAR-10 and ImageNet [16]. ",
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+    {
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+        "text": "Compared methods. We compared our method against the following baselines: (1) Local, pFedGP full model on each client with a private network and no collaboration with other clients; (2) FedAvg [51], a standard FL model with no personalization component; (3) FOLA [47], a Bayesian method that used a multivariate Gaussian product mechanism to aggregate local models; (4) FedPer [4], a PFL approach that learns a personal classifier for each client on top of a shared feature extractor; (5) LG-FedAvg [44], a PFL method that uses local feature extractor per client and global output layers; (6) pFedMe [72], a PFL method which adds a Moreau-envelopes loss term; (7) FedU [18], a recent multi-task learning approach for PFL that learns a model per client; (8) pFedHN [65], a recent PFL approach that uses a hypernetwork to generate client-specific networks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Training protocol. We follow the training strategy proposed in [65]. We limit the training process to 1000 communication rounds, in each we sample five clients uniformly at random for model updates. The training procedure is different in the FOLA and pFedHN baselines, so we used an equivalent communication cost. In LG-FedAvg, we made an extra 200 communication rounds after a pre-training stage with the FedAvg model for 1000 communication rounds. In the local model, we performed 100 epochs of training for each client. In all experiments, we used a LeNet-based network [40] having two convolution layers followed by two fully connected layers and an additional linear layer. We tuned the hyperparameters of all methods using a pre-allocated held-out validation set. Full experimental details are given in Appendix D. ",
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+    {
+        "type": "text",
+        "text": "6.1 Standard PFL setting ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first evaluated all methods in a standard PFL setting [65, 72]. We varied the total number of clients in the system from 50 to 500 and we set the number of classes per client to two/ten/four for CIFAR-10/CIFAR-100/CINIC-10 respectively. Since the total number of samples in the system is fixed, the number of samples per client changed accordingly. For each client, the same classes appeared in the training and test set. ",
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+        "type": "text",
+        "text": "The results are presented in Table 1. They show that: (1) The performance of the local baseline is significantly impaired when the number of samples per client decreases, emphasizing the importance of federated learning in the presence of limited local data. (2) FedAvg and FOLA, which do not use personalized FL, perform poorly in this heterogeneous setup. (3) pFedGP outperforms or is on par with previous state-of-the-art approaches when local data is sufficient (e.g., 50 clients on all datasets). When the data per client becomes limited, pFedGP achieves significant improvements over competing methods; note the $9 \\%$ and $2 1 \\%$ difference in CIFAR-100 over 100 and 500 clients, respectively. (4) pFedGP-IP-compute often achieves comparable results to pFedGP and is often superior to pFedGP-IP-data. We believe that it can be attributed to the fact that in pFedGP-IP-compute the training data take an active part in the GP inference formulas (Eq. 6), while in pFedGP-IP-data the data impact in a weak manner only through the loss function. (5) pFedGP-IP-data is especially helpful when few samples per class are available, e.g., CIFAR-100 with 500 clients. That last point is further illustrated by decoupling the effect of the number of clients from that of the training set size. To illustrate that, in Appendix E.2 we fixed the number of clients and varied the number of training samples per class. From this experiment, we deduced that both factors (individually) contribute to pFedGP success. ",
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+        "image_caption": [
+            "Figure 3: Reliability diagrams on CIFAR-100 with 50 clients. Diagonal indicates perfect calibration. Each plot also shows the expected & maximum calibration error (ECE & MCE) and the Brier Score (BRI). Lower is better. "
+        ],
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+        "text": "A desired property from PFL classifiers is the ability to provide uncertainty estimation. For example, in decision support systems, such as in healthcare applications, the decision-maker should have an accurate estimation of the classifier confidence in the prediction. Here, we quantify the uncertainty through calibration. Figure 3 compares all methods both visually and using common metrics [7, 25, 55] on the CIFAR-100 dataset with 50 clients. Expected calibration error (ECE) measures the weighted average between the classifier confidence and accuracy. Maximum calibration error (MCE) takes the maximum instead of the average. And, Brier score (BRI) [7] measures the average squared error between the labels and the prediction prob",
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+        "table_caption": [
+            "Table 2: Test accuracy ( $\\pm$ SEM) over 100 clients on noisy CIFAR-100. We also provide the relative accuracy decrease $( \\% )$ w.r.t. the performance on the original CIFAR-100 data (see Table 1). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Accuracy</td><td>Decrease (%)</td></tr><tr><td>FedPer [4]</td><td>28.1 ± 0.9</td><td>-35.6</td></tr><tr><td>LG-FedAvg [44]</td><td>26.9 ± 0.9</td><td>-28.3</td></tr><tr><td>pFedme [72]</td><td>33.2 ± 0.6</td><td>-30.4</td></tr><tr><td>FedU[18]</td><td>35.0 ± 0.2</td><td>-2.8</td></tr><tr><td>pFedHN[65]</td><td>38.9 ± 0.5</td><td>-25.7</td></tr><tr><td>Ours</td><td></td><td></td></tr><tr><td>pFedGP-IP-data</td><td>45.0± 0.3</td><td>-23.1</td></tr><tr><td>pFedGP-IP-compute</td><td>47.1 ± .05</td><td>-21.2</td></tr><tr><td>pFedGP</td><td>49.5 ± 0.1</td><td>-19.2</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "abilities. The figure shows that pFedGP classifiers are best calibrated across all metrics in almost all cases. We note that with temperature scaling, the calibration of the baseline methods can be improved [25]; however, choosing the right temperature requires optimization over a separate validation set, which our model does not need. Additional calibration results, including temperature scaling, are presented in Appendix E.9. ",
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+    {
+        "type": "text",
+        "text": "6.2 PFL with input noise ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In real-world federated systems, the clients may employ different measurement devices for data collection (cameras, sensors, etc.), resulting in different input noise characteristics per client. Here, we investigate pFedGP performance in this type of personalization. To simulate that, we partitioned CIFAR-10/100 to 100 clients similar to the protocol described in Section 6.1, we defined 57 unique distributions of image corruption noise [29], and we assigned a noise model to each client. Then for each example in each client, we sampled a corruption noise according to the noise model allocated to that client. Here we show the results for the noisy CIFAR-100 dataset in Table 2. Further details on the perturbations performed and result for the noisy CIFAR-10 are given in the Appendix2. We observe a significant gap in favor of the pFedGP variants compared to baseline methods. Note that using global inducing points is slightly less beneficial in this case since they are defined globally and therefore are not tied to a specific noise type as the real client data is. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "6.3 Generalization to out-of-distribution (OOD) novel clients ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "FL are dynamic systems. For example, novel clients may enter the system after the model was trained, possibly with a data distribution shift. Adapting to a new OOD client is both challenging and important for real-world FL systems. To evaluate pFedGP in this scenario, we followed the learning protocol proposed in [65]. We partitioned the CIFAR-10 dataset into two groups. The data in the first group was distributed between 90 clients for model training. The remaining data from the second group was distributed between an additional 10 clients that were excluded during training. Within each group, we set the class probabilities in each client by sam",
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+        "type": "image",
+        "img_path": "images/8e392c0adfd99111f0d066660fe0fb5834a68cb130b237f21eca7eb8685cd8b0.jpg",
+        "image_caption": [
+            "Figure 4: Generalization to novel clients on CIFAR-10. "
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+    {
+        "type": "text",
+        "text": "pling from a Dirichlet distribution with the same $\\alpha$ parameter. For the training group, we set $\\alpha = 0 . 1$ , trained the shared model using these clients, and froze it. Then, we evaluated the models on the second group by varying $\\alpha \\in \\{ . 1 , . 2 5 , . 5 , . 7 5 , 1 \\}$ , on the remaining 10 clients. As $\\alpha$ moves away from 0.1 the distribution shift between the two groups increases, resulting in more challenging OOD clients. Figure 4 reports the generalization gap as a function of the Dirichlet parameter $\\alpha$ . The generalization gap is computed by taking the difference between the average test accuracy of the (ten) novel clients and the average test accuracy of the (ninety) clients used for training. From the figure, here as well, pFedGP achieves the best generalization performance for all values of $\\alpha$ . Moreover, unlike baseline methods, pFedGP does not require any parameter tuning. Several baselines were excluded from the figure since they had a large generalization gap. ",
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+        "type": "text",
+        "text": "7 Conclusion ",
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+        "type": "text",
+        "text": "In this study, we proposed pFedGP, a novel method for PFL. pFedGP learns a kernel function, parameterized by a NN, that is shared between all clients using a personal GP classifier on each client. We proposed three variants for pFedGP, a full model approach that generally shows the best performance and two extensions to it. The first is most beneficial when the number of examples per class are small while the second allows controlling the computational requirements of the model. We also derived PAC-Bayes generalization bound on novel clients and empirically showed that it gives non-vacuous guarantees. pFedGP provides well-calibrated predictions, generalizes well to OOD novel clients, and consistently outperforms competing methods. ",
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+        "type": "text",
+        "text": "Broader impact: Our method shares the standard communication procedure of FL approaches, where no private data is directly communicated across the different nodes in the system. This protocol does not explicitly guarantee that no private information can be inferred at this time. As we show, pFedGP is particularly useful for clients with little data, and for clients that have strongly different distribution. This has great potential to improve client personalization in real-world systems, and do better at handling less common data. The latter is of great interest for decision support systems in sensitive domains such as health care or legal. ",
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+        "type": "text",
+        "text": "Acknowledgements ",
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+        "type": "text",
+        "text": "This study was funded by a grant to GC from the Israel Science Foundation (ISF 737/2018), and by an equipment grant to GC and Bar-Ilan University from the Israel Science Foundation (ISF 2332/18). IA was funded by a grant from the Israeli innovation authority, through the AVATAR consortium. ",
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+        "text": "References   \n[1] Sawsan Abdulrahman, Hanine Tout, Hakima Ould-Slimane, Azzam Mourad, Chamseddine Talhi, and Mohsen Guizani. A survey on federated learning: The journey from centralized to distributed on-site learning and beyond. IEEE Internet of Things Journal, 8(7):5476–5497, 2021.   \n[2] Idan Achituve, Aviv Navon, Yochai Yemini, Gal Chechik, and Ethan Fetaya. GP-Tree: A Gaussian process classifier for few-shot incremental learning. In Proceedings of the 38th International Conference on Machine Learning, pages 54–65. PMLR, 2021.   \n[3] Naman Agarwal, Ananda Theertha Suresh, Felix Xinnan X Yu, Sanjiv Kumar, and Brendan McMahan. cpSGD: Communication-efficient and differentially-private distributed SGD. In Advances in Neural Information Processing Systems, pages 7564–7575, 2018.   \n[4] Manoj Ghuhan Arivazhagan, Vinay Aggarwal, Aaditya Kumar Singh, and Sunav Choudhary. Federated learning with personalization layers. arXiv preprint arXiv:1912.00818, 2019.   \n[5] David Arthur and Sergei Vassilvitskii. k-means $^ { + + }$ the advantages of careful seeding. In Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, pages 1027–1035, 2007.   \n[6] Harkirat Singh Behl, Atılım Güne¸s Baydin, and Philip HS Torr. Alpha MAML: Adaptive model-agnostic meta-learning. arXiv preprint arXiv:1905.07435, 2019.   \n[7] Glenn W Brier. Verification of forecasts expressed in terms of probability. Monthly weather review, 78(1):1–3, 1950.   \n[8] Thang D Bui, Cuong V Nguyen, Siddharth Swaroop, and Richard E Turner. Partitioned variational inference: A unified framework encompassing federated and continual learning. arXiv preprint arXiv:1811.11206, 2018.   \n[9] Dongqi Cai, Qipeng Wang, Yuanqiang Liu, Yunxin Liu, Shangguang Wang, and Mengwei Xu. Towards ubiquitous learning: A first measurement of on-device training performance. In Proceedings of the 5th International Workshop on Embedded and Mobile Deep Learning, pages 31–36, 2021.   \n[10] Roberto Calandra, Jan Peters, Carl Edward Rasmussen, and Marc Peter Deisenroth. Manifold Gaussian processes for regression. In 2016 International Joint Conference on Neural Networks (IJCNN), pages 3338–3345. IEEE, 2016.   \n[11] Shuxiao Chen, Qinqing Zheng, Qi Long, and Weijie J Su. A theorem of the alternative for personalized federated learning. arXiv preprint arXiv:2103.01901, 2021.   \n[12] Luca Corinzia, Ami Beuret, and Joachim M Buhmann. Variational federated multi-task learning. arXiv preprint arXiv:1906.06268, 2019.   \n[13] Xinyan Dai, Xiao Yan, Kaiwen Zhou, Han Yang, Kelvin KW Ng, James Cheng, and Yu Fan. Hyper-sphere quantization: Communication-efficient SGD for federated learning. arXiv preprint arXiv:1911.04655, 2019.   \n[14] Zhongxiang Dai, Bryan Kian Hsiang Low, and Patrick Jaillet. Federated Bayesian optimization via Thompson sampling. Advances in Neural Information Processing Systems, 33, 2020.   \n[15] Luke N Darlow, Elliot J Crowley, Antreas Antoniou, and Amos J Storkey. CINIC-10 is not Imagenet or CIFAR-10. arXiv preprint arXiv:1810.03505, 2018.   \n[16] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A largescale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.   \n[17] Yuyang Deng, Mohammad Mahdi Kamani, and Mehrdad Mahdavi. Adaptive personalized federated learning. arXiv preprint arXiv:2003.13461, 2020.   \n[18] Canh T Dinh, Tung T Vu, Nguyen H Tran, Minh N Dao, and Hongyu Zhang. FedU: A unified framework for federated multi-task learning with Laplacian regularization. arXiv preprint arXiv:2102.07148, 2021.   \n[19] Randal Douc, Eric Moulines, and David Stoffer. Nonlinear time series: Theory, methods and applications with R examples. CRC press, 2014.   \n[20] John C Duchi, Michael I Jordan, and Martin J Wainwright. Privacy aware learning. Journal of the ACM (JACM), 61(6):1–57, 2014.   \n[21] Alireza Fallah, Aryan Mokhtari, and A. Ozdaglar. Personalized federated learning: A metalearning approach. Advances in Neural Information Processing Systems, 2020.   \n[22] Alireza Fallah, Aryan Mokhtari, and Asuman Ozdaglar. On the convergence theory of gradientbased model-agnostic meta-learning algorithms. In International Conference on Artificial Intelligence and Statistics, pages 1082–1092. PMLR, 2020.   \n[23] Théo Galy-Fajou, Florian Wenzel, Christian Donner, and Manfred Opper. Multi-class Gaussian process classification made conjugate: Efficient inference via data augmentation. In Uncertainty in Artificial Intelligence, pages 755–765. PMLR, 2020.   \n[24] Théo Galy-Fajou, Florian Wenzel, and Manfred Opper. Automated augmented conjugate inference for non-conjugate gaussian process models. In International Conference on Artificial Intelligence and Statistics, pages 3025–3035. PMLR, 2020.   \n[25] Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In International Conference on Machine Learning, pages 1321–1330. PMLR, 2017.   \n[26] Farzin Haddadpour and Mehrdad Mahdavi. On the convergence of local descent methods in federated learning. arXiv preprint arXiv:1910.14425, 2019.   \n[27] Filip Hanzely and Peter Richtárik. Federated learning of a mixture of global and local models. arXiv preprint arXiv:2002.05516, 2020.   \n[28] 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, pages 770–778, 2016.   \n[29] Dan Hendrycks and Thomas Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. In International Conference on Learning Representations, 2019.   \n[30] James Hensman, Alexander Matthews, and Zoubin Ghahramani. Scalable variational Gaussian process classification. In Artificial Intelligence and Statistics, pages 351–360. PMLR, 2015.   \n[31] T. H. Hsu, Hang Qi, and M. Brown. Measuring the effects of non-identical data distribution for federated visual classification. ArXiv, abs/1909.06335, 2019.   \n[32] Yutao Huang, Lingyang Chu, Zirui Zhou, Lanjun Wang, Jiangchuan Liu, Jian Pei, and Yong Zhang. Personalized cross-silo federated learning on non-IID data. In Proceedings of the AAAI Conference on Artificial Intelligence, 2021.   \n[33] Yihan Jiang, Jakub Konecnˇ y, Keith Rush, and Sreeram Kannan. Improving federated learning \\` personalization via model agnostic meta learning. arXiv preprint arXiv:1909.12488, 2019.   \n[34] Peter Kairouz, H Brendan McMahan, Brendan Avent, Aurélien Bellet, Mehdi Bennis, Arjun Nitin Bhagoji, Keith Bonawitz, Zachary Charles, Graham Cormode, Rachel Cummings, et al. Advances and open problems in federated learning. arXiv preprint arXiv:1912.04977, 2019.   \n[35] Sai Praneeth Karimireddy, Satyen Kale, Mehryar Mohri, Sashank Reddi, Sebastian Stich, and Ananda Theertha Suresh. SCAFFOLD: Stochastic controlled averaging for federated learning. In International Conference on Machine Learning, pages 5132–5143. PMLR, 2020.   \n[36] Rahif Kassab and Osvaldo Simeone. Federated generalized Bayesian learning via distributed Stein variational gradient descent. arXiv preprint arXiv:2009.06419, 2020.   \n[37] Jakub Konecnˇ y, H Brendan McMahan, Daniel Ramage, and Peter Richtárik. Federated optimiza- \\` tion: Distributed machine learning for on-device intelligence. arXiv preprint arXiv:1610.02527, 2016.   \n[38] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.   \n[39] V. Kulkarni, Milind Kulkarni, and A. Pant. Survey of personalization techniques for federated learning. 2020 Fourth World Conference on Smart Trends in Systems, Security and Sustainability (WorldS4), pages 794–797, 2020.   \n[40] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   \n[41] Tian Li, Anit Kumar Sahu, Maziar Sanjabi, Manzil Zaheer, Ameet Talwalkar, and Virginia Smith. On the convergence of federated optimization in heterogeneous networks. arXiv preprint arXiv:1812.06127, 2018.   \n[42] Tian Li, Anit Kumar Sahu, Ameet Talwalkar, and Virginia Smith. Federated learning: Challenges, methods, and future directions. IEEE Signal Processing Magazine, 37(3):50–60, 2020.   \n[43] Zhenguo Li, Fengwei Zhou, Fei Chen, and Hang Li. Meta-SGD: Learning to learn quickly for few-shot learning. arXiv preprint arXiv:1707.09835, 2017.   \n[44] Paul Pu Liang, Terrance Liu, Liu Ziyin, Nicholas B Allen, Randy P Auerbach, David Brent, Ruslan Salakhutdinov, and Louis-Philippe Morency. Think locally, act globally: Federated learning with local and global representations. arXiv preprint arXiv:2001.01523, 2020.   \n[45] Tao Lin, Sebastian U Stich, Kumar Kshitij Patel, and Martin Jaggi. Don’t use large mini-batches, use local SGD. In International Conference on Learning Representations, 2019.   \n[46] Scott W Linderman, Matthew J Johnson, and Ryan P Adams. Dependent multinomial models made easy: stick breaking with the Pólya-Gamma augmentation. In Proceedings of the 28th International Conference on Neural Information Processing Systems, pages 3456–3464, 2015.   \n[47] Liangxi Liu and Feng Zheng. A Bayesian federated learning framework with multivariate gaussian product. arXiv preprint arXiv:2102.01936, 2021.   \n[48] Wesley Maddox, Shuai Tang, Pablo Moreno, Andrew Gordon Wilson, and Andreas Damianou. Fast adaptation with linearized neural networks. In International Conference on Artificial Intelligence and Statistics, pages 2737–2745. PMLR, 2021.   \n[49] Y. Mansour, M. Mohri, J. Ro, and A. T. Suresh. Three approaches for personalization with applications to federated learning. ArXiv, abs/2002.10619, 2020.   \n[50] David A McAllester. PAC-Bayesian stochastic model selection. Machine Learning, 51(1):5–21, 2003.   \n[51] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas. Communication-efficient learning of deep networks from decentralized data. In Artificial Intelligence and Statistics, pages 1273–1282. PMLR, 2017.   \n[52] H Brendan McMahan, Daniel Ramage, Kunal Talwar, and Li Zhang. Learning differentially private recurrent language models. In International Conference on Learning Representations, 2018.   \n[53] Thomas Peter Minka. A family of algorithms for approximate Bayesian inference. PhD thesis, Massachusetts Institute of Technology, 2001.   \n[54] Viraaji Mothukuri, Reza M Parizi, Seyedamin Pouriyeh, Yan Huang, Ali Dehghantanha, and Gautam Srivastava. A survey on security and privacy of federated learning. Future Generation Computer Systems, 115:619–640, 2021.   \n[55] Mahdi Pakdaman Naeini, Gregory Cooper, and Milos Hauskrecht. Obtaining well calibrated probabilities using Bayesian binning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 29, 2015.   \n[56] Sebastian W. Ober, Carl E. Rasmussen, and Mark van der Wilk. The promises and pitfalls of deep kernel learning. CoRR, 2021.   \n[57] Nicholas G. Polson, James G. Scott, and Jesse Windle. Bayesian inference for logistic models using Pólya–Gamma latent variables. Journal of the American Statistical Association, pages 1339–1349, 2013.   \n[58] Joaquin Quinonero-Candela and Carl Edward Rasmussen. A unifying view of sparse approximate Gaussian process regression. The Journal of Machine Learning Research, 6:1939–1959, 2005.   \n[59] Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian Processes for Machine Learning. The MIT Press, 2006.   \n[60] David Reeb, Andreas Doerr, Sebastian Gerwinn, and Barbara Rakitsch. Learning Gaussian processes by minimizing PAC-Bayesian generalization bounds. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 3341–3351, 2018.   \n[61] Amirhossein Reisizadeh, Aryan Mokhtari, Hamed Hassani, Ali Jadbabaie, and Ramtin Pedarsani. FedPAQ: A communication-efficient federated learning method with periodic averaging and quantization. In International Conference on Artificial Intelligence and Statistics, pages 2021– 2031. PMLR, 2020.   \n[62] Ryan Rifkin and Aldebaro Klautau. In defense of one-vs-all classification. The Journal of Machine Learning Research, 5:101–141, 2004.   \n[63] Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4510–4520, 2018.   \n[64] Matthias Seeger. PAC-Bayesian generalisation error bounds for Gaussian process classification. Journal of machine learning research, 3(Oct):233–269, 2002.   \n[65] Aviv Shamsian, Aviv Navon, Ethan Fetaya, and Gal Chechik. Personalized federated learning using hypernetworks. In International Conference on Machine Learning. PMLR, 2021.   \n[66] Reza Shokri and Vitaly Shmatikov. Privacy-preserving deep learning. In Proceedings of the 22nd ACM SIGSAC conference on computer and communications security, pages 1310–1321, 2015.   \n[67] Virginia Smith, Chao-Kai Chiang, Maziar Sanjabi, and Ameet Talwalkar. Federated multi-task learning. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 4427–4437, 2017.   \n[68] Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 4080–4090, 2017.   \n[69] Jake Snell and Richard Zemel. Bayesian few-shot classification with one-vs-each Pólya-Gamma augmented Gaussian processes. In International Conference on Learning Representations, 2021.   \n[70] Edward Snelson and Zoubin Ghahramani. Sparse Gaussian processes using pseudo-inputs. In Advances in Neural Information Processing Systems, pages 1257–1264. MIT Press, 2006.   \n[71] Sebastian U Stich. Local SGD converges fast and communicates little. In International Conference on Learning Representations, 2018.   \n[72] Canh T Dinh, Nguyen Tran, and Tuan Dung Nguyen. Personalized federated learning with Moreau envelopes. Advances in Neural Information Processing Systems, 33, 2020.   \n[73] Alysa Ziying Tan, Han Yu, Lizhen Cui, and Qiang Yang. Towards personalized federated learning. arXiv preprint arXiv:2103.00710, 2021.   \n[74] Minxue Tang, Xuefei Ning, Yitu Wang, Yu Wang, and Yiran Chen. Fedgp: Correlation-based active client selection for heterogeneous federated learning. arXiv preprint arXiv:2103.13822, 2021.   \n[75] Carlos Villacampa-Calvo, Bryan Zaldívar, Eduardo C Garrido-Merchán, and Daniel HernándezLobato. Multi-class Gaussian process classification with noisy inputs. Journal of Machine Learning Research, 22(36):1–52, 2021.   \n[76] Hongyi Wang, Mikhail Yurochkin, Yuekai Sun, Dimitris Papailiopoulos, and Yasaman Khazaeni. Federated learning with matched averaging. In International Conference on Learning Representations, 2019.   \n[77] Jianyu Wang and Gauri Joshi. Cooperative SGD: A unified framework for the design and analysis of communication-efficient SGD algorithms. In ICML Workshop on Coding Theory for Machine Learning, 2019.   \n[78] Florian Wenzel, Théo Galy-Fajou, Christian Donner, Marius Kloft, and Manfred Opper. Efficient Gaussian process classification using Pólya-Gamma data augmentation. In The AAAI Conference on Artificial Intelligence, pages 5417–5424. AAAI Press, 2019.   \n[79] Christopher KI Williams and David Barber. Bayesian classification with Gaussian processes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(12):1342–1351, 1998.   \n[80] Andrew Gordon Wilson, Zhiting Hu, Ruslan Salakhutdinov, and Eric P. Xing. Deep kernel learning. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2016.   \n[81] Andrew Gordon Wilson, Zhiting Hu, Ruslan Salakhutdinov, and Eric P Xing. Stochastic variational deep kernel learning. In Proceedings of the 30th International Conference on Neural Information Processing Systems, pages 2594–2602, 2016.   \n[82] Qiang Yang, Yang Liu, Tianjian Chen, and Yongxin Tong. Federated machine learning: Concept and applications. ACM Transactions on Intelligent Systems and Technology (TIST), 10(2):1–19, 2019.   \n[83] Feng Yin, Zhidi Lin, Qinglei Kong, Yue Xu, Deshi Li, Sergios Theodoridis, and Shuguang Robert Cui. FedLoc: Federated learning framework for data-driven cooperative localization and location data processing. IEEE Open Journal of Signal Processing, 1:187–215, 2020.   \n[84] Mikhail Yurochkin, Mayank Agarwal, Soumya Ghosh, Kristjan Greenewald, Nghia Hoang, and Yasaman Khazaeni. Bayesian nonparametric federated learning of neural networks. In International Conference on Machine Learning, pages 7252–7261. PMLR, 2019.   \n[85] Chen Zhang, Yu Xie, Hang Bai, Bin Yu, Weihong Li, and Yuan Gao. A survey on federated learning. Knowledge-Based Systems, 216:106775, 2021.   \n[86] Michael Zhang, Karan Sapra, Sanja Fidler, Serena Yeung, and Jose M Alvarez. Personalized federated learning with first order model optimization. arXiv preprint arXiv:2012.08565, 2020.   \n[87] Yue Zhao, Meng Li, Liangzhen Lai, Naveen Suda, Damon Civin, and Vikas Chandra. Federated learning with non-IID data. arXiv preprint arXiv:1806.00582, 2018.   \n[88] Fan Zhou and Guojing Cong. On the convergence properties of a K-step averaging stochastic gradient descent algorithm for nonconvex optimization. In Proceedings of the 27th International Joint Conference on Artificial Intelligence, pages 3219–3227, 2018.   \n[89] Pan Zhou, Xiaotong Yuan, Huan Xu, Shuicheng Yan, and Jiashi Feng. Efficient meta learning via minibatch proximal update. Advances in Neural Information Processing Systems, 32:1534–1544, 2019.   \n[90] Wennan Zhu, Peter Kairouz, Brendan McMahan, Haicheng Sun, and Wei Li. Federated heavy hitters discovery with differential privacy. In International Conference on Artificial Intelligence and Statistics, pages 3837–3847. PMLR, 2020. ",
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parse/train/oioB7Te7Bo/oioB7Te7Bo.md

@@ -0,0 +1,416 @@
+# The Values Encoded in Machine Learning Research
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Machine learning (ML) currently exerts an outsized influence on the world, in  
+2 creasingly affecting communities and institutional practices. It is therefore critical   
+3 that we question vague conceptions of the field as value-neutral or universally   
+4 beneficial, and investigate what specific values the field is advancing. In this paper,   
+5 we present a rigorous examination of the values the field advances by quantitatively   
+6 and qualitatively analysing 100 highly cited ML papers published at premier ML   
+7 conferences, ICML and NeurIPS. We annotate key features of papers which reveal   
+8 their values: how they justify their choice of project, which aspects they uplift,   
+9 their consideration of potential negative consequences, and their institutional affili  
+10 ations and funding sources. We find that societal needs are typically very loosely   
+11 connected to the choice of project, if mentioned at all, and that consideration of   
+12 negative consequences is extremely rare. We identify 67 values that are uplifted in   
+13 these papers, and, of these, we find that papers most frequently justify and assess   
+14 themselves based on performance, generalization, efficiency, researcher understand  
+15 ing, novelty, and building on previous work. We present extensive textual evidence   
+16 and analysis of how these values are concretized. Notably, we find that each of   
+17 these top values is being defined and applied with assumptions and implications   
+18 generally supporting the centralization of power. Finally, we find increasingly close   
+19 ties between these highly cited papers and tech companies and elite universities.
+
+# 20 1 Introduction
+
+21 Over the past few decades, ML has risen from a relatively obscure research area to an extremely   
+22 influential discipline, actively being deployed in myriad applications and contexts around the world.   
+23 The objectives and values of ML research are influenced by many factors, including the personal   
+24 preferences of researchers and reviewers, other work in science and engineering, the interests   
+25 of academic institutions, funding agencies, and companies, and larger institutional and systemic   
+26 pressures, including systems of oppression impacting who is able to do research. Together these   
+27 forces influence what research gets done and who benefits from this research. As such, it is important   
+28 to document and understand the values of the field: what the field is prioritizing and working toward.   
+29 To this end, we perform a comprehensive analysis of 100 highly cited NeurIPS and ICML papers   
+30 from four recent years spanning more than a decade.
+
+31 Our key contributions are as follows:
+
+32 (1) We develop a fine-grained annotation scheme for the detection of values in research papers,   
+33 including identifying a list of 67 values uplifted in ML research. To our knowledge, our annotation   
+34 scheme is the first of its kind, and opens the door to further qualitative and quantitative analyses.   
+35 (2) We use our annotation scheme to annotate 100 influential papers and extract their value commit  
+36 ments, which reflect and shape the values of the field more broadly. Like the annotation scheme itself,
+
+37 the resulting repository of annotated papers is valuable not only in the context of this paper, but also as foundation for further qualitative or quantitative research.1 38
+
+39 (3) We perform extensive textual analysis to understand some of the dominant values: performance,   
+40 accuracy, state-of-the-art (SOTA), quantitative results, generalization, efficiency, building on previous   
+41 work, and novelty $( \ S \bar { \mathsf { E } } )$ . Our analysis indicates that while these values may seem on their face to be   
+42 purely technical, they are nevertheless socially and politically charged: specifically, we argue that   
+43 these values are defined and operationalized in ways that centralize power, i.e., disproportionally   
+44 benefit and empower the already powerful, such as large corporations, while negatively impacting   
+45 society’s least advantaged.   
+46 (4) We present a quantitative analysis of the affiliations and funding sources of these most influential   
+47 papers $\bar { ( \ S 6 ) }$ . We find substantive and increasing presence of big tech corporations. For example, in   
+48 2008/09, $24 \%$ of these top cited papers had corporate affiliated authors, and in 2018/19 this statistic   
+49 almost tripled, to $71 \%$ . Moreover, of these corporations connected to influential papers, the presence   
+50 of "big-tech" firms, such as Google and Microsoft, increased more than fivefold, from $11 \%$ to $58 \%$ .
+
+# 51 2 Methodology
+
+52 To understand the values of ML research, we examined the most highly cited papers from NeurIPS   
+53 and ICML from the years 2008, 2009, 2018, and 2019. We chose to focus on highly cited papers   
+54 because they reflect and shape the values of the discipline, drawing from NeurIPS and ICML because   
+55 they the most prestigious of the long-running ML conferences. 2 Acceptance to these conferences   
+56 is a valuable commodity used to evaluate researchers, and submitted papers are explicitly written   
+57 so as to win the approval of the community, particularly the reviewers who will be drawn from that   
+58 community. As such, these papers effectively reveal the values that authors believe are most valued   
+59 by that community. Citations largely indicate the approval of the community, and help to position   
+60 these papers as influential exemplars of ML research. To avoid detecting only short-lived trends and   
+61 enable comparisons over time, we drew papers from two recent years (2018/19) and from ten years   
+62 earlier (2008/09). We focused on conference papers because they tend to follow a standard format   
+63 and allow limited space, meaning that researchers must make hard choices about what to emphasize.   
+64 Collectively, we annotated 100 papers, analyzing over 3,500 sentences drawn from them. In the   
+65 context of qualitative content analysis, this is a significant effort which allows us to meaningfully   
+66 comment on the values central to ML.   
+67 In more detail, we began by creating an annotation scheme (see below), and then used it to manually   
+68 annotate each paper, examining the abstract, introduction, discussion, and conclusion: (1) We   
+69 examined the chain of reasoning by which each paper justified its contributions, which we call the   
+70 justificatory chain, rating the extent to which papers used technical or societal problems to justify or   
+71 motivate their contributions. (2) We carefully read the text of these sections, individually annotating   
+72 any and all values from our list that were uplifted or exhibited by each sentence.3 (3) We documented   
+73 the extent to which the paper included a discussion of potential negative impacts.   
+74 Manual annotation was necessary, both to create the list of values, and to obtain and understand   
+75 the values present in each paper. Automated approaches, such as keyword searches, would run   
+76 the risk of systematically skewing the results towards values which are easy to identify, potentially   
+77 missing or mischaracterizing values which are exhibited in more nuanced ways, or those which were   
+78 not anticipated. The qualitative approach was key for analyzing the values as well, as it requires a   
+79 subtle understanding of how the values function in the text and understanding of taken for granted   
+80 assumptions underlying the values, which methods such as keyword matching would fail to capture.   
+81 To assess consistency, $40 \%$ of the papers were annotated by two annotators. The intercoder consensus   
+82 on values in these papers achieved a Cohen kappa coefficient of $61 \%$ , which indicates substantial   
+83 agreement $\mathbb { \lVert 3 9 \rVert }$ . Furthermore, we used several established strategies to increase consistency, including   
+84 recoding data coded early in the process $\mathbb { \left| \left[ 2 3 \right] \right| }$ and conducting frequent discussions and assessments of   
+85 the coding process, code list, and annotation scheme $\pmb { \Vert 2 4 \Vert }$ .   
+86 To create the list of values (see Figure $\mathbb { D }$ , we followed best practices in manual content analysis. (1)   
+87 We began with a list of values we expected to be relevant based on prior knowledge, augmenting   
+88 this list with seven ethical principles from existing literature [5, 16]. (2) We randomly selected a   
+89 subset of 10 papers for initial annotation, searching for the values on the list sentence by sentence   
+90 and adding new values as needed. (3) Through discussion, we revisited all values and produced a   
+91 final list. (4) We annotated the full set of papers using this list of values, meeting regularly to discuss   
+92 difficult examples. (5) For the final analysis presented here, we combined some related values into   
+93 clusters (via consensus), such that they could be discussed together (but are treated separately in   
+94 the appendix). Formally stated, we establish our codes (short phrases that represent the relevant   
+95 essence of information, in this case the list of values) using the an inductive-deductive approach. The   
+96 deductive component involves starting with codes established in existing literature, which ensures   
+97 we note and can speak to values of interest, including established ethical principles. The inductive   
+98 component involves the discovery of codes from the data, and impedes inappropriately biased or   
+99 pre-conceived findings by focusing on emergent codes [7, 24].   
+100 The composition of our team also confers validity to our work. We are a diverse team, including   
+101 undergraduate, graduate, and post-graduate researchers from machine learning, NLP, robotics, cogni  
+102 tive science, and philosophy. This diversity minimizes intra-disciplinary biases, affords the unique   
+103 combination of expertise required to read the values in ML papers, allows meaningful engagement   
+104 with relevant work in other fields, and enables best practices including continually clarifying the   
+105 procedure, ensuring agreement, vetting consistency, reannotating, and discussing themes $\dot { \lVert 2 4 \rVert }$ Other   
+106 methods of manual annotation, such as crowd sourcing, lack these advantages.
+
+![](images/52d8fdcb8e79c9b5ea20204bfbc049877e6720924c6d5f974a93b5f1c6f83c5d.jpg)  
+Figure 1: Proportion of annotated papers that uplifted each value.
+
+# 107 3 Quantitative Summary
+
+108 In Figure $\bigtriangledown$ we plot the prevalence of values in 100 annotated papers. The top values are: performance   
+109 $87 \%$ of papers), building on past work $(7 9 \% )$ , generalization $( 7 9 \% )$ , efficiency $(73 \% )$ , quantitative   
+110 evidence $(72 \% )$ , and novelty $( 6 3 \% )$ . Values related to user rights and stated in ethical principles   
+111 appeared very rarely if at all: none of the papers mentioned autonomy, justice, or respect for persons.   
+112 In Table $\mathbb { L }$ (top), we show the distribution of justification scores. Most papers only justify how they   
+113 achieve their internal, technical goal; $71 \%$ don’t make any mention of societal need or impact, and   
+114 only $3 \%$ make what we considered to be a rigorous attempt at connecting their research to societal   
+115 needs. In Table $^ 1$ (bottom), we show the distribution of negative impact discussion scores. One   
+116 annotated paper included a discussion of negative impacts and a second mentioned the possibility;   
+117 none of the remaining 98 papers contained any reference to potential negative impacts. In Figure ${ \overset { \cdot } { 3 } } ,$   
+118 we show stated ties (funding and affiliations) of paper authors to different institutions. Comparing   
+119 papers written in 08/09 to those authored 18/19, ties to corporations nearly doubled to $79 \%$ of all   
+120 annotated papers, ties to big tech multiplied over fivefold to $58 \%$ , while ties to universities declined   
+121 to $81 \%$ , putting corporations nearly on par with universities in the most cited ML research. In the   
+122 next sections, we present extensive qualitative examples and analysis of our findings, with additional   
+123 analyses in the Appendix.
+
+Table 1: Annotation scheme and results for justificatory chain (top) and negative impacts (bottom).   
+
+<table><tr><td>Justificatory Chain Condition</td><td>% of Papers</td></tr><tr><td>Doesn&#x27;t rigorously justify how it achieves technical goal</td><td>1%</td></tr><tr><td>Justifies how it achieves technical goal but no mention of societal need</td><td>71%</td></tr><tr><td>States but does not justify how it connects to a societal need</td><td>16%</td></tr><tr><td>States and somewhat justifies how it connects to a societal need</td><td>9%</td></tr><tr><td>States and rigorously justifies how it connects to a a societal need</td><td>3%</td></tr><tr><td>Negative Impacts Condition</td><td>% of Papers</td></tr><tr><td>Doesn&#x27;t mention negative potential</td><td>98%</td></tr><tr><td>Mentions but does not discuss negative potential</td><td>1%</td></tr><tr><td>Discusses negative potential</td><td>1%</td></tr><tr><td>Deepens our understanding of negative potential</td><td>0%</td></tr></table>
+
+# 124 4 Qualitative Analysis of Justifications and Negative Potential
+
+# 4.1 Justificatory Chain
+
+Papers typically motivate their projects by appealing to the needs of the ML research community, but rarely mention potential societal benefits. Research-driven needs of the ML community include researcher understanding (e.g., understanding the effect of pre-training on performance/robustness, theoretically understanding multi-layer networks) as well as more practical research problems (e.g., improving efficiency of models for large datasets, creating a new benchmark for NLP tasks). Some papers do appeal to needs of the broader society, such as building models with realistic assumptions, catering to more languages, or understanding the world. However, even when societal needs are mentioned as part of the justification of the project, the connection is often loose. Almost no papers explain how their project is meant to promote a social need they identify by giving the kind of rigorous justification that is typically expected of and given for technical contributions.
