Patrick Kidger, Terry Lyons

The classical Universal Approximation Theorem holds for neural networks of arbitrary width and bounded depth. Here we consider the natural ‘dual’ scenario for networks of bounded width and arbitrary depth. Precisely, let $n$ be the number of inputs neurons, $m$ be the number of output neurons, and let $\rho$ be any nonaffine continuous function, with a continuous nonzero derivative at some point. Then we show that the class of neural networks of arbitrary depth, width $n+m+2$, and activation function $\rho$, is dense in $C(K;{\mathbb{R}}^m)$ for $K\subseteq {\mathbb{R}}^n$ with $K$ compact. This covers every activation function possible to use in practice, and also includes polynomial activation functions, which is unlike the classical version of the theorem, and provides a qualitative difference between deep narrow networks and shallow wide networks. We then consider several extensions of this result. In particular we consider nowhere differentiable activation functions, density in noncompact domains with respect to the $L^p$-norm, and how the width may be reduced to just $n+m+1$ for ‘most’ activation functions.

Zhixian Lei, Kyle Luh, Prayaag Venkat, Fred Zhang

We study the algorithmic problem of estimating the mean of a heavy-tailed random vector in R^d, given n i.i.d. samples. The goal is to design an efficient estimator that attains the optimal sub-gaussian error bound, only assuming that the random vector has bounded mean and covariance. Polynomial-time solutions to this problem are known but have high runtime due to their use of semi-definite programming (SDP). Moreover, conceptually, it remains open whether convex relaxation is truly necessary for this problem. In this work, we show that it is possible to go beyond SDP and achieve better computational efficiency. In particular, we provide a spectral algorithm that achieves the optimal statistical performance and runs in time O ( n^2 d ), improving upon the previous fastest runtime O( n^{3.5}+ n^2 d ) by Cherapanamjeri et.al. (COLT ’19). Our algorithm is spectral in that it only requires (approximate) eigenvector computations, which can be implemented very efficiently by, for example, power iteration or the Lanczos method. At the core of our algorithm is a novel connection between the furthest hyperplane problem introduced by Karnin et. al. (COLT ’12) and a structural lemma on heavy-tailed distributions by Lugosi and Mendelson (Ann. Stat. ’19). This allows us to iteratively reduce the estimation error at a geometric rate using only the information derived from the top singular vector of the data matrix, leading to a significantly faster running time.

Jeongyeol Kwon, Constantine Caramanis

We consider the problem of spherical Gaussian Mixture models with $k\ge 3$ components when the components are well separated. A fundamental previous result established that separation of $\Omega (\sqrt{\log k})$ is necessary and sufficient for identifiability of the parameters with \textit{polynomial} sample complexity (Regev and Vijayaraghavan, 2017). In the same context, we show that $\tilde{O}(kd/{ϵ}^2)$ samples suffice for any $ϵ\lesssim 1/k$, closing the gap from polynomial to linear, and thus giving the first optimal sample upper bound for the parameter estimation of well-separated Gaussian mixtures. We accomplish this by proving a new result for the Expectation-Maximization (EM) algorithm: we show that EM converges locally, under separation $\Omega (\sqrt{\log k})$. The previous best-known guarantee required $\Omega (\sqrt{k})$ separation (Yan, et al., 2017). Unlike prior work, our results do not assume or use prior knowledge of the (potentially different) mixing weights or variances of the Gaussian components. Furthermore, our results show that the finite-sample error of EM does not depend on non-universal quantities such as pairwise distances between means of Gaussian components.

Chung-Wei Lee, Haipeng Luo, Mengxiao Zhang

We study {\it small-loss} bounds for adversarial multi-armed bandits with graph feedback, that is, adaptive regret bounds that depend on the loss of the best arm or related quantities, instead of the total number of rounds. We derive the first small-loss bound for general strongly observable graphs, resolving an open problem of Lykouris et al. (2018). Specifically, we develop an algorithm with regret $\tilde{\mathcal{O}}(\sqrt{\kappa L_{\ast }})$ where $\kappa$ is the clique partition number and $L_{\ast }$ is the loss of the best arm, and for the special case of self-aware graphs where every arm has a self-loop, we improve the regret to $\tilde{\mathcal{O}}(\min \{\sqrt{\alpha T},\sqrt{\kappa L_{\ast }}\})$ where $\alpha \le \kappa$ is the independence number. Our results significantly improve and extend those by Lykouris et al. (2018) who only consider self-aware undirected graphs. Furthermore, we also take the first attempt at deriving small-loss bounds for weakly observable graphs. We first prove that no typical small-loss bounds are achievable in this case, and then propose algorithms with alternative small-loss bounds in terms of the loss of some specific subset of arms. A surprising side result is that $\tilde{\mathcal{O}}(\sqrt{T})$ regret is achievable even for weakly observable graphs as long as the best arm has a self-loop. Our algorithms are based on the Online Mirror Descent framework but require a suite of novel techniques that might be of independent interest. Moreover, all our algorithms can be made parameter-free without the knowledge of the environment.

Nadav Merlis, Shie Mannor

The Combinatorial Multi-Armed Bandit problem is a sequential decision-making problem in which an agent selects a set of arms on each round, observes feedback for each of these arms and aims to maximize a known reward function of the arms it chose. While previous work proved regret upper bounds in this setting for general reward functions, only a few works provided matching lower bounds, all for specific reward functions. In this work, we prove regret lower bounds for combinatorial bandits that hold under mild assumptions for all smooth reward functions. We derive both problem-dependent and problem-independent bounds and show that the recently proposed Gini-weighted smoothness parameter (Merlis and Mannor, 2019) also determines the lower bounds for monotone reward functions. Notably, this implies that our lower bounds are tight up to log-factors.