+
+# 36 4.2 Negative Potential
+
+Two of the 100 papers discussed potential harms, whereas the remaining 98 did not mention them at all. The lack of discussion of potential harms is especially striking for papers which deal with socially contentious application areas, such as surveillance and misinformation. For example, the annotated corpus includes a paper advancing the identification of people in images, a paper advancing face-swapping, and a paper advancing video synthesis. These papers contained no mention of the well-studied negative potential of facial surveillance, DeepFakes, or misleading videos, respectively.
+
+143 Furthermore, among the two papers that do mention negative potential, the discussions were mostly   
+144 abstract and hypothetical, rather than grounded in the negative potential of their specific contributions.   
+145 For example, authors may acknowledge "possible unwanted social biases" when applying the model   
+146 to a real-world setting, without discussing the social biases encoded in the authors’ proposed model.
+
+# 147 5 Stated values
+
+The dominant values in ML research, e.g., accuracy or efficiency, may seem purely technical. However, the following analysis of several of these values shows how they can become politically loaded in the process of prioritizing and operationalizing them: sensitivity to the way that they are operationalized, and to the fact that they are uplifted at all, reveals value-laden assumptions that are often taken for granted and may negatively impact the broader society. 4 We thus challenge a conception of prevalent values as politically neural by considering alternatives to their dominant conceptualization that may be equally or more intellectually interesting or more socially beneficial. We have encouraged ourselves, and now encourage the reader, to remember that values once held to be intrinsic, obvious, or definitional have been in many cases transformed over time.
+
+Table 2: Random examples of performance, the most common emergent value.   
+
+<table><tr><td>&quot;Our model significantly outperforms SVM&#x27;s,and it also outperforms convolutional neural nets when given additional unlabeled data produced by small translations of the training images.&quot;</td></tr><tr><td>&quot;We show in simulations on synthetic examples and on the IEDB MHC-I binding dataset,that our approach outperforms well-known convex methods for multi-task learning,as wellas related non-convex methods dedicated to the same problem.&quot;</td></tr><tr><td>&quot;Furthermore, the learning accuracy and performance of our LGP approach willbe compared with other important standard methods in Section 4,e.g.,LWPR [8],standard GPR[1],sparse online Gaussian process regression (OGP) [5] and U-support vector regresson (v-SVR) [11], respectively.&quot;</td></tr></table>
+
+157 To provide a sense of what the values we discuss look like in context, we include three randomly   
+158 selected examples of sentences annotated for each (Tables 2-5), with additional examples in the   
+159 Appendix. Note that most sentences are annotated with multiple values, but this is not shown here.5
+
+# 60 5.1 Performance
+
+Performance, accuracy, and achieving SOTA form the most common cluster of related values in annotated papers. While it might seem intrinsic for the field to care about performance, it is important to remember that models are not simply "well-performing" or "accurate" in the abstract but always in relation to and as quantified by some metric on some dataset. Examining prevalent choices of operationalization reveals political aspects of performance values. First, we find that performance values are consistently and unquestioningly operationalized as correctness averaged across individual predictions, giving equal weight to each instance. However, choosing to use equal weights when averaging is a value-laden move which might deprioritize those underrepresentated in the data or world, as well as societal and evaluee needs and preferences. Extensive research in ML fairness and related fields has considered alternatives, but we found no such discussions among the most-cited papers we examined.
+
+172 Choices of datasets are revealing. They are often driven purely by past work, so as to demonstrate   
+173 improvement over a previous baseline (see also $\ S 5 . 4 )$ . Another common justification for using a   
+174 certain dataset is applicability to the "real world". Assumptions about how to characterize the real   
+175 world may also be value-laden. One common assumption is the availability of very large datasets.   
+176 However, presupposing the availability of large datasets is power centralizing because it encodes   
+177 favoritism to those with resources to obtain and process them $\mathbb { \lVert 1 5 \rVert }$ . Further overlooked assumptions   
+178 include that the real world is binary or discrete, and that datasets come with a predefined ground-truth   
+179 label for each example, presuming that a true label always exists "out there" independent of those   
+180 carving it out, defining and labelling it. This contrasts against marginalized scholars’ calls for   
+181 ML models that allow for non-binaries, plural truths, contextual truths, and many ways of being   
+182 [12, 18, 26].   
+183 The prioritization of performance values also requires scrutiny. Valuing these properties is so   
+184 entrenched in the field that generic success terms, such as "success", "progress", or "improvement"   
+185 are often used as synonyms for performance and accuracy. However, one might alternatively invoke   
+186 generic success to mean increasingly safe, consensual, or participatory ML that reckons with impacted   
+187 communities and the environment. In fact, "performance" itself is a general success term that could   
+188 have been associated with properties other than accuracy and SOTA.
+
+# 5.2 Generalization
+
+A common way of appraising the merits of one’s work in ML is to claim that it generalizes well. Typically, generalization is understood in terms of performance or accuracy: a model generalizes when it achieves good performance on a range of samples, datasets, domains, or applications. Uplifting generalization raises two kinds of questions. First, which datasets, domains, or applications show that the model generalizes well? Typically, a paper shows that a model generalizes by showing that it performs well on multiple tasks or datasets. However, the choice of particular tasks and datasets is
+
+Table 3: Random examples of generalization, the third most common emergent value.   
+
+<table><tr><td>&quot;The range of applications that come with generative models are vast, where audio synthesis [55] and semi-supervised classification [38,31,44] are examples hereof.&quot;</td></tr><tr><td>&quot;Furthermore,the infinite limit could conceivably make sense in deep learning,since over-parametrization sems to help optimization a lot and doesn&#x27;t hurt generalization much [Zhang et al.,2O17]: deep neural nets with millions of parameters work well even for datasets with 5Ok training examples.&quot;</td></tr><tr><td>&quot;Combining the optimization and generalization results,we uncover a broad class of learnable functions, including linear functions,two-layer neural networks with polynomial activation(z） = z2l or cosine activation, etc.&quot;</td></tr></table>
+
+Table 4: Random examples of efficiency, the fourth most common emergent value.   
+
+<table><tr><td>&quot;Our model allows for controllable yet efcient generation of an entire news article-not just the body, but also the title, news source, publication date,and author list.&quot;</td></tr><tr><td>&quot;We show that Bayesian PMF models can be efciently trained using Markov chain Monte Carlo methods by applying them to the Netflix dataset,which consists of over 1OO million movie ratings.&quot;</td></tr><tr><td>&quot;In particular,our EffcientNet-B7 surpasses the best existing GPipe accuracy (Huang et al.,2018), but using 8.4x fewer parameters and running 6.1x faster on inference.&quot;</td></tr></table>
+
+196 rarely justified; the choice of tasks can often seem arbitrary, and authors rarely present evidence that   
+97 their results will generalize to more realistic settings, or help to directly address societal needs.
+
+Second, uplifting generalization itself reveals substantive assumptions. The prizing of generalization means that there is an incentive to harvest many datasets from a variety of domains, and to treat these as the only datasets that matter for that space of problems. Generalization thus prioritizes distilling every scenario down to a common set of representations or affordances, rather than treating each setting as unique. Critical scholars have advocated for valuing context, which stands at the opposite side of striving for generalization $\textcircled { 1 1 4 } \textcircled { 1 }$ . Others have argued that this kind of totalizing lens (in which model developers have unlimited power to determine how the world is represented) leads to representational harms, due to applying a single representational framework to everything [13, 1].
+
+Finally, the belief that generalization is even possible implicitly assumes a conservative approach in which new data will be sufficiently similar to previously seen data. When used in the context of ML, the assumption that the future resembles the past is also normative and often problematic as past societal stereotypes and injustice can be encoded in the process [33]. Furthermore, to the extent that predictions are performative $\pmb { \Vert 3 5 \Vert }$ , especially predictions that are enacted, those ML models which are deployed to the world will contribute to shaping social patterns. Yet, no papers attempt to counteract this quality or acknowledge its presence.
+
+# 5.3 Efficiency
+
+Efficiency is another common value in ML research. Abstractly, saying that a model is efficient typically means saying that the model uses less of some resource, such as time, memory, energy, or number of labeled examples. In practice however, efficiency is commonly referenced to imply scalability: a more efficient inference method allows you to do inference in much larger models or on larger datasets, using the same amount of resources. This is reflected in our value annotations, where $72 \%$ of papers mention valuing efficiency, but only $14 \%$ of those value requiring few resources. In this way, valuing efficiency facilitates and encourages the most powerful actors to scale up their computation to ever higher orders of magnitude, making their models even less accessible to those without resources to use them and decreasing the ability to compete with them. Alternative usages of efficiency could encode accessibility instead of scalability, aiming to create more equitable conditions for ML research.
+
+# 5.4 Novelty and Building on Past Work
+
+26 Most authors devote space in the introduction to positioning their paper in relation to past work, and   
+27 describing what is novel. Mentioning past work serves to signal awareness of related publications, to
+
+Table 5: Random examples of building on past work and novelty, the second and sixth most common emergent values, respectively.   
+
+<table><tr><td rowspan=1 colspan=1>Building on past work</td></tr><tr><td rowspan=1 colspan=1>&quot;Recent work points towards sample complexity as a possible reason for the small gains in robustness:Schmidt et al.[41] show that in a simple model,learning a classifier with non-trivial adversarially robustaccuracy requires substantially more samples than achieving good ‘standard’ accuracy.&quot;</td></tr><tr><td rowspan=1 colspan=1>&quot;Experiments indicate that our method is much faster than state of the art solvers such as Pegasos,TRON,SVMperf, and a recent primal coordinate descent implementation.&quot;</td></tr><tr><td rowspan=1 colspan=1>&quot;There is a large literature on GP (response surface) optimization.&quot;</td></tr><tr><td rowspan=1 colspan=1>Novelty</td></tr><tr><td rowspan=1 colspan=1>&quot;In this paper, we propose a video-to-video synthesis approach under the generative adversarial learningframework.&quot;</td></tr><tr><td rowspan=1 colspan=1>&quot;Third, we propose a novel method for the listwise approach, which we callListMLE.&quot;</td></tr><tr><td rowspan=1 colspan=1>&quot;The distinguishing feature of our work is the use of Markov chain Monte Carlo (MCMC) methods forapproximate inference in this model.&quot;</td></tr></table>
+
+![](images/887531813864994c3359d4b12573ca6a6640ff202555ab5ab80fca239ee35054.jpg)  
+Figure 2: Corporate and Big Tech author affiliations.
+
+establish the new work as relevant to the community, and to provide the basis upon which to make claims about what is new. Novelty is sometimes suggested implicitly (e.g., "we develop" or "we propose"), but frequently it is emphasized explicitly (e.g. "a new algorithm" or "a novel approach").
+
+This combined focus on novelty and building on recent work establishes a continuity of ideas, and might be expected to contribute to the self-correcting nature of science [29]. However, this is not always the case $\scriptstyle { \left[ \left[ 2 1 \right] \right] }$ and attention to the ways novelty and building on past work are implemented reveals value commitments. In particular, we find a clear emphasis on technical novelty, rather than critique of past work, or demonstration of measurable progress on societal problems, as has previously been observed $\mathbb { \left[ \left| 4 0 \right| \right] }$ . Although introductions sometimes point out limitations of past work (so as to further emphasize the contributions of their own paper), they are rarely explicitly critical of other papers in terms of methods or goals. Indeed, papers uncritically reuse the same datasets for years or decades to benchmark their algorithms, even if those datasets fail to represent more realistic contexts in which their algorithms will be used $\textcircled { 6 }$ . Novelty is denied to work that rectifies socially harmful aspects of existing datasets in tandem with strong pressure to benchmark on them and thereby perpetuate their use, enforcing a fundamentally conservative bent to ML research.
+
+# 243 6 Corporate Affiliations and Funding
+
+Our analysis shows substantive and increasing corporate presence in the most highly-cited papers. In 2008/09, $24 \%$ of the top cited papers had corporate affiliated authors, and in 2018/19 this statistic almost tripled, to $71 \%$ . Furthermore, we also find a much greater concentration of a few large tech firms, such as Google and Microsoft, with the presence of these "big tech" firms $\mathbb { H }$ increasing more than fivefold, from $11 \%$ to $58 \%$ (see Figure $\textcircled{2}$ . The number of most influential papers with corporate ties, by author affiliation or funding, published dramatically increased from $43 \%$ in 2008/09 to $79 \%$ in 2018/19. In addition, we found paramount domination of elite universities in our analysis as shown in Figure $\textcircled{3}$ Of the total papers with university affiliations, we found $82 \%$ were from elite universities (defined as the top 50 universities by QS World University Rankings, following
+
+![](images/9a93dad3606c3c8fd648d86c21f18c9f40bdc11021b4dddf8c1f100b998c4a4b.jpg)  
+Figure 3: Corporate affiliations and funding ties. Non-N.A. Universities are those outside the U.S. and Canada.
+
+253 past work $\pmb { \| 4 \| }$ ). These findings are consistent with previous work indicating a pronounced corporate   
+254 presence in ML research. In an analysis of 171,394 peer-reviewed papers from 57 major computer   
+255 science conferences, Ahmed and Wahed [4] show that the share of papers that have at least one   
+256 corporate affiliated co-author increased from $10 \%$ in 2005 for both ICML and NeurIPS to $30 \%$ and   
+257 $3 5 \%$ respectively in 2019. Our analysis shows that corporate presence is even more pronounced in   
+258 those papers from ICML and NeurIPS that end up receiving the most citations.   
+259 The influence of powerful players in ML research is consistent with field-wide value commitments   
+260 that centralize power. Others have also argued for causal connections. For example, Abdalla and   
+261 Abdalla $\left[ \left[ 2 \right] \right]$ argue that the strategies that big tech uses to sway and influence academic and public   
+262 discourse, closely resemble that of Big Tobacco. Moreover, examining the prevalent values of   
+263 big tech, critiques have repeatedly pointed out that objectives such as efficiency, scale, and wealth   
+264 accumulation [33, 34, 19] drive the industry at large, often at the expense of individuals rights, respect   
+265 for persons, consideration of negative impacts, beneficence, and justice. The top stated values of ML   
+266 that we presented in this paper such as performance, generalization, and efficiency not only enable   
+267 and facilitate the realization of big tech’s objectives, they also suppress values such as beneficence,   
+268 justice, and inclusion. A "state-of-the-art" large image dataset, for example, is instrumental for   
+269 large scale models, further benefiting ML researchers and big tech in possession of huge computing   
+270 power. A large image dataset that considers negative consequences and is built on the foundations   
+271 of individual rights and respect for persons, on the other hand, is one that would start with gaining   
+272 informed consent from the data subject and is considerate of contextual norms over scalability $\bar { \| 1 9 \| }$ .   
+273 However, in the current climate where values such as efficiency and scale are a priority, informed   
+274 consent is perceived as costly and time consuming, evading social needs.
+
+# 275 7 Discussion
+
+ML research is often perceived as value-neutral, and emphasis is placed on positive applications or potential. This fits into a historical strain of thinking which has tended to frame technology as "neutral", based on the notion that new technologies can be unpredictably applied for both beneficial and harmful purposes $\mathbb { \oplus 3 } \mathbb { \mathbb { I } }$ . Ironically, this claim of neutrality frequently serves as an insulation from critiques of AI and as a permission to emphasize the benefits of AI [38, 41]. Although it is rare to see anyone explicitly argue in print that ML is neutral, related ideas are part of contemporary conversation, including these canonical claims: long term impacts are too difficult to predict; sociological impacts are outside the expertise or purview of ML researchers $\mathbb { \left. 2 0 \right. }$ ; critiques of AI are really misdirected critiques of those deploying AI with bad data ("garbage in, garbage out"), again outside the purview of many AI researchers; and proposals such as broader impact statements represent merely a "bureaucratic constraint" [3]. A recent qualitative analysis of broader impact statements from NeurIPS 2020 similarly observed that these statements leaned towards positive consequences (often mentioning negative consequences only briefly and in some cases not at all), emphasized uncertainty about how a technology might be used, or simply omit any discussion of societal consequences altogether [31].
+
+Importantly, there is a foundational understanding in Science, Technology, and Society Studies 292 (STSS), Critical Theory, and Philosophy of Science that science and technologies are inherently 293 value-laden, and these values are encoded in technological artifacts, many times in contrast to a field’s 294 formal research criteria, espoused consequences, or ethics guidelines [44, 10, 8]. There is a long
+
+295 tradition of exposing and critiquing such values in technology and computer science. Foundationally,   
+296 Winner $[ \textcircled { 4 4 } ]$ introduced several ways technology can encode political values. This work is closely   
+297 related to Rogaway $\textcircled { 1 3 7 }$ , who notes that cryptography has political and moral dimensions and argues   
+298 for a cryptography that better addresses societal needs. Weizenbaum $\mathbb { \lVert \rVert }$ argued in 1976 that the   
+299 computer has from the beginning been a fundamentally conservative force which solidified existing   
+300 power. In place of fundamental social changes, the computer renders technical solutions that allow   
+301 existing power hierarchies to remain intact.   
+302 Our paper extends these critiques to the field of ML. It is a part of a rich space of interdisciplinary   
+303 critiques and alternative lenses used to examine the field. Works such as $\overline { { \mathbb { B } 0 } } \overline { { \mathbb { Q } } }$ critique AI, ML, and   
+304 data using a decolonial lens, noting how these technologies replicate colonial power relationships   
+305 and values, and propose decolonial values and methods. Others $[ \mathbb { R } , \mathbb { 3 } 2 , \mathbb { 1 4 } ]$ examine technology and   
+306 data science from an anti-racist and intersectional feminist lens, discussing how our infrastructure   
+307 has largely been built by and for white men; D’Ignazio and Klein $\pmb { \Vert 4 \Vert }$ present a set of alternative   
+308 principles and methodologies for an intersectional feminist data science. Similarly, Kalluri $\pmb { \mathbb { \left\| 2 2 \right\| } }$   
+309 denotes that the core values of ML are closely aligned with the values of the most privileged and   
+310 outlines a vision where ML models are used to shift power from the most to the least powerful. Dotan   
+311 and Milli [15] argue that the rise of deep learning is value-laden, promoting the centralization of   
+312 power among other political values. Many researchers, as well as organizations such as Data for   
+313 Black Lives, the Algorithmic Justice League, Indigenous AI, Black in AI, and Queer in AI, work on   
+314 continuing to uncover particular ways technology in general and ML in particular can encode and   
+315 amplify racist, sexist, queerphobic, transphobic, and otherwise marginalizing values [11, 36].   
+316 We present this work in part in order to expose the contingency of the present state of the field; it could   
+317 be otherwise. For individuals, communities, and institutions wading through difficult-to-pin-down   
+318 values of the field, as well as those striving toward alternative values, it is a useful tool to have a   
+319 characterization of the way the field is now, for understanding, shaping, dismantling, or transforming   
+320 what is, and for articulating and bringing about alternative visions.   
+21 As with all methods, our chosen approach (careful reading of important sections of highly-cited   
+22 papers) has limitations. Most notably, this approach does not automatically scale or generalize to   
+23 other data, which limits our ability to draw strong conclusions about other conferences or different   
+24 years. Similarly, this approach is less reproducible than fully automated approaches, and for both   
+25 our final list of values and specific annotation of individual sentences, different researchers might   
+26 make somewhat different choices. However, given the overwhelming presence of certain values, the   
+27 high agreement rate among annotators, and the similarity of observations made by our team, we   
+28 strongly believe other researchers taking a similar approach would reach similar conclusions about   
+29 what values are most frequently uplifted by the most influential papers in this field. Lastly, we cannot   
+30 claim to have identified every relevant value in ML. However, by including important ethical values   
+31 identified by past work, and specifically looking for these, we can confidently assert their relative   
+32 absence in this set of papers, which we take to be representative of influential work in ML.
+
+# 333 8 Conclusion and Future Work
+
+334 We reject the vague conceptualization of the discipline of ML as value-neutral. Instead, we argue   
+335 that the discipline of ML is inherently value-laden. Our analysis of highly influential papers in the   
+336 discipline shows that the discipline not only favors the needs of research communities and large firms   
+337 over broader social needs, but also that it takes this favoritism for granted. The favoritism manifests   
+338 in the choice of projects, the lack of consideration of potential negative impacts, and the prioritization   
+339 and operationalization of values such as accuracy, generalization, efficiency, and novelty. All of   
+340 these overwhelmingly disfavor societal needs, usually without any discussion or acknowledgment.   
+341 Moreover, we uncover an overwhelming and increasing presence of big tech and elite universities in   
+342 highly cited papers, which is consistent with a system of power-centralizing value-commitments.   
+343 The upshot is that the discipline of ML is not value-neutral. It is socially and politically loaded,   
+344 valuing and promoting conservative needs at the cost of individuals rights, respect for persons and   
+345 justice; it increasingly concentrates power in the hands of few already powerful actors; it poses   
+346 a threat to society’s most marginalized by neglecting the potential harms of socially contentions   
+347 applications of ML.
+
+48 References [1] Mohsen Abbasi, Sorelle A. Friedler, Carlos Scheidegger, and Suresh Venkatasubramanian. Fairness in representation: Quantifying stereotyping as a representational harm. In Proceedings of the 2019 SIAM International Conference on Data Mining, 2019. [2] Mohamed Abdalla and Moustafa Abdalla. The grey hoodie project: Big tobacco, big tech, and the threat on academic integrity. arXiv preprint arXiv:2009.13676, 2020. [3] Grace Abuhamad and Claudel Rheault. Like a researcher stating broader impact for the very first time. arXiv preprint arXiv:2011.13032, 2020. [4] Nur Ahmed and Muntasir Wahed. The de-democratization of AI: Deep learning and the compute divide in artificial intelligence research. arXiv preprint arXiv:2010.15581, 2020. [5] Michael Bailey, David Dittrich, Erin Kenneally, and Doug Maughan. The Menlo Report: Ethical Principles Guiding Information and Communication Technology Research. Technical report, U.S. Department of Homeland Security, Aug 2012. [6] Emily M Bender, Timnit Gebru, Angelina McMillan-Major, and Shmargaret Shmitchell. On the dangers of stochastic parrots: Can language models be too big? Proceedings of FAccT, 2021. [7] Mariette Bengtsson. How to plan and perform a qualitative study using content analysis. NursingPlus Open, 2:8–14, 2016. [8] Ruha Benjamin. Race After Technology: Abolitionist Tools for the New Jim Code. Wiley, 2019. [9] Abeba Birhane. Algorithmic colonization of Africa. SCRIPTed, 17(2), 2020. [10] Geoffrey C Bowker and Susan Leigh Star. Sorting things out: Classification and its consequences. MIT press, 2000. [11] Joy Buolamwini and Timnit Gebru. Gender shades: Intersectional accuracy disparities in commercial gender classification. In Proceedings of the Conference on Fairness, Accountability and Transparency, 2018.   
+72 [12] Sasha Costanza-Chock. Design justice, AI, and escape from the matrix of domination. Journal of Design and Science, 2018.   
+74 [13] Kate Crawford. The trouble with bias. NeurIPS Keynote, 2017.   
+75 [14] Catherine D’Ignazio and Lauren F Klein. Data Feminism. MIT Press, 2020.   
+76 [15] Ravit Dotan and Smitha Milli. Value-laden disciplinary shifts in machine learning. arXiv preprint arXiv:1912.01172, 2019.   
+78 [16] Luciano Floridi and Josh Cowls. A unified framework of five principles for AI in society. Harvard Data Science Review, 1(1), 2019. [17] Suzanne Fricke. Semantic scholar. Journal of the Medical Library Association: JMLA, 106(1):145, 2018. [18] Foad Hamidi, Morgan Klaus Scheuerman, and Stacy M Branham. Gender recognition or gender reductionism? The social implications of embedded gender recognition systems. In Proceedings CHI, 2018.   
+85 [19] Alex Hanna and Tina M. Park. Against scale: Provocations and resistances to scale thinking. arXiv preprint arXiv:2010.08850, 2020. [20] Kenneth Holstein, Jennifer Wortman Vaughan, Hal Daumé, Miro Dudik, and Hanna Wallach. Improving fairness in machine learning systems: What do industry practitioners need? In Proceedings of CHI, 2019.   
+90 [21] John P. A. Ioannidis. Why science is not necessarily self-correcting. Perspectives on Psychological Science, 7(6):645–654, 2012.
+
+392 [22] Pratyusha Kalluri. Don’t ask if artificial intelligence is good or fair, ask how it shifts power.   
+393 Nature, 583(7815):169–169, 2020.   
+394 [23] Laura Krefting. Rigor in qualitative research: The assessment of trustworthiness. American   
+395 Journal of Occupational Therapy, 45(3):214–222, 03 1991.   
+396 [24] Klaus Krippendorff. Content Analysis: An Introduction to its Methodology. Sage Publications,   
+397 2018.   
+398 [25] Thomas S. Kuhn. Objectivity, value judgment, and theory choice. In The Essential Tension:   
+399 Selected Studies in Scientific Tradition and Change, pages 320–39. University of Chicago Press,   
+400 1977.   
+401 [26] Jason Edward Lewis, Angie Abdilla, Noelani Arista, Kaipulaumakaniolono Baker, Scott Be  
+402 nesiinaabandan, Michelle Brown, Melanie Cheung, Meredith Coleman, Ashley Cordes, Joel   
+403 Davison, et al. Indigenous protocol and artificial intelligence position paper. 2020.   
+404 [27] Helen E. Longino. Cognitive and non-cognitive values in science: Rethinking the dichotomy.   
+405 In Lynn Hankinson Nelson and Jack Nelson, editors, Feminism, Science, and the Philosophy of   
+406 Science, pages 39–58. Springer Netherlands, 1996.   
+407 [28] Ernan McMullin. Values in science. In Proceedings of the Biennial Meeting of the Philosophy   
+408 of Science Association, 1982.   
+409 [29] Robert K Merton. The Sociology of Science: Theoretical and Empirical Investigations. Univer  
+410 sity of Chicago press, 1973.   
+411 [30] Shakir Mohamed, Marie-Therese Png, and William Isaac. Decolonial AI: Decolonial rheory   
+412 as sociotechnical foresight in artificial intelligence. Philosophy & Technology, 33:659––684,   
+413 2020.   
+414 [31] Priyanka Nanayakkara, Jessica Hullman, and Nicholas Diakopoulos. Unpacking the expressed   
+415 consequences of AI research in broader impact statements. CoRR, abs/2105.04760, 2021.   
+416 [32] Safiya Umoja Noble. Algorithms of oppression: How search engines reinforce racism. NYU   
+417 Press, 2018.   
+418 [33] Cathy O’Neil. Weapons of Math Destruction: How Big Data Increases Inequality and Threatens   
+419 Democracy. Broadway Books, 2016.   
+420 [34] Frank Pasquale. The black box society. Harvard University Press, 2015.   
+421 [35] Juan Perdomo, Tijana Zrnic, Celestine Mendler-Dünner, and Moritz Hardt. Performative   
+422 prediction. In Proceedings of ICML, 2020.   
+423 [36] Vinay Uday Prabhu and Abeba Birhane. Large image datasets: A Pyrrhic win for computer   
+424 vision? arXiv preprint arXiv:2006.16923, 2020.   
+425 [37] Phillip Rogaway. The moral character of cryptographic work. IACR Cryptol. ePrint Arch., 2015,   
+426 2015.   
+427 [38] Daniela Rus. Rise of the robots: Are you ready? Financial Times Magazine, March 2018.   
+428 [39] Anthony J. Viera and Joanne M. Garrett. Understanding interobserver agreement: The kappa   
+429 statistic. Family Medicine, 37(5):360–3, 2005.   
+430 [40] Kiri Wagstaff. Machine learning that matters. In Proceedings of ICML, 2012.   
+431 [41] Joseph Weizenbaum. On the impact of the computer on society. Science, 176(4035):609–614,   
+432 1972.   
+433 [42] Joseph Weizenbaum. Computer Power and Human Reason: From Judgment to Calculation.   
+434 WH Freeman & Co, 1976.   
+435 [43] Langdon Winner. Autonomous Technology: Technics-out-of-Control as a Theme in Political   
+436 Thought. MIT Press, 1977.   
+437 [44] Langdon Winner. Do artifacts have politics? Daedalus, 109(1):121–136, 1980.
+
+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] See Discussion.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] Included in the Appendix.   
+(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? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Included in supplementary zipfile for experiments in appendix.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Included in appendix.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No]   
+(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] Included in appendix.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] Full listing of annotated papers is given in the appendix.   
+(b) Did you mention the license of the assets? [Yes] See Footnote 1.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Included in supplementary zipfile.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] Discussed in Appendix A.2. Additional Methodological Details.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] Discussed in Appendix A.2. Additional Methodological Details.
+
+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]

parse/train/rkl8sJBYvH/rkl8sJBYvH.md

@@ -0,0 +1,282 @@
+# HARNESSING THE POWER OF INFINITELY WIDE DEEP NETS ON SMALL-DATA TASKS
+
+Sanjeev Arora Princeton University arora@cs.princeton.edu
+
+Simon S. Du   
+Institute for Advanced Study   
+ssdu@ias.edu   
+Zhiyuan Li   
+Princeton University   
+zhiyuanli@cs.princeton.edu
+
+Ruslan Salakhutdinov Carnegie Mellon University rsalakhu@cs.cmu.edu
+
+Ruosong Wang Carnegie Mellon University ruosongw@andrew.cmu.edu
+
+Dingli Yu   
+Princeton University   
+dingliy@cs.princeton.edu
+
+# ABSTRACT
+
+Recent research shows that the following two models are equivalent: (a) infinitely wide neural networks (NNs) trained under $\ell _ { 2 }$ loss by gradient descent with infinitesimally small learning rate (b) kernel regression with respect to so-called Neural Tangent Kernels (NTKs) (Jacot et al., 2018). An efficient algorithm to compute the NTK, as well as its convolutional counterparts, appears in Arora et al. (2019a), which allowed studying performance of infinitely wide nets on datasets like CIFAR-10. However, super-quadratic running time of kernel methods makes them best suited for small-data tasks. We report results suggesting neural tangent kernels perform strongly on low-data tasks.
+
+1. On a standard testbed of classification/regression tasks from the UCI database, NTK SVM beats the previous gold standard, Random Forests (RF), and also the corresponding finite nets.   
+2. On CIFAR-10 with 10 – 640 training samples, Convolutional NTK consistently beats ResNet-34 by $1 \% - 3 \%$ .   
+3. On VOC07 testbed for few-shot image classification tasks on ImageNet with transfer learning (Goyal et al., 2019), replacing the linear SVM currently used with a Convolutional NTK SVM consistently improves performance.   
+4. Comparing the performance of NTK with the finite-width net it was derived from, NTK behavior starts at lower net widths than suggested by theoretical analysis(Arora et al., 2019a). NTK’s efficacy may trace to lower variance of output.
+
+# 1 INTRODUCTION
+
+Modern neural networks (NNs) have way more parameters than training data points, which allow them to achieve near-zero training error while simultaneously — for some reason yet to be understood — have low generalization error (Zhang et al., 2016). This motivated formal study of highly overparametrized networks, including networks whose width (i.e., number of nodes in layers, or number of channels in convolutional layers) goes to infinity. A recent line of theoretical results shows that with $\ell _ { 2 }$ loss and infinitesimal learning rate, in the limit of infinite width the trajectory of training converges to kernel regression with a particular kernel, neural tangent kernel (NTK) (Jacot et al., 2018). For convolutional networks, the kernel is CNTK. See Section 2 for more discussions. Arora et al. (2019a) gave an algorithm to exactly compute the kernel corresponding to the infinite limit of various realistic NN architectures with convolutions and pooling layers, allowing them to compute performance on CIFAR-10, which revealed that the infinite networks have 6 to $8 \%$ higher error than their finite counterparts. This is still fairly good performance for a fixed kernel.
+
+Ironically, while the above-mentioned analysis, at first sight, appears to reduce the study of a complicated model — deep networks — to an older, simpler model — kernel regression — in practice the simpler model is computationally less efficient because running time of kernel regression can be quadratic in the number of data points!1 Thus computing using CNTK kernel on large datasets like ImageNet currently appears infeasible. Even on CIFAR-10, it seems infeasible to incorporate data augmentation.
+
+However, kernel classifiers are very efficient on small datasets. Here NTKs could conceivably be practical while at the same time bringing some of the power of deep networks to these settings. We recall that recently Olson et al. (2018) showed that multilayer neural networks can be reasonably effective on small datasets, specifically on a UCI testbed of tasks with as few as dozens of training examples. Of course, this required some hyperparameter tuning, although they noted that such tuning is also needed for the champion method, Random Forests (RF), which multilayer neural networks could not beat.
+
+It is thus natural to check if NTK — corresponding to infinitely wide fully-connected networks — performs well in such small-data tasks2. Convex objectives arising from kernels have stable solvers with minimal hyperparameter tuning. Furthermore, random initialization in deep network training seems to lead to higher variance in the output, which can hurt performance in small-data settings. Can NTK’s do better? Below we will see that in the setup of Olson et al. (2018), NTK predictors indeed outperforms corresponding finite deep networks, and also slightly beats the earlier gold standard, Random Forests. This suggests NTK predictors should belong in any list of off-theshelf machine learning methods.
+
+Following are low-data settings where we used NTKs and CNTKs:
+
+• In the testbed of 90 classification tasks from UCI database, NTK predictor achieves superior, and arguably the strongest classification performance. This is verified via several standard statistical tests, including Friedman Rank, Average Accuracy, Percentage of the Maximum Accuracy (PMA) and probability of achieving $9 0 \% / 9 5 \%$ maximum accuracy (P90 and P95), performed to compare performances of different classifiers on 90 datasets from UCI database. (The authors plan to release the code, to allow off-the-shelf use of this method. It does not require GPUs.) • We find the performance of NN is close to that of NTK. On every dataset from UCI database, the difference between the classification accuracy of NN and that of NTK is within $5 \%$ . On the other hand, on some datasets, the difference between classification accuracy of NN (or NTK) and that of other classifiers like RF can be as high as $2 0 \%$ . This indicates in low-data settings, NTK is indeed a good description of NN. Furthermore, we find NTK is more stable (smaller variance), which seems to help it achieve better accuracy on small datasets (cf. Figure 2b). • CNTK is useful in computer vision tasks with small-data. On CIFAR-10, we compare CNTK with ResNet using 10 - 640 training samples and find CNTK can beat ResNet by $1 \% - 3 \%$ . We further study few-shot image classification task on VOC07 dataset. The standard method is to first use a pre-trained network, e.g., ResNet-50 trained on ImageNet, to extract features and then directly apply a linear classifier on the extracted features (Goyal et al., 2019). Here we replace the linear classifier with CNTK and obtain better classification accuracy in various setups.
+
+Paper organization. Section 2 discusses related work. Section 3 reviews the derivation of NTK. Section 4 presents experiments using NN and NTK on UCI datasets. Section 5 presents experiments using CNN and CNTK on small CIFAR-10 datasets. Section 6 presents experiments using CNTK for the few-shot learning setting. Additional technical details are presented in appendix.
+
+# 2 RELATED WORK
+
+Our paper is inspired by Fernandez-Delgado et al. (2014) which conducted extensive experiments ´ on UCI dataset. Their conclusion is random forest performs the best, which is followed by the SVM with Gaussian kernel. Therefore, RF may be considered as a reference (“gold-standard”) to compare with new classifiers. Olson et al. (2018) followed this testing strategy to evaluate the performance of modern neural networks concluding that modern neural networks, even though being highly overparameterized, still give reasonable performances on these small datasets, though not as strong as RFs. Our paper follows the same testing strategy to evaluate the performance of NTK.
+
+The focus of this paper, neural tangent kernel is induced from a neural network architecture. The connection between infinitely wide neural networks and kernel methods is not new (Neal, 1996; Williams, 1997; Roux & Bengio, 2007; Hazan & Jaakkola, 2015; Lee et al., 2018; Matthews et al., 2018; Novak et al., 2019; Garriga-Alonso et al., 2019; Cho & Saul, 2009; Daniely et al., 2016; Daniely, 2017). However, these kernels correspond to neural network where only the last layer is trained. Neural tangent kernel, first proposed by Jacot et al. (2018), is fundamentally different as NTKs correspond to infinitely wide NNs with all layer being trained. Theoretically, a line of work study the optimization and generalization behavior of ultra-wide NNs (Allen-Zhu et al., 2018b;a; Arora et al., 2019b; Du et al., 2018b;a; Li & Liang, 2018; Zou et al., 2018; Yang, 2019). Recently, Arora et al. (2019a) gave non-asymptotic perturbation bound between the NN predictor trained by gradient descent and the NTK predictor. Empirically, Lee et al. (2019) verified on small scale data, NTK is a good approximation to NN. However, Arora et al. (2019a) showed on large scale dataset, NN can outperform NTK which may due to the effect of finite-width and/or optimization procedure.
+
+Generalization to architectures other than fully-connected NN and CNN are recently proposed (Yang, 2019; Du et al., 2019; Bietti & Mairal, 2019). Du et al. (2019) showed graph neural tangent kernel (GNTK) can achieve better performance than its counter part, graph neural network (GNN), on datasets with up to 5000 samples.
+
+# 3 NEURAL NETWORK AND NEURAL TANGENT KERNEL
+
+Since NTK is induced by a NN architecture, we first define a NN formally. Let $x \in \mathbb { R } ^ { d }$ be the input, and denote $g ^ { ( 0 ) } ( x ) = x$ and $d _ { 0 } = d$ for notational convenience. We define an $L$ -hidden-layer fully-connected neural network recursively:
+
+$$
+f ^ { ( h ) } ( \boldsymbol { x } ) = W ^ { ( h ) } g ^ { ( h - 1 ) } ( \boldsymbol { x } ) \in \mathbb { R } ^ { d _ { h } } , \quad g ^ { ( h ) } ( \boldsymbol { x } ) = \sqrt { \frac { c _ { \sigma } } { d _ { h } } } \sigma \left( f ^ { ( h ) } ( \boldsymbol { x } ) \right) \in \mathbb { R } ^ { d _ { h } } , \qquad h = 1 , 2 , \ldots , L ,
+$$
+
+where $W ^ { ( h ) } \in \mathbb { R } ^ { d _ { h } \times d _ { h - 1 } }$ is the weight matrix in the $h$ -th layer $\mathit { h } \in [ L ] )$ , $\sigma : \mathbb { R }  \mathbb { R }$ is a coordinatewise activation function, and $c _ { \sigma }$ is a scaling factor.3 In this paper, for NN we will consider $\sigma$ being ReLU or ELU (Clevert et al., 2015) and for NTK we will only consider kernel functions induced by NNs with ReLU activation. The last layer of the neural network is
+
+$$
+\begin{array} { l } { { f ( w , x ) = f ^ { ( L + 1 ) } ( x ) = W ^ { ( L + 1 ) } \cdot g ^ { ( L ) } ( x ) \ ~ } } \\ { { = W ^ { ( L + 1 ) } \cdot \sqrt { \frac { C _ { \sigma } } { d _ { L } } } \sigma \left( W ^ { ( L ) } \cdot \sqrt { \frac { C _ { \sigma } } { d _ { L - 1 } } } \sigma \left( W ^ { ( L - 1 ) } \cdot \cdot \sqrt { \frac { C _ { \sigma } } { d _ { 1 } } } \sigma \left( W ^ { ( 1 ) } x \right) \right) \right) , } } \end{array}
+$$
+
+where $W ^ { ( L + 1 ) } \in \mathbb { R } ^ { 1 \times d _ { L } }$ is the weights in the final layer, and we let $w = \left( W ^ { ( 1 ) } , \dots , W ^ { ( L + 1 ) } \right)$ be all parameters in the network. All the weights are initialized to be i.i.d. $\mathcal { N } ( 0 , 1 )$ random variables. From now on, by NTK initialization we mean a neural network with parameterization defined in Equation 2 with all weighted being initialized to be i.i.d. $\mathcal { N } ( 0 , 1 )$ .
+
+When the hidden widths $d _ { 1 , } d _ { 2 } , \ldots , d _ { L }  \infty$ , certain limiting behavior emerges along the gradient trajectory. Let $x , x ^ { \prime } \in \mathbb { R } ^ { d }$ be two data points, the covariance kernel of the $h$ -th layer’s outputs, $\Sigma ^ { ( h ) } ( x , x ^ { \prime } ) = f ^ { ( h ) } ( x ) \cdot f ^ { ( h ) } ( x ^ { \prime } )$ , can be recursively defined in an analytical form:
+
+$$
+\begin{array} { r l } & { \Sigma ^ { ( 0 ) } ( x , x ^ { \prime } ) = x ^ { \top } x ^ { \prime } , } \\ & { \Lambda ^ { ( h ) } ( x , x ^ { \prime } ) = \left( \sum _ { \Sigma ^ { ( h - 1 ) } ( x ^ { \prime } , x ) } ^ { ( h - 1 ) } \ \Sigma ^ { ( h - 1 ) } ( x , x ^ { \prime } ) \right) \in \mathbb { R } ^ { 2 \times 2 } , } \\ & { \Sigma ^ { ( h ) } ( x , x ^ { \prime } ) = c _ { \sigma } \mathbb { E } _ { ( u , v ) \sim \mathcal { N } \left( 0 , \Lambda ^ { ( h ) } \right) } \left[ \sigma \left( u \right) \sigma \left( v \right) \right] , } \end{array}
+$$
+
+Table 1: Comparisons of different classifiers on $9 0 \ \mathrm { U C I }$ datasets. P90/P95: the number of datasets a classifier achieves $9 0 \% / 9 5 \%$ or more of the maximum accuracy, divided by the total number of datasets. PMA: average percentage of the maximum accuracy.   
+
+<table><tr><td rowspan=1 colspan=1>Classifier</td><td rowspan=1 colspan=1>Friedman Rank</td><td rowspan=1 colspan=1> Average Accuracy</td><td rowspan=1 colspan=1>P90</td><td rowspan=1 colspan=1>P95</td><td rowspan=1 colspan=1>PMA</td></tr><tr><td rowspan=1 colspan=1>NTK</td><td rowspan=1 colspan=1>28.34</td><td rowspan=1 colspan=1>81.95%±14.10%</td><td rowspan=1 colspan=1>88.89%</td><td rowspan=1 colspan=1>72.22%</td><td rowspan=1 colspan=1>95.72% ±5.17%</td></tr><tr><td rowspan=1 colspan=1>NN (He init)</td><td rowspan=1 colspan=1>40.97</td><td rowspan=1 colspan=1>80.88%±14.96%</td><td rowspan=1 colspan=1>81.11%</td><td rowspan=1 colspan=1>65.56%</td><td rowspan=1 colspan=1>94.34% ±7.22%</td></tr><tr><td rowspan=1 colspan=1>NN (NTK init)</td><td rowspan=1 colspan=1>38.06</td><td rowspan=1 colspan=1>81.02%±14.47%</td><td rowspan=1 colspan=1>85.56%</td><td rowspan=1 colspan=1>60.00%</td><td rowspan=1 colspan=1>94.55% ±5.89%</td></tr><tr><td rowspan=1 colspan=1>RF</td><td rowspan=1 colspan=1>33.51</td><td rowspan=1 colspan=1>81.56% ±13.90%</td><td rowspan=1 colspan=1>85.56%</td><td rowspan=1 colspan=1>67.78%</td><td rowspan=1 colspan=1>95.25% ±5.30%</td></tr><tr><td rowspan=1 colspan=1>Gaussian Kernel</td><td rowspan=1 colspan=1>35.76</td><td rowspan=1 colspan=1>81.03% ± 15.09%</td><td rowspan=1 colspan=1>85.56%</td><td rowspan=1 colspan=1>72.22%</td><td rowspan=1 colspan=1>94.56% ±8.22%</td></tr><tr><td rowspan=1 colspan=1>Polynomial Kernel</td><td rowspan=1 colspan=1>38.44</td><td rowspan=1 colspan=1>78.21% ± 20.30%</td><td rowspan=1 colspan=1>80.00%</td><td rowspan=1 colspan=1>62.22%</td><td rowspan=1 colspan=1>91.29% ±18.05%</td></tr></table>
+
+for $h \in [ L ]$ . Crucially, this analytical form holds not only at the initialization, but also holds during the training (when gradient descent with small learning rate is used as the optimization routine).
+
+Formally, NTK is defined as the limiting gradient kernel
+
+$$
+\Theta \left( x , x ^ { \prime } \right) \triangleq \left. \frac { \partial f ( w , x ) } { \partial w } , \frac { \partial f ( w , x ^ { \prime } ) } { \partial w } \right. = \sum _ { h = 1 } ^ { L + 1 } \left. \frac { \partial f ( w , x ) } { \partial W ^ { ( h ) } } , \frac { \partial f ( w , x ^ { \prime } ) } { \partial W ^ { ( h ) } } \right. .
+$$
+
+Again, one can obtain a recursive formula
+
+$$
+\begin{array} { l } { \displaystyle \dot { \Sigma } ^ { ( h ) } ( x , x ^ { \prime } ) = c _ { \sigma } \mathbb { E } _ { ( u , v ) \sim \mathcal { N } \left( 0 , \Lambda ^ { ( h ) } \right) } \left[ \dot { \sigma } ( u ) \dot { \sigma } ( v ) \right] , h = 1 , \ldots , L + 1 } \\ { \displaystyle \Theta ( x , x ^ { \prime } ) = \sum _ { h = 1 } ^ { L + 1 } \left( \Sigma ^ { ( h - 1 ) } ( x , x ^ { \prime } ) \cdot \prod _ { h ^ { \prime } = h } ^ { L + 1 } \dot { \Sigma } ^ { ( h ^ { \prime } ) } ( x , x ^ { \prime } ) \right) , } \end{array}
+$$
+
+where we let $\dot { \Sigma } ^ { ( L + 1 ) } ( x , x ^ { \prime } ) = 1$ for convenience. It is easy to check if we fix the first $L ^ { \prime }$ layers and only train the remaining $( L + 1 - L ^ { \prime } )$ layers, then the resulting NTK is $\Theta ( x , x ^ { \prime } ) ~ =$ $\begin{array} { r } { \sum _ { h = L ^ { \prime } + 1 } ^ { L + 1 } \Big ( \Sigma ^ { ( h - 1 ) } ( x , x ^ { \prime } ) \cdot \prod _ { h ^ { \prime } = h } ^ { L + 1 } \dot { \Sigma } ^ { ( h ^ { \prime } ) } ( x , x ^ { \prime } ) \Big ) } \end{array}$ . Note when $L ^ { \prime } = L$ , then the resulting NTK is $\Sigma ^ { ( L ) } \left( x , x ^ { \prime } \right)$ , which is the NNGP kernel (Lee et al., 2018). $L ^ { \prime }$ can be viewed as a hyperparameter of NTK classifier, and in our UCI experiment we tune $L ^ { \prime }$ . Given a kernel function, one can directly use it for downstream classification tasks (Scholkopf & Smola, 2001).
+
+# 4 EXPERIMENTS ON UCI DATASETS
+
+In this section, we present our experimental results on UCI datasets which follow the setup of Fernandez-Delgado et al. (2014) with extensive comparisons of classifiers, including random for- ´ est, kernel SVM, multilayer neural networks, etc. Section 4.2 discusses the performance of NTK through detailed comparisons with other classifiers tested by Fernandez-Delgado et al. (2014). Sec- ´ tion 4.2 compares NTK classifier and the corresponding NN classifier and verifies how similar their predictions are. See Table 6 in Appendix A for a summary of datasets we used. The detailed experiment setup, including the choices the datasets, training / test splitting and ranges of hyperparameters, is described in Appendix A. We note that usual methods of obtaining confidence bounds in these low-data settings are somewhat heuristic.
+
+# 4.1 OVERALL PERFORMANCE COMPARISONS
+
+Table 1 lists the performance of 6 classifiers under various metrics: the three top classifiers identified in Fernandez-Delgado et al. (2014), namely, RF, Gaussian kernel and polynomial kernel, along ´ with our new methods NTK, NN with He initialization and NN with NTK initialization. Table 1 shows NTK is the best classifier under all metrics, followed by RF, the best classifier identified in Fernandez-Delgado et al. (2014). Now we interpret each metric in more details. ´
+
+Friedman Ranking and Average Accuracy. NTK is the best (Friedman Rank 28.34, Average Accuracy $8 1 . 9 5 \%$ ), followed by RF (Friedman Rank 33.51, Average Accuracy $8 1 . 5 6 \%$ ) and then followed by SVM with Gaussian kernel (Friedman Rank 35.76, Average Accuracy $8 1 . 0 3 \%$ ). The difference between NTK and RF is significant (-5.17 in Friedman Rank and $+ 0 . 3 9 \%$ in Average Accuracy), just as the superiority of RF is significant compared to other classifiers as claimed in Fernandez-Delgado et al. (2014). NN (with either He initialization or NTK initialization) per- ´ forms significantly better than the polynomial kernel in terms of the Average Accuracy $( 8 0 . 8 8 \%$ and $8 1 . 0 2 \%$ vs. $7 8 . 2 1 \%$ ), but in terms of Friedman Rank, NN with He initialization is worse than polynomial kernel (40.97 vs. 38.44) and NN with NTK initialization is slightly better than polynomial kernel (38.06 vs. 38.44). On many datasets where most classifiers have similar performances, NN’s rank is high as well, whereas on other datasets, NN is significantly better than most classifiers, including SVM with polynomial kernel. Therefore, NN enjoys higher Average Accuracy but suffers higher Friedman Rank. For example, on OZONE dataset, NN with NTK initialization is only $0 . 2 5 \%$ worse than polynomial kernel but their difference in terms of rank is 56. It is also interesting to see that NN with NTK initialization performs better than NN with He initialization (38.06 vs. 40.97 in Friedman Rank and $8 1 . 0 2 \%$ vs. $8 0 . 8 8 \%$ in Average Accuracy).
+
+P90/P95 and PMA. This measures, for a given classifier, the fraction of datasets on which it achieves more than $9 0 \% / 9 5 \%$ of the maximum accuracy among all classifiers. NTK is one of the best classifiers (ties with Gaussian kernel on P95), which shows NTK can consistently achieve superior classification performance across a broad range of datasets. Lastly, we consider the Percentage of the Maximum Accuracy (PMA). NTK achieves the best average PMA followed by RF whose PMA is $0 . 4 7 \%$ below that of NTK and other classifiers are all below $9 4 . 6 \%$ . An interesting observation is that NTK, NN with NTK initialization and RF have small standard deviation $5 . 1 7 \%$ , $5 . 8 9 \%$ and $5 . 3 0 \%$ ) whereas all other classifiers have much larger standard deviation.
+
+# 4.2 PAIRWISE COMPARISONS
+
+NTK vs. RF. In Figure 1a, we compare NTK with RF. There are 42 datasets that NTK outperforms RF and 40 datasets that RF outperforms NTK.4 The mean difference is $2 . 5 1 \%$ , which is statistically significant by a Wilcoxon signed rank test. We see NTK and RF perform similarly when the Bayes error rate is low with NTK being slightly better. There are some exceptions. For example, on the BALANCE-SCALE dataset, NTK achieves $98 \%$ accuracy whereas RF only achieves $8 4 . 1 \%$ . When the Bayes error rate is high, either classifier can be significantly better than the other.
+
+NTK vs. Gaussian Kernel. The gap between NTK and Gaussian kernel is more significant. As shown in Figure 1b, NTK generally performs better than Gaussian kernel no matter Bayes error rate is low or high. There are 43 datasets that NTK outperforms Gaussian kernel and there are 34 datasets that Gaussian kernel outperforms NTK. The mean difference is $2 . 2 2 \%$ , which is also statistically significant by a Wilcoxon signed rank test. On BALLOONS, HEART-SWITZERLAND, PITTSBURGBRIDGES-MATERIAL, TEACHING, TRAINS datasets, NTK is better than Gaussian kernel by at least $11 \%$ in terms of accuracy. These five datasets all have less than 200 samples, which shows that NTK can perform much better than Gaussian kernel when the number of samples is small.
+
+NTK vs. NN In this section we compare NTK with NN. The goals are (i) comparing the performance and (ii) verifying NTK is a good approximation to NN.
+
+In Figure 2a, we compare the performance between NTK and NN with He initialization. For most datasets, these two classifiers perform similarly. However, there are a few datasets that NTK performs significantly better. There are 50 datasets that NTK outperforms NN with He initialization and there are 27 datasets that NN with He initialization outperforms NTK. The mean difference is $1 . 9 6 \%$ , which is also statistically significant by a Wilcoxon signed rank test.
+
+In Figure 2b, we compare the performance between NTK and NN with NTK initialization. Recall that for NN with NTK initialization, when the width goes to infinity, the predictor is just NTK.
+
+![](images/d126f2441334a9de8d571de9748b55c004d12bf48f4fd1f950caf6db87fec70a.jpg)  
+Figure 1: Performance comparisons between NTK and other classifiers.
+
+![](images/43526e3362e86e82c4736a424b1b71f3926b21a9d95fac8e2356d9024dc966ba.jpg)  
+Figure 2: Performance Comparisons between NTK and NN.
+
+Therefore, we expect these two predictors give similar performance. Figure 2b verifies our conjecture. There is no dataset the one classifier is significantly better than the other. We do not have the same observation in Figure 1a, 1b or 2a. Nevertheless, NTK often performs better than NN. There are 52 datasets that NTK outperforms NN with He initialization and there are 25 datasets NN with He initialization outperforms NTK. The mean difference is $1 . 5 4 \%$ , which is also statistically significant by a Wilcoxon signed rank test.
+
+# 5 EXPERIMENTS ON SMALL CIFAR-10 DATASET
+
+In this section, we study the performance of CNTK on subsampled CIFAR-10 dataset. We randomly choose $n$ samples from CIFAR-10 training set, use them to train CNTK and ResNet-34, and test both classifiers on the whole test set. In our experiments, we vary $n$ from 10 to 1280, and the number of convolutional layers of CNTK varies from 5 to 14. See Appendix B for detailed experiment setup.
+
+The results are reported in Table 2. It can be observed that in this setting CNTK consistently outperforms ResNet-34. The largest gap occurs at $n = 3 2 0$ where 14-layer CNTK achieves $3 \dot { 6 } . 5 7 \%$ accuracy and ResNet achieves $3 3 . 1 5 \%$ . The smallest improvement occurs at $n = 1 0$ : $1 5 . 3 3 \%$ vs.
+
+Table 2: Performance of ResNet-34 and CNTK on small CIFAR-10 Dataset.   
+
+<table><tr><td rowspan=1 colspan=1>n</td><td rowspan=1 colspan=1>ResNet</td><td rowspan=1 colspan=1>5-layer CNTK</td><td rowspan=1 colspan=1>8-layer CNTK</td><td rowspan=1 colspan=1>11-layer CNTK</td><td rowspan=1 colspan=1>14-layer CNTK</td></tr><tr><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>14.59% ±1.99%</td><td rowspan=1 colspan=1>15.08%±2.43%</td><td rowspan=1 colspan=1>15.24% ± 2.44%</td><td rowspan=1 colspan=1>15.31% ±2.38%</td><td rowspan=1 colspan=1>15.33% ±2.43%</td></tr><tr><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>17.50% ± 2.47%</td><td rowspan=1 colspan=1>18.03% ±1.91%</td><td rowspan=1 colspan=1>18.50% ±2.03%</td><td rowspan=1 colspan=1>18.69% ±2.07%</td><td rowspan=1 colspan=1>18.79% ±2.13%</td></tr><tr><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>19.52% ±1.39%</td><td rowspan=1 colspan=1>20.83% ±1.68%</td><td rowspan=1 colspan=1>21.07% ±1.80%</td><td rowspan=1 colspan=1>21.23% ±1.86%</td><td rowspan=1 colspan=1>21.34% ± 1.91%</td></tr><tr><td rowspan=1 colspan=1>80</td><td rowspan=1 colspan=1>23.32% ±1.61%</td><td rowspan=1 colspan=1>24.82% ±1.75%</td><td rowspan=1 colspan=1>25.18% ±1.80%</td><td rowspan=1 colspan=1>25.40% ±1.84%</td><td rowspan=1 colspan=1>25.48% ± 1.91%</td></tr><tr><td rowspan=1 colspan=1>160</td><td rowspan=1 colspan=1>28.30% ±1.38%</td><td rowspan=1 colspan=1>29.63% ±1.13%</td><td rowspan=1 colspan=1>30.17% ±1.11%</td><td rowspan=1 colspan=1>30.46% ±1.15%</td><td rowspan=1 colspan=1>30.48% ± 1.17%</td></tr><tr><td rowspan=1 colspan=1>320</td><td rowspan=1 colspan=1>33.15%±1.20%</td><td rowspan=1 colspan=1>35.26% ± 0.97%</td><td rowspan=1 colspan=1>36.05% ± 0.92%</td><td rowspan=1 colspan=1>36.44% ± 0.91%</td><td rowspan=1 colspan=1>36.57% ± 0.88%</td></tr><tr><td rowspan=1 colspan=1>640</td><td rowspan=1 colspan=1>41.66% ±1.09%</td><td rowspan=1 colspan=1>41.24%± 0.78%</td><td rowspan=1 colspan=1>42.10% ± 0.74%</td><td rowspan=1 colspan=1>42.44% ± 0.72%</td><td rowspan=1 colspan=1>42.63% ± 0.68%</td></tr><tr><td rowspan=1 colspan=1>1280</td><td rowspan=1 colspan=1>49.14% ± 1.31%</td><td rowspan=1 colspan=1>47.21% ± 0.49%</td><td rowspan=1 colspan=1>48.22% ± 0.49%</td><td rowspan=1 colspan=1>48.67% ± 0.57%</td><td rowspan=1 colspan=1>48.86% ± 0.68%</td></tr></table>
+
+Table 3: Performance of linear SVM and CNTK with different number of convolutional layers on the 11-20th classes in VOC07. The cost value $C$ is tuned on the first 10 classes. Feature extracted from conv5 in ResNet-50.   
+
+<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linear SVM</td><td rowspan=1 colspan=1>1-layerCNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layer CNTK</td><td rowspan=1 colspan=1>4-layer CNTK</td><td rowspan=1 colspan=1>5-layer CNTK</td><td rowspan=1 colspan=1>6-layerCNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>49.46 ± 4.71</td><td rowspan=1 colspan=1>50.04± 4.28</td><td rowspan=1 colspan=1>49.94 ± 4.19</td><td rowspan=1 colspan=1>49.66 ± 4.19</td><td rowspan=1 colspan=1>49.24 ± 4.20</td><td rowspan=1 colspan=1>48.66± 4.26</td><td rowspan=1 colspan=1>48.00± 4.41</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>64.39 ± 5.02</td><td rowspan=1 colspan=1>65.81 ±5.28</td><td rowspan=1 colspan=1>65.89±5.23</td><td rowspan=1 colspan=1>65.65± 5.17</td><td rowspan=1 colspan=1>65.30±5.10</td><td rowspan=1 colspan=1>64.76±5.07</td><td rowspan=1 colspan=1>64.12 ±5.00</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>70.38 ±3.28</td><td rowspan=1 colspan=1>71.60 ± 3.51</td><td rowspan=1 colspan=1>71.56± 3.65</td><td rowspan=1 colspan=1>71.32 ± 3.74</td><td rowspan=1 colspan=1>70.96 ± 3.84</td><td rowspan=1 colspan=1>70.51 ± 3.92</td><td rowspan=1 colspan=1>69.98±3.95</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>72.43± 3.21</td><td rowspan=1 colspan=1>73.65±3.29</td><td rowspan=1 colspan=1>73.59 ±3.28</td><td rowspan=1 colspan=1>73.35 ± 3.23</td><td rowspan=1 colspan=1>73.00±3.20</td><td rowspan=1 colspan=1>72.56± 3.20</td><td rowspan=1 colspan=1>72.09±3.20</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>74.49 ±3.08</td><td rowspan=1 colspan=1>75.87 ± 2.88</td><td rowspan=1 colspan=1>75.69± 2.89</td><td rowspan=1 colspan=1>75.45± 2.89</td><td rowspan=1 colspan=1>75.13 ± 2.85</td><td rowspan=1 colspan=1>74.68 ± 2.88</td><td rowspan=1 colspan=1>74.24 ± 2.85</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>76.59 ± 2.15</td><td rowspan=1 colspan=1>77.64± 2.37</td><td rowspan=1 colspan=1>77.57 ± 2.43</td><td rowspan=1 colspan=1>77.39 ± 2.50</td><td rowspan=1 colspan=1>77.13 ± 2.56</td><td rowspan=1 colspan=1>76.87± 2.60</td><td rowspan=1 colspan=1>76.52 ± 2.63</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>77.11 ± 1.25</td><td rowspan=1 colspan=1>78.64± 1.35</td><td rowspan=1 colspan=1>78.57 ± 1.39</td><td rowspan=1 colspan=1>78.39 ± 1.41</td><td rowspan=1 colspan=1>78.12 ± 1.42</td><td rowspan=1 colspan=1>77.80 ± 1.41</td><td rowspan=1 colspan=1>77.43 ± 1.41</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>79.57± 0.87</td><td rowspan=1 colspan=1>80.42±0.88</td><td rowspan=1 colspan=1>80.38±0.86</td><td rowspan=1 colspan=1>80.23±0.86</td><td rowspan=1 colspan=1>79.99 ±0.88</td><td rowspan=1 colspan=1>79.72 ± 0.89</td><td rowspan=1 colspan=1>79.41 ± 0.92</td></tr></table>
+
+$1 4 . 5 9 \%$ . When the number of training data is large, i.e., $n = 1 2 8 0$ , ResNet can outperform CNTK. It is also interesting to see that 14-layer CNTK is the best performing CNTK for all values of $n$ , suggesting depth has a significant effect on this task.
+
+# 6 EXPERIMENTS ON FEW-SHOT LEARNING
+
+In this section, we test the ability of NTK as a drop-in classifier to replace the linear classifier in the few-shot learning setting. Using linear SVM on top of extracted features is arguably the most widely used strategy in few-shot learning as linear SVM is easy and fast to train whereas more complicated strategies like fine-tuning or training a neural network on top of the extracted feature have more randomness and may overfit due to the small size of the training set. NTK has the same benefits (easy and fast to train, no randomness) as the linear classifier but also allows some non-linearity in the design. Note since we consider image classification tasks, we use CNTK in experiments in this section.
+
+Experiment Setup. We mostly follow the settings of Goyal et al. (2019). Features are extracted from layer conv1, conv2, conv3, conv4, conv5 in ResNet-50 (He et al., 2016) trained on ImageNet (Deng et al., 2009), and we use these features for VOC07 classification task. The number of positive examples $k$ varies from 1 to 8. For each value of $k$ , for each class in the VOC07 dataset, we choose $k$ positive examples and $1 9 k$ negative examples. For each $k \in \{ 1 , 2 , \dots , 8 \}$ and for each class, we randomly choose 10 independent sets with $2 0 k$ training samples in each set. We report the mean and standard deviation of mAP on the test split of VOC07 dataset. This setting has been used in numbers of previous few-shot learning papers (Goyal et al., 2019; Zhang et al., 2017).
+
+Table 4: Performance of linear SVM and CNTK with different number of convolutional layers on the 11-20th classes in VOC07. The cost value $C$ is tuned on the first 10 classes. Feature extracted from conv4 in ResNet-50.   
+
+<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linear SVM</td><td rowspan=1 colspan=1>1-layer CNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layer CNTK</td><td rowspan=1 colspan=1>4-layerCNTK</td><td rowspan=1 colspan=1>5-layerCNTK</td><td rowspan=1 colspan=1>6-layerCNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>23.25± 2.89</td><td rowspan=1 colspan=1>24.54 ± 3.39</td><td rowspan=1 colspan=1>24.57± 3.46</td><td rowspan=1 colspan=1>24.57 ± 3.50</td><td rowspan=1 colspan=1>24.52 ± 3.51</td><td rowspan=1 colspan=1>24.43± 3.50</td><td rowspan=1 colspan=1>24.34±3.49</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>32.26 ±3.85</td><td rowspan=1 colspan=1>33.81± 4.25</td><td rowspan=1 colspan=1>33.77 ± 4.33</td><td rowspan=1 colspan=1>33.68± 4.40</td><td rowspan=1 colspan=1>33.53± 4.46</td><td rowspan=1 colspan=1>33.31 ± 4.51</td><td rowspan=1 colspan=1>33.04± 4.55</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>38.41 ± 2.74</td><td rowspan=1 colspan=1>40.06± 2.70</td><td rowspan=1 colspan=1>40.10 ± 2.47</td><td rowspan=1 colspan=1>40.02 ± 2.41</td><td rowspan=1 colspan=1>39.87 ± 2.37</td><td rowspan=1 colspan=1>39.65± 2.35</td><td rowspan=1 colspan=1>39.36± 2.29</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>39.88± 2.08</td><td rowspan=1 colspan=1>42.20±1.78</td><td rowspan=1 colspan=1>41.80 ± 2.13</td><td rowspan=1 colspan=1>42.64± 2.00</td><td rowspan=1 colspan=1>42.51 ± 2.02</td><td rowspan=1 colspan=1>42.32 ± 2.03</td><td rowspan=1 colspan=1>42.08± 2.03</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>42.46 ± 2.70</td><td rowspan=1 colspan=1>44.71± 3.12</td><td rowspan=1 colspan=1>44.69 ± 2.98</td><td rowspan=1 colspan=1>44.72 ± 3.01</td><td rowspan=1 colspan=1>44.69 ± 3.04</td><td rowspan=1 colspan=1>44.57 ± 3.07</td><td rowspan=1 colspan=1>44.40±3.10</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>45.71 ± 3.27</td><td rowspan=1 colspan=1>49.02 ±2.63</td><td rowspan=1 colspan=1>48.63± 2.69</td><td rowspan=1 colspan=1>48.97±2.68</td><td rowspan=1 colspan=1>48.92 ± 2.71</td><td rowspan=1 colspan=1>48.79 ± 2.74</td><td rowspan=1 colspan=1>48.63± 2.77</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>47.97 ± 3.25</td><td rowspan=1 colspan=1>50.89±3.50</td><td rowspan=1 colspan=1>50.43±3.48</td><td rowspan=1 colspan=1>50.50±3.47</td><td rowspan=1 colspan=1>50.48 ±3.44</td><td rowspan=1 colspan=1>50.39 ±3.44</td><td rowspan=1 colspan=1>50.24±3.41</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>49.81 ± 2.18</td><td rowspan=1 colspan=1>52.32± 2.65</td><td rowspan=1 colspan=1>52.06± 2.32</td><td rowspan=1 colspan=1>52.23 ± 2.38</td><td rowspan=1 colspan=1>52.31 ± 2.40</td><td rowspan=1 colspan=1>52.26± 2.45</td><td rowspan=1 colspan=1>52.14± 2.50</td></tr></table>
+
+Table 5: Performance of linear SVM and CNTK with different number of convolutional layers on the 11-20th classes in VOC07. The cost value $C$ is tuned on the first 10 classes. Feature extracted from conv3. in ResNet-50   
+
+<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linear SVM</td><td rowspan=1 colspan=1>1-layer CNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layerCNTK</td><td rowspan=1 colspan=1>4-layer CNTK</td><td rowspan=1 colspan=1>5-layerCNTK</td><td rowspan=1 colspan=1>6-layerCNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>14.38 ± 1.98</td><td rowspan=1 colspan=1>15.42 ± 1.90</td><td rowspan=1 colspan=1>15.36 ± 1.93</td><td rowspan=1 colspan=1>15.30 ± 1.97</td><td rowspan=1 colspan=1>15.24 ± 2.02</td><td rowspan=1 colspan=1>15.17 ± 2.05</td><td rowspan=1 colspan=1>15.10± 2.08</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>16.67 ± 1.50</td><td rowspan=1 colspan=1>18.48 ± 1.58</td><td rowspan=1 colspan=1>18.39 ± 1.55</td><td rowspan=1 colspan=1>18.28 ± 1.52</td><td rowspan=1 colspan=1>18.18 ± 1.49</td><td rowspan=1 colspan=1>18.09 ± 1.47</td><td rowspan=1 colspan=1>17.99 ± 1.45</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>19.56 ± 1.04</td><td rowspan=1 colspan=1>21.79±1.69</td><td rowspan=1 colspan=1>21.73 ± 1.63</td><td rowspan=1 colspan=1>21.67 ± 1.60</td><td rowspan=1 colspan=1>21.61 ± 1.55</td><td rowspan=1 colspan=1>21.54± 1.50</td><td rowspan=1 colspan=1>21.47 ± 1.46</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>20.53 ± 1.62</td><td rowspan=1 colspan=1>23.39±2.13</td><td rowspan=1 colspan=1>23.36±2.18</td><td rowspan=1 colspan=1>23.29 ± 2.22</td><td rowspan=1 colspan=1>23.20± 2.24</td><td rowspan=1 colspan=1>23.11 ± 2.26</td><td rowspan=1 colspan=1>23.03 ± 2.27</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>21.51 ± 1.68</td><td rowspan=1 colspan=1>25.13±2.36</td><td rowspan=1 colspan=1>25.09± 2.37</td><td rowspan=1 colspan=1>25.03± 2.36</td><td rowspan=1 colspan=1>24.96± 2.36</td><td rowspan=1 colspan=1>24.87±2.36</td><td rowspan=1 colspan=1>24.78 ± 2.37</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>23.51 ± 2.39</td><td rowspan=1 colspan=1>26.51± 2.39</td><td rowspan=1 colspan=1>26.26±2.43</td><td rowspan=1 colspan=1>26.10± 2.47</td><td rowspan=1 colspan=1>25.97 ± 2.47</td><td rowspan=1 colspan=1>25.87 ± 2.47</td><td rowspan=1 colspan=1>25.78± 2.46</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>24.33 ± 1.59</td><td rowspan=1 colspan=1>27.98 ±2.46</td><td rowspan=1 colspan=1>27.75± 2.52</td><td rowspan=1 colspan=1>28.24±2.22</td><td rowspan=1 colspan=1>28.20±2.22</td><td rowspan=1 colspan=1>28.14 ± 2.21</td><td rowspan=1 colspan=1>28.08± 2.21</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>25.31 ± 2.07</td><td rowspan=1 colspan=1>27.76 ± 2.87</td><td rowspan=1 colspan=1>28.52 ± 2.65</td><td rowspan=1 colspan=1>28.52 ± 2.69</td><td rowspan=1 colspan=1>28.49± 2.70</td><td rowspan=1 colspan=1>28.44± 2.72</td><td rowspan=1 colspan=1>28.38 ± 2.74</td></tr></table>
+
+We take the extracted features as the input to CNTK. We tried CNTK with 0-6 convolution layers, a global average pooling layer and a fully-connected layer. We normalize the data in the feature space, and finally use SVM to train the classifiers. Note without the convolution layer, it is equivalent to directly applying linear SVM after global average pooling and normalization. We use sklearn.svm.LinearSVC to train linear SVMs, and sklearn.svm.SVC to train kernel SVMs (for CNTK). To train SVM, we choose the cost value $C$ from $2 ^ { [ - 1 9 , - 4 ] } \cup 1 0 ^ { [ - 7 , 2 ] }$ and set the class weight ratio to be $2 : 1$ for positive/negative classes as in Goyal et al. (2019).
+
+Since the number of given samples is usually small, Goyal et al. (2019) chooses to report the performance of the best cost value $C$ . In our experiments, we use a more standard method to perform cross-validation. We use the first 10 classes of VOC07 to tune $C$ , and report the performance of selected $C$ in the other 10 classes in Table 3-5. We also report the performance of the best $C$ as in Goyal et al. (2019), in Tables 7-9 in the appendix for completeness.
+
+Discussions. First, we find CNTK is a strong drop-in classifier to replace the linear classifier in few-shot learning. Tables 3-9 clearly demonstrate that CNTK is consistently better than linear classifier. Note Tables 3-9 only show the prediction accuracy in an average sense. In fact, we find that on every randomly sampled training set, CNTK always gives a better performance. We conjecture that CNTK gives better performance because the non-linearity in CNTK helps prediction. This is verified by looking at the performance gain of CNTK for different feature extractors. Note Conv5 is often considered to be most useful for linear classification. There, CNTK only outperforms the linear classifier by about $1 \%$ . On the other hand, with Conv3 and Conv4 features, CNTK can outperform linear classifier by around $2 \%$ and sometimes by $3 \% { - } 4 \%$ (last two rows in Table 5). We believe this happens because Conv3 and Conv4 correspond to middle-level features, and thus non-linearity is indeed beneficial for better accuracy.
+
+We also observe that CNTK often performs the best with a single convolutional layer. This is expected since features extracted by ResNet-50 already produce a good representation. Nevertheless, we find for middle-level features Conv3 with $k = 7$ or 8, CNTK with 3 convolutional layers give the best performance. We believe this is because with more data, utilizing the non-linearity induced by CNTK can further boost the performance.
+
+# 7 CONCLUSION
+
+The Neural Tangent Kernel, discovered by mathematical curiosity about deep networks in the limit of infinite width, is found to yield superb performance on low-data tasks, beating extensively tuned versions of classic methods such as random forests. The (fully-connected) NTK classifiers are easy to compute (no GPU required) and thus should be a good off-the-shelf classifier in many settings. We plan to release our code to allow drop-in replacement for SVMs and linear regression.
+
+Many theoretical questions arise. Do NTK SVMs correspond to some infinite net architecture (as NTK ridge regression does)? What explains generalization in small-data settings? (This understanding is imperfect even for random forests.) Finally one can derive NTK corresponding to other architectures, e.g., recurrent neural tangent kernel (RNTK) induced by recurrent neural networks. It would be an interesting future research direction to test their performance on benchmark tasks.
+
+# ACKNOWLEDGMENTS
+
+S. Arora, Z. Li and D. Yu are supported by NSF, ONR, Simons Foundation, Schmidt Foundation, Amazon Research, DARPA and SRC. S. S. Du is supported by National Science Foundation (Grant No. DMS-1638352) and the Infosys Membership. R. Salakhutdinov and R. Wang are supported in part by NSF IIS-1763562, AFRL CogDeCON FA875018C0014, and DARPA SAGAMORE HR00111990016. Part of this work was done while S. S. Du was visiting Google Brain Princeton and R. Wang was visiting Princeton University. The authors would like to thank Amazon Web Services for providing compute time for the experiments in this paper, and NVIDIA for GPU support. We thank Priya Goyal for providing experiment details of Goyal et al. (2019). We thank Xiaolong Wang for discussing the few-shot learning task.
+
+# REFERENCES
+
+Zeyuan Allen-Zhu, Yuanzhi Li, and Yingyu Liang. Learning and generalization in overparameterized neural networks, going beyond two layers. arXiv preprint arXiv:1811.04918, 2018a.
+
+Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via overparameterization. arXiv preprint arXiv:1811.03962, 2018b.
+
+Sanjeev Arora, Simon S Du, Wei Hu, Zhiyuan Li, Ruslan Salakhutdinov, and Ruosong Wang. On exact computation with an infinitely wide neural net. arXiv preprint arXiv:1904.11955, 2019a.
+
+Sanjeev Arora, Simon S Du, Wei Hu, Zhiyuan Li, and Ruosong Wang. Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks. arXiv preprint arXiv:1901.08584, 2019b.
+
+Alberto Bietti and Julien Mairal. On the inductive bias of neural tangent kernels. arXiv preprint arXiv:1905.12173, 2019.
+
+Youngmin Cho and Lawrence K Saul. Kernel methods for deep learning. In Advances in neural information processing systems, pp. 342–350, 2009.
+
+Djork-Arne Clevert, Thomas Unterthiner, and Sepp Hochreiter. Fast and accurate deep network ´ learning by exponential linear units (elus). arXiv preprint arXiv:1511.07289, 2015.
+
+Amit Daniely. SGD learns the conjugate kernel class of the network. arXiv preprint arXiv:1702.08503, 2017.
+
+Amit Daniely, Roy Frostig, and Yoram Singer. Toward deeper understanding of neural networks: The power of initialization and a dual view on expressivity. In Advances In Neural Information Processing Systems, pp. 2253–2261, 2016.
+
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248–255. Ieee, 2009.
+
+Simon S Du, Jason D Lee, Haochuan Li, Liwei Wang, and Xiyu Zhai. Gradient descent finds global minima of deep neural networks. arXiv preprint arXiv:1811.03804, 2018a.
+
+Simon S Du, Xiyu Zhai, Barnabas Poczos, and Aarti Singh. Gradient descent provably optimizes over-parameterized neural networks. arXiv preprint arXiv:1810.02054, 2018b.
+
+Simon S. Du, Kangcheng Hou, Barnabas P ´ oczos, Ruslan Salakhutdinov, Ruosong Wang, and Keyulu ´ Xu. Graph neural tangent kernel: Fusing graph neural networks with graph kernels. ArXiv, abs/1905.13192, 2019.
+
+Manuel Fernandez-Delgado, Eva Cernadas, Sen ´ en Barro, and Dinani Amorim. Do we need hun- ´ dreds of classifiers to solve real world classification problems? The Journal of Machine Learning Research, 15(1):3133–3181, 2014.
+
+Adria Garriga-Alonso, Carl Edward Rasmussen, and Laurence Aitchison. Deep convolutional net- \` works as shallow gaussian processes. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id $=$ Bklfsi0cKm.
+
+Priya Goyal, Dhruv Mahajan, Abhinav Gupta, and Ishan Misra. Scaling and benchmarking selfsupervised visual representation learning. arXiv preprint arXiv:1905.01235, 2019.
+
+Tamir Hazan and Tommi Jaakkola. Steps toward deep kernel methods from infinite neural networks. arXiv preprint arXiv:1508.05133, 2015.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Arthur Jacot, Franck Gabriel, and Clement Hongler. Neural tangent kernel: Convergence and gen- ´ eralization in neural networks. In Advances in neural information processing systems, pp. 8571– 8580, 2018.
+
+Jaehoon Lee, Jascha Sohl-dickstein, Jeffrey Pennington, Roman Novak, Sam Schoenholz, and Yasaman Bahri. Deep neural networks as gaussian processes. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ B1EA-M-0Z.