Mansur Arief, Zhiyuan Huang, Guru Koushik Senthil Kumar, Yuanlu Bai, Shengyi He, Wenhao Ding, Henry Lam, Ding Zhao

Evaluating the reliability of intelligent physical systems against rare safety-critical events poses a huge testing burden for real-world applications. Simulation provides a useful platform to evaluate the extremal risks of these systems before their deployments. Importance Sampling (IS), while proven to be powerful for rare-event simulation, faces challenges in handling these learning-based systems due to their black-box nature that fundamentally undermines its efficiency guarantee, which can lead to under-estimation without diagnostically detected. We propose a framework called Deep Probabilistic Accelerated Evaluation (Deep-PrAE) to design statistically guaranteed IS, by converting black-box samplers that are versatile but could lack guarantees, into one with what we call a relaxed efficiency certificate that allows accurate estimation of bounds on the safety-critical event probability. We present the theory of Deep-PrAE that combines the dominating point concept with rare-event set learning via deep neural network classifiers, and demonstrate its effectiveness in numerical examples including the safety-testing of an intelligent driving algorithm.

Tianyi Chen, Ziye Guo, Yuejiao Sun, Wotao Yin

Stochastic gradient descent (SGD) has taken the stage as the primary workhorse for largescale machine learning. It is often used with its adaptive variants such as AdaGrad, Adam, and AMSGrad. This paper proposes an adaptive stochastic gradient descent method for distributed machine learning, which can be viewed as the communicationadaptive counterpart of the celebrated Adam method — justifying its name CADA. The key components of CADA are a set of new rules tailored for adaptive stochastic gradients that can be implemented to save communication upload. The new algorithms adaptively reuse the stale Adam gradients, thus saving communication, and still have convergence rates comparable to original Adam. In numerical experiments, CADA achieves impressive empirical performance in terms of total communication round reduction.

Ian En-Hsu Yen, Satyen Kale, Felix Yu, Daniel Holtmann-Rice, Sanjiv Kumar, Pradeep Ravikumar

For problems with large output spaces, evaluation of the loss function and its gradient are expensive, typically taking linear time in the size of the output space. Recently, methods have been developed to speed up learning via efficient data structures for Nearest-Neighbor Search (NNS) or Maximum Inner-Product Search (MIPS). However, the performance of such data structures typically degrades in high dimensions. In this work, we propose a novel technique to reduce the intractable high dimensional search problem to several much more tractable lower dimensional ones via dual decomposition of the loss function. At the same time, we demonstrate guaranteed convergence to the original loss via a greedy message passing procedure. In our experiments on multiclass and multilabel classification with hundreds of thousands of classes, as well as training skip-gram word embeddings with a vocabulary size of half a million, our technique consistently improves the accuracy of search-based gradient approximation methods and outperforms sampling-based gradient approximation methods by a large margin.

Xiaojie Wang, Rui Zhang, Yu Sun, Jianzhong Qi

In recommender systems, usually the ratings of a user to most items are missing and a critical problem is that the missing ratings are often missing not at random (MNAR) in reality. It is widely acknowledged that MNAR ratings make it difficult to accurately predict the ratings and unbiasedly estimate the performance of rating prediction. Recent approaches use imputed errors to recover the prediction errors for missing ratings, or weight observed ratings with the propensities of being observed. These approaches can still be severely biased in performance estimation or suffer from the variance of the propensities. To overcome these limitations, we first propose an estimator that integrates the imputed errors and propensities in a doubly robust way to obtain unbiased performance estimation and alleviate the effect of the propensity variance. To achieve good performance guarantees, based on this estimator, we propose joint learning of rating prediction and error imputation, which outperforms the state-of-the-art approaches on four real-world datasets.

Hugo Lebeau, Romain Couillet, Florent Chatelain

This article introduces a random matrix framework for the analysis of clustering on high-dimensional data streams, a particularly relevant setting for a more sober processing of large amounts of data with limited memory and energy resources. Assuming data ${\mathbf{x}}_1,{\mathbf{x}}_2,\dots$ arrives as a continuous flow and a small number $L$ of them can be kept in the learning pipeline, one has only access to the diagonal elements of the Gram kernel matrix: ${\left[{\mathbf{K}}_L\right]}_{i,j}=\frac{1}{p}{\mathbf{x}}_i^{\top }{\mathbf{x}}_j{1}_{|i-j|<L}$. Under a large-dimensional data regime, we derive the limiting spectral distribution of the banded kernel matrix ${\mathbf{K}}_L$ and study its isolated eigenvalues and eigenvectors, which behave in an unfamiliar way. We detail how these results can be used to perform efficient online kernel spectral clustering and provide theoretical performance guarantees. Our findings are empirically confirmed on image clustering tasks. Leveraging on optimality results of spectral methods for clustering, this work offers insights on efficient online clustering techniques for high-dimensional data.

Li Shen, Peng Sun, Yitong Wang, Wei Liu, Tong Zhang

We propose a novel algorithmic framework of Variable Metric Over-Relaxed Hybrid Proximal Extra-gradient (VMOR-HPE) method with a global convergence guarantee for the maximal monotone operator inclusion problem. Its iteration complexities and local linear convergence rate are provided, which theoretically demonstrate that a large over-relaxed step-size contributes to accelerating the proposed VMOR-HPE as a byproduct. Specifically, we find that a large class of primal and primal-dual operator splitting algorithms are all special cases of VMOR-HPE. Hence, the proposed framework offers a new insight into these operator splitting algorithms. In addition, we apply VMOR-HPE to the Karush-Kuhn-Tucker (KKT) generalized equation of linear equality constrained multi-block composite convex optimization, yielding a new algorithm, namely nonsymmetric Proximal Alternating Direction Method of Multipliers with a preconditioned Extra-gradient step in which the preconditioned metric is generated by a blockwise Barzilai-Borwein line search technique (PADMM-EBB). We also establish iteration complexities of PADMM-EBB in terms of the KKT residual. Finally, we apply PADMM-EBB to handle the nonnegative dual graph regularized low-rank representation problem. Promising results on synthetic and real datasets corroborate the efficacy of PADMM-EBB.