+
+Jaehoon Lee, Lechao Xiao, Samuel S Schoenholz, Yasaman Bahri, Jascha Sohl-Dickstein, and Jeffrey Pennington. Wide neural networks of any depth evolve as linear models under gradient descent. arXiv preprint arXiv:1902.06720, 2019.
+
+Yuanzhi Li and Yingyu Liang. Learning overparameterized neural networks via stochastic gradient descent on structured data. arXiv preprint arXiv:1808.01204, 2018.
+
+Alexander G de G Matthews, Mark Rowland, Jiri Hron, Richard E Turner, and Zoubin Ghahramani. Gaussian process behaviour in wide deep neural networks. arXiv preprint arXiv:1804.11271, 2018.
+
+Radford M Neal. Priors for infinite networks. In Bayesian Learning for Neural Networks, pp. 29–53. Springer, 1996.
+
+Roman Novak, Lechao Xiao, Yasaman Bahri, Jaehoon Lee, Greg Yang, Daniel A. Abolafia, Jeffrey Pennington, and Jascha Sohl-dickstein. Bayesian deep convolutional networks with many channels are gaussian processes. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id $=$ B1g30j0qF7.
+
+Matthew Olson, Abraham Wyner, and Richard Berk. Modern neural networks generalize on small data sets. In Advances in Neural Information Processing Systems, pp. 3619–3628, 2018.
+
+Daniel S Park, Jascha Sohl-Dickstein, Quoc V Le, and Samuel L Smith. The effect of network width on stochastic gradient descent and generalization: an empirical study. arXiv preprint arXiv:1905.03776, 2019.
+
+Nicolas Le Roux and Yoshua Bengio. Continuous neural networks. In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, volume 2 of Proceedings of Machine Learning Research, pp. 404–411, San Juan, Puerto Rico, 2007.
+
+Bernhard Scholkopf and Alexander J Smola. Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT press, 2001.
+
+Christopher KI Williams. Computing with infinite networks. In Advances in neural information processing systems, pp. 295–301, 1997.
+
+Greg Yang. Scaling limits of wide neural networks with weight sharing: Gaussian process behavior, gradient independence, and neural tangent kernel derivation. arXiv preprint arXiv:1902.04760, 2019.
+
+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.
+
+Richard Zhang, Phillip Isola, and Alexei A Efros. Split-brain autoencoders: Unsupervised learning by cross-channel prediction. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1058–1067, 2017.
+
+Difan Zou, Yuan Cao, Dongruo Zhou, and Quanquan Gu. Stochastic gradient descent optimizes over-parameterized deep ReLU networks. arXiv preprint arXiv:1811.08888, 2018.
+
+# A ADDITIONAL EXPERIMENTAL DETAILS ON UCI
+
+In this section, we describe our experiment setup.
+
+Dataset Selection Since we only wish to test on small datasets, we only select UCI datasets with number of samples smaller than 5000. Furthermore, we only use datasets without explicit training / testing splitting from the link: http://persoal.citius.usc.es/manuel.fernandez. delgado/papers/jmlr/data.tar.gz, which are the pre-processed datasets by Fernandez- ´ Delgado et al. (2014). These datasets are originally from UCI datasets (including most of the datasets before March, 2013) and 4 real-world datasets not included in the UCI repository (see Fernandez- ´ Delgado et al. (2014) for details). The datasets are pre-processed to be classification problems and all categorical features are turned into numerical features and normalized along the samples for each feature. The reason to discard datasets with explicit splitting is that we found there is an obvious distributional shift between training and testing data. The pre-processed data provided from the link shows that training and testing data are normalized with different mean and standard deviation. For example, on the AUDIOLOGY-STD dataset, the support of the first feature is $\{ - 0 . 7 6 1 5 2 6 , 1 . 3 0 5 4 7 \}$ in training and $\{ - 1 . 2 , 0 . 8 \}$ in testing; on the ANNEALING dataset, the support of the first feature is $\{ - 0 . 6 7 0 2 7 4 , - 0 . 2 2 3 4 2 5 , 3 . 7 9 8 2 2 \}$ in training and $\{ - 0 . 1 1 1 1 1 , 0 , 1 \}$ in testing. See Table 6 for a summary of the datasets.
+
+Table 6: Dataset Summary   
+
+<table><tr><td></td><td># samples</td><td>#features</td><td>#classes</td></tr><tr><td>min</td><td>10</td><td>3</td><td>2</td></tr><tr><td>25%</td><td>178</td><td>8</td><td>2</td></tr><tr><td>50%</td><td>583</td><td>16</td><td>3</td></tr><tr><td>75%</td><td>1022</td><td>32</td><td>6</td></tr><tr><td>max</td><td>5000</td><td>262</td><td>100</td></tr></table>
+
+Performance Comparison Details We follow the comparison setup in Fernandez-Delgado et al. ´ (2014) that we report 4-fold cross-validation. For hyperparameters, we tune them with the same validation methodology in Fernandez-Delgado et al. (2014): all available training samples are ran- ´ domly split into one training and one test set, while imposing that each class has the same number of training and test samples. Then the parameter with best validation accuracy is selected. It is possible to give confidence bounds for this parameter tuning scheme, but they are worse than standard ones for separated training/validation/testing data. For NTK and NN classifiers we train them on these 90 datasets. For other classifiers, we use the results from Fernandez-Delgado et al. (2014). ´
+
+NTK Specification We calculate NTK induced fully-connected neural networks with $L$ layers where $L ^ { \prime }$ bottom layers are fixed, and then use $C$ -support vector classification implemented by sklearn.svm. We tune hyperparameters $L$ from 1 to 5, $L ^ { \prime }$ from 0 to $L - 1$ , and cost value $C$ as powers of ten from $- 2$ to 4. The number of kernels used is 15, so the total number of parameter combinations is 105. Note this number is much less than the number of hyperparameter of Gaussian Kernel reported in Fernandez-Delgado et al. (2014) where they tune hyperparameters with 500 ´ combinations.
+
+NN Specification We use fully-connected NN with $L$ layers, 512 number of hidden nodes per layer and use gradient descent to train the neural network. We tune hyperparameters $L$ from 1 to 5, with $/$ without batch normalization and learning rate 0.1 or 1. We run gradient descent for 2000 epochs.5 We treat NN with He initialization and NTK initialization as two classifiers and report their results separately.
+
+# B ADDITIONAL EXPERIMENTAL DETAILS ON SMALL CIFAR-10 DATASETS
+
+We randomly choose $n / 1 0$ samples from each class of CIFAR-10 training set and test classifiers on the whole testing set. $n$ varies from 10 to 1280. For each $n$ , we repeat 20 times and report the mean accuracy and its standard deviation for each classifier.
+
+The number of convolution layers of CNTK ranges from 5-14. After convolutional layers, we apply a global pooling layer and a fully connected layer. We refer readers to Arora et al. (2019a) for exact formulas of CNTK. We normalize the kernel such that each sample has unit length in feature space.
+
+We use ResNet-34 with width 64,128,256 and default hyperparameters: learning rate 0.1, momentum 0.9, weight decay 0.0005. We decay the learning rate by 10 at the epoch of 80 and 120, with 160 training epochs in total. The training batch size is the minimum of the size of the whole training dataset and 160. We report the best testing accuracy among epochs.
+
+# C ADDITIONAL RESULTS IN FEW-SHOT LEARNING
+
+Tables 7-9 show the performance of the best $C$ as has been done in Goyal et al. (2019).   
+
+<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linearSVM</td><td rowspan=1 colspan=1>1-layer CNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layer CNTK</td><td rowspan=1 colspan=1>4-layer CNTK</td><td rowspan=1 colspan=1>5-layerCNTK</td><td rowspan=1 colspan=1>6-layerCNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>52.63 ± 5.35</td><td rowspan=1 colspan=1>53.16± 4.96</td><td rowspan=1 colspan=1>53.08±4.89</td><td rowspan=1 colspan=1>52.83 ± 4.85</td><td rowspan=1 colspan=1>52.44 ± 4.85</td><td rowspan=1 colspan=1>51.88 ± 4.85</td><td rowspan=1 colspan=1>51.21 ± 4.88</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>65.08± 2.69</td><td rowspan=1 colspan=1>66.25±3.09</td><td rowspan=1 colspan=1>66.29±3.09</td><td rowspan=1 colspan=1>66.11 ± 3.07</td><td rowspan=1 colspan=1>65.79±3.04</td><td rowspan=1 colspan=1>65.33± 3.00</td><td rowspan=1 colspan=1>64.77± 2.99</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>71.78 ± 1.85</td><td rowspan=1 colspan=1>72.76 ± 1.76</td><td rowspan=1 colspan=1>72.72 ± 1.73</td><td rowspan=1 colspan=1>72.49 ± 1.71</td><td rowspan=1 colspan=1>72.14 ± 1.73</td><td rowspan=1 colspan=1>71.71 ± 1.76</td><td rowspan=1 colspan=1>71.17 ± 1.78</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>74.40 ± 1.86</td><td rowspan=1 colspan=1>75.37 ± 1.95</td><td rowspan=1 colspan=1>75.33 ± 1.95</td><td rowspan=1 colspan=1>75.14 ± 1.96</td><td rowspan=1 colspan=1>74.85 ± 1.96</td><td rowspan=1 colspan=1>74.48 ± 1.99</td><td rowspan=1 colspan=1>74.05 ± 1.97</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>75.94 ± 2.51</td><td rowspan=1 colspan=1>77.01± 2.39</td><td rowspan=1 colspan=1>76.91 ± 2.39</td><td rowspan=1 colspan=1>76.72 ± 2.39</td><td rowspan=1 colspan=1>76.47 ± 2.37</td><td rowspan=1 colspan=1>76.13 ± 2.39</td><td rowspan=1 colspan=1>75.76 ± 2.39</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>76.39 ± 1.27</td><td rowspan=1 colspan=1>77.15 ± 1.47</td><td rowspan=1 colspan=1>77.14 ± 1.50</td><td rowspan=1 colspan=1>77.02 ± 1.55</td><td rowspan=1 colspan=1>76.83 ± 1.58</td><td rowspan=1 colspan=1>76.62 ± 1.60</td><td rowspan=1 colspan=1>76.35 ± 1.61</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>78.18±0.86</td><td rowspan=1 colspan=1>79.42± 0.85</td><td rowspan=1 colspan=1>79.36 ± 0.87</td><td rowspan=1 colspan=1>79.22 ± 0.88</td><td rowspan=1 colspan=1>79.00±0.91</td><td rowspan=1 colspan=1>78.75± 0.93</td><td rowspan=1 colspan=1>78.50±0.94</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>79.78 ± 0.65</td><td rowspan=1 colspan=1>80.47 ± 0.59</td><td rowspan=1 colspan=1>80.45± 0.61</td><td rowspan=1 colspan=1>80.33 ± 0.63</td><td rowspan=1 colspan=1>80.14± 0.65</td><td rowspan=1 colspan=1>79.92 ± 0.67</td><td rowspan=1 colspan=1>79.66±0.70</td></tr></table>
+
+Table 7: Performance of linear SVM and CNTK with different number of convolutional layers on all classes in VOC07 with the best $C$ . Feature extracted from conv5 in ResNet-50   
+
+<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linear SVM</td><td rowspan=1 colspan=1>1-layer CNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layer CNTK</td><td rowspan=1 colspan=1>4-layer CNTK</td><td rowspan=1 colspan=1>5-layer CNTK</td><td rowspan=1 colspan=1>6-layer CNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>24.18± 2.64</td><td rowspan=1 colspan=1>25.15 ± 2.93</td><td rowspan=1 colspan=1>25.13±3.00</td><td rowspan=1 colspan=1>25.06±3.05</td><td rowspan=1 colspan=1>24.95 ±3.10</td><td rowspan=1 colspan=1>24.80 ±3.13</td><td rowspan=1 colspan=1>24.60±3.14</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>31.59 ± 2.64</td><td rowspan=1 colspan=1>32.88 ± 2.91</td><td rowspan=1 colspan=1>32.81 ± 2.94</td><td rowspan=1 colspan=1>32.67 ± 2.96</td><td rowspan=1 colspan=1>32.48± 2.98</td><td rowspan=1 colspan=1>32.24± 2.99</td><td rowspan=1 colspan=1>31.94 ± 3.01</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>37.34 ± 2.18</td><td rowspan=1 colspan=1>38.69 ± 2.39</td><td rowspan=1 colspan=1>38.64±2.26</td><td rowspan=1 colspan=1>38.53±2.28</td><td rowspan=1 colspan=1>38.36 ± 2.31</td><td rowspan=1 colspan=1>38.12 ± 2.33</td><td rowspan=1 colspan=1>37.83± 2.35</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>39.67 ± 2.31</td><td rowspan=1 colspan=1>41.34 ± 2.33</td><td rowspan=1 colspan=1>41.39 ± 2.28</td><td rowspan=1 colspan=1>41.33 ± 2.24</td><td rowspan=1 colspan=1>41.21 ± 2.22</td><td rowspan=1 colspan=1>41.02 ±2.20</td><td rowspan=1 colspan=1>40.77±2.18</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>41.74 ± 1.20</td><td rowspan=1 colspan=1>43.48±1.40</td><td rowspan=1 colspan=1>43.48±1.24</td><td rowspan=1 colspan=1>43.44 ± 1.25</td><td rowspan=1 colspan=1>43.32 ± 1.27</td><td rowspan=1 colspan=1>43.12 ± 1.30</td><td rowspan=1 colspan=1>42.89 ± 1.33</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>44.68 ± 1.84</td><td rowspan=1 colspan=1>46.77± 1.82</td><td rowspan=1 colspan=1>46.42 ± 1.91</td><td rowspan=1 colspan=1>46.44 ± 1.90</td><td rowspan=1 colspan=1>46.39± 1.91</td><td rowspan=1 colspan=1>46.25 ± 1.93</td><td rowspan=1 colspan=1>46.08± 1.94</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>47.71 ± 1.84</td><td rowspan=1 colspan=1>50.07 ± 1.97</td><td rowspan=1 colspan=1>49.74 ± 1.95</td><td rowspan=1 colspan=1>49.63± 1.81</td><td rowspan=1 colspan=1>49.56 ± 1.81</td><td rowspan=1 colspan=1>49.41 ± 1.82</td><td rowspan=1 colspan=1>49.19 ± 1.81</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>48.91 ± 1.90</td><td rowspan=1 colspan=1>51.32±1.97</td><td rowspan=1 colspan=1>50.98 ± 1.87</td><td rowspan=1 colspan=1>51.09 ± 1.84</td><td rowspan=1 colspan=1>51.09 ± 1.81</td><td rowspan=1 colspan=1>51.00 ± 1.78</td><td rowspan=1 colspan=1>50.84± 1.76</td></tr></table>
+
+Table 8: Performance of linear SVM and CNTK with different number of convolutional layers on all classes in VOC07 with the best $C$ . Feature extracted from conv4 in ResNet-50   
+Table 9: Performance of linear SVM and CNTK with different number of convolutional layers on all classes in VOC07 with the best $C$ . Feature extracted from conv3 in ResNet-50   
+
+<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linear SVM</td><td rowspan=1 colspan=1>1-layerCNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layer CNTK</td><td rowspan=1 colspan=1>4-layer CNTK</td><td rowspan=1 colspan=1>5-layerCNTK</td><td rowspan=1 colspan=1>6-layerCNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>14.72 ± 1.49</td><td rowspan=1 colspan=1>15.01 ± 1.27</td><td rowspan=1 colspan=1>14.96 ± 1.28</td><td rowspan=1 colspan=1>14.92 ± 1.30</td><td rowspan=1 colspan=1>14.87 ± 1.32</td><td rowspan=1 colspan=1>14.81 ± 1.34</td><td rowspan=1 colspan=1>14.75 ± 1.36</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>16.67 ± 1.04</td><td rowspan=1 colspan=1>17.52 ± 1.05</td><td rowspan=1 colspan=1>17.45 ± 1.03</td><td rowspan=1 colspan=1>17.36 ± 1.02</td><td rowspan=1 colspan=1>17.28 ± 1.00</td><td rowspan=1 colspan=1>17.20± 0.99</td><td rowspan=1 colspan=1>17.11 ± 0.97</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>19.21 ± 1.31</td><td rowspan=1 colspan=1>20.65±1.68</td><td rowspan=1 colspan=1>20.63± 1.65</td><td rowspan=1 colspan=1>20.58± 1.63</td><td rowspan=1 colspan=1>20.53± 1.61</td><td rowspan=1 colspan=1>20.47 ± 1.59</td><td rowspan=1 colspan=1>20.41± 1.57</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>20.87 ± 1.49</td><td rowspan=1 colspan=1>22.28±1.80</td><td rowspan=1 colspan=1>22.27 ± 1.81</td><td rowspan=1 colspan=1>22.22 ± 1.83</td><td rowspan=1 colspan=1>22.16 ± 1.83</td><td rowspan=1 colspan=1>22.09 ± 1.83</td><td rowspan=1 colspan=1>22.03± 1.82</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>21.98 ± 1.17</td><td rowspan=1 colspan=1>23.45±1.14</td><td rowspan=1 colspan=1>23.45±1.12</td><td rowspan=1 colspan=1>23.43± 1.08</td><td rowspan=1 colspan=1>23.38 ± 1.07</td><td rowspan=1 colspan=1>23.32 ± 1.05</td><td rowspan=1 colspan=1>23.25± 1.04</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>23.02 ± 1.56</td><td rowspan=1 colspan=1>24.44 ± 1.61</td><td rowspan=1 colspan=1>24.29 ± 1.66</td><td rowspan=1 colspan=1>24.27 ± 1.82</td><td rowspan=1 colspan=1>24.25 ± 1.81</td><td rowspan=1 colspan=1>24.21 ± 1.80</td><td rowspan=1 colspan=1>24.18± 1.78</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>24.38± 1.52</td><td rowspan=1 colspan=1>25.96 ±1.80</td><td rowspan=1 colspan=1>25.97± 1.70</td><td rowspan=1 colspan=1>25.98± 1.70</td><td rowspan=1 colspan=1>25.96 ± 1.69</td><td rowspan=1 colspan=1>25.92 ± 1.67</td><td rowspan=1 colspan=1>25.88± 1.66</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>25.28 ± 1.39</td><td rowspan=1 colspan=1>27.06± 1.61</td><td rowspan=1 colspan=1>27.15 ± 1.58</td><td rowspan=1 colspan=1>27.18±1.56</td><td rowspan=1 colspan=1>27.17 ± 1.53</td><td rowspan=1 colspan=1>27.13 ± 1.51</td><td rowspan=1 colspan=1>27.09± 1.50</td></tr></table>

parse/train/rkl8sJBYvH/rkl8sJBYvH_content_list.json

@@ -0,0 +1,1457 @@
+[
+    {
+        "type": "text",
+        "text": "HARNESSING THE POWER OF INFINITELY WIDE DEEP NETS ON SMALL-DATA TASKS ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sanjeev Arora Princeton University arora@cs.princeton.edu ",
+        "bbox": [
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+            170,
+            401,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Simon S. Du   \nInstitute for Advanced Study   \nssdu@ias.edu   \nZhiyuan Li   \nPrinceton University   \nzhiyuanli@cs.princeton.edu ",
+        "bbox": [
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+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            233,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ruslan Salakhutdinov Carnegie Mellon University rsalakhu@cs.cmu.edu ",
+        "bbox": [
+            625,
+            233,
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+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ruosong Wang Carnegie Mellon University ruosongw@andrew.cmu.edu ",
+        "bbox": [
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+            296,
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+            338
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dingli Yu   \nPrinceton University   \ndingliy@cs.princeton.edu ",
+        "bbox": [
+            501,
+            296,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            376,
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+            390
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent research shows that the following two models are equivalent: (a) infinitely wide neural networks (NNs) trained under $\\ell _ { 2 }$ loss by gradient descent with infinitesimally small learning rate (b) kernel regression with respect to so-called Neural Tangent Kernels (NTKs) (Jacot et al., 2018). An efficient algorithm to compute the NTK, as well as its convolutional counterparts, appears in Arora et al. (2019a), which allowed studying performance of infinitely wide nets on datasets like CIFAR-10. However, super-quadratic running time of kernel methods makes them best suited for small-data tasks. We report results suggesting neural tangent kernels perform strongly on low-data tasks. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. On a standard testbed of classification/regression tasks from the UCI database, NTK SVM beats the previous gold standard, Random Forests (RF), and also the corresponding finite nets.   \n2. On CIFAR-10 with 10 – 640 training samples, Convolutional NTK consistently beats ResNet-34 by $1 \\% - 3 \\%$ .   \n3. On VOC07 testbed for few-shot image classification tasks on ImageNet with transfer learning (Goyal et al., 2019), replacing the linear SVM currently used with a Convolutional NTK SVM consistently improves performance.   \n4. Comparing the performance of NTK with the finite-width net it was derived from, NTK behavior starts at lower net widths than suggested by theoretical analysis(Arora et al., 2019a). NTK’s efficacy may trace to lower variance of output. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            726,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Modern neural networks (NNs) have way more parameters than training data points, which allow them to achieve near-zero training error while simultaneously — for some reason yet to be understood — have low generalization error (Zhang et al., 2016). This motivated formal study of highly overparametrized networks, including networks whose width (i.e., number of nodes in layers, or number of channels in convolutional layers) goes to infinity. A recent line of theoretical results shows that with $\\ell _ { 2 }$ loss and infinitesimal learning rate, in the limit of infinite width the trajectory of training converges to kernel regression with a particular kernel, neural tangent kernel (NTK) (Jacot et al., 2018). For convolutional networks, the kernel is CNTK. See Section 2 for more discussions. Arora et al. (2019a) gave an algorithm to exactly compute the kernel corresponding to the infinite limit of various realistic NN architectures with convolutions and pooling layers, allowing them to compute performance on CIFAR-10, which revealed that the infinite networks have 6 to $8 \\%$ higher error than their finite counterparts. This is still fairly good performance for a fixed kernel. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ironically, while the above-mentioned analysis, at first sight, appears to reduce the study of a complicated model — deep networks — to an older, simpler model — kernel regression — in practice the simpler model is computationally less efficient because running time of kernel regression can be quadratic in the number of data points!1 Thus computing using CNTK kernel on large datasets like ImageNet currently appears infeasible. Even on CIFAR-10, it seems infeasible to incorporate data augmentation. ",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, kernel classifiers are very efficient on small datasets. Here NTKs could conceivably be practical while at the same time bringing some of the power of deep networks to these settings. We recall that recently Olson et al. (2018) showed that multilayer neural networks can be reasonably effective on small datasets, specifically on a UCI testbed of tasks with as few as dozens of training examples. Of course, this required some hyperparameter tuning, although they noted that such tuning is also needed for the champion method, Random Forests (RF), which multilayer neural networks could not beat. ",
+        "bbox": [
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+            194,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "It is thus natural to check if NTK — corresponding to infinitely wide fully-connected networks — performs well in such small-data tasks2. Convex objectives arising from kernels have stable solvers with minimal hyperparameter tuning. Furthermore, random initialization in deep network training seems to lead to higher variance in the output, which can hurt performance in small-data settings. Can NTK’s do better? Below we will see that in the setup of Olson et al. (2018), NTK predictors indeed outperforms corresponding finite deep networks, and also slightly beats the earlier gold standard, Random Forests. This suggests NTK predictors should belong in any list of off-theshelf machine learning methods. ",
+        "bbox": [
+            173,
+            297,
+            825,
+            410
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Following are low-data settings where we used NTKs and CNTKs: ",
+        "bbox": [
+            174,
+            416,
+            612,
+            431
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• In the testbed of 90 classification tasks from UCI database, NTK predictor achieves superior, and arguably the strongest classification performance. This is verified via several standard statistical tests, including Friedman Rank, Average Accuracy, Percentage of the Maximum Accuracy (PMA) and probability of achieving $9 0 \\% / 9 5 \\%$ maximum accuracy (P90 and P95), performed to compare performances of different classifiers on 90 datasets from UCI database. (The authors plan to release the code, to allow off-the-shelf use of this method. It does not require GPUs.) • We find the performance of NN is close to that of NTK. On every dataset from UCI database, the difference between the classification accuracy of NN and that of NTK is within $5 \\%$ . On the other hand, on some datasets, the difference between classification accuracy of NN (or NTK) and that of other classifiers like RF can be as high as $2 0 \\%$ . This indicates in low-data settings, NTK is indeed a good description of NN. Furthermore, we find NTK is more stable (smaller variance), which seems to help it achieve better accuracy on small datasets (cf. Figure 2b). • CNTK is useful in computer vision tasks with small-data. On CIFAR-10, we compare CNTK with ResNet using 10 - 640 training samples and find CNTK can beat ResNet by $1 \\% - 3 \\%$ . We further study few-shot image classification task on VOC07 dataset. The standard method is to first use a pre-trained network, e.g., ResNet-50 trained on ImageNet, to extract features and then directly apply a linear classifier on the extracted features (Goyal et al., 2019). Here we replace the linear classifier with CNTK and obtain better classification accuracy in various setups. ",
+        "bbox": [
+            173,
+            438,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Paper organization. Section 2 discusses related work. Section 3 reviews the derivation of NTK. Section 4 presents experiments using NN and NTK on UCI datasets. Section 5 presents experiments using CNN and CNTK on small CIFAR-10 datasets. Section 6 presents experiments using CNTK for the few-shot learning setting. Additional technical details are presented in appendix. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            784,
+            344,
+            799
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our paper is inspired by Fernandez-Delgado et al. (2014) which conducted extensive experiments ´ on UCI dataset. Their conclusion is random forest performs the best, which is followed by the SVM with Gaussian kernel. Therefore, RF may be considered as a reference (“gold-standard”) to compare with new classifiers. Olson et al. (2018) followed this testing strategy to evaluate the performance of modern neural networks concluding that modern neural networks, even though being highly overparameterized, still give reasonable performances on these small datasets, though not as strong as RFs. Our paper follows the same testing strategy to evaluate the performance of NTK. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The focus of this paper, neural tangent kernel is induced from a neural network architecture. The connection between infinitely wide neural networks and kernel methods is not new (Neal, 1996; Williams, 1997; Roux & Bengio, 2007; Hazan & Jaakkola, 2015; Lee et al., 2018; Matthews et al., 2018; Novak et al., 2019; Garriga-Alonso et al., 2019; Cho & Saul, 2009; Daniely et al., 2016; Daniely, 2017). However, these kernels correspond to neural network where only the last layer is trained. Neural tangent kernel, first proposed by Jacot et al. (2018), is fundamentally different as NTKs correspond to infinitely wide NNs with all layer being trained. Theoretically, a line of work study the optimization and generalization behavior of ultra-wide NNs (Allen-Zhu et al., 2018b;a; Arora et al., 2019b; Du et al., 2018b;a; Li & Liang, 2018; Zou et al., 2018; Yang, 2019). Recently, Arora et al. (2019a) gave non-asymptotic perturbation bound between the NN predictor trained by gradient descent and the NTK predictor. Empirically, Lee et al. (2019) verified on small scale data, NTK is a good approximation to NN. However, Arora et al. (2019a) showed on large scale dataset, NN can outperform NTK which may due to the effect of finite-width and/or optimization procedure. ",
+        "bbox": [
+            174,
+            166,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Generalization to architectures other than fully-connected NN and CNN are recently proposed (Yang, 2019; Du et al., 2019; Bietti & Mairal, 2019). Du et al. (2019) showed graph neural tangent kernel (GNTK) can achieve better performance than its counter part, graph neural network (GNN), on datasets with up to 5000 samples. ",
+        "bbox": [
+            174,
+            353,
+            823,
+            410
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 NEURAL NETWORK AND NEURAL TANGENT KERNEL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            429,
+            650,
+            446
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Since NTK is induced by a NN architecture, we first define a NN formally. Let $x \\in \\mathbb { R } ^ { d }$ be the input, and denote $g ^ { ( 0 ) } ( x ) = x$ and $d _ { 0 } = d$ for notational convenience. We define an $L$ -hidden-layer fully-connected neural network recursively: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dfc977ee98e884d1febea301e3cff5f6d871ce35f281de651f0a7ab89755acee.jpg",
+        "text": "$$\nf ^ { ( h ) } ( \\boldsymbol { x } ) = W ^ { ( h ) } g ^ { ( h - 1 ) } ( \\boldsymbol { x } ) \\in \\mathbb { R } ^ { d _ { h } } , \\quad g ^ { ( h ) } ( \\boldsymbol { x } ) = \\sqrt { \\frac { c _ { \\sigma } } { d _ { h } } } \\sigma \\left( f ^ { ( h ) } ( \\boldsymbol { x } ) \\right) \\in \\mathbb { R } ^ { d _ { h } } , \\qquad h = 1 , 2 , \\ldots , L ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            183,
+            507,
+            812,
+            542
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $W ^ { ( h ) } \\in \\mathbb { R } ^ { d _ { h } \\times d _ { h - 1 } }$ is the weight matrix in the $h$ -th layer $\\mathit { h } \\in [ L ] )$ , $\\sigma : \\mathbb { R }  \\mathbb { R }$ is a coordinatewise activation function, and $c _ { \\sigma }$ is a scaling factor.3 In this paper, for NN we will consider $\\sigma$ being ReLU or ELU (Clevert et al., 2015) and for NTK we will only consider kernel functions induced by NNs with ReLU activation. The last layer of the neural network is ",
+        "bbox": [
+            176,
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+            608
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/77788e1901c07ad994f2221802ae96d85b0a5fe6f39266b47028f4cd2c5b1ce1.jpg",
+        "text": "$$\n\\begin{array} { l } { { f ( w , x ) = f ^ { ( L + 1 ) } ( x ) = W ^ { ( L + 1 ) } \\cdot g ^ { ( L ) } ( x ) \\ ~ } } \\\\ { { = W ^ { ( L + 1 ) } \\cdot \\sqrt { \\frac { C _ { \\sigma } } { d _ { L } } } \\sigma \\left( W ^ { ( L ) } \\cdot \\sqrt { \\frac { C _ { \\sigma } } { d _ { L - 1 } } } \\sigma \\left( W ^ { ( L - 1 ) } \\cdot \\cdot \\sqrt { \\frac { C _ { \\sigma } } { d _ { 1 } } } \\sigma \\left( W ^ { ( 1 ) } x \\right) \\right) \\right) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            218,
+            611,
+            777,
+            669
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $W ^ { ( L + 1 ) } \\in \\mathbb { R } ^ { 1 \\times d _ { L } }$ is the weights in the final layer, and we let $w = \\left( W ^ { ( 1 ) } , \\dots , W ^ { ( L + 1 ) } \\right)$ be all parameters in the network. All the weights are initialized to be i.i.d. $\\mathcal { N } ( 0 , 1 )$ random variables. From now on, by NTK initialization we mean a neural network with parameterization defined in Equation 2 with all weighted being initialized to be i.i.d. $\\mathcal { N } ( 0 , 1 )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "When the hidden widths $d _ { 1 , } d _ { 2 } , \\ldots , d _ { L }  \\infty$ , certain limiting behavior emerges along the gradient trajectory. Let $x , x ^ { \\prime } \\in \\mathbb { R } ^ { d }$ be two data points, the covariance kernel of the $h$ -th layer’s outputs, $\\Sigma ^ { ( h ) } ( x , x ^ { \\prime } ) = f ^ { ( h ) } ( x ) \\cdot f ^ { ( h ) } ( x ^ { \\prime } )$ , can be recursively defined in an analytical form: ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/7a1e5cf0fa97a6f81b1819d3903c93c508f36c361a91748ec42a712a551a774b.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\Sigma ^ { ( 0 ) } ( x , x ^ { \\prime } ) = x ^ { \\top } x ^ { \\prime } , } \\\\ & { \\Lambda ^ { ( h ) } ( x , x ^ { \\prime } ) = \\left( \\sum _ { \\Sigma ^ { ( h - 1 ) } ( x ^ { \\prime } , x ) } ^ { ( h - 1 ) } \\ \\Sigma ^ { ( h - 1 ) } ( x , x ^ { \\prime } ) \\right) \\in \\mathbb { R } ^ { 2 \\times 2 } , } \\\\ & { \\Sigma ^ { ( h ) } ( x , x ^ { \\prime } ) = c _ { \\sigma } \\mathbb { E } _ { ( u , v ) \\sim \\mathcal { N } \\left( 0 , \\Lambda ^ { ( h ) } \\right) } \\left[ \\sigma \\left( u \\right) \\sigma \\left( v \\right) \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "table",
+        "img_path": "images/a5eb307469c18140811776dcfb56cc7963122cad1dc00aedf7a960191e7d95fe.jpg",
+        "table_caption": [
+            "Table 1: Comparisons of different classifiers on $9 0 \\ \\mathrm { U C I }$ datasets. P90/P95: the number of datasets a classifier achieves $9 0 \\% / 9 5 \\%$ or more of the maximum accuracy, divided by the total number of datasets. PMA: average percentage of the maximum accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Classifier</td><td rowspan=1 colspan=1>Friedman Rank</td><td rowspan=1 colspan=1> Average Accuracy</td><td rowspan=1 colspan=1>P90</td><td rowspan=1 colspan=1>P95</td><td rowspan=1 colspan=1>PMA</td></tr><tr><td rowspan=1 colspan=1>NTK</td><td rowspan=1 colspan=1>28.34</td><td rowspan=1 colspan=1>81.95%±14.10%</td><td rowspan=1 colspan=1>88.89%</td><td rowspan=1 colspan=1>72.22%</td><td rowspan=1 colspan=1>95.72% ±5.17%</td></tr><tr><td rowspan=1 colspan=1>NN (He init)</td><td rowspan=1 colspan=1>40.97</td><td rowspan=1 colspan=1>80.88%±14.96%</td><td rowspan=1 colspan=1>81.11%</td><td rowspan=1 colspan=1>65.56%</td><td rowspan=1 colspan=1>94.34% ±7.22%</td></tr><tr><td rowspan=1 colspan=1>NN (NTK init)</td><td rowspan=1 colspan=1>38.06</td><td rowspan=1 colspan=1>81.02%±14.47%</td><td rowspan=1 colspan=1>85.56%</td><td rowspan=1 colspan=1>60.00%</td><td rowspan=1 colspan=1>94.55% ±5.89%</td></tr><tr><td rowspan=1 colspan=1>RF</td><td rowspan=1 colspan=1>33.51</td><td rowspan=1 colspan=1>81.56% ±13.90%</td><td rowspan=1 colspan=1>85.56%</td><td rowspan=1 colspan=1>67.78%</td><td rowspan=1 colspan=1>95.25% ±5.30%</td></tr><tr><td rowspan=1 colspan=1>Gaussian Kernel</td><td rowspan=1 colspan=1>35.76</td><td rowspan=1 colspan=1>81.03% ± 15.09%</td><td rowspan=1 colspan=1>85.56%</td><td rowspan=1 colspan=1>72.22%</td><td rowspan=1 colspan=1>94.56% ±8.22%</td></tr><tr><td rowspan=1 colspan=1>Polynomial Kernel</td><td rowspan=1 colspan=1>38.44</td><td rowspan=1 colspan=1>78.21% ± 20.30%</td><td rowspan=1 colspan=1>80.00%</td><td rowspan=1 colspan=1>62.22%</td><td rowspan=1 colspan=1>91.29% ±18.05%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for $h \\in [ L ]$ . Crucially, this analytical form holds not only at the initialization, but also holds during the training (when gradient descent with small learning rate is used as the optimization routine). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Formally, NTK is defined as the limiting gradient kernel ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/96f3474c5d0851ec6929df80d738ee26617cfb1610f92ac78e6c36d1f6ba2e52.jpg",
+        "text": "$$\n\\Theta \\left( x , x ^ { \\prime } \\right) \\triangleq \\left. \\frac { \\partial f ( w , x ) } { \\partial w } , \\frac { \\partial f ( w , x ^ { \\prime } ) } { \\partial w } \\right. = \\sum _ { h = 1 } ^ { L + 1 } \\left. \\frac { \\partial f ( w , x ) } { \\partial W ^ { ( h ) } } , \\frac { \\partial f ( w , x ^ { \\prime } ) } { \\partial W ^ { ( h ) } } \\right. .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Again, one can obtain a recursive formula ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f151518b148bde5e933ec54d89f38a02ea7ebc3c70d3272d9324cf9c8ab74235.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\dot { \\Sigma } ^ { ( h ) } ( x , x ^ { \\prime } ) = c _ { \\sigma } \\mathbb { E } _ { ( u , v ) \\sim \\mathcal { N } \\left( 0 , \\Lambda ^ { ( h ) } \\right) } \\left[ \\dot { \\sigma } ( u ) \\dot { \\sigma } ( v ) \\right] , h = 1 , \\ldots , L + 1 } \\\\ { \\displaystyle \\Theta ( x , x ^ { \\prime } ) = \\sum _ { h = 1 } ^ { L + 1 } \\left( \\Sigma ^ { ( h - 1 ) } ( x , x ^ { \\prime } ) \\cdot \\prod _ { h ^ { \\prime } = h } ^ { L + 1 } \\dot { \\Sigma } ^ { ( h ^ { \\prime } ) } ( x , x ^ { \\prime } ) \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where we let $\\dot { \\Sigma } ^ { ( L + 1 ) } ( x , x ^ { \\prime } ) = 1$ for convenience. It is easy to check if we fix the first $L ^ { \\prime }$ layers and only train the remaining $( L + 1 - L ^ { \\prime } )$ layers, then the resulting NTK is $\\Theta ( x , x ^ { \\prime } ) ~ =$ $\\begin{array} { r } { \\sum _ { h = L ^ { \\prime } + 1 } ^ { L + 1 } \\Big ( \\Sigma ^ { ( h - 1 ) } ( x , x ^ { \\prime } ) \\cdot \\prod _ { h ^ { \\prime } = h } ^ { L + 1 } \\dot { \\Sigma } ^ { ( h ^ { \\prime } ) } ( x , x ^ { \\prime } ) \\Big ) } \\end{array}$ . Note when $L ^ { \\prime } = L$ , then the resulting NTK is $\\Sigma ^ { ( L ) } \\left( x , x ^ { \\prime } \\right)$ , which is the NNGP kernel (Lee et al., 2018). $L ^ { \\prime }$ can be viewed as a hyperparameter of NTK classifier, and in our UCI experiment we tune $L ^ { \\prime }$ . Given a kernel function, one can directly use it for downstream classification tasks (Scholkopf & Smola, 2001). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ON UCI DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we present our experimental results on UCI datasets which follow the setup of Fernandez-Delgado et al. (2014) with extensive comparisons of classifiers, including random for- ´ est, kernel SVM, multilayer neural networks, etc. Section 4.2 discusses the performance of NTK through detailed comparisons with other classifiers tested by Fernandez-Delgado et al. (2014). Sec- ´ tion 4.2 compares NTK classifier and the corresponding NN classifier and verifies how similar their predictions are. See Table 6 in Appendix A for a summary of datasets we used. The detailed experiment setup, including the choices the datasets, training / test splitting and ranges of hyperparameters, is described in Appendix A. We note that usual methods of obtaining confidence bounds in these low-data settings are somewhat heuristic. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1 OVERALL PERFORMANCE COMPARISONS ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+    {
+        "type": "text",