Yifang Chen, Chung-Wei Lee, Haipeng Luo, Chen-Yu Wei

We propose the first contextual bandit algorithm that is parameter-free, efficient, and optimal in terms of dynamic regret. Specifically, our algorithm achieves $\mathcal{O}(\min \{\sqrt{KST},K^{\frac{1}{3}}{\Delta }^{\frac{1}{3}}{T}^{\frac{2}{3}}\})$ dynamic regret for a contextual bandit problem with $T$ rounds, $K$ actions, $S$ switches and $\Delta$ total variation in data distributions. Importantly, our algorithm is adaptive and does not need to know $S$ or $\Delta$ ahead of time, and can be implemented efficiently assuming access to an ERM oracle. Our results strictly improve the $\mathcal{O}(\min \{S^{\frac{1}{4}}{T}^{\frac{3}{4}},{\Delta }^{\frac{1}{5}}{T}^{\frac{4}{5}}\})$ bound of (Luo et al., 2018), and greatly generalize and improve the $\mathcal{O}(\sqrt{ST})$ result of (Auer et al., 2018) that holds only for the two-armed bandit problem without contextual information. The key novelty of our algorithm is to introduce {\it replay phases}, in which the algorithm acts according to its previous decisions for a certain amount of time in order to detect non-stationarity while maintaining a good balance between exploration and exploitation.

Swaminathan Gurumurthy, Sumit Kumar, Katia Sycara

Meta-Reinforcement learning approaches aim to develop learning procedures that can adapt quickly to a distribution of tasks with the help of a few examples. Developing efficient exploration strategies capable of finding the most useful samples becomes critical in such settings. Existing approaches towards finding efficient exploration strategies add auxiliary objectives to promote exploration by the pre-update policy, however, this makes the adaptation using a few gradient steps difficult as the pre-update (exploration) and post-update (exploitation) policies are often quite different. Instead, we propose to explicitly model a separate exploration policy for the task distribution. Having two different policies gives more flexibility in training the exploration policy and also makes adaptation to any specific task easier. We show that using self-supervised or supervised learning objectives for adaptation allows for more efficient inner-loop updates and also demonstrate the superior performance of our model compared to prior works in this domain.

Mengye Ren, Wenyuan Zeng, Bin Yang, Raquel Urtasun

Deep neural networks have been shown to be very powerful modeling tools for many supervised learning tasks involving complex input patterns. However, they can also easily overfit to training set biases and label noises. In addition to various regularizers, example reweighting algorithms are popular solutions to these problems, but they require careful tuning of additional hyperparameters, such as example mining schedules and regularization hyperparameters. In contrast to past reweighting methods, which typically consist of functions of the cost value of each example, in this work we propose a novel meta-learning algorithm that learns to assign weights to training examples based on their gradient directions. To determine the example weights, our method performs a meta gradient descent step on the current mini-batch example weights (which are initialized from zero) to minimize the loss on a clean unbiased validation set. Our proposed method can be easily implemented on any type of deep network, does not require any additional hyperparameter tuning, and achieves impressive performance on class imbalance and corrupted label problems where only a small amount of clean validation data is available.

Edith Cohen, Ofir Geri, Tamas Sarlos, Uri Stemmer

Common datasets have the form of elements with keys (e.g., transactions and products) and the goal is to perform analytics on the aggregated form of key and frequency pairs. A weighted sample of keys by (a function of) frequency is a highly versatile summary that provides a sparse set of representative keys and supports approximate evaluations of query statistics. We propose private weighted sampling (PWS): A method that sanitizes a weighted sample as to ensure element-level differential privacy, while retaining its utility to the maximum extent possible. PWS maximizes the reporting probabilities of keys and estimation quality of a broad family of statistics. PWS improves over the state of the art even for the well-studied special case of private histograms, when no sampling is performed. We empirically observe significant performance gains of 20%-300% increase in key reporting for common Zipfian frequency distributions and accurate estimation with x2-8 lower frequencies. PWS is applied as a post-processing of a non-private sample, without requiring the original data. Therefore, it can be a seamless addition to existing implementations, such as those optimizes for distributed or streamed data. We believe that due to practicality and performance, PWS may become a method of choice in applications where privacy is desired.

Yixing Zhang, Xiuyuan Cheng, Galen Reeves

The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d+2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d+2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.

Shengjie Wang, Tianyi Zhou, Jeff Bilmes

The gradient of a deep neural network (DNN) w.r.t. the input provides information that can be used to explain the output prediction in terms of the input features and has been widely studied to assist in interpreting DNNs. In a linear model (i.e., g(x) = wx + b), the gradient corresponds to the weights w. Such a model can reasonably locally-linearly approximate a smooth nonlinear DNN, and hence the weights of this local model are the gradient. The bias b, however, is usually overlooked in attribution methods. In this paper, we observe that since the bias in a DNN also has a non-negligible contribution to the correctness of predictions, it can also play a significant role in understanding DNN behavior. We propose a backpropagation-type algorithm “bias back-propagation (BBp)” that starts at the output layer and iteratively attributes the bias of each layer to its input nodes as well as combining the resulting bias term of the previous layer. Together with the backpropagation of the gradient generating w, we can fully recover the locally linear model g(x) = wx + b. In experiments, we show that BBp can generate complementary and highly interpretable explanations.

Ahmed Touati, Pierre-Luc Bacon, Doina Precup, Pascal Vincent

Off-policy learning is key to scaling up reinforcement learning as it allows to learn about a target policy from the experience generated by a different behavior policy. Unfortunately, it has been challenging to combine off-policy learning with function approximation and multi-step bootstrapping in a way that leads to both stable and efficient algorithms. In this work, we show that the Tree Backup and Retrace algorithms are unstable with linear function approximation, both in theory and in practice with specific examples. Based on our analysis, we then derive stable and efficient gradient-based algorithms using a quadratic convex-concave saddle-point formulation. By exploiting the problem structure proper to these algorithms, we are able to provide convergence guarantees and finite-sample bounds. The applicability of our new analysis also goes beyond Tree Backup and Retrace and allows us to provide new convergence rates for the GTD and GTD2 algorithms without having recourse to projections or Polyak averaging.

Mark Braverman, Jieming Mao, Jon Schneider, S. Matthew Weinberg

We study a strategic version of the multi-armed bandit problem, where each arm is an individual strategic agent and we, the principal, pull one arm each round. When pulled, the arm receives some private reward $v_a$ and can choose an amount $x_a$ to pass on to the principal (keeping $v_a-x_a$ for itself). All non-pulled arms get reward $0$. Each strategic arm tries to maximize its own utility over the course of $T$ rounds. Our goal is to design an algorithm for the principal incentivizing these arms to pass on as much of their private rewards as possible. When private rewards are stochastically drawn each round ($v_a^t\leftarrow D_a$), we show that: \begin{itemize} \item Algorithms that perform well in the classic adversarial multi-armed bandit setting necessarily perform poorly: For all algorithms that guarantee low regret in an adversarial setting, there exist distributions $D_1,\dots ,D_k$ and an $o(T)$-approximate Nash equilibrium for the arms where the principal receives reward $o(T)$. \item There exists an algorithm for the principal that induces a game among the arms where each arm has a dominant strategy. Moreover, for every $o(T)$-approximate Nash equilibrium, the principal receives expected reward $\mu ’T-o(T)$, where $\mu ’$ is the second-largest of the means $\mathbb{E}[D_a]$. This algorithm maintains its guarantee if the arms are non-strategic ($x_a=v_a$), and also if there is a mix of strategic and non-strategic arms. \end{itemize}

Masashi Okada, Tadahiro Taniguchi

In recent studies on model-based reinforcement learning (MBRL), incorporating uncertainty in forward dynamics is a state-of-the-art strategy to enhance learning performance, making MBRLs competitive to cutting-edge modelfree methods, especially in simulated robotics tasks. Probabilistic ensembles with trajectory sampling (PETS) is a leading type of MBRL, which employs Bayesian inference to dynamics modeling and model predictive control (MPC) with stochastic optimization via the cross entropy method (CEM). In this paper, we propose a novel extension to the uncertainty-aware MBRL. Our main contributions are twofold: Firstly, we introduce a variational inference MPC (VI-MPC), which reformulates various stochastic methods, including CEM, in a Bayesian fashion. Secondly, we propose a novel instance of the framework, called probabilistic action ensembles with trajectory sampling (PaETS). As a result, our Bayesian MBRL can involve multimodal uncertainties both in dynamics and optimal trajectories. In comparison to PETS, our method consistently improves asymptotic performance on several challenging locomotion tasks.

Abhishek Gupta, Vikash Kumar, Corey Lynch, Sergey Levine, Karol Hausman

We present relay policy learning, a method for imitation and reinforcement learning that can solve multi-stage, long-horizon robotic tasks. This general and universally-applicable, two-phase approach consists of an imitation learning stage resulting in goal-conditioned hierarchical policies that can be easily improved using fine-tuning via reinforcement learning in the subsequent phase. Our method, while not necessarily perfect at imitation learning, is very amenable to further improvement via environment interaction allowing it to scale to challenging long-horizon tasks. In particular, we simplify the long-horizon policy learning problem by using a novel data-relabeling algorithm for learning goal-conditioned hierarchical policies, where the low-level only acts for a fixed number of steps, regardless of the goal achieved. While we rely on demonstration data to bootstrap policy learning, we do not assume access to demonstrations of specific tasks. Instead, our approach can leverage unstructured and unsegmented demonstrations of semantically meaningful behaviors that are not only less burdensome to provide, but also can greatly facilitate further improvement using reinforcement learning. We demonstrate the effectiveness of our method on a number of multi-stage, long-horizon manipulation tasks in a challenging kitchen simulation environment.

Marco Schmalhofer

We show a simple perceptron-like algorithm to learn origin-centered halfspaces in ${\mathbb{R}}^n$ with accuracy $1-ϵ$ and confidence $1-\delta$ in time $\mathcal{O}\left(\frac{n^2}{ϵ}\left(\log \frac{1}{ϵ}+\log \frac{1}{\delta }\right)\right)$ using $\mathcal{O}\left(\frac{n}{ϵ}\left(\log \frac{1}{ϵ}+\log \frac{1}{\delta }\right)\right)$ labeled examples drawn uniformly from the unit $n$-sphere. This improves upon algorithms given in Baum(1990), Long(1994) and Servedio(1999). The time and sample complexity of our algorithm match the lower bounds given in Long(1995) up to logarithmic factors.

Gil I Shamir

We address the problem of the achievable regret rates with online logistic regression. We derive lower bounds with logarithmic regret under $L_1$, $L_2$, and $L_{\infty }$ constraints on the parameter values. The bounds are dominated by $d/2\log T$, where $T$ is the horizon and $d$ is the dimensionality of the parameter space. We show their achievability for $d=o(T^{1/3})$ in all these cases with Bayesian methods, that achieve them up to a $d/2\log d$ term. Interesting different behaviors are shown for larger dimensionality. Specifically, on the negative side, if $d=\Omega (\sqrt{T})$, any algorithm is guaranteed regret of $\Omega (d\log T)$ (greater than $\Theta (\sqrt{T})$) under $L_{\infty }$ constraints on the parameters (and the example features). On the positive side, under $L_1$ constraints on the parameters, there exist Bayesian algorithms that can achieve regret that is sub-linear in $d$ for the asymptotically larger values of $d$. For $L_2$ constraints, it is shown that for large enough $d$, the regret remains linear in $d$ but no longer logarithmic in $T$. Adapting the \emph{redundancy-capacity\/} theorem from information theory, we demonstrate a principled methodology based on grids of parameters to derive lower bounds. Grids are also utilized to derive some upper bounds. Our results strengthen results by Kakade and Ng (2005) and Foster et al. (2018) for upper bounds for this problem, introduce novel lower bounds, and adapt a methodology that can be used to obtain such bounds for other related problems. They also give a novel characterization of the asymptotic behavior when the dimension of the parameter space is allowed to grow with $T$. They additionally strengthen connections to the information theory literature, demonstrating that the actual regret for logistic regression depends on the richness of the parameter class, where even within this problem, richer classes lead to greater regret.