+        "text": "Table 1 lists the performance of 6 classifiers under various metrics: the three top classifiers identified in Fernandez-Delgado et al. (2014), namely, RF, Gaussian kernel and polynomial kernel, along ´ with our new methods NTK, NN with He initialization and NN with NTK initialization. Table 1 shows NTK is the best classifier under all metrics, followed by RF, the best classifier identified in Fernandez-Delgado et al. (2014). Now we interpret each metric in more details. ´ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Friedman Ranking and Average Accuracy. NTK is the best (Friedman Rank 28.34, Average Accuracy $8 1 . 9 5 \\%$ ), followed by RF (Friedman Rank 33.51, Average Accuracy $8 1 . 5 6 \\%$ ) and then followed by SVM with Gaussian kernel (Friedman Rank 35.76, Average Accuracy $8 1 . 0 3 \\%$ ). The difference between NTK and RF is significant (-5.17 in Friedman Rank and $+ 0 . 3 9 \\%$ in Average Accuracy), just as the superiority of RF is significant compared to other classifiers as claimed in Fernandez-Delgado et al. (2014). NN (with either He initialization or NTK initialization) per- ´ forms significantly better than the polynomial kernel in terms of the Average Accuracy $( 8 0 . 8 8 \\%$ and $8 1 . 0 2 \\%$ vs. $7 8 . 2 1 \\%$ ), but in terms of Friedman Rank, NN with He initialization is worse than polynomial kernel (40.97 vs. 38.44) and NN with NTK initialization is slightly better than polynomial kernel (38.06 vs. 38.44). On many datasets where most classifiers have similar performances, NN’s rank is high as well, whereas on other datasets, NN is significantly better than most classifiers, including SVM with polynomial kernel. Therefore, NN enjoys higher Average Accuracy but suffers higher Friedman Rank. For example, on OZONE dataset, NN with NTK initialization is only $0 . 2 5 \\%$ worse than polynomial kernel but their difference in terms of rank is 56. It is also interesting to see that NN with NTK initialization performs better than NN with He initialization (38.06 vs. 40.97 in Friedman Rank and $8 1 . 0 2 \\%$ vs. $8 0 . 8 8 \\%$ in Average Accuracy). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "P90/P95 and PMA. This measures, for a given classifier, the fraction of datasets on which it achieves more than $9 0 \\% / 9 5 \\%$ of the maximum accuracy among all classifiers. NTK is one of the best classifiers (ties with Gaussian kernel on P95), which shows NTK can consistently achieve superior classification performance across a broad range of datasets. Lastly, we consider the Percentage of the Maximum Accuracy (PMA). NTK achieves the best average PMA followed by RF whose PMA is $0 . 4 7 \\%$ below that of NTK and other classifiers are all below $9 4 . 6 \\%$ . An interesting observation is that NTK, NN with NTK initialization and RF have small standard deviation $5 . 1 7 \\%$ , $5 . 8 9 \\%$ and $5 . 3 0 \\%$ ) whereas all other classifiers have much larger standard deviation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 PAIRWISE COMPARISONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "NTK vs. RF. In Figure 1a, we compare NTK with RF. There are 42 datasets that NTK outperforms RF and 40 datasets that RF outperforms NTK.4 The mean difference is $2 . 5 1 \\%$ , which is statistically significant by a Wilcoxon signed rank test. We see NTK and RF perform similarly when the Bayes error rate is low with NTK being slightly better. There are some exceptions. For example, on the BALANCE-SCALE dataset, NTK achieves $98 \\%$ accuracy whereas RF only achieves $8 4 . 1 \\%$ . When the Bayes error rate is high, either classifier can be significantly better than the other. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "NTK vs. Gaussian Kernel. The gap between NTK and Gaussian kernel is more significant. As shown in Figure 1b, NTK generally performs better than Gaussian kernel no matter Bayes error rate is low or high. There are 43 datasets that NTK outperforms Gaussian kernel and there are 34 datasets that Gaussian kernel outperforms NTK. The mean difference is $2 . 2 2 \\%$ , which is also statistically significant by a Wilcoxon signed rank test. On BALLOONS, HEART-SWITZERLAND, PITTSBURGBRIDGES-MATERIAL, TEACHING, TRAINS datasets, NTK is better than Gaussian kernel by at least $11 \\%$ in terms of accuracy. These five datasets all have less than 200 samples, which shows that NTK can perform much better than Gaussian kernel when the number of samples is small. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "NTK vs. NN In this section we compare NTK with NN. The goals are (i) comparing the performance and (ii) verifying NTK is a good approximation to NN. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "In Figure 2a, we compare the performance between NTK and NN with He initialization. For most datasets, these two classifiers perform similarly. However, there are a few datasets that NTK performs significantly better. There are 50 datasets that NTK outperforms NN with He initialization and there are 27 datasets that NN with He initialization outperforms NTK. The mean difference is $1 . 9 6 \\%$ , which is also statistically significant by a Wilcoxon signed rank test. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Figure 2b, we compare the performance between NTK and NN with NTK initialization. Recall that for NN with NTK initialization, when the width goes to infinity, the predictor is just NTK. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/d126f2441334a9de8d571de9748b55c004d12bf48f4fd1f950caf6db87fec70a.jpg",
+        "image_caption": [
+            "Figure 1: Performance comparisons between NTK and other classifiers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/43526e3362e86e82c4736a424b1b71f3926b21a9d95fac8e2356d9024dc966ba.jpg",
+        "image_caption": [
+            "Figure 2: Performance Comparisons between NTK and NN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Therefore, we expect these two predictors give similar performance. Figure 2b verifies our conjecture. There is no dataset the one classifier is significantly better than the other. We do not have the same observation in Figure 1a, 1b or 2a. Nevertheless, NTK often performs better than NN. There are 52 datasets that NTK outperforms NN with He initialization and there are 25 datasets NN with He initialization outperforms NTK. The mean difference is $1 . 5 4 \\%$ , which is also statistically significant by a Wilcoxon signed rank test. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ON SMALL CIFAR-10 DATASET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            785,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we study the performance of CNTK on subsampled CIFAR-10 dataset. We randomly choose $n$ samples from CIFAR-10 training set, use them to train CNTK and ResNet-34, and test both classifiers on the whole test set. In our experiments, we vary $n$ from 10 to 1280, and the number of convolutional layers of CNTK varies from 5 to 14. See Appendix B for detailed experiment setup. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The results are reported in Table 2. It can be observed that in this setting CNTK consistently outperforms ResNet-34. The largest gap occurs at $n = 3 2 0$ where 14-layer CNTK achieves $3 \\dot { 6 } . 5 7 \\%$ accuracy and ResNet achieves $3 3 . 1 5 \\%$ . The smallest improvement occurs at $n = 1 0$ : $1 5 . 3 3 \\%$ vs. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/207f8114544127e360c28f891ce370ac70f056f63861d5b88121973c906977cc.jpg",
+        "table_caption": [
+            "Table 2: Performance of ResNet-34 and CNTK on small CIFAR-10 Dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>n</td><td rowspan=1 colspan=1>ResNet</td><td rowspan=1 colspan=1>5-layer CNTK</td><td rowspan=1 colspan=1>8-layer CNTK</td><td rowspan=1 colspan=1>11-layer CNTK</td><td rowspan=1 colspan=1>14-layer CNTK</td></tr><tr><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>14.59% ±1.99%</td><td rowspan=1 colspan=1>15.08%±2.43%</td><td rowspan=1 colspan=1>15.24% ± 2.44%</td><td rowspan=1 colspan=1>15.31% ±2.38%</td><td rowspan=1 colspan=1>15.33% ±2.43%</td></tr><tr><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>17.50% ± 2.47%</td><td rowspan=1 colspan=1>18.03% ±1.91%</td><td rowspan=1 colspan=1>18.50% ±2.03%</td><td rowspan=1 colspan=1>18.69% ±2.07%</td><td rowspan=1 colspan=1>18.79% ±2.13%</td></tr><tr><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>19.52% ±1.39%</td><td rowspan=1 colspan=1>20.83% ±1.68%</td><td rowspan=1 colspan=1>21.07% ±1.80%</td><td rowspan=1 colspan=1>21.23% ±1.86%</td><td rowspan=1 colspan=1>21.34% ± 1.91%</td></tr><tr><td rowspan=1 colspan=1>80</td><td rowspan=1 colspan=1>23.32% ±1.61%</td><td rowspan=1 colspan=1>24.82% ±1.75%</td><td rowspan=1 colspan=1>25.18% ±1.80%</td><td rowspan=1 colspan=1>25.40% ±1.84%</td><td rowspan=1 colspan=1>25.48% ± 1.91%</td></tr><tr><td rowspan=1 colspan=1>160</td><td rowspan=1 colspan=1>28.30% ±1.38%</td><td rowspan=1 colspan=1>29.63% ±1.13%</td><td rowspan=1 colspan=1>30.17% ±1.11%</td><td rowspan=1 colspan=1>30.46% ±1.15%</td><td rowspan=1 colspan=1>30.48% ± 1.17%</td></tr><tr><td rowspan=1 colspan=1>320</td><td rowspan=1 colspan=1>33.15%±1.20%</td><td rowspan=1 colspan=1>35.26% ± 0.97%</td><td rowspan=1 colspan=1>36.05% ± 0.92%</td><td rowspan=1 colspan=1>36.44% ± 0.91%</td><td rowspan=1 colspan=1>36.57% ± 0.88%</td></tr><tr><td rowspan=1 colspan=1>640</td><td rowspan=1 colspan=1>41.66% ±1.09%</td><td rowspan=1 colspan=1>41.24%± 0.78%</td><td rowspan=1 colspan=1>42.10% ± 0.74%</td><td rowspan=1 colspan=1>42.44% ± 0.72%</td><td rowspan=1 colspan=1>42.63% ± 0.68%</td></tr><tr><td rowspan=1 colspan=1>1280</td><td rowspan=1 colspan=1>49.14% ± 1.31%</td><td rowspan=1 colspan=1>47.21% ± 0.49%</td><td rowspan=1 colspan=1>48.22% ± 0.49%</td><td rowspan=1 colspan=1>48.67% ± 0.57%</td><td rowspan=1 colspan=1>48.86% ± 0.68%</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/e7b58a719d8bf463a57dac5870fc10a3f3366477e896624d566827db6e23e907.jpg",
+        "table_caption": [
+            "Table 3: Performance of linear SVM and CNTK with different number of convolutional layers on the 11-20th classes in VOC07. The cost value $C$ is tuned on the first 10 classes. Feature extracted from conv5 in ResNet-50. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linear SVM</td><td rowspan=1 colspan=1>1-layerCNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layer CNTK</td><td rowspan=1 colspan=1>4-layer CNTK</td><td rowspan=1 colspan=1>5-layer CNTK</td><td rowspan=1 colspan=1>6-layerCNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>49.46 ± 4.71</td><td rowspan=1 colspan=1>50.04± 4.28</td><td rowspan=1 colspan=1>49.94 ± 4.19</td><td rowspan=1 colspan=1>49.66 ± 4.19</td><td rowspan=1 colspan=1>49.24 ± 4.20</td><td rowspan=1 colspan=1>48.66± 4.26</td><td rowspan=1 colspan=1>48.00± 4.41</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>64.39 ± 5.02</td><td rowspan=1 colspan=1>65.81 ±5.28</td><td rowspan=1 colspan=1>65.89±5.23</td><td rowspan=1 colspan=1>65.65± 5.17</td><td rowspan=1 colspan=1>65.30±5.10</td><td rowspan=1 colspan=1>64.76±5.07</td><td rowspan=1 colspan=1>64.12 ±5.00</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>70.38 ±3.28</td><td rowspan=1 colspan=1>71.60 ± 3.51</td><td rowspan=1 colspan=1>71.56± 3.65</td><td rowspan=1 colspan=1>71.32 ± 3.74</td><td rowspan=1 colspan=1>70.96 ± 3.84</td><td rowspan=1 colspan=1>70.51 ± 3.92</td><td rowspan=1 colspan=1>69.98±3.95</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>72.43± 3.21</td><td rowspan=1 colspan=1>73.65±3.29</td><td rowspan=1 colspan=1>73.59 ±3.28</td><td rowspan=1 colspan=1>73.35 ± 3.23</td><td rowspan=1 colspan=1>73.00±3.20</td><td rowspan=1 colspan=1>72.56± 3.20</td><td rowspan=1 colspan=1>72.09±3.20</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>74.49 ±3.08</td><td rowspan=1 colspan=1>75.87 ± 2.88</td><td rowspan=1 colspan=1>75.69± 2.89</td><td rowspan=1 colspan=1>75.45± 2.89</td><td rowspan=1 colspan=1>75.13 ± 2.85</td><td rowspan=1 colspan=1>74.68 ± 2.88</td><td rowspan=1 colspan=1>74.24 ± 2.85</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>76.59 ± 2.15</td><td rowspan=1 colspan=1>77.64± 2.37</td><td rowspan=1 colspan=1>77.57 ± 2.43</td><td rowspan=1 colspan=1>77.39 ± 2.50</td><td rowspan=1 colspan=1>77.13 ± 2.56</td><td rowspan=1 colspan=1>76.87± 2.60</td><td rowspan=1 colspan=1>76.52 ± 2.63</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>77.11 ± 1.25</td><td rowspan=1 colspan=1>78.64± 1.35</td><td rowspan=1 colspan=1>78.57 ± 1.39</td><td rowspan=1 colspan=1>78.39 ± 1.41</td><td rowspan=1 colspan=1>78.12 ± 1.42</td><td rowspan=1 colspan=1>77.80 ± 1.41</td><td rowspan=1 colspan=1>77.43 ± 1.41</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>79.57± 0.87</td><td rowspan=1 colspan=1>80.42±0.88</td><td rowspan=1 colspan=1>80.38±0.86</td><td rowspan=1 colspan=1>80.23±0.86</td><td rowspan=1 colspan=1>79.99 ±0.88</td><td rowspan=1 colspan=1>79.72 ± 0.89</td><td rowspan=1 colspan=1>79.41 ± 0.92</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "$1 4 . 5 9 \\%$ . When the number of training data is large, i.e., $n = 1 2 8 0$ , ResNet can outperform CNTK. It is also interesting to see that 14-layer CNTK is the best performing CNTK for all values of $n$ , suggesting depth has a significant effect on this task. ",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ON FEW-SHOT LEARNING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we test the ability of NTK as a drop-in classifier to replace the linear classifier in the few-shot learning setting. Using linear SVM on top of extracted features is arguably the most widely used strategy in few-shot learning as linear SVM is easy and fast to train whereas more complicated strategies like fine-tuning or training a neural network on top of the extracted feature have more randomness and may overfit due to the small size of the training set. NTK has the same benefits (easy and fast to train, no randomness) as the linear classifier but also allows some non-linearity in the design. Note since we consider image classification tasks, we use CNTK in experiments in this section. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experiment Setup. We mostly follow the settings of Goyal et al. (2019). Features are extracted from layer conv1, conv2, conv3, conv4, conv5 in ResNet-50 (He et al., 2016) trained on ImageNet (Deng et al., 2009), and we use these features for VOC07 classification task. The number of positive examples $k$ varies from 1 to 8. For each value of $k$ , for each class in the VOC07 dataset, we choose $k$ positive examples and $1 9 k$ negative examples. For each $k \\in \\{ 1 , 2 , \\dots , 8 \\}$ and for each class, we randomly choose 10 independent sets with $2 0 k$ training samples in each set. We report the mean and standard deviation of mAP on the test split of VOC07 dataset. This setting has been used in numbers of previous few-shot learning papers (Goyal et al., 2019; Zhang et al., 2017). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8406328e7172d77e3491c98f9b3421570da4ab48260156a35c4fc488254c5201.jpg",
+        "table_caption": [
+            "Table 4: Performance of linear SVM and CNTK with different number of convolutional layers on the 11-20th classes in VOC07. The cost value $C$ is tuned on the first 10 classes. Feature extracted from conv4 in ResNet-50. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linear SVM</td><td rowspan=1 colspan=1>1-layer CNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layer CNTK</td><td rowspan=1 colspan=1>4-layerCNTK</td><td rowspan=1 colspan=1>5-layerCNTK</td><td rowspan=1 colspan=1>6-layerCNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>23.25± 2.89</td><td rowspan=1 colspan=1>24.54 ± 3.39</td><td rowspan=1 colspan=1>24.57± 3.46</td><td rowspan=1 colspan=1>24.57 ± 3.50</td><td rowspan=1 colspan=1>24.52 ± 3.51</td><td rowspan=1 colspan=1>24.43± 3.50</td><td rowspan=1 colspan=1>24.34±3.49</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>32.26 ±3.85</td><td rowspan=1 colspan=1>33.81± 4.25</td><td rowspan=1 colspan=1>33.77 ± 4.33</td><td rowspan=1 colspan=1>33.68± 4.40</td><td rowspan=1 colspan=1>33.53± 4.46</td><td rowspan=1 colspan=1>33.31 ± 4.51</td><td rowspan=1 colspan=1>33.04± 4.55</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>38.41 ± 2.74</td><td rowspan=1 colspan=1>40.06± 2.70</td><td rowspan=1 colspan=1>40.10 ± 2.47</td><td rowspan=1 colspan=1>40.02 ± 2.41</td><td rowspan=1 colspan=1>39.87 ± 2.37</td><td rowspan=1 colspan=1>39.65± 2.35</td><td rowspan=1 colspan=1>39.36± 2.29</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>39.88± 2.08</td><td rowspan=1 colspan=1>42.20±1.78</td><td rowspan=1 colspan=1>41.80 ± 2.13</td><td rowspan=1 colspan=1>42.64± 2.00</td><td rowspan=1 colspan=1>42.51 ± 2.02</td><td rowspan=1 colspan=1>42.32 ± 2.03</td><td rowspan=1 colspan=1>42.08± 2.03</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>42.46 ± 2.70</td><td rowspan=1 colspan=1>44.71± 3.12</td><td rowspan=1 colspan=1>44.69 ± 2.98</td><td rowspan=1 colspan=1>44.72 ± 3.01</td><td rowspan=1 colspan=1>44.69 ± 3.04</td><td rowspan=1 colspan=1>44.57 ± 3.07</td><td rowspan=1 colspan=1>44.40±3.10</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>45.71 ± 3.27</td><td rowspan=1 colspan=1>49.02 ±2.63</td><td rowspan=1 colspan=1>48.63± 2.69</td><td rowspan=1 colspan=1>48.97±2.68</td><td rowspan=1 colspan=1>48.92 ± 2.71</td><td rowspan=1 colspan=1>48.79 ± 2.74</td><td rowspan=1 colspan=1>48.63± 2.77</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>47.97 ± 3.25</td><td rowspan=1 colspan=1>50.89±3.50</td><td rowspan=1 colspan=1>50.43±3.48</td><td rowspan=1 colspan=1>50.50±3.47</td><td rowspan=1 colspan=1>50.48 ±3.44</td><td rowspan=1 colspan=1>50.39 ±3.44</td><td rowspan=1 colspan=1>50.24±3.41</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>49.81 ± 2.18</td><td rowspan=1 colspan=1>52.32± 2.65</td><td rowspan=1 colspan=1>52.06± 2.32</td><td rowspan=1 colspan=1>52.23 ± 2.38</td><td rowspan=1 colspan=1>52.31 ± 2.40</td><td rowspan=1 colspan=1>52.26± 2.45</td><td rowspan=1 colspan=1>52.14± 2.50</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/0e729b38069cc634adcd8e002a78dc67648d766c94e5d9b1372e33fcd00f7cc0.jpg",
+        "table_caption": [
+            "Table 5: Performance of linear SVM and CNTK with different number of convolutional layers on the 11-20th classes in VOC07. The cost value $C$ is tuned on the first 10 classes. Feature extracted from conv3. in ResNet-50 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linear SVM</td><td rowspan=1 colspan=1>1-layer CNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layerCNTK</td><td rowspan=1 colspan=1>4-layer CNTK</td><td rowspan=1 colspan=1>5-layerCNTK</td><td rowspan=1 colspan=1>6-layerCNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>14.38 ± 1.98</td><td rowspan=1 colspan=1>15.42 ± 1.90</td><td rowspan=1 colspan=1>15.36 ± 1.93</td><td rowspan=1 colspan=1>15.30 ± 1.97</td><td rowspan=1 colspan=1>15.24 ± 2.02</td><td rowspan=1 colspan=1>15.17 ± 2.05</td><td rowspan=1 colspan=1>15.10± 2.08</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>16.67 ± 1.50</td><td rowspan=1 colspan=1>18.48 ± 1.58</td><td rowspan=1 colspan=1>18.39 ± 1.55</td><td rowspan=1 colspan=1>18.28 ± 1.52</td><td rowspan=1 colspan=1>18.18 ± 1.49</td><td rowspan=1 colspan=1>18.09 ± 1.47</td><td rowspan=1 colspan=1>17.99 ± 1.45</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>19.56 ± 1.04</td><td rowspan=1 colspan=1>21.79±1.69</td><td rowspan=1 colspan=1>21.73 ± 1.63</td><td rowspan=1 colspan=1>21.67 ± 1.60</td><td rowspan=1 colspan=1>21.61 ± 1.55</td><td rowspan=1 colspan=1>21.54± 1.50</td><td rowspan=1 colspan=1>21.47 ± 1.46</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>20.53 ± 1.62</td><td rowspan=1 colspan=1>23.39±2.13</td><td rowspan=1 colspan=1>23.36±2.18</td><td rowspan=1 colspan=1>23.29 ± 2.22</td><td rowspan=1 colspan=1>23.20± 2.24</td><td rowspan=1 colspan=1>23.11 ± 2.26</td><td rowspan=1 colspan=1>23.03 ± 2.27</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>21.51 ± 1.68</td><td rowspan=1 colspan=1>25.13±2.36</td><td rowspan=1 colspan=1>25.09± 2.37</td><td rowspan=1 colspan=1>25.03± 2.36</td><td rowspan=1 colspan=1>24.96± 2.36</td><td rowspan=1 colspan=1>24.87±2.36</td><td rowspan=1 colspan=1>24.78 ± 2.37</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>23.51 ± 2.39</td><td rowspan=1 colspan=1>26.51± 2.39</td><td rowspan=1 colspan=1>26.26±2.43</td><td rowspan=1 colspan=1>26.10�� 2.47</td><td rowspan=1 colspan=1>25.97 ± 2.47</td><td rowspan=1 colspan=1>25.87 ± 2.47</td><td rowspan=1 colspan=1>25.78± 2.46</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>24.33 ± 1.59</td><td rowspan=1 colspan=1>27.98 ±2.46</td><td rowspan=1 colspan=1>27.75± 2.52</td><td rowspan=1 colspan=1>28.24±2.22</td><td rowspan=1 colspan=1>28.20±2.22</td><td rowspan=1 colspan=1>28.14 ± 2.21</td><td rowspan=1 colspan=1>28.08± 2.21</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>25.31 ± 2.07</td><td rowspan=1 colspan=1>27.76 ± 2.87</td><td rowspan=1 colspan=1>28.52 ± 2.65</td><td rowspan=1 colspan=1>28.52 ± 2.69</td><td rowspan=1 colspan=1>28.49± 2.70</td><td rowspan=1 colspan=1>28.44± 2.72</td><td rowspan=1 colspan=1>28.38 ± 2.74</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We take the extracted features as the input to CNTK. We tried CNTK with 0-6 convolution layers, a global average pooling layer and a fully-connected layer. We normalize the data in the feature space, and finally use SVM to train the classifiers. Note without the convolution layer, it is equivalent to directly applying linear SVM after global average pooling and normalization. We use sklearn.svm.LinearSVC to train linear SVMs, and sklearn.svm.SVC to train kernel SVMs (for CNTK). To train SVM, we choose the cost value $C$ from $2 ^ { [ - 1 9 , - 4 ] } \\cup 1 0 ^ { [ - 7 , 2 ] }$ and set the class weight ratio to be $2 : 1$ for positive/negative classes as in Goyal et al. (2019). ",
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+    {
+        "type": "text",
+        "text": "Since the number of given samples is usually small, Goyal et al. (2019) chooses to report the performance of the best cost value $C$ . In our experiments, we use a more standard method to perform cross-validation. We use the first 10 classes of VOC07 to tune $C$ , and report the performance of selected $C$ in the other 10 classes in Table 3-5. We also report the performance of the best $C$ as in Goyal et al. (2019), in Tables 7-9 in the appendix for completeness. ",
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+    {
+        "type": "text",
+        "text": "Discussions. First, we find CNTK is a strong drop-in classifier to replace the linear classifier in few-shot learning. Tables 3-9 clearly demonstrate that CNTK is consistently better than linear classifier. Note Tables 3-9 only show the prediction accuracy in an average sense. In fact, we find that on every randomly sampled training set, CNTK always gives a better performance. We conjecture that CNTK gives better performance because the non-linearity in CNTK helps prediction. This is verified by looking at the performance gain of CNTK for different feature extractors. Note Conv5 is often considered to be most useful for linear classification. There, CNTK only outperforms the linear classifier by about $1 \\%$ . On the other hand, with Conv3 and Conv4 features, CNTK can outperform linear classifier by around $2 \\%$ and sometimes by $3 \\% { - } 4 \\%$ (last two rows in Table 5). We believe this happens because Conv3 and Conv4 correspond to middle-level features, and thus non-linearity is indeed beneficial for better accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also observe that CNTK often performs the best with a single convolutional layer. This is expected since features extracted by ResNet-50 already produce a good representation. Nevertheless, we find for middle-level features Conv3 with $k = 7$ or 8, CNTK with 3 convolutional layers give the best performance. We believe this is because with more data, utilizing the non-linearity induced by CNTK can further boost the performance. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The Neural Tangent Kernel, discovered by mathematical curiosity about deep networks in the limit of infinite width, is found to yield superb performance on low-data tasks, beating extensively tuned versions of classic methods such as random forests. The (fully-connected) NTK classifiers are easy to compute (no GPU required) and thus should be a good off-the-shelf classifier in many settings. We plan to release our code to allow drop-in replacement for SVMs and linear regression. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Many theoretical questions arise. Do NTK SVMs correspond to some infinite net architecture (as NTK ridge regression does)? What explains generalization in small-data settings? (This understanding is imperfect even for random forests.) Finally one can derive NTK corresponding to other architectures, e.g., recurrent neural tangent kernel (RNTK) induced by recurrent neural networks. It would be an interesting future research direction to test their performance on benchmark tasks. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "S. Arora, Z. Li and D. Yu are supported by NSF, ONR, Simons Foundation, Schmidt Foundation, Amazon Research, DARPA and SRC. S. S. Du is supported by National Science Foundation (Grant No. DMS-1638352) and the Infosys Membership. R. Salakhutdinov and R. Wang are supported in part by NSF IIS-1763562, AFRL CogDeCON FA875018C0014, and DARPA SAGAMORE HR00111990016. Part of this work was done while S. S. Du was visiting Google Brain Princeton and R. Wang was visiting Princeton University. The authors would like to thank Amazon Web Services for providing compute time for the experiments in this paper, and NVIDIA for GPU support. We thank Priya Goyal for providing experiment details of Goyal et al. (2019). We thank Xiaolong Wang for discussing the few-shot learning task. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            329,
+            285,
+            344
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Zeyuan Allen-Zhu, Yuanzhi Li, and Yingyu Liang. Learning and generalization in overparameterized neural networks, going beyond two layers. arXiv preprint arXiv:1811.04918, 2018a. ",
+        "bbox": [
+            176,
+            352,
+            821,
+            380
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Zeyuan Allen-Zhu, Yuanzhi Li, and Zhao Song. A convergence theory for deep learning via overparameterization. arXiv preprint arXiv:1811.03962, 2018b. ",
+        "bbox": [
+            174,
+            387,
+            820,
+            416
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Sanjeev Arora, Simon S Du, Wei Hu, Zhiyuan Li, Ruslan Salakhutdinov, and Ruosong Wang. On exact computation with an infinitely wide neural net. arXiv preprint arXiv:1904.11955, 2019a. ",
+        "bbox": [
+            174,
+            424,
+            821,
+            453
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Sanjeev Arora, Simon S Du, Wei Hu, Zhiyuan Li, and Ruosong Wang. Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks. arXiv preprint arXiv:1901.08584, 2019b. ",
+        "bbox": [
+            173,
+            460,
+            823,
+            502
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alberto Bietti and Julien Mairal. On the inductive bias of neural tangent kernels. arXiv preprint arXiv:1905.12173, 2019. ",
+        "bbox": [
+            173,
+            511,
+            823,
+            540
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Youngmin Cho and Lawrence K Saul. Kernel methods for deep learning. In Advances in neural information processing systems, pp. 342–350, 2009. ",
+        "bbox": [
+            174,
+            547,
+            821,
+            577
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Djork-Arne Clevert, Thomas Unterthiner, and Sepp Hochreiter. Fast and accurate deep network ´ learning by exponential linear units (elus). arXiv preprint arXiv:1511.07289, 2015. ",
+        "bbox": [
+            174,
+            583,
+            821,
+            613
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Amit Daniely. SGD learns the conjugate kernel class of the network. arXiv preprint arXiv:1702.08503, 2017. ",
+        "bbox": [
+            174,
+            621,
+            823,
+            648
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Amit Daniely, Roy Frostig, and Yoram Singer. Toward deeper understanding of neural networks: The power of initialization and a dual view on expressivity. In Advances In Neural Information Processing Systems, pp. 2253–2261, 2016. ",
+        "bbox": [
+            176,
+            656,
+            823,
+            700
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248–255. Ieee, 2009. ",
+        "bbox": [
+            174,
+            707,
+            823,
+            750
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Simon S Du, Jason D Lee, Haochuan Li, Liwei Wang, and Xiyu Zhai. Gradient descent finds global minima of deep neural networks. arXiv preprint arXiv:1811.03804, 2018a. ",
+        "bbox": [
+            173,
+            757,
+            821,
+            786
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Simon S Du, Xiyu Zhai, Barnabas Poczos, and Aarti Singh. Gradient descent provably optimizes over-parameterized neural networks. arXiv preprint arXiv:1810.02054, 2018b. ",
+        "bbox": [
+            173,
+            794,
+            820,
+            824
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Simon S. Du, Kangcheng Hou, Barnabas P ´ oczos, Ruslan Salakhutdinov, Ruosong Wang, and Keyulu ´ Xu. Graph neural tangent kernel: Fusing graph neural networks with graph kernels. ArXiv, abs/1905.13192, 2019. ",
+        "bbox": [
+            174,
+            830,
+            821,
+            873
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Manuel Fernandez-Delgado, Eva Cernadas, Sen ´ en Barro, and Dinani Amorim. Do we need hun- ´ dreds of classifiers to solve real world classification problems? The Journal of Machine Learning Research, 15(1):3133–3181, 2014. ",
+        "bbox": [
+            174,
+            882,
+            821,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Adria Garriga-Alonso, Carl Edward Rasmussen, and Laurence Aitchison. Deep convolutional net- \\` works as shallow gaussian processes. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id $=$ Bklfsi0cKm. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Priya Goyal, Dhruv Mahajan, Abhinav Gupta, and Ishan Misra. Scaling and benchmarking selfsupervised visual representation learning. arXiv preprint arXiv:1905.01235, 2019. ",
+        "bbox": [
+            174,
+            156,
+            820,
+            185
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tamir Hazan and Tommi Jaakkola. Steps toward deep kernel methods from infinite neural networks. arXiv preprint arXiv:1508.05133, 2015. ",
+        "bbox": [
+            174,
+            195,
+            820,
+            224
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "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. ",
+        "bbox": [
+            173,
+            234,
+            825,
+            277
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Arthur Jacot, Franck Gabriel, and Clement Hongler. Neural tangent kernel: Convergence and gen- ´ eralization in neural networks. In Advances in neural information processing systems, pp. 8571– 8580, 2018. ",
+        "bbox": [
+            174,
+            287,
+            825,
+            330
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jaehoon Lee, Jascha Sohl-dickstein, Jeffrey Pennington, Roman Novak, Sam Schoenholz, and Yasaman Bahri. Deep neural networks as gaussian processes. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id $\\underline { { \\underline { { \\mathbf { \\Pi } } } } } =$ B1EA-M-0Z. ",
+        "bbox": [
+            173,
+            340,
+            825,
+            396
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jaehoon Lee, Lechao Xiao, Samuel S Schoenholz, Yasaman Bahri, Jascha Sohl-Dickstein, and Jeffrey Pennington. Wide neural networks of any depth evolve as linear models under gradient descent. arXiv preprint arXiv:1902.06720, 2019. ",
+        "bbox": [
+            173,
+            406,
+            825,
+            450
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yuanzhi Li and Yingyu Liang. Learning overparameterized neural networks via stochastic gradient descent on structured data. arXiv preprint arXiv:1808.01204, 2018. ",
+        "bbox": [
+            173,
+            460,
+            823,
+            489
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alexander G de G Matthews, Mark Rowland, Jiri Hron, Richard E Turner, and Zoubin Ghahramani. Gaussian process behaviour in wide deep neural networks. arXiv preprint arXiv:1804.11271, 2018. ",
+        "bbox": [
+            174,
+            498,
+            826,
+            541
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Radford M Neal. Priors for infinite networks. In Bayesian Learning for Neural Networks, pp. 29–53. Springer, 1996. ",
+        "bbox": [
+            171,
+            551,
+            823,
+            580
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Roman Novak, Lechao Xiao, Yasaman Bahri, Jaehoon Lee, Greg Yang, Daniel A. Abolafia, Jeffrey Pennington, and Jascha Sohl-dickstein. Bayesian deep convolutional networks with many channels are gaussian processes. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id $=$ B1g30j0qF7. ",
+        "bbox": [
+            173,
+            590,
+            825,
+            648
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Matthew Olson, Abraham Wyner, and Richard Berk. Modern neural networks generalize on small data sets. In Advances in Neural Information Processing Systems, pp. 3619–3628, 2018. ",
+        "bbox": [
+            174,
+            657,
+            821,
+            688
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Daniel S Park, Jascha Sohl-Dickstein, Quoc V Le, and Samuel L Smith. The effect of network width on stochastic gradient descent and generalization: an empirical study. arXiv preprint arXiv:1905.03776, 2019. ",
+        "bbox": [
+            174,
+            695,
+            825,
+            739
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Nicolas Le Roux and Yoshua Bengio. Continuous neural networks. In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, volume 2 of Proceedings of Machine Learning Research, pp. 404–411, San Juan, Puerto Rico, 2007. ",
+        "bbox": [
+            173,
+            750,
+            826,
+            792
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Bernhard Scholkopf and Alexander J Smola. Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT press, 2001. ",
+        "bbox": [
+            169,
+            803,
+            823,
+            833
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Christopher KI Williams. Computing with infinite networks. In Advances in neural information processing systems, pp. 295–301, 1997. ",
+        "bbox": [
+            171,
+            842,
+            823,
+            871
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Greg Yang. Scaling limits of wide neural networks with weight sharing: Gaussian process behavior, gradient independence, and neural tangent kernel derivation. arXiv preprint arXiv:1902.04760, 2019. ",
+        "bbox": [
+            174,
+            882,
+            825,
+            922
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Richard Zhang, Phillip Isola, and Alexei A Efros. Split-brain autoencoders: Unsupervised learning by cross-channel prediction. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1058–1067, 2017. ",
+        "bbox": [
+            176,
+            140,
+            823,
+            184
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Difan Zou, Yuan Cao, Dongruo Zhou, and Quanquan Gu. Stochastic gradient descent optimizes over-parameterized deep ReLU networks. arXiv preprint arXiv:1811.08888, 2018. ",
+        "bbox": [
+            171,
+            193,
+            823,
+            222
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A ADDITIONAL EXPERIMENTAL DETAILS ON UCI ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            606,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe our experiment setup. ",
+        "bbox": [
+            173,
+            137,
+            498,
+            151
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Dataset Selection Since we only wish to test on small datasets, we only select UCI datasets with number of samples smaller than 5000. Furthermore, we only use datasets without explicit training / testing splitting from the link: http://persoal.citius.usc.es/manuel.fernandez. delgado/papers/jmlr/data.tar.gz, which are the pre-processed datasets by Fernandez- ´ Delgado et al. (2014). These datasets are originally from UCI datasets (including most of the datasets before March, 2013) and 4 real-world datasets not included in the UCI repository (see Fernandez- ´ Delgado et al. (2014) for details). The datasets are pre-processed to be classification problems and all categorical features are turned into numerical features and normalized along the samples for each feature. The reason to discard datasets with explicit splitting is that we found there is an obvious distributional shift between training and testing data. The pre-processed data provided from the link shows that training and testing data are normalized with different mean and standard deviation. For example, on the AUDIOLOGY-STD dataset, the support of the first feature is $\\{ - 0 . 7 6 1 5 2 6 , 1 . 3 0 5 4 7 \\}$ in training and $\\{ - 1 . 2 , 0 . 8 \\}$ in testing; on the ANNEALING dataset, the support of the first feature is $\\{ - 0 . 6 7 0 2 7 4 , - 0 . 2 2 3 4 2 5 , 3 . 7 9 8 2 2 \\}$ in training and $\\{ - 0 . 1 1 1 1 1 , 0 , 1 \\}$ in testing. See Table 6 for a summary of the datasets. ",
+        "bbox": [
+            174,
+            174,
+            825,
+            382
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/a8f7280014c5716817cfc23a40eb922d31222638dfa3505820897e1c873b48fa.jpg",
+        "table_caption": [
+            "Table 6: Dataset Summary "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td># samples</td><td>#features</td><td>#classes</td></tr><tr><td>min</td><td>10</td><td>3</td><td>2</td></tr><tr><td>25%</td><td>178</td><td>8</td><td>2</td></tr><tr><td>50%</td><td>583</td><td>16</td><td>3</td></tr><tr><td>75%</td><td>1022</td><td>32</td><td>6</td></tr><tr><td>max</td><td>5000</td><td>262</td><td>100</td></tr></table>",
+        "bbox": [
+            349,
+            401,
+            647,
+            489
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Performance Comparison Details We follow the comparison setup in Fernandez-Delgado et al. ´ (2014) that we report 4-fold cross-validation. For hyperparameters, we tune them with the same validation methodology in Fernandez-Delgado et al. (2014): all available training samples are ran- ´ domly split into one training and one test set, while imposing that each class has the same number of training and test samples. Then the parameter with best validation accuracy is selected. It is possible to give confidence bounds for this parameter tuning scheme, but they are worse than standard ones for separated training/validation/testing data. For NTK and NN classifiers we train them on these 90 datasets. For other classifiers, we use the results from Fernandez-Delgado et al. (2014). ´ ",
+        "bbox": [
+            173,
+            551,
+            825,
+            662
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "NTK Specification We calculate NTK induced fully-connected neural networks with $L$ layers where $L ^ { \\prime }$ bottom layers are fixed, and then use $C$ -support vector classification implemented by sklearn.svm. We tune hyperparameters $L$ from 1 to 5, $L ^ { \\prime }$ from 0 to $L - 1$ , and cost value $C$ as powers of ten from $- 2$ to 4. The number of kernels used is 15, so the total number of parameter combinations is 105. Note this number is much less than the number of hyperparameter of Gaussian Kernel reported in Fernandez-Delgado et al. (2014) where they tune hyperparameters with 500 ´ combinations. ",
+        "bbox": [
+            174,
+            685,
+            825,
+            782