Chen-Yu Wei, Haipeng Luo, Alekh Agarwal

We initiate the study of learning in contextual bandits with the help of loss predictors. The main question we address is whether one can improve over the minimax regret $\mathcal{O}(\sqrt{T})$ for learning over $T$ rounds, when the total error of the predicted losses relative to the realized losses, denoted as $\mathcal{E}\le T$, is relatively small. We provide a complete answer to this question, with upper and lower bounds for various settings: adversarial and stochastic environments, known and unknown $\mathcal{E}$, and single and multiple predictors. We show several surprising results, such as 1) the optimal regret is $\mathcal{O}(\min \{\sqrt{T},\sqrt{\mathcal{E}}{T}^{\frac{1}{4}}\})$ when $\mathcal{E}$ is known, in contrast to the standard and better bound $\mathcal{O}(\sqrt{\mathcal{E}})$ for non-contextual problems (such as multi-armed bandits); 2) the same bound cannot be achieved if $\mathcal{E}$ is unknown, but as a remedy, $\mathcal{O}(\sqrt{\mathcal{E}}{T}^{\frac{1}{3}})$ is achievable; 3) with $M$ predictors, a linear dependence on $M$ is necessary, even though logarithmic dependence is possible for non-contextual problems. We also develop several novel algorithmic techniques to achieve matching upper bounds, including 1) a key \emph{action remapping} technique for optimal regret with known $\mathcal{E}$, 2) computationally efficient implementation of Catoni’s robust mean estimator via an ERM oracle in the stochastic setting with optimal regret, 3) an underestimator for $\mathcal{E}$ via estimating the histogram with bins of exponentially increasing size for the stochastic setting with unknown $\mathcal{E}$, and 4) a self-referential scheme for learning with multiple predictors, all of which might be of independent interest.

Qiaomin Xie, Yudong Chen, Zhaoran Wang, Zhuoran Yang

In this work, we develop provably efficient reinforcement learning algorithms for two-player zero-sum Markov games with simultaneous moves. We consider a family of Markov games where the reward function and transition kernel possess a linear structure. Two settings are studied: In the offline setting, we control both players and the goal is to find the Nash Equilibrium efficiently by minimizing the worst-case duality gap. In the online setting, we control a single player and play against an arbitrary opponent; the goal is to minimize the regret. For both settings, we propose an optimistic variant of the least-squares minimax value iteration algorithm. We show that our algorithm is computationally efficient and provably achieves an $\tilde{O}(\sqrt{d^3{H}^{3}T})$ upper bound on the duality gap and regret, without requiring additional assumptions on the sampling model. We highlight that our setting requires overcoming several new challenges that are absent in MDPs or turn-based Markov games. In particular, to achieve optimism under the simultaneous-move games, we construct both upper and lower confidence bounds of the value function, and then derive the optimistic policy by solving a general-sum matrix game with these bounds as the payoff matrices. As finding the Nash Equilibrium of this general-sum game is computationally hard, our algorithm instead solves for a Coarse Correlated Equilibrium (CCE), which can be obtained efficiently via linear programming. To our best knowledge, such a CCE-based mechanism for implementing optimism has not appeared in the literature and might be of interest in its own right.

Xiao-Tong Yuan, Ping Li

Given a vector $w\in {\mathbb{R}}^p$ and a positive semi-definite matrix $A\in {\mathbb{R}}^{p\times p}$, we study the expansion ratio bound for the following defined Mahalanobis hard thresholding operator of $w$: \[ \mathcal{H}_{A,k}(w):=\argmin_{\|\theta\|_0\le k} \frac{1}{2}\|\theta - w\|^2_A, \]{where} $k\le p$ is the desired sparsity level. The core contribution of this paper is to prove that for any $\bar{k}$-sparse vector $\bar{w}$ with $\bar{k}<k$, the estimation error $\Vert {\mathcal{H}}_{A,k}(w)-\bar{w}{\Vert }_A$ satisfies \[ \|\mathcal{H}_{A,k}(w) - \bar w\|^2_A \le \left(1+ \mathcal{O}\left(\kappa(A,2k) \sqrt{\frac{\bar k }{k - \bar k}}\right)\right) \|{w} - \bar w\|^2_A, \]{where} $\kappa (A,2k)$ is the restricted strong condition number of $A$ over $(2k)$-sparse subspace. This estimation error bound is nearly non-expansive when $k$ is sufficiently larger than $\bar{k}$. Specially when $A$ is the identity matrix such that $\kappa (A,2k)\equiv 1$, our bound recovers the previously known nearly non-expansive bounds for Euclidean hard thresholding operator. We further show that such a bound extends to an approximate version of ${\mathcal{H}}_{A,k}(w)$ estimated by Hard Thresholding Pursuit (HTP) algorithm. We demonstrate the applicability of these bounds to the mean squared error analysis of HTP and its novel extension based on preconditioning method. Numerical evidence is provided to support our theory and demonstrate the superiority of the proposed preconditioning HTP algorithm.

Thomas Kerdreux, Alexandre d’Aspremont, Sebastian Pokutta

The Frank-Wolfe method solves smooth constrained convex optimization problems at a generic sublinear rate of $\mathcal{O}(1/T)$, and it (or its variants) enjoys accelerated convergence rates for two fundamental classes of constraints: polytopes and strongly-convex sets. Uniformly convex sets non-trivially subsume strongly convex sets and form a large variety of \textit{curved} convex sets commonly encountered in machine learning and signal processing. For instance, the ${\ell }_p$-balls are uniformly convex for all $p>1$, but strongly convex for $p\in ]1,2]$ only. We show that these sets systematically induce accelerated convergence rates for the original Frank-Wolfe algorithm, which continuously interpolate between known rates. Our accelerated convergence rates emphasize that it is the curvature of the constraint sets – not just their strong convexity – that leads to accelerated convergence rates. These results also importantly highlight that the Frank-Wolfe algorithm is adaptive to much more generic constraint set structures, thus explaining faster empirical convergence. Finally, we also show accelerated convergence rates when the set is only locally uniformly convex around the optima and provide similar results in online linear optimization.

Shinsaku Sakaue

Motivated by, e.g., sensitivity analysis and end-to-end learning, the demand for differentiable optimization algorithms has been increasing. This paper presents a theoretically guaranteed differentiable greedy algorithm for monotone submodular function maximization. We smooth the greedy algorithm via randomization, and prove that it almost recovers original approximation guarantees in expectation for the cases of cardinality and $\kappa$-extendible system constraints. We then present how to efficiently compute gradient estimators of any expected output-dependent quantities. We demonstrate the usefulness of our method by instantiating it for various applications.

Valerii Likhosherstov, Jared Davis, Krzysztof Choromanski, Adrian Weller

We introduce an efficient approach for optimization over orthogonal groups on highly parallel computation units such as GPUs or TPUs. As in earlier work, we parametrize an orthogonal matrix as a product of Householder reflections. However, to overcome low parallelization capabilities of computing Householder reflections sequentially, we propose employing an accumulation scheme called the compact WY (or CWY) transform – a compact parallelization-friendly matrix representation for the series of Householder reflections. We further develop a novel Truncated CWY (or T-CWY) approach for Stiefel manifold parametrization which has a competitive complexity and, again, yields benefits when computed on GPUs and TPUs. We prove that our CWY and T-CWY methods lead to convergence to a stationary point of the training objective when coupled with stochastic gradient descent. We apply our methods to train recurrent neural network architectures in the tasks of neural machine translation and video prediction.

Hisham Husain, Kamil Ciosek, Ryota Tomioka

Entropic regularization of policies in Reinforcement Learning (RL) is a commonly used heuristic to ensure that the learned policy explores the state-space sufficiently before overfitting to a local optimal policy. The primary motivation for using entropy is for exploration and disambiguating optimal policies; however, the theoretical effects are not entirely understood. In this work, we study the more general regularized RL objective and using Fenchel duality; we derive the dual problem which takes the form of an adversarial reward problem. In particular, we find that the optimal policy found by a regularized objective is precisely an optimal policy of a reinforcement learning problem under a worst-case adversarial reward. Our result allows us to reinterpret the popular entropic regularization scheme as a form of robustification. Furthermore, due to the generality of our results, we apply to other existing regularization schemes. Our results thus give insights into the effects of regularization of policies and deepen our understanding of exploration through robust rewards at large.

Sattar Vakili, Kia Khezeli, Victor Picheny

Consider the sequential optimization of an expensive to evaluate and possibly non-convex objective function $f$ from noisy feedback, that can be considered as a continuum-armed bandit problem. Upper bounds on the regret performance of several learning algorithms (GP-UCB, GP-TS, and their variants) are known under both a Bayesian (when $f$ is a sample from a Gaussian process (GP)) and a frequentist (when $f$ lives in a reproducing kernel Hilbert space) setting. The regret bounds often rely on the maximal information gain ${\gamma }_T$ between $T$ observations and the underlying GP (surrogate) model. We provide general bounds on ${\gamma }_T$ based on the decay rate of the eigenvalues of the GP kernel, whose specialisation for commonly used kernels improves the existing bounds on ${\gamma }_T$, and subsequently the regret bounds relying on ${\gamma }_T$ under numerous settings. For the Matérn family of kernels, where the lower bounds on ${\gamma }_T$, and regret under the frequentist setting, are known, our results close a huge polynomial in $T$ gap between the upper and lower bounds (up to logarithmic in $T$ factors).

Daniel Hsu, Vidya Muthukumar, Ji Xu

The support vector machine (SVM) is a well-established classification method whose name refers to the particular training examples, called support vectors, that determine the maximum margin separating hyperplane. The SVM classifier is known to enjoy good generalization properties when the number of support vectors is small compared to the number of training examples. However, recent research has shown that in sufficiently high-dimensional linear classification problems, the SVM can generalize well despite a proliferation of support vectors where all training examples are support vectors. In this paper, we identify new deterministic equivalences for this phenomenon of support vector proliferation, and use them to (1) substantially broaden the conditions under which the phenomenon occurs in high-dimensional settings, and (2) prove a nearly matching converse result.

Ziping Xu, Amirhossein Meisami, Ambuj Tewari

This paper studies the decision making problem with Funnel Structure. Funnel structure, a well-known concept in the marketing field, occurs in those systems where the decision maker interacts with the environment in a layered manner receiving far fewer observations from deep layers than shallow ones. For example, in the email marketing campaign application, the layers correspond to Open, Click and Purchase events. Conversions from Click to Purchase happen very infrequently because a purchase cannot be made unless the link in an email is clicked on. We formulate this challenging decision making problem as a contextual bandit with funnel structure and develop a multi-task learning algorithm that mitigates the lack of sufficient observations from deeper layers. We analyze both the prediction error and the regret of our algorithms. We verify our theory on prediction errors through a simple simulation. Experiments on both a simulated environment and an environment based on real-world data from a major email marketing company show that our algorithms offer significant improvement over previous methods.