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "NN Specification We use fully-connected NN with $L$ layers, 512 number of hidden nodes per layer and use gradient descent to train the neural network. We tune hyperparameters $L$ from 1 to 5, with $/$ without batch normalization and learning rate 0.1 or 1. We run gradient descent for 2000 epochs.5 We treat NN with He initialization and NTK initialization as two classifiers and report their results separately. ",
+        "bbox": [
+            174,
+            804,
+            825,
+            875
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL EXPERIMENTAL DETAILS ON SMALL CIFAR-10 DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            815,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We randomly choose $n / 1 0$ samples from each class of CIFAR-10 training set and test classifiers on the whole testing set. $n$ varies from 10 to 1280. For each $n$ , we repeat 20 times and report the mean accuracy and its standard deviation for each classifier. ",
+        "bbox": [
+            174,
+            133,
+            823,
+            175
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The number of convolution layers of CNTK ranges from 5-14. After convolutional layers, we apply a global pooling layer and a fully connected layer. We refer readers to Arora et al. (2019a) for exact formulas of CNTK. We normalize the kernel such that each sample has unit length in feature space. ",
+        "bbox": [
+            174,
+            183,
+            825,
+            224
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We use ResNet-34 with width 64,128,256 and default hyperparameters: learning rate 0.1, momentum 0.9, weight decay 0.0005. We decay the learning rate by 10 at the epoch of 80 and 120, with 160 training epochs in total. The training batch size is the minimum of the size of the whole training dataset and 160. We report the best testing accuracy among epochs. ",
+        "bbox": [
+            174,
+            231,
+            825,
+            287
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL RESULTS IN FEW-SHOT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            308,
+            611,
+            324
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/381824b3c1342a199c6c2be9a630d898f111d500738fbe35c50880e52e89e4a0.jpg",
+        "table_caption": [
+            "Tables 7-9 show the performance of the best $C$ as has been done in Goyal et al. (2019). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linearSVM</td><td rowspan=1 colspan=1>1-layer CNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layer CNTK</td><td rowspan=1 colspan=1>4-layer CNTK</td><td rowspan=1 colspan=1>5-layerCNTK</td><td rowspan=1 colspan=1>6-layerCNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>52.63 ± 5.35</td><td rowspan=1 colspan=1>53.16± 4.96</td><td rowspan=1 colspan=1>53.08±4.89</td><td rowspan=1 colspan=1>52.83 ± 4.85</td><td rowspan=1 colspan=1>52.44 ± 4.85</td><td rowspan=1 colspan=1>51.88 ± 4.85</td><td rowspan=1 colspan=1>51.21 ± 4.88</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>65.08± 2.69</td><td rowspan=1 colspan=1>66.25±3.09</td><td rowspan=1 colspan=1>66.29±3.09</td><td rowspan=1 colspan=1>66.11 ± 3.07</td><td rowspan=1 colspan=1>65.79±3.04</td><td rowspan=1 colspan=1>65.33± 3.00</td><td rowspan=1 colspan=1>64.77± 2.99</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>71.78 ± 1.85</td><td rowspan=1 colspan=1>72.76 ± 1.76</td><td rowspan=1 colspan=1>72.72 ± 1.73</td><td rowspan=1 colspan=1>72.49 ± 1.71</td><td rowspan=1 colspan=1>72.14 ± 1.73</td><td rowspan=1 colspan=1>71.71 ± 1.76</td><td rowspan=1 colspan=1>71.17 ± 1.78</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>74.40 ± 1.86</td><td rowspan=1 colspan=1>75.37 ± 1.95</td><td rowspan=1 colspan=1>75.33 ± 1.95</td><td rowspan=1 colspan=1>75.14 ± 1.96</td><td rowspan=1 colspan=1>74.85 ± 1.96</td><td rowspan=1 colspan=1>74.48 ± 1.99</td><td rowspan=1 colspan=1>74.05 ± 1.97</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>75.94 ± 2.51</td><td rowspan=1 colspan=1>77.01± 2.39</td><td rowspan=1 colspan=1>76.91 ± 2.39</td><td rowspan=1 colspan=1>76.72 ± 2.39</td><td rowspan=1 colspan=1>76.47 ± 2.37</td><td rowspan=1 colspan=1>76.13 ± 2.39</td><td rowspan=1 colspan=1>75.76 ± 2.39</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>76.39 ± 1.27</td><td rowspan=1 colspan=1>77.15 ± 1.47</td><td rowspan=1 colspan=1>77.14 ± 1.50</td><td rowspan=1 colspan=1>77.02 ± 1.55</td><td rowspan=1 colspan=1>76.83 ± 1.58</td><td rowspan=1 colspan=1>76.62 ± 1.60</td><td rowspan=1 colspan=1>76.35 ± 1.61</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>78.18±0.86</td><td rowspan=1 colspan=1>79.42± 0.85</td><td rowspan=1 colspan=1>79.36 ± 0.87</td><td rowspan=1 colspan=1>79.22 ± 0.88</td><td rowspan=1 colspan=1>79.00±0.91</td><td rowspan=1 colspan=1>78.75± 0.93</td><td rowspan=1 colspan=1>78.50±0.94</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>79.78 ± 0.65</td><td rowspan=1 colspan=1>80.47 ± 0.59</td><td rowspan=1 colspan=1>80.45± 0.61</td><td rowspan=1 colspan=1>80.33 ± 0.63</td><td rowspan=1 colspan=1>80.14± 0.65</td><td rowspan=1 colspan=1>79.92 ± 0.67</td><td rowspan=1 colspan=1>79.66±0.70</td></tr></table>",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/b12ff60ea59bdc372a6ca2221cd5677606889c3a603fb70e7897767ac09237a3.jpg",
+        "table_caption": [
+            "Table 7: Performance of linear SVM and CNTK with different number of convolutional layers on all classes in VOC07 with the best $C$ . Feature extracted from conv5 in ResNet-50 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linear SVM</td><td rowspan=1 colspan=1>1-layer CNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layer CNTK</td><td rowspan=1 colspan=1>4-layer CNTK</td><td rowspan=1 colspan=1>5-layer CNTK</td><td rowspan=1 colspan=1>6-layer CNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>24.18± 2.64</td><td rowspan=1 colspan=1>25.15 ± 2.93</td><td rowspan=1 colspan=1>25.13±3.00</td><td rowspan=1 colspan=1>25.06±3.05</td><td rowspan=1 colspan=1>24.95 ±3.10</td><td rowspan=1 colspan=1>24.80 ±3.13</td><td rowspan=1 colspan=1>24.60±3.14</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>31.59 ± 2.64</td><td rowspan=1 colspan=1>32.88 ± 2.91</td><td rowspan=1 colspan=1>32.81 ± 2.94</td><td rowspan=1 colspan=1>32.67 ± 2.96</td><td rowspan=1 colspan=1>32.48± 2.98</td><td rowspan=1 colspan=1>32.24± 2.99</td><td rowspan=1 colspan=1>31.94 ± 3.01</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>37.34 ± 2.18</td><td rowspan=1 colspan=1>38.69 ± 2.39</td><td rowspan=1 colspan=1>38.64±2.26</td><td rowspan=1 colspan=1>38.53±2.28</td><td rowspan=1 colspan=1>38.36 ± 2.31</td><td rowspan=1 colspan=1>38.12 ± 2.33</td><td rowspan=1 colspan=1>37.83± 2.35</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>39.67 ± 2.31</td><td rowspan=1 colspan=1>41.34 ± 2.33</td><td rowspan=1 colspan=1>41.39 ± 2.28</td><td rowspan=1 colspan=1>41.33 ± 2.24</td><td rowspan=1 colspan=1>41.21 ± 2.22</td><td rowspan=1 colspan=1>41.02 ±2.20</td><td rowspan=1 colspan=1>40.77±2.18</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>41.74 ± 1.20</td><td rowspan=1 colspan=1>43.48±1.40</td><td rowspan=1 colspan=1>43.48±1.24</td><td rowspan=1 colspan=1>43.44 ± 1.25</td><td rowspan=1 colspan=1>43.32 ± 1.27</td><td rowspan=1 colspan=1>43.12 ± 1.30</td><td rowspan=1 colspan=1>42.89 ± 1.33</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>44.68 ± 1.84</td><td rowspan=1 colspan=1>46.77± 1.82</td><td rowspan=1 colspan=1>46.42 ± 1.91</td><td rowspan=1 colspan=1>46.44 ± 1.90</td><td rowspan=1 colspan=1>46.39± 1.91</td><td rowspan=1 colspan=1>46.25 ± 1.93</td><td rowspan=1 colspan=1>46.08± 1.94</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>47.71 ± 1.84</td><td rowspan=1 colspan=1>50.07 ± 1.97</td><td rowspan=1 colspan=1>49.74 ± 1.95</td><td rowspan=1 colspan=1>49.63± 1.81</td><td rowspan=1 colspan=1>49.56 ± 1.81</td><td rowspan=1 colspan=1>49.41 ± 1.82</td><td rowspan=1 colspan=1>49.19 ± 1.81</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>48.91 ± 1.90</td><td rowspan=1 colspan=1>51.32±1.97</td><td rowspan=1 colspan=1>50.98 ± 1.87</td><td rowspan=1 colspan=1>51.09 ± 1.84</td><td rowspan=1 colspan=1>51.09 ± 1.81</td><td rowspan=1 colspan=1>51.00 ± 1.78</td><td rowspan=1 colspan=1>50.84± 1.76</td></tr></table>",
+        "bbox": [
+            178,
+            530,
+            821,
+            633
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/5039e7f5e2c06f67acd75c550e473de348ee1d0840df2bb8ba10bf200d98cdd6.jpg",
+        "table_caption": [
+            "Table 8: Performance of linear SVM and CNTK with different number of convolutional layers on all classes in VOC07 with the best $C$ . Feature extracted from conv4 in ResNet-50 ",
+            "Table 9: Performance of linear SVM and CNTK with different number of convolutional layers on all classes in VOC07 with the best $C$ . Feature extracted from conv3 in ResNet-50 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>k</td><td rowspan=1 colspan=1>linear SVM</td><td rowspan=1 colspan=1>1-layerCNTK</td><td rowspan=1 colspan=1>2-layer CNTK</td><td rowspan=1 colspan=1>3-layer CNTK</td><td rowspan=1 colspan=1>4-layer CNTK</td><td rowspan=1 colspan=1>5-layerCNTK</td><td rowspan=1 colspan=1>6-layerCNTK</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>14.72 ± 1.49</td><td rowspan=1 colspan=1>15.01 ± 1.27</td><td rowspan=1 colspan=1>14.96 ± 1.28</td><td rowspan=1 colspan=1>14.92 ± 1.30</td><td rowspan=1 colspan=1>14.87 ± 1.32</td><td rowspan=1 colspan=1>14.81 ± 1.34</td><td rowspan=1 colspan=1>14.75 ± 1.36</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>16.67 ± 1.04</td><td rowspan=1 colspan=1>17.52 ± 1.05</td><td rowspan=1 colspan=1>17.45 ± 1.03</td><td rowspan=1 colspan=1>17.36 ± 1.02</td><td rowspan=1 colspan=1>17.28 ± 1.00</td><td rowspan=1 colspan=1>17.20± 0.99</td><td rowspan=1 colspan=1>17.11 ± 0.97</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>19.21 ± 1.31</td><td rowspan=1 colspan=1>20.65±1.68</td><td rowspan=1 colspan=1>20.63± 1.65</td><td rowspan=1 colspan=1>20.58± 1.63</td><td rowspan=1 colspan=1>20.53± 1.61</td><td rowspan=1 colspan=1>20.47 ± 1.59</td><td rowspan=1 colspan=1>20.41± 1.57</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>20.87 ± 1.49</td><td rowspan=1 colspan=1>22.28±1.80</td><td rowspan=1 colspan=1>22.27 ± 1.81</td><td rowspan=1 colspan=1>22.22 ± 1.83</td><td rowspan=1 colspan=1>22.16 ± 1.83</td><td rowspan=1 colspan=1>22.09 ± 1.83</td><td rowspan=1 colspan=1>22.03± 1.82</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>21.98 ± 1.17</td><td rowspan=1 colspan=1>23.45±1.14</td><td rowspan=1 colspan=1>23.45±1.12</td><td rowspan=1 colspan=1>23.43± 1.08</td><td rowspan=1 colspan=1>23.38 ± 1.07</td><td rowspan=1 colspan=1>23.32 ± 1.05</td><td rowspan=1 colspan=1>23.25± 1.04</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>23.02 ± 1.56</td><td rowspan=1 colspan=1>24.44 ± 1.61</td><td rowspan=1 colspan=1>24.29 ± 1.66</td><td rowspan=1 colspan=1>24.27 ± 1.82</td><td rowspan=1 colspan=1>24.25 ± 1.81</td><td rowspan=1 colspan=1>24.21 ± 1.80</td><td rowspan=1 colspan=1>24.18± 1.78</td></tr><tr><td rowspan=1 colspan=1>7</td><td rowspan=1 colspan=1>24.38± 1.52</td><td rowspan=1 colspan=1>25.96 ±1.80</td><td rowspan=1 colspan=1>25.97± 1.70</td><td rowspan=1 colspan=1>25.98± 1.70</td><td rowspan=1 colspan=1>25.96 ± 1.69</td><td rowspan=1 colspan=1>25.92 ± 1.67</td><td rowspan=1 colspan=1>25.88± 1.66</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>25.28 ± 1.39</td><td rowspan=1 colspan=1>27.06± 1.61</td><td rowspan=1 colspan=1>27.15 ± 1.58</td><td rowspan=1 colspan=1>27.18±1.56</td><td rowspan=1 colspan=1>27.17 ± 1.53</td><td rowspan=1 colspan=1>27.13 ± 1.51</td><td rowspan=1 colspan=1>27.09± 1.50</td></tr></table>",
+        "bbox": [
+            176,
+            694,
+            821,
+            797
+        ],
+        "page_idx": 12
+    }
+]

parse/train/ry1arUgCW/ry1arUgCW.md

@@ -0,0 +1,327 @@
+# DORA THE EXPLORER: DIRECTED OUTREACHING REINFORCEMENT ACTION-SELECTION
+
+Leshem Choshen∗ School of Computer Science and Engineering and Department of Cognitive Sciences The Hebrew University of Jerusalem leshem.choshen@mail.huji.ac.il
+
+Lior Fox∗ The Edmond and Lily Safra Center for Brain Sciences The Hebrew University of Jerusalem lior.fox@mail.huji.ac.il
+
+# Yonatan Loewenstein
+
+The Edmond and Lily Safra Center for Brain Sciences, Departments of Neurobiology and Cognitive Sciences and the Federmann Center for the Study of Rationality The Hebrew University of Jerusalem yonatan@huji.ac.il
+
+# ABSTRACT
+
+Exploration is a fundamental aspect of Reinforcement Learning, typically implemented using stochastic action-selection. Exploration, however, can be more efficient if directed toward gaining new world knowledge. Visit-counters have been proven useful both in practice and in theory for directed exploration. However, a major limitation of counters is their locality. While there are a few model-based solutions to this shortcoming, a model-free approach is still missing. We propose $E$ -values, a generalization of counters that can be used to evaluate the propagating exploratory value over state-action trajectories. We compare our approach to commonly used RL techniques, and show that using $E$ -values improves learning and performance over traditional counters. We also show how our method can be implemented with function approximation to efficiently learn continuous MDPs. We demonstrate this by showing that our approach surpasses state of the art performance in the Freeway Atari 2600 game.
+
+# 1 INTRODUCTION
+
+”If there’s a place you gotta go - I’m the one you need to know.“ (Map, Dora The Explorer)
+
+We consider Reinforcement Learning in a Markov Decision Process (MDP). An MDP is a fivetuple $M = ( \mathcal { S } , \mathcal { A } , P , R , \gamma )$ where $s$ is a set of states and $\mathcal { A }$ is a set of actions. The dynamics of the process is given by $P \left( s ^ { \prime } | s , a \right)$ which denotes the transition probability from state $s$ to state $s ^ { \prime }$ following action $a$ . Each such transition also has a distribution $R \left( r | s , a \right)$ from which the reward for such transitions is sampled. Given a policy $\pi : { \mathcal { S } }  A$ , a function – possibly stochastic – deciding which actions to take in each of the states, the state-action value function $Q ^ { \pi } : S \times \mathcal { A }  \mathbb { R }$ satisfies:
+
+$$
+Q ^ { \pi } \left( s , a \right) = \underset { r , s ^ { \prime } \sim R \times P \left( \cdot | s , a \right) } { \mathbb { E } } \left[ r + \gamma Q ^ { \pi } \left( s ^ { \prime } , \pi \left( s ^ { \prime } \right) \right) \right]
+$$
+
+where $\gamma$ is the discount factor. The agent’s goal is to find an optimal policy $\pi ^ { * }$ that maximizes $Q ^ { \pi } \left( s , \pi \left( s \right) \right)$ . For brevity, $Q ^ { \pi ^ { * } } \triangleq Q ^ { * }$ . There are two main approaches for learning $\pi ^ { * }$ . The first is a model-based approach, where the agent learns an internal model of the MDP (namely $P$ and $R$ ). Given a model, the optimal policy could be found using dynamic programming methods such as Value Iteration (Sutton $\&$ Barto, 1998). The alternative is a model-free approach, where the agent learns only the value function of states or state-action pairs, without learning a model (Kaelbling et al., 1996)1.
+
+The ideas put forward in this paper are relevant to any model-free learning of MDPs. For concreteness, we focus on a particular example, $Q$ -Learning (Watkins & Dayan, 1992; Sutton & Barto, 1998). $Q$ -Learning is a common method for learning $Q ^ { * }$ , where the agent iteratively updates its values of $Q \left( s , a \right)$ by performing actions and observing their outcomes. At each step the agent takes action $a _ { t }$ then it is transferred from $s _ { t }$ to $s _ { t + 1 }$ and observe reward $r$ . Then it applies the update rule regulated by a learning rate $\alpha$ :
+
+$$
+Q \left( s _ { t } , a _ { t } \right) \gets \left( 1 - \alpha \right) Q \left( s _ { t } , a _ { t } \right) + \alpha \left( r + \gamma \operatorname* { m a x } _ { a } Q \left( s _ { t + 1 } , a \right) \right) .
+$$
+
+# 1.1 EXPLORATION AND EXPLOITATION
+
+Balancing between Exploration and Exploitation is a major challenge in Reinforcement Learning. Seemingly, the agent may want to choose the alternative associated with the highest expected reward, a behavior known as exploitation. However, in that case it may fail to learn that there are better options. Therefore exploration, namely the taking of new actions and the visit of new states, may also be beneficial. It is important to note that exploitation is also inherently relevant for learning, as we want the agent to have better estimations of the values of valuable state-actions and we care less about the exact values of actions that the agent already knows to be clearly inferior.
+
+Formally, to guarantee convergence to $Q ^ { * }$ , the Q-Learning algorithm must visit each state-action pair infinitely many times. A naive random walk exploration is sufficient for converging asymptotically. However, such random exploration has two major limitations when the learning process is finite. First, the agent would not utilize its current knowledge about the world to guide its exploration. For example, an action with a known disastrous outcome will be explored over and over again. Second, the agent would not be biased in favor of exploring unvisited trajectories more than the visited ones – hence ”wasting” exploration resources on actions and trajectories which are already well known to it.
+
+A widely used method for dealing with the first problem is the $\epsilon$ -greedy schema (Sutton & Barto, 1998), in which with probability $1 - \epsilon$ the agent greedily chooses the best action (according to current estimation), and with probability $\epsilon$ it chooses a random action. Another popular alternative, emphasizing the preference to learn about actions associated with higher rewards, is to draw actions from a Boltzmann Distribution (Softmax) over the learned $Q$ values, regulated by a Temperature parameter. While such approaches lead to more informed exploration that is based on learning experience, they still fail to address the second issue, namely they are not directed (Thrun, 1992) towards gaining more knowledge, not biasing actions in the direction of unexplored trajectories.
+
+Another important approach in the study of efficient exploration is based on Sample Complexity of Exploration as defined in the PAC-MDP literature (Kakade et al., 2003). Relevant to our work is Delayed Q Learning (Strehl et al., 2006), a model-free algorithm that has theoretical PAC-MDP guarantees. However, to ensure these theoretical guarantees this algorithm uses a conservative exploration which might be impractical (see also (Kolter & $\mathrm { N g }$ , 2009) and Appendix B).
+
+# 1.2 CURRENT DIRECTED EXPLORATION AND ITS LIMITATIONS
+
+In order to achieve directed exploration, the estimation of an exploration value of the different stateactions (often termed exploration bonus) is needed. The most commonly used exploration bonus is based on counting (Thrun, 1992) – for each pair $( s , a )$ , store a counter $\displaystyle \dot { C } \left( s , a \right)$ that indicates how many times the agent performed action $a$ at state $s$ so far. Counter-based methods are widely used both in practice and in theory (Kolter & $\mathrm { N g }$ , 2009; Strehl & Littman, 2008; Guez et al., 2012; Busoniu et al., 2008). Other options for evaluating exploration include recency and value difference (or error) measures (Thrun, 1992; Tokic & Palm, 2011). While all of these exploration measures can be used for directed exploration, their major limitation in a model-free settings is that the exploratory value of a state-action pair is evaluated with respect only to its immediate outcome, one step ahead. It seems desirable to determine the exploratory value of an action not only by how much new immediate knowledge the agent gains from it, but also by how much more new knowledge could be gained from a trajectory starting with it. The goal of this work is to develop a measure for such exploratory values of state-action pairs, in a model-free settings.
+
+# 2 LEARNING EXPLORATION VALUES
+
+# 2.1 PROPAGATING EXPLORATION VALUES
+
+The challenge discussed in 1.2 is in fact similar to that of learning the value functions. The value of a state-action represents not only the immediate reward, but also the temporally discounted sum of expected rewards over a trajectory starting from this state and action. Similarly, the ”exploration-value” of a state-action should represent not only the immediate knowledge gained but also the expected future gained knowledge. This suggests that a similar approach to that used for value-learning might be appropriate for learning the exploration values as well, using exploration bonus as the immediate reward. However, because it is reasonable to require exploration bonus to decrease over repetitions of the same trajectories, a naive implementation would violate the Markovian property.
+
+This challenge has been addressed in a model-based setting: The idea is to use at every step the current estimate of the parameters of the MDP in order to compute, using dynamic programming, the future exploration bonus (Little & Sommer, 2014). However, this solution cannot be implemented in a model-free setting. Therefore, a satisfying approach for propagating directed exploration in model-free reinforcement learning is still missing. In this section, we propose such an approach.
+
+# 2.2 $E$ -VALUES
+
+We propose a novel approach for directed exploration, based on two parallel MDPs. One MDP is the original MDP, which is used to estimate the value function. The second MDP is identical except for one important difference. We posit that there are no rewards associated with any of the state-actions. Thus, the true value of all state-action pairs is 0. We will use an RL algorithm to ”learn” the ”actionvalues” in this new MDP which we denote as $E$ -values. We will show that these $E$ -values represent the missing knowledge and thus can be used for propagating directed exploration. This will be done by initializing $E$ -values to 1. These positive initial conditions will subsequently result in an optimistic bias that will lead to directed exploration, by giving high estimations only to state-action pairs from which an optimistic outcome has not yet been excluded by the agent’s experience.
+
+Formally, given an MDP $M = ( S , \mathcal { A } , P , R , \gamma )$ we construct a new MDP $M ^ { \prime } = ( { \cal S } , { \cal A } , { \cal P } , { \bf 0 } , { \gamma _ { E } } )$ with 0 denoting the identically zero function, and $0 \leq \gamma _ { E } < 1$ is a discount parameter. The agent now learns both $Q$ and $E$ values concurrently, while initially $E \left( s , a \right) = 1$ for all $s , a$ . Clearly, $E ^ { * } = \mathbf { 0 }$ . However intuitively, the value of $E \left( s , a \right)$ at a given timestep during training stands for the knowledge, or uncertainty, that the agent has regarding this state-action pair. Eventually, after enough exploration, there is no additional knowledge left to discover which corresponds to $E ( s , a )  E ^ { * } ( s , a ) = 0$ .
+
+For learning $E$ , we use the SARSA algorithm (Rummery & Niranjan, 1994; Sutton & Barto, 1998) which differs from Watkin’s $Q$ -Learning by being on-policy, following the update rule:
+
+Where $\alpha _ { E }$ is the learning rate. For simplicity, we will assume throughout the paper that $\alpha _ { E } = \alpha$
+
+Note that this learning rule updates the $E$ -values based on $E \left( { { s } _ { t + 1 } } , { { a } _ { t + 1 } } \right)$ rather than $\operatorname* { m a x } _ { a } E \left( s _ { t + 1 } , a \right)$ , thus not considering potentially highly informative actions which are never selected. This is important for guaranteeing that exploration values will decrease when repeating the same trajectory (as we will show below). Maintaining these additional updates doesn’t affect the asymptotic space/time complexity of the learning algorithm, since it is simply performing the same updates of a standard $Q$ -Learning process twice.
+
+# 2.3 $E$ -VALUES AS GENERALIZED COUNTERS
+
+The logarithm of $E$ -Values can be thought of as a generalization of visit counters, with propagation of the values along state-action pairs. To see this, let us examine the case of $\gamma _ { E } = 0$ in which there is no propagation from future states. In this case, the update rule is given by:
+
+$$
+E \left( s , a \right) \gets \left( 1 - \alpha \right) E \left( s , a \right) + \alpha \left( 0 + \gamma _ { E } E \left( s ^ { \prime } , a ^ { \prime } \right) \right) = \left( 1 - \alpha \right) E \left( s , a \right)
+$$
+
+So after being visited $n$ times, the value of the state-action pair is $\left( 1 - \alpha \right) ^ { n }$ , where $\alpha$ is the learning rate. By taking a logarithm transformation, we can see that $\log _ { 1 - \alpha } \left( E \right) = n$ . In addition, when $s$ is a terminal state with one action, $\log _ { 1 - \alpha } \left( E \right) = n$ − for any value of $\gamma _ { E }$ .
+
+![](images/f13f4bf01e18a22e6b593a229b961a5bfed1d2c33d323718c50f6a7881e7d985.jpg)  
+Figure 1: Left: Tree MDP, with $k$ leaves. Tree: $\log _ { 1 - \alpha } E \left( s , { \mathrm { s t a r t } } \right)$ as function of visit cycles, for different trees of $k$ leaves (color coded). For each $k$ , a cycle consists of visiting all leaves, hence $k$ visits of the start action. $\log _ { 1 - \alpha } E$ behaves as a generalized counter, where each cycle contributes approximately one generalized visit.
+
+When $\gamma _ { E } > 0$ and for non-terminal states, $E$ will decrease more slowly and therefore $\log _ { 1 - \alpha } E$ −will increase more slowly than a counter. The exact rate will depend on the MDP, the policy and the specific value of $\gamma _ { E }$ . Crucially, for state-actions which lead to many potential states, each visit contributes less to the generalized counter, because more visits are required to exhaust the potential outcomes of the action. To gain more insight, consider the MDP depicted in Figure 1 left, a tree with the root as initial state and the leaves as terminal states. If actions are chosen sequentially, one leaf after the other, we expect that each complete round of choices (which will result with $k$ actual visits of the ${ \bf \Phi } _ { s }$ , start) pair) will be roughly equivalent to one generalized counter. Simulation of this and other simple MDPs show that $E$ -values behave in accordance with such intuitions (see Figure 1 right).
+
+An important property of $E$ -values is that they decrease over repetitions. Formally, by completing a trajectory of the form $s _ { 0 } , a _ { 0 } , \ldots , s _ { n } , a _ { n } , s _ { 0 } , a _ { 0 }$ in the MDP, the maximal value of $E \left( { { s } _ { i } } , { { a } _ { i } } \right)$ will decrease. To see this, assume that $E \left( { { s } _ { i } } , { { a } _ { i } } \right)$ was maximal, and consider its value after the update:
+
+$$
+E \left( s _ { i } , a _ { i } \right) \gets \left( 1 - \alpha \right) E \left( s _ { i } , a _ { i } \right) + \alpha \gamma _ { E } E \left( s _ { i + 1 } , a _ { i + 1 } \right)
+$$
+
+Because $\gamma _ { E } < 1$ and $E \left( s _ { i + 1 } , a _ { i + 1 } \right) \leq E \left( s _ { i } , a _ { i } \right)$ , we get that after the update, the value of $E \left( s _ { i } , a _ { i } \right)$ decreased. For any non-maximal $( s _ { j } , a _ { j } )$ , its value after the update is a convex combination of its previous value and $\gamma _ { E } E \left( s _ { k } , a _ { k } \right)$ which is not larger than its composing terms, which in turn are smaller than the maximal $E$ -value.
+
+# 3 APPLYING $E$ -VALUES
+
+The logarithm of $E$ -values can be considered as a generalization of counters. As such, algorithms that utilize counters can be generalized to incorporate $E$ -values. Here we consider two such generalizations.
+
+# 3.1 $E$ -VALUES AS REWARD EXPLORATION BONUS
+
+In model-based RL, counters have been used to create an augmented reward function. Motivated by this result, augmenting the reward with a counter-based exploration bonus has also been used in model-free RL (Storck et al., 1995; Bellemare et al., 2016). $E$ -Values can naturally generalize this approach, by replacing the standard counter with its corresponding generalized counter $( \log _ { 1 - \alpha } E )$ .
+
+To demonstrate the advantage of using $E$ -values over standard counters, we tested an $\epsilon$ -greedy agent with an exploration bonus of $\frac { 1 } { \log _ { 1 - \alpha } E }$ added to the observed reward on the bridge MDP (Figure 2). To measure the learning progress and its convergence, we calculated the mean square error $\mathbb { E } _ { P ( s , a | \pi ^ { * } ) } \left[ \left( Q \left( s , a \right) - Q ^ { * } \left( s , a \right) \right) ^ { 2 } \right]$ , where the average is over the probability of state-action pairs when following the optimal policy $\pi ^ { * }$ . We varied the value of $\gamma _ { E }$ from 0 – resulting effectively in standard counters – to $\gamma _ { E } = 0 . 9$ . Our results (Figure 3) show that adding the exploration bonus to the reward leads to faster learning. Moreover, the larger the value of $\gamma _ { E }$ in this example the faster the learning, demonstrating that generalized counters significantly outperforming standard counters.
+
+![](images/8065bd1d0577a346121c1dafad546ce14b15bb53ca4c31ce28c701ddc8216df8.jpg)  
+Figure 2: Bridge MDP
+
+![](images/fb180738f53361e6aace8ff042e5df6cf910f3fd73e24b94ff9d4ec95e363341.jpg)  
+Figure 3: MSE between $Q$ and $Q ^ { * }$ on optimal policy per episode. Convergence of $\epsilon$ -greedy on the short bridge environment $k = 5$ ) with and without exploration bonuses added to the reward. Note the logarithmic scale of the abscissa.
+
+# 3.2 $E$ -VALUES AND ACTION-SELECTION RULES
+
+Another way in which counters can be used to assist exploration is by adding them to the estimated $Q$ -values. In this framework, action-selection is a function not only of the $Q$ -values but also of the counters. Several such action-selection rules have been proposed (Thrun, 1992; Meuleau & Bourgine, 1999; Kolter & Ng, 2009). These usually take the form of a deterministic policy that maximizes some combination of the estimated $Q$ -value with a counter-based exploration bonus. It is easy to generalize such rules using $E$ -values – simply replace the counters $C$ by the generalized counters $\log _ { 1 - \alpha } \left( E \right)$ .
+
+# 3.2.1 DETERMINIZATION OF STOCHASTIC DECISION RULES
+
+Here, we consider a special family of action-selection rules that are derived as deterministic equivalents of standard stochastic rules. Stochastic action-selection rules are commonly used in RL. In their simple form they include rules such as the $\epsilon$ -greedy or Softmax exploration described above. In this framework, exploratory behavior is achieved by stochastic action selection, independent of past choices. At first glance, it might be unclear how $E ^ { }$ -values can contribute or improve such rules. We now turn to show that, by using counters, for every stochastic rule there exist equivalent deterministic rules. Once turned to deterministic counter-based rules, it is again possible improve them using $E$ -values.
+
+The stochastic action-selection rules determine the frequency of choosing the different actions in the limit of a large number of repetitions, while abstracting away the specific order of choices. This fact is a key to understanding the relation between deterministic and stochastic rules. An equivalence of two such rules can only be an in-the-limit equivalence, and can be seen as choosing a specific realization of sample from the distribution. Therefore, in order to derive a deterministic equivalent of a given stochastic rule, we only have to make sure that the frequencies of actions selected under both rules are equal in the limit of infinitely many steps. As the probability for each action is likely to depend on the current $Q$ -values, we have to consider fixed $Q$ -values to define this equivalence.
+
+We prove that given a stochastic action-selection rule $f \left( a | s \right)$ , every deterministic policy that does not choose an action that was visited too many times until now (with respect to the expected number according to the probability distribution) is a determinization of $f$ . Formally, lets assume that given a certain $Q$ function and state $s$ we wish a certain ratio between different choices of actions $a \in A$ to hold. We denote the frequency of this ratio $f _ { Q } \left( a | s \right)$ . For brevity we assume $s$ and $Q$ are constants and denote $f _ { Q } \left( a | s \right) = { \bar { f } } \left( a \right)$ . We also assume a counter $C \left( s , a \right)$ is kept denoting the number of choices of $a$ in $s$ . For brevity we denote $C \left( s , a \right) = C \left( a \right)$ and $\begin{array} { r } { \sum _ { a } C \left( s , a \right) = C } \end{array}$ . When we look at the counters after $T$ steps we use subscript $C _ { T } \left( a \right)$ . Following this notation, note that $C _ { T } = T$ .
+
+Theorem 3.1. For any sub-linear function $b \left( t \right)$ and for any deterministic policy which chooses at step $T$ an action a such that $\begin{array} { r } { \frac { C _ { T } \left( a \right) } { T } - f \left( a \right) \leq b \left( t \right) } \end{array}$ it holds that $\forall a \in { \mathcal { A } }$
+
+$$
+\operatorname* { l i m } _ { T  \infty } \frac { C _ { T } ( a ) } { T } = f ( a )
+$$
+
+Proof. For a full proof of the theorem see Appendix A in the supplementary materials
+
+The result above is not a vacuous truth – we now provide two possible determinization rules that achieves it. One rule is straightforward from the theorem, using $b = \mathbf { 0 }$ , choosing $\begin{array} { r } { \arg \operatorname* { m i n } _ { a } \frac { C ( a ) } { C } - } \end{array}$ $f \left( a \right)$ . Another rule follows the probability ratio between the stochastic policy and the empirical distribution: arg maxa C(a) . We denote this determinization $L L L$ , because when generalized counters are used instead of counters it becomes a $\begin{array} { r } { \mathrm { r g } \operatorname* { m a x } _ { a } \log f \left( s , a \right) - \log \log _ { 1 - \alpha } E \left( s , a \right) . } \end{array}$ .
+
+Now we can replace the visit counters $C \left( s , a \right)$ with the generalized counters $\log _ { 1 - \alpha } \left( E \left( s , a \right) \right)$ to create Directed Outreaching Reinforcement Action-Selection – DORA the explorer. By this, we can transform any stochastic or counter-based action-selection rule into a deterministic rule in which exploration propagates over the states and the expected trajectories to follow.
+
+Input: Stochastic action-selection rule $f$ , learning rate $\alpha$ , Exploration discount factor $\gamma _ { E }$ initialize $Q \left( s , a \right) = 0$ , $E \left( s , a \right) = 1$ ;
+
+foreach episode do init $s$ ; while not terminated do Choose $a = \arg \operatorname* { m a x } _ { x } \log f _ { Q } \left( x | s \right) - \log \log _ { 1 - \alpha } E \left( s , x \right) ;$ Observe transitions $( s , a , r , s ^ { \prime } , a ^ { \prime } )$ ; $Q ( s , a )  ( 1 - \alpha ) Q ( s , a ) + \alpha ( r + \gamma \operatorname* { m a x } _ { x } Q ( s ^ { \prime } , x ) ) ;$ $E \left( s , a \right) \gets \left( 1 - \alpha \right) E \left( s , a \right) + \alpha \gamma _ { E } E \left( s ^ { \prime } , a ^ { \prime } \right)$ ; end   
+end
+
+Algorithm 1: DORA algorithm using $L L L$ determinization for stochastic policy $f$
+
+# 3.3 RESULTS – FINITE MDPS
+
+To test this algorithm, the first set of experiments were done on Bridge environments of various lengths $k$ (Figure 2). We considered the following agents: $\epsilon$ -greedy, Softmax and their respective $L L L$ determinizations (as described in 3.2.1) using both counters and $E$ -values. In addition, we compared a more standard counter-based agent in the form of a UCB-like algorithm (Auer et al., 2002) following an action-selection rule with exploration bonus of $\sqrt { \frac { \log t } { C } }$ log t . We tested two variants of this algorithm, using ordinary visit counters and $E$ -values. Each agent’s hyperparameters $\dot { \epsilon }$ and temperature) were fitted separately to optimize learning. For stochastic agents, we averaged the results over 50 trials for each execution. Unless stated otherwise, $\gamma _ { E } = 0 . 9$ .
+
+We also used a normalized version of the bridge environment, where all rewards are between 0 and 1, to compare DORA with the Delayed $Q$ -Learning algorithm (Strehl et al., 2006).
+
+Our results (Figure 4) demonstrate that $E$ -value based agents outperform both their counter-based and their stochastic equivalents on the bridge problem. As shown in Figure 4, Stochastic and counter-based $\epsilon$ -greedy agents, as well as the standard UCB fail to converge. $E$ -value agents are the first to reach low error values, indicating that they learn faster. Similar results were achieved on other gridworld environments, such as the Cliff problem (Sutton & Barto, 1998) (not shown). We also achieved competitive results with respect to Delayed $Q$ Learning (see supplementary B and Figure 7 there).
+
+![](images/c8182ddcd866e022dfedaf0bf5e5888a7c6102523bb6ca82822ffd3bdca883e5.jpg)  
+Figure 4: MSE between $Q$ and $Q ^ { * }$ on optimal policy per episode. Convergence measure of all agents, long bridge environment $k = 1 5$ ). $E$ -values agents are the first to converge, suggesting their superior learning abilities.
+
+The success of $E$ -values based learning relative to counter based learning implies that the use of $E$ -values lead to more efficient exploration. If this is indeed the case, we expect $E$ -values to better represent the agent’s missing knowledge than visit counters during learning. To test this hypothesis we studied the behavior of an $E$ -value $L L L$ Softmax on a shorter bridge environment $k = 5$ ). For a given state-action pair, a measure of the missing knowledge is the normalized distance between its estimated value $( Q )$ and its optimal-policy value $( Q ^ { * } )$ . We recorded $C$ , $\log _ { 1 - \alpha } \left( E \right)$ and $\left| \frac { Q - Q ^ { * } } { Q ^ { * } } \right|$ for each $s , a$ at the end of each episode. Generally, this measure of missing knowledge is expected to be a monotonously-decreasing function of the number of visits $( C )$ . This is indeed true, as depicted in Figure 5 (left). However, considering all state-action pairs, visit counters do not capture well the amount of missing knowledge, as the convergence level depends not only on the counter but also on the identity of the state-action it counts. By contrast, considering the convergence level as a function of the generalized counter (Figure 5, right) reveals a strikingly different pattern. Independently of the state-action identity, the convergence level is a unique function of the generalized counter. These results demonstrate that generalized counters are a useful measure of the amount of missing knowledge.
+
+# 4 $E$ -VALUES WITH FUNCTION APPROXIMATION
+
+So far we discussed $E$ -values in the tabular case, relying on finite (and small) state and action spaces. However, a main motivation for using model-free approach is that it can be successfully applied in large MDPs where tabular methods are intractable. In this case (in particular for continuous MDPs), achieving directed exploration is a non-trivial task. Because revisiting a state or a state-action pair is unlikely, and because it is intractable to store individual values for all state-action pairs, counterbased methods cannot be directly applied. In fact, most implementations in these cases adopt simple exploration strategies such as $\epsilon$ -greedy or softmax (Bellemare et al., 2016).
+
+There are standard model-free techniques to estimate value function in function-approximation scenarios. Because learning $E$ -values is simply learning another value-function, the same techniques can be applied for learning $E$ -values in these scenarios. In this case, the concept of visit-count – or a generalized visit-count – will depend on the representation of states used by the approximating function.
+
+To test whether $E$ -values can serve as generalized visit-counters in the function-approximation case, we used a linear approximation architecture on the MountainCar problem (Moore, 1990) (Appendix
+
+C). To dissociate $Q$ and $E$ -values, actions were chosen by an $\epsilon$ -greedy agent independently of $E$ - values. As shown in Appendix C, $E$ -values are an effective way for counting both visits and generalized visits in continuous MDPs. For completeness, we also compared the performance of $L L L$ agents to stochastic agents on a sparse-reward MountainCar problem, and found that $L L L$ agents learns substantially faster than the stochastic agents (Appendix D).
+
+# 4.1 RESULTS – FUNCTION APPROXIMATION
+
+To show our approach scales to complex problems, we used the Freeway Atari 2600 game, which is known as a hard exploration problem (Bellemare et al., 2016). We trained a neural network with two streams to predict the $Q$ and $E$ -values. First, we trained the network using standard DQN technique (Mnih et al., 2015), which ignores the $\mathrm { E }$ -values. Second, we trained the network while adding an exploration bonus of $\frac { \beta } { \sqrt { - \log E } }$ to the reward (In all reported simulations, $\beta = 0 . 0 5$ ). In both cases, action-selection was performed by an $\epsilon$ -greedy rule, as in Bellemare et al. (2016).
+
+Note that the exploration bonus requires $0 \textless E < 1$ . To satisfy this requirement, we applied a logistic activation fucntion on the output of the last layer of the $E$ -value stream, and initialized the weights of this layer to 0. As a result, the $E$ -values were initialized at 0.5 and satisfied $0 < E < 1$ throughout the training. In comparison, no non-linearity was applied in the last layer of the $Q$ -value stream and the weights were randmoly initialized.
+
+We compared our approach to a DQN baseline, as well as to the density model counters suggested by (Bellemare et al., 2016). The baseline used here does not utilize additional enhancements (such as Double DQN and Monte-Carlo return) which were used in (Bellemare et al., 2016). Our results, depicted in Figure 6, demonstrate that the use of $E$ -values outperform both DQN and density model counters baselines. In addition, our approach results in better performance than in (Bellemare et al., 2016) (with the mentioned enhancements), converging in approximately $2 \cdot 1 0 ^ { 6 }$ steps, instead of $1 0 \cdot 1 0 ^ { 6 }$ steps2.
+
+# 5 RELATED WORK
+
+The idea of using reinforcement-learning techniques to estimate exploration can be traced back to Storck et al. (1995) and Meuleau & Bourgine (1999) who also analyzed propagation of uncertainties and exploration values. These works followed a model-based approach, and did not fully deal with the problem of non-Markovity arising from using exploration bonus as the immediate reward. A related approach was used by Little & Sommer (2014), where exploration was investigated by information-theoretic measures. Such interpretation of exploration can also be found in other works (Schmidhuber (1991); Sun et al. (2011); Houthooft et al. (2016)).
+
+Efficient exploration in model-free RL was also analyzed in PAC-MDP framework, most notably the Delayed $Q$ Learning algorithm by Strehl et al. (2006). For further discussion and comparison of our approach with Delayed $Q$ Learning, see 1.1 and Appendix B.
+
+In terms of generalizing Counter-based methods, there has been some works on using counter-like notions for exploration in continuous MDPs (Nouri & Littman, 2009). A more direct attempt was recently proposed by Bellemare et al. (2016). This generalization provides a way to implement visit counters in large, continuous state and action spaces by using density models. Our generalization is different, as it aims first on generalizing the notion of visit counts themselves, from actual counters to ”propagating counters”. In addition, our approach does not depend on any estimated model – which might be an advantage in domains for which good density models are not available. Nevertheless, we believe that an interesting future work will be comparing between the approach suggested by Bellemare et al. (2016) and our approach, in particular for the case of $\gamma _ { E } = 0$ .
+
+![](images/4683886049710d08ce9ffa88cfe453f359909fa4c66364220b7f4836779d18b2.jpg)  
+Figure 5: Convergence of $Q$ to $Q ^ { * }$ for individual state-action pairs (each denoted by a different color), with respect to counters (left) and generalized counters (right). Results obtained from $E$ - Value $L L L$ Softmax on the short bridge environment $k = 5$ ). Triangle markers indicate pairs with ”east” actions, which constitute the optimal policy of crossing the bridge. Circle markers indicate state-action pairs that are not part of the optimal policy. Generalized counters are a useful measure of the amount of missing knowledge.
+
+![](images/bf544dda638c5414e55a0ce2a52e60c33ae561e2ab24fdfdb38f55bab0f6b677.jpg)  
+Figure 6: Results on Freeway game. All agents used $\epsilon$ -greedy action-selection rule without exploration bonus (DQN, blue), with a bonus term based on density model counters (Density, orange) added to the reward, or with bonus term based on $E$ -values (black).
+
+# 6 ACKNOWLEDGMENTS
+
+We thank Nadav Cohen, Leo Joskowicz, Ron Meir, Michal Moshkovitz, and Jeff Rosenschein for discussions. This work was supported by the Israel Science Foundation (Grant No. 757/16) and the Gatsby Charitable Foundation.
+
+# REFERENCES
+
+Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. Finite-time analysis of the multiarmed bandit problem. Machine learning, 47(2-3):235–256, 2002.
+
+Marc Bellemare, Sriram Srinivasan, Georg Ostrovski, Tom Schaul, David Saxton, and Remi Munos. Unifying count-based exploration and intrinsic motivation. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett (eds.), Advances in Neural Information Processing Systems 29, pp. 1471–1479. Curran Associates, Inc., 2016.
+
+Lucian Busoniu, Robert Babuska, and Bart De Schutter. A comprehensive survey of multiagent reinforcement learning. IEEE Transactions on Systems, Man, And Cybernetics-Part C: Applications and Reviews, 38 (2), 2008, 2008.
+
+Arthur Guez, David Silver, and Peter Dayan. Efficient bayes-adaptive reinforcement learning using sample-based search. In Advances in Neural Information Processing Systems, pp. 1025–1033, 2012.
+
+Rein Houthooft, Xi Chen, Yan Duan, John Schulman, Filip De Turck, and Pieter Abbeel. Vime: Variational information maximizing exploration. In Advances in Neural Information Processing Systems, pp. 1109–1117, 2016.
+
+Leslie Pack Kaelbling, Michael L Littman, and Andrew W Moore. Reinforcement learning: A survey. Journal of artificial intelligence research, 4:237–285, 1996.
+
+Sham Machandranath Kakade et al. On the sample complexity of reinforcement learning. PhD thesis, 2003.
+
+J Zico Kolter and Andrew Y Ng. Near-bayesian exploration in polynomial time. In Proceedings of the 26th Annual International Conference on Machine Learning, pp. 513–520. ACM, 2009.
+
+Daniel Y Little and Friedrich T Sommer. Learning and exploration in action-perception loops. Closing the Loop Around Neural Systems, pp. 295, 2014.
+
+Nicolas Meuleau and Paul Bourgine. Exploration of multi-state environments: Local measures and back-propagation of uncertainty. Machine Learning, 35(2):117–154, 1999.
+
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, 2015.
+
+Andrew William Moore. Efficient memory-based learning for robot control. 1990.
+
+Ali Nouri and Michael L Littman. Multi-resolution exploration in continuous spaces. In Advances in neural information processing systems, pp. 1209–1216, 2009.
+
+Gavin A Rummery and Mahesan Niranjan. On-line Q-learning using connectionist systems. University of Cambridge, Department of Engineering, 1994.
+
+Jurgen Schmidhuber. Curious model-building control systems. In ¨ Neural Networks, 1991. 1991 IEEE International Joint Conference on, pp. 1458–1463. IEEE, 1991.
+
+Jan Storck, Sepp Hochreiter, and Jurgen Schmidhuber. Reinforcement driven information acquisi- ¨ tion in non-deterministic environments. In Proceedings of the international conference on artificial neural networks, Paris, volume 2, pp. 159–164. Citeseer, 1995.
+
+Alexander L Strehl and Michael L Littman. An analysis of model-based interval estimation for markov decision processes. Journal of Computer and System Sciences, 74(8):1309–1331, 2008.
+
+Alexander L Strehl, Lihong Li, Eric Wiewiora, John Langford, and Michael L Littman. Pac modelfree reinforcement learning. In Proceedings of the 23rd international conference on Machine learning, pp. 881–888. ACM, 2006.
+
+Yi Sun, Faustino Gomez, and Jurgen Schmidhuber. Planning to be surprised: Optimal bayesian ¨ exploration in dynamic environments. In International Conference on Artificial General Intelligence, pp. 41–51. Springer, 2011.
+
+Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 1998.
+
+Sebastian B. Thrun. Efficient exploration in reinforcement learning, 1992.
+
+Michel Tokic and Gunther Palm. Value-difference based exploration: adaptive control between ¨ epsilon-greedy and softmax. In KI 2011: Advances in Artificial Intelligence, pp. 335–346. Springer, 2011.
+
+Thomas J Walsh, Istvan Szita, Carlos Diuk, and Michael L Littman. Exploring compact ´ reinforcement-learning representations with linear regression. In Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, pp. 591–598. AUAI Press, 2009.
+
+Christopher JCH Watkins and Peter Dayan. Q-learning. Machine learning, 8(3-4):279–292, 1992.
+
+# A PROOF OF THE DETERMINIZATION THEOREM
+
+The proof for the determinization mentioned in the paper is achieved based on the following lemmata.
+
+Lemma A.1. The absolute sum of positive and negative differences between the empiric distribution (deterministic frequency) and goal distribution (non-deterministic frequency) is equal.
+
+$$
+\sum _ { a : f \left( a \right) \geq \frac { C \left( a \right) } { C } } f \left( a \right) - \frac { C \left( a \right) } { C } = - \sum _ { a : f \left( a \right) < \frac { C \left( a \right) } { C } } f \left( a \right) - \frac { C \left( a \right) } { C }
+$$
+
+Proof. Straightforward from the observation that
+
+$$
+\sum _ { a } f \left( a \right) = \sum _ { a } { \frac { C \left( a \right) } { C } } = 1
+$$
+
+Lemma A.2. For any $t$
+
+$$
+\operatorname* { m a x } _ { a } \left\{ { \frac { C _ { t } \left( a \right) } { t } } - f \left( a \right) \right\} \leq { \frac { 1 + b \left( t \right) } { t } }
+$$
+
+Proof. The proof of A.2 is done by induction. For $t = 1$
+
+$$
+\forall a \in A : { \frac { C _ { t } \left( a \right) } { t } } - f \left( a \right) = \operatorname* { m a x } _ { a } \left\{ { \frac { C _ { t } \left( a \right) } { t } } - f \left( a \right) \right\}
+$$
+
+Hence we look at $a \in A$ .
+
+$$
+\begin{array} { l } { \displaystyle \frac { C _ { t } \left( a \right) } { t } - f \left( a \right) \leq \frac { C _ { t } \left( a \right) } { t } \qquad } \\ { \displaystyle \leq \frac { 1 + b \left( 1 \right) } { 1 } } \end{array}
+$$
+
+assume the claim is true for $t = T$ then for $t = T + 1$ There exists $a$ such that $C _ { T } \left( a \right) / T - f \left( a \right) \leq$ $b \left( t \right)$ which the algorithm chooses for this $a$ . For it
+
+$$
+\begin{array} { c } { \displaystyle \frac { C _ { T + 1 } \left( a \right) } { T + 1 } - f \left( a \right) = \displaystyle \frac { C _ { T } \left( a \right) + 1 } { T + 1 } - f \left( a \right) } \\ { = \displaystyle \frac { C _ { T } \left( a \right) } { T + 1 } - f \left( a \right) + \displaystyle \frac { 1 } { T + 1 } } \\ { = \displaystyle \frac { C _ { T } \left( a \right) - \left( T + 1 \right) f \left( a \right) } { T + 1 } + \displaystyle \frac { 1 } { T + 1 } } \\ { \leq \displaystyle \frac { 1 + b \left( t \right) } { T + 1 } } \end{array}
+$$
+
+It also holds that $\forall a ^ { \prime } \in A$ s.t. $\boldsymbol { a } ^ { \prime } \neq \boldsymbol { a }$
+
+$$
+\begin{array} { l } { \displaystyle \frac { C _ { T + 1 } \left( a \right) } { T + 1 } - f \left( a \right) = \displaystyle \frac { C _ { T } \left( a \right) } { T + 1 } - f \left( a \right) } \\ { \displaystyle = \frac { C _ { T } \left( a \right) - \left( T + 1 \right) f \left( a \right) } { T + 1 } } \\ { \displaystyle < \frac { C _ { T } \left( a \right) - T f \left( a \right) } { T + 1 } } \\ { \displaystyle \le \frac { 1 + b \left( t \right) } { T + 1 } } \end{array}
+$$
+
+Proof of 3.1. It holds from A.2 together with A.1 that in the step $t$ in the worst case all but one of the actions have $\begin{array} { r } { \frac { C _ { t } ( a ) } { t } - f ( a ) = \frac { 1 } { t } } \end{array}$ and the last action has $\begin{array} { r } { f ( { a } ) - \frac { { \bf \bar { \Phi } } _ { C _ { t } } ( { a } ) } { t } = - \frac { | A | - 1 } { t } } \end{array}$ . So by the bound on sum of positives and negatives we get:
+
+$$
+\operatorname* { l i m } _ { T \to \infty } \frac { C _ { T } ( a ) } { T } = f ( a )
+$$
+
+![](images/5e5df4cbb49ca8a5cd3567df92afb642680d55e68a45e5bea1d8417745ca9038.jpg)  
+Figure 7: Normalized MSE between $Q$ and $Q ^ { * }$ on optimal policy per episode. Convergence of $E$ -value $L L L$ and Delayed $Q$ -Learning on, normalized bridge environment $( k = 1 5 )$ ). MSE was noramlized for each agent to enable comparison.
+
+# B COMPARISON WITH DELAYED $Q$ -LEARNING
+
+Because Delayed $Q$ learning initializes its values optimistically, which result in a high MSE, we normalized the MSE of the two agents (separately) to enable comparison. Notably, to achieve this performance by the Delayed $Q$ Learning, we had to manually choose a low value for $m$ (in Figure 7, $m = 1 0$ ), the hyperparameter regulating the number of visits required before any update. This is an order of magnitude smaller than the theoretical value required for even moderate PAC-requirements in the usual notion of $\epsilon , \delta$ , such $m$ also implies learning in orders of magnitudes slower. In fact, for this limit of $m  1$ the algorithm is effectively quite similar to a ”Vanilla” $Q$ -Learning with an optimistic initialization, which is possible due to the assumption made by the algorithm that all rewards are between 0 and 1. In fact, several exploration schemes relying on optimism in the face of uncertainty were proposed (Walsh et al., 2009). However, because our approach separate reward values and exploratory values, we are able to use optimism for the latter without assuming any prior knowledge about the first – while still achieving competitive results to an optimistic initialization based on prior knowledge.
+
+# C EVALUATING $E$ -VALUES DYNAMICS IN FUNCTION-APPROXIMATION
+
+To gain insight into the relation between E-values and number of visits, we used the linearapproximation architecture on the MountainCar problem. Note that when using $E$ -values, they are generally correlated with visit counts both because visits result in update of the $E$ -values through learning and because $E$ -values affect visits through the exploration bonus (or action-selection rule). To dissociate the two, $Q$ -values and $E$ -values were learned in parallel in these simulation, but actionselection was independent of the $E$ -values. Rather, actions were chosen by an $\epsilon$ -greedy agent. To estimate visit-counts, we recorded the entire set of visited states, and computed the empirical visits histogram by binning the two-dimensional state-space. For each state, its visit counter estimator $\tilde { C } \left( s \right)$ is the value of the matching bin in the histogram for this state. In addition, we recorded the learned model (weights vector for $E$ -values) and computed the $E$ -values map by sampling a state for each bin, and calculating its $E$ -values using the model. For simplicity, we consider here the resolution of states alone, summing over all 3 actions for each state. That is, we compare $\tilde { C } \left( s \right)$ to $\begin{array} { r } { \sum _ { a } \log _ { 1 - \alpha } E \left( s , a \right) = C _ { E } \left( s \right) } \end{array}$ . Figure 8 depicts the empirical visits histogram (left) and the estimated $E$ -values for the case of $\gamma _ { E } = 0$ after the complete training. The results of the analysis show that, roughly speaking, those regions in the state space that were more often visited, were also associated with a higher $C _ { E } \left( s \right)$ .
+
+![](images/29be4dababb04f926c7bc56c2288e4c2f190df6e5fc2dd283325a9cf757ca8cc.jpg)  
+Figure 8: Empirical visits histogram (left) and learned $C _ { E }$ (right) after training, $\gamma _ { E } = 0$ .
+
+To better understand these results, we considered smaller time-windows in the learning process. Specifically, Figure 9 depicts the empirical visit histogram (left), and the corresponding $C _ { E } \left( s \right)$ (right) in the first 10 episodes, in which visits were more centrally distributed. Figure 10 depicts the change in the empirical visit histogram (left), and change in the corresponding $C _ { E }$ (s) (right) in the last 10 episodes of the training, in which visits were distributed along a spiral (forming an nearoptimal behavior). These results demonstrate high similarity between visit-counts and the $E$ -value representation of them, indicating that $E$ -values are good proxies of visit counters.
+
+![](images/422f2bbf6cebd9e935380b9c12833a721be8090514550426a0bc6d466169d7f2.jpg)  
+Figure 9: Empirical visits histogram (left) and learned $C _ { E }$ (right) in the first 10 training episodes, $\gamma _ { E } = 0$ .
+
+![](images/aba80c86fa21a948f82e2b97dbc136f39feeddaa5557444f3fac938c60957d06.jpg)  
+Figure 10: Difference in empirical visits histogram (left) and learned $C _ { E }$ (right) in the last 10 training episodes, $\gamma _ { E } = 0$ .
+
+The results depicted in Figures 9 and 10 were achieved with $\gamma _ { E } = 0$ . For $\gamma _ { E } > 0$ , we expect the generalized counters (represented by $E$ -values) to account not for standard visits but for ”generalized visits”, weighting the trajectories starting in each state. We repeated the analysis of Figure 10 for the case of $\gamma _ { E } ~ = ~ 0 . 9 9$ . Results, depicted in Figure 11, shows that indeed for terminal or nearterminal states (where position $> 0 . 5$ ) generalized visits, measured by difference in their generalized counters, are higher – comparing to far-from terminal states – than the empirical visits of these states (comparing to far-from terminal states).
+
+![](images/e3625613ff30956da8419837612a93f4282737cd2d9ca78d9d8ffd9088dc3c8b.jpg)  
+Figure 11: Difference in empirical visits histogram (left) and learned $C _ { E }$ (right) in the last 10 training episodes, $\gamma _ { E } = 0 . 9 9$ . Note that the results are based on a different simulation than those in Figure 10.
+
+To quantify the relation between visits and $E$ -values, we densely sampled the (achievable) statespace to generate many examples of states. For each sampled state, we computed the correlation coefficient between $C _ { E } \left( s \right)$ and $\tilde { C } \left( s \right)$ throughout the learning process (snapshots taken each 10 episodes). The values $\tilde { C } \left( s \right)$ were estimated by the empirical visits histogram (value of the bin corresponding to the sampled state) calculated based on visits history up to each snapshot. Figure 12, depicting the histogram of correlation coefficients between the two measures, demonstrating strong positive correlations between empirical visit-counters and generalized counters represented by $E$ -values. These results indicate that $E$ -values are an effective way for counting effective visits in continuous MDPs. Note that the number of model parameters used to estimate $E \left( s , a \right)$ in this case is much smaller than the size of the table we would have to use in order to track state-action counters in such binning resolution.
+
+![](images/844d10739b90fffa50f60dca6f008d682d5e38f2255e51d95449e6b785a38cdd.jpg)  
+Figure 12: Histogram of correlation coefficients between empirical visit counters and $C _ { E }$ throughout training, per state $( \gamma _ { E } = 0$ ).
+
+# D RESULTS ON CONTINUOUS MDPS – MOUNTAINCAR
+
+To test the performance of $E$ -values based agents, simulations were performed using the MountainCar environment. The version of the problem considered here is with sparse and delayed reward, meaning that there is a constant reward of 0 unless reaching a goal state which provides a reward of magnitude 1. Episode length was limited to 1000 steps. We used linear approximation with tilecoding features (Sutton & Barto, 1998), learning the weights vectors for $Q$ and $E$ in parallel. To guarantee that $E$ -values are uniformly initialized and are kept between 0 and 1 throughout learning, we initialized the weights vector for $E$ -values to 0 and added a logistic non-linearity to the results of the standard linear approximation. In contrast, the $Q$ -values weights vector was initialized at random, and there was no non-linearity. We compared the performance of several agents. The first two used only $Q$ -values, with a softmax or an $\epsilon$ -greedy action-selection rules. The other two agents are the DORA variants using both $Q$ and $E$ values, following the $L L L$ determinization for softmax either with $\gamma _ { E } = 0$ or with $\gamma _ { E } = 0 . 9 9$ . Parameters for each agent (temperature and $\dot { \epsilon }$ ) were fitted separately to maximize performance. The results depicted in Figure 13 demonstrate that using $E$ -values with $\gamma _ { E } > 0$ lead to better performance in the MountainCar problem
+
+In addition we tested our approach using (relatively simple) neural networks. We trained two neural networks in parallel (unlike the two-streams single network used for Atari simulations), for predicting $Q$ and $E$ values. In this architecture, the same technique of 0 initializing and a logistic non-linearity was applied to the last linear of the $E$ -network. Similarly to the linear approximation approach, $E$ -values based agents outperform their $\epsilon$ -greedy and softmax counterparts (not shown).
+
+![](images/8812ff7dda21f6500540aa61c9cbd2827e94be23caa1d5d207a09cd0c73067f3.jpg)  
+Figure 13: Probability of reaching goal on MountainCar (computed by averaging over 50 simulations of each agent), as a function of training episodes. While Softmax exploration fails to solve the problem within 1000 episodes, LLL $E$ -values agents with generalized counters $( \gamma _ { E } > 0 ) ,$ ) quickly reach high success rates.

parse/train/ry1arUgCW/ry1arUgCW_content_list.json

@@ -0,0 +1,1710 @@
+[
+    {
+        "type": "text",
+        "text": "DORA THE EXPLORER: DIRECTED OUTREACHING REINFORCEMENT ACTION-SELECTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            825,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Leshem Choshen∗ School of Computer Science and Engineering and Department of Cognitive Sciences The Hebrew University of Jerusalem leshem.choshen@mail.huji.ac.il ",
+        "bbox": [
+            183,
+            170,
+            482,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Lior Fox∗ The Edmond and Lily Safra Center for Brain Sciences The Hebrew University of Jerusalem lior.fox@mail.huji.ac.il ",
+        "bbox": [
+            500,
+            170,
+            856,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yonatan Loewenstein ",
+        "text_level": 1,
+        "bbox": [
+            184,
+            262,
+            334,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The Edmond and Lily Safra Center for Brain Sciences, Departments of Neurobiology and Cognitive Sciences and the Federmann Center for the Study of Rationality The Hebrew University of Jerusalem yonatan@huji.ac.il ",
+        "bbox": [
+            184,
+            275,
+            542,
+            344
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            381,
+            544,
+            396
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Exploration is a fundamental aspect of Reinforcement Learning, typically implemented using stochastic action-selection. Exploration, however, can be more efficient if directed toward gaining new world knowledge. Visit-counters have been proven useful both in practice and in theory for directed exploration. However, a major limitation of counters is their locality. While there are a few model-based solutions to this shortcoming, a model-free approach is still missing. We propose $E$ -values, a generalization of counters that can be used to evaluate the propagating exploratory value over state-action trajectories. We compare our approach to commonly used RL techniques, and show that using $E$ -values improves learning and performance over traditional counters. We also show how our method can be implemented with function approximation to efficiently learn continuous MDPs. We demonstrate this by showing that our approach surpasses state of the art performance in the Freeway Atari 2600 game. ",
+        "bbox": [
+            233,
+            410,
+            764,
+            590
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            614,
+            336,
+            630
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "”If there’s a place you gotta go - I’m the one you need to know.“ (Map, Dora The Explorer) ",
+        "bbox": [
+            259,
+            638,
+            763,
+            666
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We consider Reinforcement Learning in a Markov Decision Process (MDP). An MDP is a fivetuple $M = ( \\mathcal { S } , \\mathcal { A } , P , R , \\gamma )$ where $s$ is a set of states and $\\mathcal { A }$ is a set of actions. The dynamics of the process is given by $P \\left( s ^ { \\prime } | s , a \\right)$ which denotes the transition probability from state $s$ to state $s ^ { \\prime }$ following action $a$ . Each such transition also has a distribution $R \\left( r | s , a \\right)$ from which the reward for such transitions is sampled. Given a policy $\\pi : { \\mathcal { S } }  A$ , a function – possibly stochastic – deciding which actions to take in each of the states, the state-action value function $Q ^ { \\pi } : S \\times \\mathcal { A }  \\mathbb { R }$ satisfies: ",
+        "bbox": [
+            174,
+            676,
+            825,
+            761
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f41c4d6fad21e97fa46b3ca145df371994c07574682cb646c6d043dddb0a2e05.jpg",
+        "text": "$$\nQ ^ { \\pi } \\left( s , a \\right) = \\underset { r , s ^ { \\prime } \\sim R \\times P \\left( \\cdot | s , a \\right) } { \\mathbb { E } } \\left[ r + \\gamma Q ^ { \\pi } \\left( s ^ { \\prime } , \\pi \\left( s ^ { \\prime } \\right) \\right) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            762,
+            661,
+            789
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "where $\\gamma$ is the discount factor. The agent’s goal is to find an optimal policy $\\pi ^ { * }$ that maximizes $Q ^ { \\pi } \\left( s , \\pi \\left( s \\right) \\right)$ . For brevity, $Q ^ { \\pi ^ { * } } \\triangleq Q ^ { * }$ . There are two main approaches for learning $\\pi ^ { * }$ . The first is a model-based approach, where the agent learns an internal model of the MDP (namely $P$ and $R$ ). Given a model, the optimal policy could be found using dynamic programming methods such as Value Iteration (Sutton $\\&$ Barto, 1998). The alternative is a model-free approach, where the agent learns only the value function of states or state-action pairs, without learning a model (Kaelbling et al., 1996)1. ",
+        "bbox": [
+            174,
+            790,
+            825,
+            890
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The ideas put forward in this paper are relevant to any model-free learning of MDPs. For concreteness, we focus on a particular example, $Q$ -Learning (Watkins & Dayan, 1992; Sutton & Barto, 1998). $Q$ -Learning is a common method for learning $Q ^ { * }$ , where the agent iteratively updates its values of $Q \\left( s , a \\right)$ by performing actions and observing their outcomes. At each step the agent takes action $a _ { t }$ then it is transferred from $s _ { t }$ to $s _ { t + 1 }$ and observe reward $r$ . Then it applies the update rule regulated by a learning rate $\\alpha$ : ",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3f60660c39207c8d1ef09579e943d7de0106164ce2e74baa63832e1411f92cac.jpg",
+        "text": "$$\nQ \\left( s _ { t } , a _ { t } \\right) \\gets \\left( 1 - \\alpha \\right) Q \\left( s _ { t } , a _ { t } \\right) + \\alpha \\left( r + \\gamma \\operatorname* { m a x } _ { a } Q \\left( s _ { t + 1 } , a \\right) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            200,
+            705,
+            228
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.1 EXPLORATION AND EXPLOITATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            250,
+            459,
+            265
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Balancing between Exploration and Exploitation is a major challenge in Reinforcement Learning. Seemingly, the agent may want to choose the alternative associated with the highest expected reward, a behavior known as exploitation. However, in that case it may fail to learn that there are better options. Therefore exploration, namely the taking of new actions and the visit of new states, may also be beneficial. It is important to note that exploitation is also inherently relevant for learning, as we want the agent to have better estimations of the values of valuable state-actions and we care less about the exact values of actions that the agent already knows to be clearly inferior. ",
+        "bbox": [
+            174,
+            279,
+            825,
+            377
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Formally, to guarantee convergence to $Q ^ { * }$ , the Q-Learning algorithm must visit each state-action pair infinitely many times. A naive random walk exploration is sufficient for converging asymptotically. However, such random exploration has two major limitations when the learning process is finite. First, the agent would not utilize its current knowledge about the world to guide its exploration. For example, an action with a known disastrous outcome will be explored over and over again. Second, the agent would not be biased in favor of exploring unvisited trajectories more than the visited ones – hence ”wasting” exploration resources on actions and trajectories which are already well known to it. ",
+        "bbox": [
+            174,
+            383,
+            825,
+            494
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A widely used method for dealing with the first problem is the $\\epsilon$ -greedy schema (Sutton & Barto, 1998), in which with probability $1 - \\epsilon$ the agent greedily chooses the best action (according to current estimation), and with probability $\\epsilon$ it chooses a random action. Another popular alternative, emphasizing the preference to learn about actions associated with higher rewards, is to draw actions from a Boltzmann Distribution (Softmax) over the learned $Q$ values, regulated by a Temperature parameter. While such approaches lead to more informed exploration that is based on learning experience, they still fail to address the second issue, namely they are not directed (Thrun, 1992) towards gaining more knowledge, not biasing actions in the direction of unexplored trajectories. ",
+        "bbox": [
+            174,
+            501,
+            825,
+            613
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another important approach in the study of efficient exploration is based on Sample Complexity of Exploration as defined in the PAC-MDP literature (Kakade et al., 2003). Relevant to our work is Delayed Q Learning (Strehl et al., 2006), a model-free algorithm that has theoretical PAC-MDP guarantees. However, to ensure these theoretical guarantees this algorithm uses a conservative exploration which might be impractical (see also (Kolter & $\\mathrm { N g }$ , 2009) and Appendix B). ",
+        "bbox": [
+            174,
+            619,
+            825,
+            690
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.2 CURRENT DIRECTED EXPLORATION AND ITS LIMITATIONS",
+        "text_level": 1,
+        "bbox": [
+            176,
+            714,
+            622,
+            728
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In order to achieve directed exploration, the estimation of an exploration value of the different stateactions (often termed exploration bonus) is needed. The most commonly used exploration bonus is based on counting (Thrun, 1992) – for each pair $( s , a )$ , store a counter $\\displaystyle \\dot { C } \\left( s , a \\right)$ that indicates how many times the agent performed action $a$ at state $s$ so far. Counter-based methods are widely used both in practice and in theory (Kolter & $\\mathrm { N g }$ , 2009; Strehl & Littman, 2008; Guez et al., 2012; Busoniu et al., 2008). Other options for evaluating exploration include recency and value difference (or error) measures (Thrun, 1992; Tokic & Palm, 2011). While all of these exploration measures can be used for directed exploration, their major limitation in a model-free settings is that the exploratory value of a state-action pair is evaluated with respect only to its immediate outcome, one step ahead. It seems desirable to determine the exploratory value of an action not only by how much new immediate knowledge the agent gains from it, but also by how much more new knowledge could be gained from a trajectory starting with it. The goal of this work is to develop a measure for such exploratory values of state-action pairs, in a model-free settings. ",
+        "bbox": [
+            174,
+            743,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 LEARNING EXPLORATION VALUES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            495,
+            118
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 PROPAGATING EXPLORATION VALUES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            132,
+            478,
+            146
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The challenge discussed in 1.2 is in fact similar to that of learning the value functions. The value of a state-action represents not only the immediate reward, but also the temporally discounted sum of expected rewards over a trajectory starting from this state and action. Similarly, the ”exploration-value” of a state-action should represent not only the immediate knowledge gained but also the expected future gained knowledge. This suggests that a similar approach to that used for value-learning might be appropriate for learning the exploration values as well, using exploration bonus as the immediate reward. However, because it is reasonable to require exploration bonus to decrease over repetitions of the same trajectories, a naive implementation would violate the Markovian property. ",
+        "bbox": [
+            174,
+            157,
+            825,
+            270
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This challenge has been addressed in a model-based setting: The idea is to use at every step the current estimate of the parameters of the MDP in order to compute, using dynamic programming, the future exploration bonus (Little & Sommer, 2014). However, this solution cannot be implemented in a model-free setting. Therefore, a satisfying approach for propagating directed exploration in model-free reinforcement learning is still missing. In this section, we propose such an approach. ",
+        "bbox": [
+            174,
+            276,
+            825,
+            347
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 $E$ -VALUES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            362,
+            290,
+            376
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose a novel approach for directed exploration, based on two parallel MDPs. One MDP is the original MDP, which is used to estimate the value function. The second MDP is identical except for one important difference. We posit that there are no rewards associated with any of the state-actions. Thus, the true value of all state-action pairs is 0. We will use an RL algorithm to ”learn” the ”actionvalues” in this new MDP which we denote as $E$ -values. We will show that these $E$ -values represent the missing knowledge and thus can be used for propagating directed exploration. This will be done by initializing $E$ -values to 1. These positive initial conditions will subsequently result in an optimistic bias that will lead to directed exploration, by giving high estimations only to state-action pairs from which an optimistic outcome has not yet been excluded by the agent’s experience. ",
+        "bbox": [
+            174,
+            387,
+            825,
+            513
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Formally, given an MDP $M = ( S , \\mathcal { A } , P , R , \\gamma )$ we construct a new MDP $M ^ { \\prime } = ( { \\cal S } , { \\cal A } , { \\cal P } , { \\bf 0 } , { \\gamma _ { E } } )$ with 0 denoting the identically zero function, and $0 \\leq \\gamma _ { E } < 1$ is a discount parameter. The agent now learns both $Q$ and $E$ values concurrently, while initially $E \\left( s , a \\right) = 1$ for all $s , a$ . Clearly, $E ^ { * } = \\mathbf { 0 }$ . However intuitively, the value of $E \\left( s , a \\right)$ at a given timestep during training stands for the knowledge, or uncertainty, that the agent has regarding this state-action pair. Eventually, after enough exploration, there is no additional knowledge left to discover which corresponds to $E ( s , a )  E ^ { * } ( s , a ) = 0$ . ",
+        "bbox": [
+            173,
+            520,
+            825,
+            618
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For learning $E$ , we use the SARSA algorithm (Rummery & Niranjan, 1994; Sutton & Barto, 1998) which differs from Watkin’s $Q$ -Learning by being on-policy, following the update rule: ",
+        "bbox": [
+            173,
+            625,
+            826,
+            654
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Where $\\alpha _ { E }$ is the learning rate. For simplicity, we will assume throughout the paper that $\\alpha _ { E } = \\alpha$ ",
+        "bbox": [
+            176,
+            672,
+            792,
+            686
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that this learning rule updates the $E$ -values based on $E \\left( { { s } _ { t + 1 } } , { { a } _ { t + 1 } } \\right)$ rather than $\\operatorname* { m a x } _ { a } E \\left( s _ { t + 1 } , a \\right)$ , thus not considering potentially highly informative actions which are never selected. This is important for guaranteeing that exploration values will decrease when repeating the same trajectory (as we will show below). Maintaining these additional updates doesn’t affect the asymptotic space/time complexity of the learning algorithm, since it is simply performing the same updates of a standard $Q$ -Learning process twice. ",
+        "bbox": [
+            173,
+            693,
+            825,
+            777
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 $E$ -VALUES AS GENERALIZED COUNTERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            792,
+            500,
+            808
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The logarithm of $E$ -Values can be thought of as a generalization of visit counters, with propagation of the values along state-action pairs. To see this, let us examine the case of $\\gamma _ { E } = 0$ in which there is no propagation from future states. In this case, the update rule is given by: ",
+        "bbox": [
+            173,
+            819,
+            825,
+            861
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/505a135ca1ca7edb121ce44e05dfb1833b51b60172045b185ef210691434e14b.jpg",
+        "text": "$$\nE \\left( s , a \\right) \\gets \\left( 1 - \\alpha \\right) E \\left( s , a \\right) + \\alpha \\left( 0 + \\gamma _ { E } E \\left( s ^ { \\prime } , a ^ { \\prime } \\right) \\right) = \\left( 1 - \\alpha \\right) E \\left( s , a \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            862,
+            732,
+            878
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "So after being visited $n$ times, the value of the state-action pair is $\\left( 1 - \\alpha \\right) ^ { n }$ , where $\\alpha$ is the learning rate. By taking a logarithm transformation, we can see that $\\log _ { 1 - \\alpha } \\left( E \\right) = n$ . In addition, when $s$ is a terminal state with one action, $\\log _ { 1 - \\alpha } \\left( E \\right) = n$ − for any value of $\\gamma _ { E }$ . ",
+        "bbox": [
+            174,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/f13f4bf01e18a22e6b593a229b961a5bfed1d2c33d323718c50f6a7881e7d985.jpg",
+        "image_caption": [
+            "Figure 1: Left: Tree MDP, with $k$ leaves. Tree: $\\log _ { 1 - \\alpha } E \\left( s , { \\mathrm { s t a r t } } \\right)$ as function of visit cycles, for different trees of $k$ leaves (color coded). For each $k$ , a cycle consists of visiting all leaves, hence $k$ visits of the start action. $\\log _ { 1 - \\alpha } E$ behaves as a generalized counter, where each cycle contributes approximately one generalized visit. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            103,
+            792,
+            314
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When $\\gamma _ { E } > 0$ and for non-terminal states, $E$ will decrease more slowly and therefore $\\log _ { 1 - \\alpha } E$ −will increase more slowly than a counter. The exact rate will depend on the MDP, the policy and the specific value of $\\gamma _ { E }$ . Crucially, for state-actions which lead to many potential states, each visit contributes less to the generalized counter, because more visits are required to exhaust the potential outcomes of the action. To gain more insight, consider the MDP depicted in Figure 1 left, a tree with the root as initial state and the leaves as terminal states. If actions are chosen sequentially, one leaf after the other, we expect that each complete round of choices (which will result with $k$ actual visits of the ${ \\bf \\Phi } _ { s }$ , start) pair) will be roughly equivalent to one generalized counter. Simulation of this and other simple MDPs show that $E$ -values behave in accordance with such intuitions (see Figure 1 right). ",
+        "bbox": [
+            173,
+            404,
+            825,
+            544
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "An important property of $E$ -values is that they decrease over repetitions. Formally, by completing a trajectory of the form $s _ { 0 } , a _ { 0 } , \\ldots , s _ { n } , a _ { n } , s _ { 0 } , a _ { 0 }$ in the MDP, the maximal value of $E \\left( { { s } _ { i } } , { { a } _ { i } } \\right)$ will decrease. To see this, assume that $E \\left( { { s } _ { i } } , { { a } _ { i } } \\right)$ was maximal, and consider its value after the update: ",
+        "bbox": [
+            174,
+            550,
+            825,
+            593
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/013c660a504e8fda65e7210acc0b95f1568f5f731622247a386b7407f0574e0b.jpg",
+        "text": "$$\nE \\left( s _ { i } , a _ { i } \\right) \\gets \\left( 1 - \\alpha \\right) E \\left( s _ { i } , a _ { i } \\right) + \\alpha \\gamma _ { E } E \\left( s _ { i + 1 } , a _ { i + 1 } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            598,
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+            616
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Because $\\gamma _ { E } < 1$ and $E \\left( s _ { i + 1 } , a _ { i + 1 } \\right) \\leq E \\left( s _ { i } , a _ { i } \\right)$ , we get that after the update, the value of $E \\left( s _ { i } , a _ { i } \\right)$ decreased. For any non-maximal $( s _ { j } , a _ { j } )$ , its value after the update is a convex combination of its previous value and $\\gamma _ { E } E \\left( s _ { k } , a _ { k } \\right)$ which is not larger than its composing terms, which in turn are smaller than the maximal $E$ -value. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 APPLYING $E$ -VALUES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            698,
+            388,
+            714
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The logarithm of $E$ -values can be considered as a generalization of counters. As such, algorithms that utilize counters can be generalized to incorporate $E$ -values. Here we consider two such generalizations. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 $E$ -VALUES AS REWARD EXPLORATION BONUS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            789,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In model-based RL, counters have been used to create an augmented reward function. Motivated by this result, augmenting the reward with a counter-based exploration bonus has also been used in model-free RL (Storck et al., 1995; Bellemare et al., 2016). $E$ -Values can naturally generalize this approach, by replacing the standard counter with its corresponding generalized counter $( \\log _ { 1 - \\alpha } E )$ . ",