Feynman Liang, Nimar Arora, Nazanin Tehrani, Yucen Li, Michael Tingley, Erik Meijer

In order to construct accurate proposers for Metropolis-Hastings Markov Chain Monte Carlo, we integrate ideas from probabilistic graphical models and neural networks in an open-source framework we call Lightweight Inference Compilation (LIC). LIC implements amortized inference within an open-universe declarative probabilistic programming language (PPL). Graph neural networks are used to parameterize proposal distributions as functions of Markov blankets, which during “compilation” are optimized to approximate single-site Gibbs sampling distributions. Unlike prior work in inference compilation (IC), LIC forgoes importance sampling of linear execution traces in favor of operating directly on Bayesian networks. Through using a declarative PPL, the Markov blankets of nodes (which may be non-static) are queried at inference-time to produce proposers Experimental results show LIC can produce proposers which have less parameters, greater robustness to nuisance random variables, and improved posterior sampling in a Bayesian logistic regression and n-schools inference application.

Dimitri Bouche, Marianne Clausel, François Roueff, Florence d’Alché-Buc

To address functional-output regression, we introduce projection learning (PL), a novel dictionary-based approach that learns to predict a function that is expanded on a dictionary while minimizing an empirical risk based on a functional loss. PL makes it possible to use non orthogonal dictionaries and can then be combined with dictionary learning; it is thus much more flexible than expansion-based approaches relying on vectorial losses. This general method is instantiated with reproducing kernel Hilbert spaces of vector-valued functions as kernel-based projection learning (KPL). For the functional square loss, two closed-form estimators are proposed, one for fully observed output functions and the other for partially observed ones. Both are backed theoretically by an excess risk analysis. Then, in the more general setting of integral losses based on differentiable ground losses, KPL is implemented using first-order optimization for both fully and partially observed output functions. Eventually, several robustness aspects of the proposed algorithms are highlighted on a toy dataset; and a study on two real datasets shows that they are competitive compared to other nonlinear approaches. Notably, using the square loss and a learnt dictionary, KPL enjoys a particularily attractive trade-off between computational cost and performances.

Tianjian Huang, Prajwal Singhania, Maziar Sanjabi, Pabitra Mitra, Meisam Razaviyayn

Quantization of the parameters of machine learning models, such as deep neural networks, requires solving constrained optimization problems, where the constraint set is formed by the Cartesian product of many simple discrete sets. For such optimization problems, we study the performance of the Alternating Direction Method of Multipliers for Quantization (ADMM-Q) algorithm, which is a variant of the widely-used ADMM method applied to our discrete optimization problem. We establish the convergence of the iterates of ADMM-Q to certain stationary points. To the best of our knowledge, this is the first analysis of an ADMM-type method for problems with discrete variables/constraints. Based on our theoretical insights, we develop a few variants of ADMM-Q that can handle inexact update rules, and have improved performance via the use of "soft projection" and "injecting randomness to the algorithm". We empirically evaluate the efficacy of our proposed approaches.

Emanuele Dolera, Stefano Favaro, Stefano Peluchetti

The count-min sketch (CMS) is a randomized data structure that provides estimates of tokens’ frequencies in a large data stream using a compressed representation of the data by random hashing. In this paper, we rely on a recent Bayesian nonparametric (BNP) view on the CMS to develop a novel learning-augmented CMS under power-law data streams. We assume that tokens in the stream are drawn from an unknown discrete distribution, which is endowed with a normalized inverse Gaussian process (NIGP) prior. Then, using distributional properties of the NIGP, we compute the posterior distribution of a token’s frequency in the stream, given the hashed data, and in turn corresponding BNP estimates. Applications to synthetic and real data show that our approach achieves a remarkable performance in the estimation of low-frequency tokens. This is known to be a desirable feature in the context of natural language processing, where it is indeed common in the context of the power-law behaviour of the data.

Tomohiro Hayase, Ryo Karakida

The Fisher information matrix (FIM) is fundamental to understanding the trainability of deep neural nets (DNN), since it describes the parameter space’s local metric. We investigate the spectral distribution of the conditional FIM, which is the FIM given a single sample, by focusing on fully-connected networks achieving dynamical isometry. Then, while dynamical isometry is known to keep specific backpropagated signals independent of the depth, we find that the parameter space’s local metric linearly depends on the depth even under the dynamical isometry. More precisely, we reveal that the conditional FIM’s spectrum concentrates around the maximum and the value grows linearly as the depth increases. To examine the spectrum, considering random initialization and the wide limit, we construct an algebraic methodology based on the free probability theory. As a byproduct, we provide an analysis of the solvable spectral distribution in two-hidden-layer cases. Lastly, experimental results verify that the appropriate learning rate for the online training of DNNs is in inverse proportional to depth, which is determined by the conditional FIM’s spectrum.

Ilkay Yildiz, Jennifer Dy, Deniz Erdogmus, Susan Ostmo, J. Peter Campbell, Michael F. Chiang, Stratis Ioannidis

Learning from ranking observations arises in many domains, and siamese deep neural networks have shown excellent inference performance in this setting. However, SGD does not scale well, as an epoch grows exponentially with the ranking observation size. We show that a spectral algorithm can be combined with deep learning methods to significantly accelerate training. We combine a spectral estimate of Plackett-Luce ranking scores with a deep model via the Alternating Directions Method of Multipliers with a Kullback-Leibler proximal penalty. Compared to a state-of-the-art siamese network, our algorithms are up to 175 times faster and attain better predictions by up to 26% Top-1 Accuracy and 6% Kendall-Tau correlation over five real-life ranking datasets.