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the advantage of using $E$ -values over standard counters, we tested an $\\epsilon$ -greedy agent with an exploration bonus of $\\frac { 1 } { \\log _ { 1 - \\alpha } E }$ added to the observed reward on the bridge MDP (Figure 2). To measure the learning progress and its convergence, we calculated the mean square error $\\mathbb { E } _ { P ( s , a | \\pi ^ { * } ) } \\left[ \\left( Q \\left( s , a \\right) - Q ^ { * } \\left( s , a \\right) \\right) ^ { 2 } \\right]$ , where the average is over the probability of state-action pairs when following the optimal policy $\\pi ^ { * }$ . We varied the value of $\\gamma _ { E }$ from 0 – resulting effectively in standard counters – to $\\gamma _ { E } = 0 . 9$ . Our results (Figure 3) show that adding the exploration bonus to the reward leads to faster learning. Moreover, the larger the value of $\\gamma _ { E }$ in this example the faster the learning, demonstrating that generalized counters significantly outperforming standard counters. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/8065bd1d0577a346121c1dafad546ce14b15bb53ca4c31ce28c701ddc8216df8.jpg",
+        "image_caption": [
+            "Figure 2: Bridge MDP "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/fb180738f53361e6aace8ff042e5df6cf910f3fd73e24b94ff9d4ec95e363341.jpg",
+        "image_caption": [
+            "Figure 3: MSE between $Q$ and $Q ^ { * }$ on optimal policy per episode. Convergence of $\\epsilon$ -greedy on the short bridge environment $k = 5$ ) with and without exploration bonuses added to the reward. Note the logarithmic scale of the abscissa. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 $E$ -VALUES AND ACTION-SELECTION RULES ",
+        "text_level": 1,
+        "bbox": [
+            179,
+            530,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Another way in which counters can be used to assist exploration is by adding them to the estimated $Q$ -values. In this framework, action-selection is a function not only of the $Q$ -values but also of the counters. Several such action-selection rules have been proposed (Thrun, 1992; Meuleau & Bourgine, 1999; Kolter & Ng, 2009). These usually take the form of a deterministic policy that maximizes some combination of the estimated $Q$ -value with a counter-based exploration bonus. It is easy to generalize such rules using $E$ -values – simply replace the counters $C$ by the generalized counters $\\log _ { 1 - \\alpha } \\left( E \\right)$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2.1 DETERMINIZATION OF STOCHASTIC DECISION RULES",
+        "text_level": 1,
+        "bbox": [
+            174,
+            670,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here, we consider a special family of action-selection rules that are derived as deterministic equivalents of standard stochastic rules. Stochastic action-selection rules are commonly used in RL. In their simple form they include rules such as the $\\epsilon$ -greedy or Softmax exploration described above. In this framework, exploratory behavior is achieved by stochastic action selection, independent of past choices. At first glance, it might be unclear how $E ^ { }$ -values can contribute or improve such rules. We now turn to show that, by using counters, for every stochastic rule there exist equivalent deterministic rules. Once turned to deterministic counter-based rules, it is again possible improve them using $E$ -values. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The stochastic action-selection rules determine the frequency of choosing the different actions in the limit of a large number of repetitions, while abstracting away the specific order of choices. This fact is a key to understanding the relation between deterministic and stochastic rules. An equivalence of two such rules can only be an in-the-limit equivalence, and can be seen as choosing a specific realization of sample from the distribution. Therefore, in order to derive a deterministic equivalent of a given stochastic rule, we only have to make sure that the frequencies of actions selected under both rules are equal in the limit of infinitely many steps. As the probability for each action is likely to depend on the current $Q$ -values, we have to consider fixed $Q$ -values to define this equivalence. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We prove that given a stochastic action-selection rule $f \\left( a | s \\right)$ , every deterministic policy that does not choose an action that was visited too many times until now (with respect to the expected number according to the probability distribution) is a determinization of $f$ . Formally, lets assume that given a certain $Q$ function and state $s$ we wish a certain ratio between different choices of actions $a \\in A$ to hold. We denote the frequency of this ratio $f _ { Q } \\left( a | s \\right)$ . For brevity we assume $s$ and $Q$ are constants and denote $f _ { Q } \\left( a | s \\right) = { \\bar { f } } \\left( a \\right)$ . We also assume a counter $C \\left( s , a \\right)$ is kept denoting the number of choices of $a$ in $s$ . For brevity we denote $C \\left( s , a \\right) = C \\left( a \\right)$ and $\\begin{array} { r } { \\sum _ { a } C \\left( s , a \\right) = C } \\end{array}$ . When we look at the counters after $T$ steps we use subscript $C _ { T } \\left( a \\right)$ . Following this notation, note that $C _ { T } = T$ . ",
+        "bbox": [
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+            103,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1. For any sub-linear function $b \\left( t \\right)$ and for any deterministic policy which chooses at step $T$ an action a such that $\\begin{array} { r } { \\frac { C _ { T } \\left( a \\right) } { T } - f \\left( a \\right) \\leq b \\left( t \\right) } \\end{array}$ it holds that $\\forall a \\in { \\mathcal { A } }$ ",
+        "bbox": [
+            171,
+            219,
+            825,
+            252
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0443c92949651144ca1b76bdc86f2a98fd4c2effaa44355ac160887b5a7b22ad.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { T  \\infty } \\frac { C _ { T } ( a ) } { T } = f ( a )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            424,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof. For a full proof of the theorem see Appendix A in the supplementary materials ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The result above is not a vacuous truth – we now provide two possible determinization rules that achieves it. One rule is straightforward from the theorem, using $b = \\mathbf { 0 }$ , choosing $\\begin{array} { r } { \\arg \\operatorname* { m i n } _ { a } \\frac { C ( a ) } { C } - } \\end{array}$ $f \\left( a \\right)$ . Another rule follows the probability ratio between the stochastic policy and the empirical distribution: arg maxa C(a) . We denote this determinization $L L L$ , because when generalized counters are used instead of counters it becomes a $\\begin{array} { r } { \\mathrm { r g } \\operatorname* { m a x } _ { a } \\log f \\left( s , a \\right) - \\log \\log _ { 1 - \\alpha } E \\left( s , a \\right) . } \\end{array}$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Now we can replace the visit counters $C \\left( s , a \\right)$ with the generalized counters $\\log _ { 1 - \\alpha } \\left( E \\left( s , a \\right) \\right)$ to create Directed Outreaching Reinforcement Action-Selection – DORA the explorer. By this, we can transform any stochastic or counter-based action-selection rule into a deterministic rule in which exploration propagates over the states and the expected trajectories to follow. ",
+        "bbox": [
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+            422,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Input: Stochastic action-selection rule $f$ , learning rate $\\alpha$ , Exploration discount factor $\\gamma _ { E }$ initialize $Q \\left( s , a \\right) = 0$ , $E \\left( s , a \\right) = 1$ ; ",
+        "bbox": [
+            174,
+            489,
+            754,
+            518
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "foreach episode do init $s$ ; while not terminated do Choose $a = \\arg \\operatorname* { m a x } _ { x } \\log f _ { Q } \\left( x | s \\right) - \\log \\log _ { 1 - \\alpha } E \\left( s , x \\right) ;$ Observe transitions $( s , a , r , s ^ { \\prime } , a ^ { \\prime } )$ ; $Q ( s , a )  ( 1 - \\alpha ) Q ( s , a ) + \\alpha ( r + \\gamma \\operatorname* { m a x } _ { x } Q ( s ^ { \\prime } , x ) ) ;$ $E \\left( s , a \\right) \\gets \\left( 1 - \\alpha \\right) E \\left( s , a \\right) + \\alpha \\gamma _ { E } E \\left( s ^ { \\prime } , a ^ { \\prime } \\right)$ ; end   \nend ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1: DORA algorithm using $L L L$ determinization for stochastic policy $f$ ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 RESULTS – FINITE MDPS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            679,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To test this algorithm, the first set of experiments were done on Bridge environments of various lengths $k$ (Figure 2). We considered the following agents: $\\epsilon$ -greedy, Softmax and their respective $L L L$ determinizations (as described in 3.2.1) using both counters and $E$ -values. In addition, we compared a more standard counter-based agent in the form of a UCB-like algorithm (Auer et al., 2002) following an action-selection rule with exploration bonus of $\\sqrt { \\frac { \\log t } { C } }$ log t . We tested two variants of this algorithm, using ordinary visit counters and $E$ -values. Each agent’s hyperparameters $\\dot { \\epsilon }$ and temperature) were fitted separately to optimize learning. For stochastic agents, we averaged the results over 50 trials for each execution. Unless stated otherwise, $\\gamma _ { E } = 0 . 9$ . ",
+        "bbox": [
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+            704,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We also used a normalized version of the bridge environment, where all rewards are between 0 and 1, to compare DORA with the Delayed $Q$ -Learning algorithm (Strehl et al., 2006). ",
+        "bbox": [
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+            832,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our results (Figure 4) demonstrate that $E$ -value based agents outperform both their counter-based and their stochastic equivalents on the bridge problem. As shown in Figure 4, Stochastic and counter-based $\\epsilon$ -greedy agents, as well as the standard UCB fail to converge. $E$ -value agents are the first to reach low error values, indicating that they learn faster. Similar results were achieved on other gridworld environments, such as the Cliff problem (Sutton & Barto, 1998) (not shown). We also achieved competitive results with respect to Delayed $Q$ Learning (see supplementary B and Figure 7 there). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/c8182ddcd866e022dfedaf0bf5e5888a7c6102523bb6ca82822ffd3bdca883e5.jpg",
+        "image_caption": [
+            "Figure 4: MSE between $Q$ and $Q ^ { * }$ on optimal policy per episode. Convergence measure of all agents, long bridge environment $k = 1 5$ ). $E$ -values agents are the first to converge, suggesting their superior learning abilities. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The success of $E$ -values based learning relative to counter based learning implies that the use of $E$ -values lead to more efficient exploration. If this is indeed the case, we expect $E$ -values to better represent the agent’s missing knowledge than visit counters during learning. To test this hypothesis we studied the behavior of an $E$ -value $L L L$ Softmax on a shorter bridge environment $k = 5$ ). For a given state-action pair, a measure of the missing knowledge is the normalized distance between its estimated value $( Q )$ and its optimal-policy value $( Q ^ { * } )$ . We recorded $C$ , $\\log _ { 1 - \\alpha } \\left( E \\right)$ and $\\left| \\frac { Q - Q ^ { * } } { Q ^ { * } } \\right|$ for each $s , a$ at the end of each episode. Generally, this measure of missing knowledge is expected to be a monotonously-decreasing function of the number of visits $( C )$ . This is indeed true, as depicted in Figure 5 (left). However, considering all state-action pairs, visit counters do not capture well the amount of missing knowledge, as the convergence level depends not only on the counter but also on the identity of the state-action it counts. By contrast, considering the convergence level as a function of the generalized counter (Figure 5, right) reveals a strikingly different pattern. Independently of the state-action identity, the convergence level is a unique function of the generalized counter. These results demonstrate that generalized counters are a useful measure of the amount of missing knowledge. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 $E$ -VALUES WITH FUNCTION APPROXIMATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            683,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "So far we discussed $E$ -values in the tabular case, relying on finite (and small) state and action spaces. However, a main motivation for using model-free approach is that it can be successfully applied in large MDPs where tabular methods are intractable. In this case (in particular for continuous MDPs), achieving directed exploration is a non-trivial task. Because revisiting a state or a state-action pair is unlikely, and because it is intractable to store individual values for all state-action pairs, counterbased methods cannot be directly applied. In fact, most implementations in these cases adopt simple exploration strategies such as $\\epsilon$ -greedy or softmax (Bellemare et al., 2016). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "There are standard model-free techniques to estimate value function in function-approximation scenarios. Because learning $E$ -values is simply learning another value-function, the same techniques can be applied for learning $E$ -values in these scenarios. In this case, the concept of visit-count – or a generalized visit-count – will depend on the representation of states used by the approximating function. ",
+        "bbox": [
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+            819,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To test whether $E$ -values can serve as generalized visit-counters in the function-approximation case, we used a linear approximation architecture on the MountainCar problem (Moore, 1990) (Appendix ",
+        "bbox": [
+            174,
+            895,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "C). To dissociate $Q$ and $E$ -values, actions were chosen by an $\\epsilon$ -greedy agent independently of $E$ - values. As shown in Appendix C, $E$ -values are an effective way for counting both visits and generalized visits in continuous MDPs. For completeness, we also compared the performance of $L L L$ agents to stochastic agents on a sparse-reward MountainCar problem, and found that $L L L$ agents learns substantially faster than the stochastic agents (Appendix D). ",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1 RESULTS – FUNCTION APPROXIMATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            203,
+            490,
+            217
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To show our approach scales to complex problems, we used the Freeway Atari 2600 game, which is known as a hard exploration problem (Bellemare et al., 2016). We trained a neural network with two streams to predict the $Q$ and $E$ -values. First, we trained the network using standard DQN technique (Mnih et al., 2015), which ignores the $\\mathrm { E }$ -values. Second, we trained the network while adding an exploration bonus of $\\frac { \\beta } { \\sqrt { - \\log E } }$ to the reward (In all reported simulations, $\\beta = 0 . 0 5$ ). In both cases, action-selection was performed by an $\\epsilon$ -greedy rule, as in Bellemare et al. (2016). ",
+        "bbox": [
+            174,
+            233,
+            825,
+            321
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Note that the exploration bonus requires $0 \\textless E < 1$ . To satisfy this requirement, we applied a logistic activation fucntion on the output of the last layer of the $E$ -value stream, and initialized the weights of this layer to 0. As a result, the $E$ -values were initialized at 0.5 and satisfied $0 < E < 1$ throughout the training. In comparison, no non-linearity was applied in the last layer of the $Q$ -value stream and the weights were randmoly initialized. ",
+        "bbox": [
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+            328,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We compared our approach to a DQN baseline, as well as to the density model counters suggested by (Bellemare et al., 2016). The baseline used here does not utilize additional enhancements (such as Double DQN and Monte-Carlo return) which were used in (Bellemare et al., 2016). Our results, depicted in Figure 6, demonstrate that the use of $E$ -values outperform both DQN and density model counters baselines. In addition, our approach results in better performance than in (Bellemare et al., 2016) (with the mentioned enhancements), converging in approximately $2 \\cdot 1 0 ^ { 6 }$ steps, instead of $1 0 \\cdot 1 0 ^ { 6 }$ steps2. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            535,
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+            551
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The idea of using reinforcement-learning techniques to estimate exploration can be traced back to Storck et al. (1995) and Meuleau & Bourgine (1999) who also analyzed propagation of uncertainties and exploration values. These works followed a model-based approach, and did not fully deal with the problem of non-Markovity arising from using exploration bonus as the immediate reward. A related approach was used by Little & Sommer (2014), where exploration was investigated by information-theoretic measures. Such interpretation of exploration can also be found in other works (Schmidhuber (1991); Sun et al. (2011); Houthooft et al. (2016)). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Efficient exploration in model-free RL was also analyzed in PAC-MDP framework, most notably the Delayed $Q$ Learning algorithm by Strehl et al. (2006). For further discussion and comparison of our approach with Delayed $Q$ Learning, see 1.1 and Appendix B. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In terms of generalizing Counter-based methods, there has been some works on using counter-like notions for exploration in continuous MDPs (Nouri & Littman, 2009). A more direct attempt was recently proposed by Bellemare et al. (2016). This generalization provides a way to implement visit counters in large, continuous state and action spaces by using density models. Our generalization is different, as it aims first on generalizing the notion of visit counts themselves, from actual counters to ”propagating counters”. In addition, our approach does not depend on any estimated model – which might be an advantage in domains for which good density models are not available. Nevertheless, we believe that an interesting future work will be comparing between the approach suggested by Bellemare et al. (2016) and our approach, in particular for the case of $\\gamma _ { E } = 0$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/4683886049710d08ce9ffa88cfe453f359909fa4c66364220b7f4836779d18b2.jpg",
+        "image_caption": [
+            "Figure 5: Convergence of $Q$ to $Q ^ { * }$ for individual state-action pairs (each denoted by a different color), with respect to counters (left) and generalized counters (right). Results obtained from $E$ - Value $L L L$ Softmax on the short bridge environment $k = 5$ ). Triangle markers indicate pairs with ”east” actions, which constitute the optimal policy of crossing the bridge. Circle markers indicate state-action pairs that are not part of the optimal policy. Generalized counters are a useful measure of the amount of missing knowledge. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/bf544dda638c5414e55a0ce2a52e60c33ae561e2ab24fdfdb38f55bab0f6b677.jpg",
+        "image_caption": [
+            "Figure 6: Results on Freeway game. All agents used $\\epsilon$ -greedy action-selection rule without exploration bonus (DQN, blue), with a bonus term based on density model counters (Density, orange) added to the reward, or with bonus term based on $E$ -values (black). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            847,
+            387,
+            862
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Nadav Cohen, Leo Joskowicz, Ron Meir, Michal Moshkovitz, and Jeff Rosenschein for discussions. This work was supported by the Israel Science Foundation (Grant No. 757/16) and the Gatsby Charitable Foundation. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            287,
+            118
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. Finite-time analysis of the multiarmed bandit problem. Machine learning, 47(2-3):235–256, 2002. ",
+        "bbox": [
+            173,
+            126,
+            825,
+            155
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Marc Bellemare, Sriram Srinivasan, Georg Ostrovski, Tom Schaul, David Saxton, and Remi Munos. Unifying count-based exploration and intrinsic motivation. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett (eds.), Advances in Neural Information Processing Systems 29, pp. 1471–1479. Curran Associates, Inc., 2016. ",
+        "bbox": [
+            174,
+            165,
+            825,
+            223
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Lucian Busoniu, Robert Babuska, and Bart De Schutter. A comprehensive survey of multiagent reinforcement learning. IEEE Transactions on Systems, Man, And Cybernetics-Part C: Applications and Reviews, 38 (2), 2008, 2008. ",
+        "bbox": [
+            173,
+            233,
+            825,
+            275
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Arthur Guez, David Silver, and Peter Dayan. Efficient bayes-adaptive reinforcement learning using sample-based search. In Advances in Neural Information Processing Systems, pp. 1025–1033, 2012. ",
+        "bbox": [
+            173,
+            286,
+            825,
+            329
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Rein Houthooft, Xi Chen, Yan Duan, John Schulman, Filip De Turck, and Pieter Abbeel. Vime: Variational information maximizing exploration. In Advances in Neural Information Processing Systems, pp. 1109–1117, 2016. ",
+        "bbox": [
+            176,
+            339,
+            823,
+            382
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Leslie Pack Kaelbling, Michael L Littman, and Andrew W Moore. Reinforcement learning: A survey. Journal of artificial intelligence research, 4:237–285, 1996. ",
+        "bbox": [
+            169,
+            392,
+            825,
+            422
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sham Machandranath Kakade et al. On the sample complexity of reinforcement learning. PhD thesis, 2003. ",
+        "bbox": [
+            174,
+            431,
+            823,
+            462
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "J Zico Kolter and Andrew Y Ng. Near-bayesian exploration in polynomial time. In Proceedings of the 26th Annual International Conference on Machine Learning, pp. 513–520. ACM, 2009. ",
+        "bbox": [
+            173,
+            472,
+            825,
+            502
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Daniel Y Little and Friedrich T Sommer. Learning and exploration in action-perception loops. Closing the Loop Around Neural Systems, pp. 295, 2014. ",
+        "bbox": [
+            173,
+            511,
+            821,
+            541
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Nicolas Meuleau and Paul Bourgine. Exploration of multi-state environments: Local measures and back-propagation of uncertainty. Machine Learning, 35(2):117–154, 1999. ",
+        "bbox": [
+            174,
+            550,
+            825,
+            580
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, 2015. ",
+        "bbox": [
+            176,
+            590,
+            823,
+            633
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Andrew William Moore. Efficient memory-based learning for robot control. 1990. ",
+        "bbox": [
+            173,
+            643,
+            715,
+            660
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ali Nouri and Michael L Littman. Multi-resolution exploration in continuous spaces. In Advances in neural information processing systems, pp. 1209–1216, 2009. ",
+        "bbox": [
+            173,
+            669,
+            820,
+            699
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gavin A Rummery and Mahesan Niranjan. On-line Q-learning using connectionist systems. University of Cambridge, Department of Engineering, 1994. ",
+        "bbox": [
+            174,
+            709,
+            823,
+            738
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jurgen Schmidhuber. Curious model-building control systems. In ¨ Neural Networks, 1991. 1991 IEEE International Joint Conference on, pp. 1458–1463. IEEE, 1991. ",
+        "bbox": [
+            173,
+            748,
+            825,
+            779
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jan Storck, Sepp Hochreiter, and Jurgen Schmidhuber. Reinforcement driven information acquisi- ¨ tion in non-deterministic environments. In Proceedings of the international conference on artificial neural networks, Paris, volume 2, pp. 159–164. Citeseer, 1995. ",
+        "bbox": [
+            173,
+            787,
+            823,
+            832
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alexander L Strehl and Michael L Littman. An analysis of model-based interval estimation for markov decision processes. Journal of Computer and System Sciences, 74(8):1309–1331, 2008. ",
+        "bbox": [
+            169,
+            842,
+            823,
+            871
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alexander L Strehl, Lihong Li, Eric Wiewiora, John Langford, and Michael L Littman. Pac modelfree reinforcement learning. In Proceedings of the 23rd international conference on Machine learning, pp. 881–888. ACM, 2006. ",
+        "bbox": [
+            176,
+            881,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yi Sun, Faustino Gomez, and Jurgen Schmidhuber. Planning to be surprised: Optimal bayesian ¨ exploration in dynamic environments. In International Conference on Artificial General Intelligence, pp. 41–51. Springer, 2011. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 1998. ",
+        "bbox": [
+            171,
+            155,
+            825,
+            170
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sebastian B. Thrun. Efficient exploration in reinforcement learning, 1992. ",
+        "bbox": [
+            173,
+            179,
+            661,
+            194
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Michel Tokic and Gunther Palm. Value-difference based exploration: adaptive control between ¨ epsilon-greedy and softmax. In KI 2011: Advances in Artificial Intelligence, pp. 335–346. Springer, 2011. ",
+        "bbox": [
+            173,
+            203,
+            825,
+            244
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Thomas J Walsh, Istvan Szita, Carlos Diuk, and Michael L Littman. Exploring compact ´ reinforcement-learning representations with linear regression. In Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, pp. 591–598. AUAI Press, 2009. ",
+        "bbox": [
+            176,
+            253,
+            825,
+            296
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Christopher JCH Watkins and Peter Dayan. Q-learning. Machine learning, 8(3-4):279–292, 1992. ",
+        "bbox": [
+            169,
+            306,
+            816,
+            321
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A PROOF OF THE DETERMINIZATION THEOREM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            588,
+            119
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The proof for the determinization mentioned in the paper is achieved based on the following lemmata. ",
+        "bbox": [
+            173,
+            132,
+            823,
+            161
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Lemma A.1. The absolute sum of positive and negative differences between the empiric distribution (deterministic frequency) and goal distribution (non-deterministic frequency) is equal. ",
+        "bbox": [
+            174,
+            164,
+            821,
+            193
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/09f714c3b2228c942a293ee4980ec318b2dbba299f6f5fabf74e5bc3ddc96551.jpg",
+        "text": "$$\n\\sum _ { a : f \\left( a \\right) \\geq \\frac { C \\left( a \\right) } { C } } f \\left( a \\right) - \\frac { C \\left( a \\right) } { C } = - \\sum _ { a : f \\left( a \\right) < \\frac { C \\left( a \\right) } { C } } f \\left( a \\right) - \\frac { C \\left( a \\right) } { C }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            193,
+            691,
+            238
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. Straightforward from the observation that ",
+        "bbox": [
+            173,
+            251,
+            498,
+            266
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a6f9854e347748d440897c3043094032892106eb59f484caead0179111023962.jpg",
+        "text": "$$\n\\sum _ { a } f \\left( a \\right) = \\sum _ { a } { \\frac { C \\left( a \\right) } { C } } = 1\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            267,
+            591,
+            306
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Lemma A.2. For any $t$ ",
+        "bbox": [
+            173,
+            328,
+            331,
+            343
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d2517e2f7cf5d3f15a456ac22ce91d2991cb02ae0c5058df61944e4f727a17c0.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { a } \\left\\{ { \\frac { C _ { t } \\left( a \\right) } { t } } - f \\left( a \\right) \\right\\} \\leq { \\frac { 1 + b \\left( t \\right) } { t } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            340,
+            611,
+            372
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. The proof of A.2 is done by induction. For $t = 1$ ",
+        "bbox": [
+            173,
+            385,
+            544,
+            400
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1401aa2672d486738b746fffaa1caab20ed5d73a34b328fff2e914607d16477c.jpg",
+        "text": "$$\n\\forall a \\in A : { \\frac { C _ { t } \\left( a \\right) } { t } } - f \\left( a \\right) = \\operatorname* { m a x } _ { a } \\left\\{ { \\frac { C _ { t } \\left( a \\right) } { t } } - f \\left( a \\right) \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            401,
+            658,
+            433
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hence we look at $a \\in A$ . ",
+        "bbox": [
+            178,
+            434,
+            343,
+            449
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5a9f796a64f5132251f959d644bbb245b9f77e7bd424210f08f2f70b2ff2596f.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { C _ { t } \\left( a \\right) } { t } - f \\left( a \\right) \\leq \\frac { C _ { t } \\left( a \\right) } { t } \\qquad } \\\\ { \\displaystyle \\leq \\frac { 1 + b \\left( 1 \\right) } { 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            449,
+            584,
+            506
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "assume the claim is true for $t = T$ then for $t = T + 1$ There exists $a$ such that $C _ { T } \\left( a \\right) / T - f \\left( a \\right) \\leq$ $b \\left( t \\right)$ which the algorithm chooses for this $a$ . For it ",
+        "bbox": [
+            173,
+            507,
+            823,
+            535
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c2df7197bf18f81a1acf9b92fba9ccb5468584de2883bee73b5dfaff03354ae3.jpg",
+        "text": "$$\n\\begin{array} { c } { \\displaystyle \\frac { C _ { T + 1 } \\left( a \\right) } { T + 1 } - f \\left( a \\right) = \\displaystyle \\frac { C _ { T } \\left( a \\right) + 1 } { T + 1 } - f \\left( a \\right) } \\\\ { = \\displaystyle \\frac { C _ { T } \\left( a \\right) } { T + 1 } - f \\left( a \\right) + \\displaystyle \\frac { 1 } { T + 1 } } \\\\ { = \\displaystyle \\frac { C _ { T } \\left( a \\right) - \\left( T + 1 \\right) f \\left( a \\right) } { T + 1 } + \\displaystyle \\frac { 1 } { T + 1 } } \\\\ { \\leq \\displaystyle \\frac { 1 + b \\left( t \\right) } { T + 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            536,
+            669,
+            655
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "It also holds that $\\forall a ^ { \\prime } \\in A$ s.t. $\\boldsymbol { a } ^ { \\prime } \\neq \\boldsymbol { a }$ ",
+        "bbox": [
+            178,
+            656,
+            419,
+            671
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4c82f7084bbcdcfd1df137c70044cd092251836cda89e00b9bd2aa636b1d2ea2.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { C _ { T + 1 } \\left( a \\right) } { T + 1 } - f \\left( a \\right) = \\displaystyle \\frac { C _ { T } \\left( a \\right) } { T + 1 } - f \\left( a \\right) } \\\\ { \\displaystyle = \\frac { C _ { T } \\left( a \\right) - \\left( T + 1 \\right) f \\left( a \\right) } { T + 1 } } \\\\ { \\displaystyle < \\frac { C _ { T } \\left( a \\right) - T f \\left( a \\right) } { T + 1 } } \\\\ { \\displaystyle \\le \\frac { 1 + b \\left( t \\right) } { T + 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            674,
+            640,
+            792
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof of 3.1. It holds from A.2 together with A.1 that in the step $t$ in the worst case all but one of the actions have $\\begin{array} { r } { \\frac { C _ { t } ( a ) } { t } - f ( a ) = \\frac { 1 } { t } } \\end{array}$ and the last action has $\\begin{array} { r } { f ( { a } ) - \\frac { { \\bf \\bar { \\Phi } } _ { C _ { t } } ( { a } ) } { t } = - \\frac { | A | - 1 } { t } } \\end{array}$ . So by the bound on sum of positives and negatives we get: ",
+        "bbox": [
+            173,
+            821,
+            826,
+            867
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/759fb5404b7603306b75f81f979cb1a6ad2bf0ea5390706b3bcb67de2cace4d3.jpg",
+        "text": "$$\n\\operatorname* { l i m } _ { T \\to \\infty } \\frac { C _ { T } ( a ) } { T } = f ( a )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            868,
+            570,
+            900
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/5e5df4cbb49ca8a5cd3567df92afb642680d55e68a45e5bea1d8417745ca9038.jpg",
+        "image_caption": [
+            "Figure 7: Normalized MSE between $Q$ and $Q ^ { * }$ on optimal policy per episode. Convergence of $E$ -value $L L L$ and Delayed $Q$ -Learning on, normalized bridge environment $( k = 1 5 )$ ). MSE was noramlized for each agent to enable comparison. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            101,
+            754,
+            349
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B COMPARISON WITH DELAYED $Q$ -LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            430,
+            576,
+            446
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Because Delayed $Q$ learning initializes its values optimistically, which result in a high MSE, we normalized the MSE of the two agents (separately) to enable comparison. Notably, to achieve this performance by the Delayed $Q$ Learning, we had to manually choose a low value for $m$ (in Figure 7, $m = 1 0$ ), the hyperparameter regulating the number of visits required before any update. This is an order of magnitude smaller than the theoretical value required for even moderate PAC-requirements in the usual notion of $\\epsilon , \\delta$ , such $m$ also implies learning in orders of magnitudes slower. In fact, for this limit of $m  1$ the algorithm is effectively quite similar to a ”Vanilla” $Q$ -Learning with an optimistic initialization, which is possible due to the assumption made by the algorithm that all rewards are between 0 and 1. In fact, several exploration schemes relying on optimism in the face of uncertainty were proposed (Walsh et al., 2009). However, because our approach separate reward values and exploratory values, we are able to use optimism for the latter without assuming any prior knowledge about the first – while still achieving competitive results to an optimistic initialization based on prior knowledge. ",
+        "bbox": [
+            173,
+            462,
+            825,
+            642
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C EVALUATING $E$ -VALUES DYNAMICS IN FUNCTION-APPROXIMATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            664,
+            769,
+            679
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To gain insight into the relation between E-values and number of visits, we used the linearapproximation architecture on the MountainCar problem. Note that when using $E$ -values, they are generally correlated with visit counts both because visits result in update of the $E$ -values through learning and because $E$ -values affect visits through the exploration bonus (or action-selection rule). To dissociate the two, $Q$ -values and $E$ -values were learned in parallel in these simulation, but actionselection was independent of the $E$ -values. Rather, actions were chosen by an $\\epsilon$ -greedy agent. To estimate visit-counts, we recorded the entire set of visited states, and computed the empirical visits histogram by binning the two-dimensional state-space. For each state, its visit counter estimator $\\tilde { C } \\left( s \\right)$ is the value of the matching bin in the histogram for this state. In addition, we recorded the learned model (weights vector for $E$ -values) and computed the $E$ -values map by sampling a state for each bin, and calculating its $E$ -values using the model. For simplicity, we consider here the resolution of states alone, summing over all 3 actions for each state. That is, we compare $\\tilde { C } \\left( s \\right)$ to $\\begin{array} { r } { \\sum _ { a } \\log _ { 1 - \\alpha } E \\left( s , a \\right) = C _ { E } \\left( s \\right) } \\end{array}$ . Figure 8 depicts the empirical visits histogram (left) and the estimated $E$ -values for the case of $\\gamma _ { E } = 0$ after the complete training. The results of the analysis show that, roughly speaking, those regions in the state space that were more often visited, were also associated with a higher $C _ { E } \\left( s \\right)$ . ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/29be4dababb04f926c7bc56c2288e4c2f190df6e5fc2dd283325a9cf757ca8cc.jpg",
+        "image_caption": [
+            "Figure 8: Empirical visits histogram (left) and learned $C _ { E }$ (right) after training, $\\gamma _ { E } = 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To better understand these results, we considered smaller time-windows in the learning process. Specifically, Figure 9 depicts the empirical visit histogram (left), and the corresponding $C _ { E } \\left( s \\right)$ (right) in the first 10 episodes, in which visits were more centrally distributed. Figure 10 depicts the change in the empirical visit histogram (left), and change in the corresponding $C _ { E }$ (s) (right) in the last 10 episodes of the training, in which visits were distributed along a spiral (forming an nearoptimal behavior). These results demonstrate high similarity between visit-counts and the $E$ -value representation of them, indicating that $E$ -values are good proxies of visit counters. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/422f2bbf6cebd9e935380b9c12833a721be8090514550426a0bc6d466169d7f2.jpg",
+        "image_caption": [
+            "Figure 9: Empirical visits histogram (left) and learned $C _ { E }$ (right) in the first 10 training episodes, $\\gamma _ { E } = 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/aba80c86fa21a948f82e2b97dbc136f39feeddaa5557444f3fac938c60957d06.jpg",
+        "image_caption": [
+            "Figure 10: Difference in empirical visits histogram (left) and learned $C _ { E }$ (right) in the last 10 training episodes, $\\gamma _ { E } = 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The results depicted in Figures 9 and 10 were achieved with $\\gamma _ { E } = 0$ . For $\\gamma _ { E } > 0$ , we expect the generalized counters (represented by $E$ -values) to account not for standard visits but for ”generalized visits”, weighting the trajectories starting in each state. We repeated the analysis of Figure 10 for the case of $\\gamma _ { E } ~ = ~ 0 . 9 9$ . Results, depicted in Figure 11, shows that indeed for terminal or nearterminal states (where position $> 0 . 5$ ) generalized visits, measured by difference in their generalized counters, are higher – comparing to far-from terminal states – than the empirical visits of these states (comparing to far-from terminal states). ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/e3625613ff30956da8419837612a93f4282737cd2d9ca78d9d8ffd9088dc3c8b.jpg",
+        "image_caption": [
+            "Figure 11: Difference in empirical visits histogram (left) and learned $C _ { E }$ (right) in the last 10 training episodes, $\\gamma _ { E } = 0 . 9 9$ . Note that the results are based on a different simulation than those in Figure 10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To quantify the relation between visits and $E$ -values, we densely sampled the (achievable) statespace to generate many examples of states. For each sampled state, we computed the correlation coefficient between $C _ { E } \\left( s \\right)$ and $\\tilde { C } \\left( s \\right)$ throughout the learning process (snapshots taken each 10 episodes). The values $\\tilde { C } \\left( s \\right)$ were estimated by the empirical visits histogram (value of the bin corresponding to the sampled state) calculated based on visits history up to each snapshot. Figure 12, depicting the histogram of correlation coefficients between the two measures, demonstrating strong positive correlations between empirical visit-counters and generalized counters represented by $E$ -values. These results indicate that $E$ -values are an effective way for counting effective visits in continuous MDPs. Note that the number of model parameters used to estimate $E \\left( s , a \\right)$ in this case is much smaller than the size of the table we would have to use in order to track state-action counters in such binning resolution. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/844d10739b90fffa50f60dca6f008d682d5e38f2255e51d95449e6b785a38cdd.jpg",
+        "image_caption": [
+            "Figure 12: Histogram of correlation coefficients between empirical visit counters and $C _ { E }$ throughout training, per state $( \\gamma _ { E } = 0$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D RESULTS ON CONTINUOUS MDPS – MOUNTAINCAR ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "To test the performance of $E$ -values based agents, simulations were performed using the MountainCar environment. The version of the problem considered here is with sparse and delayed reward, meaning that there is a constant reward of 0 unless reaching a goal state which provides a reward of magnitude 1. Episode length was limited to 1000 steps. We used linear approximation with tilecoding features (Sutton & Barto, 1998), learning the weights vectors for $Q$ and $E$ in parallel. To guarantee that $E$ -values are uniformly initialized and are kept between 0 and 1 throughout learning, we initialized the weights vector for $E$ -values to 0 and added a logistic non-linearity to the results of the standard linear approximation. In contrast, the $Q$ -values weights vector was initialized at random, and there was no non-linearity. We compared the performance of several agents. The first two used only $Q$ -values, with a softmax or an $\\epsilon$ -greedy action-selection rules. The other two agents are the DORA variants using both $Q$ and $E$ values, following the $L L L$ determinization for softmax either with $\\gamma _ { E } = 0$ or with $\\gamma _ { E } = 0 . 9 9$ . Parameters for each agent (temperature and $\\dot { \\epsilon }$ ) were fitted separately to maximize performance. The results depicted in Figure 13 demonstrate that using $E$ -values with $\\gamma _ { E } > 0$ lead to better performance in the MountainCar problem ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In addition we tested our approach using (relatively simple) neural networks. We trained two neural networks in parallel (unlike the two-streams single network used for Atari simulations), for predicting $Q$ and $E$ values. In this architecture, the same technique of 0 initializing and a logistic non-linearity was applied to the last linear of the $E$ -network. Similarly to the linear approximation approach, $E$ -values based agents outperform their $\\epsilon$ -greedy and softmax counterparts (not shown). ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/8812ff7dda21f6500540aa61c9cbd2827e94be23caa1d5d207a09cd0c73067f3.jpg",
+        "image_caption": [
+            "Figure 13: Probability of reaching goal on MountainCar (computed by averaging over 50 simulations of each agent), as a function of training episodes. While Softmax exploration fails to solve the problem within 1000 episodes, LLL $E$ -values agents with generalized counters $( \\gamma _ { E } > 0 ) ,$ ) quickly reach high success rates. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 16
+    }
+]