Yingkai Li, Yining Wang, Xi Chen, Yuan Zhou

Linear contextual bandit is a class of sequential decision-making problems with important applications in recommendation systems, online advertising, healthcare, and other machine learning-related tasks. While there is much prior research, tight regret bounds of linear contextual bandit with infinite action sets remain open. In this paper, we consider the linear contextual bandit problem with (changing) infinite action sets. We prove a regret upper bound on the order of O(\sqrt{d^2T\log T}) \poly(\log\log T) where d is the domain dimension and T is the time horizon. Our upper bound matches the previous lower bound of \Omega(\sqrt{d^2 T\log T}) in [Li et al., 2019] up to iterated logarithmic terms.

Fanghui Liu, Xiaolin Huang, Yingyi Chen, Johan Suykens

In this paper, we attempt to solve a long-lasting open question for non-positive definite (non-PD) kernels in machine learning community: can a given non-PD kernel be decomposed into the difference of two PD kernels (termed as positive decomposition)? We cast this question as a distribution view by introducing the signed measure, which transforms positive decomposition to measure decomposition: a series of non-PD kernels can be associated with the linear combination of specific finite Borel measures. In this manner, our distribution-based framework provides a sufficient and necessary condition to answer this open question. Specifically, this solution is also computationally implementable in practice to scale non-PD kernels in large sample cases, which allows us to devise the first random features algorithm to obtain an unbiased estimator. Experimental results on several benchmark datasets verify the effectiveness of our algorithm over the existing methods.

Marco Mondelli, Ramji Venkataramanan

We consider the problem of estimating a signal from measurements obtained via a generalized linear model. We focus on estimators based on approximate message passing (AMP), a family of iterative algorithms with many appealing features: the performance of AMP in the high-dimensional limit can be succinctly characterized under suitable model assumptions; AMP can also be tailored to the empirical distribution of the signal entries, and for a wide class of estimation problems, AMP is conjectured to be optimal among all polynomial-time algorithms. However, a major issue of AMP is that in many models (such as phase retrieval), it requires an initialization correlated with the ground-truth signal and independent from the measurement matrix. Assuming that such an initialization is available is typically not realistic. In this paper, we solve this problem by proposing an AMP algorithm initialized with a spectral estimator. With such an initialization, the standard AMP analysis fails since the spectral estimator depends in a complicated way on the design matrix. Our main contribution is a rigorous characterization of the performance of AMP with spectral initialization in the high-dimensional limit. The key technical idea is to define and analyze a two-phase artificial AMP algorithm that first produces the spectral estimator, and then closely approximates the iterates of the true AMP. We also provide numerical results that demonstrate the validity of the proposed approach.

Wenkai Xu, Gesine Reinert

We propose and analyse a novel nonparametric goodness-of-fit testing procedure for ex-changeable exponential random graph model (ERGM) when a single network realisation is observed. The test determines how likely it is that the observation is generated from a target unnormalised ERGM density. Our test statistics are derived of kernel Stein discrepancy, a divergence constructed via Stein’s method using functions from a reproducing kernel Hilbert space (RKHS), combined with a discrete Stein operator for ERGMs. The test is a Monte Carlo test using simulated networks from the target ERGM. We show theoretical properties for the testing procedure w.r.t a class of ERGMs. Simulation studies and real network applications are presented.

Tianyi Zhou, Shengjie Wang, Jeff Bilmes

We study a novel curriculum learning scheme where in each round, samples are selected to achieve the greatest progress and fastest learning speed towards the ground-truth on all available samples. Inspired by an analysis of optimization dynamics under gradient flow for both regression and classification, the problem reduces to selecting training samples by a score computed from samples’ residual and linear temporal dynamics. It encourages the model to focus on the samples at learning frontier, i.e., those with large loss but fast learning speed. The scores in discrete time can be estimated via already-available byproducts of training, and thus require a negligible amount of extra computation. We discuss the properties and potential advantages of the proposed dynamics optimization via current deep learning theory and empirical study. By integrating it with cyclical training of neural networks, we introduce "dynamics-optimized curriculum learning (DoCL)", which selects the training set for each step by weighted sampling based on the scores. On nine different datasets, DoCL significantly outperforms random mini-batch SGD and recent curriculum learning methods both in terms of efficiency and final performance.

Daniel de Souza, Diego Mesquita, João Paulo Gomes, César Lincoln Mattos

Gaussian Process Latent Variable Model (GPLVM) is a flexible framework to handle uncertain inputs in Gaussian Processes (GPs) and incorporate GPs as components of larger graphical models. Nonetheless, the standard GPLVM variational inference approach is tractable only for a narrow family of kernel functions. The most popular implementations of GPLVM circumvent this limitation using quadrature methods, which may become a computational bottleneck even for relatively low dimensions. For instance, the widely employed Gauss-Hermite quadrature has exponential complexity on the number of dimensions. In this work, we propose using the unscented transformation instead. Overall, this method presents comparable, if not better, performance than off-the-shelf solutions to GPLVM, and its computational complexity scales only linearly on dimension. In contrast to Monte Carlo methods, our approach is deterministic and works well with quasi-Newton methods, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. We illustrate the applicability of our method with experiments on dimensionality reduction and multistep-ahead prediction with uncertainty propagation.

Guillaume Ausset, Stephan Clémencon, François Portier

Motivated by a wide variety of applications, ranging from stochastic optimization to dimension reduction through variable selection, the problem of estimating gradients accurately is of crucial importance in statistics and learning theory. We consider here the classic regression setup, where a real valued square integrable r.v. Y is to be predicted upon observing a (possibly high dimensional) random vector X by means of a predictive function f(X) as accurately as possible in the mean-squared sense and study a nearest-neighbour-based pointwise estimate of the gradient of the optimal predictive function, the regression function m(x)=E[Y | X=x]. Under classic smoothness conditions combined with the assumption that the tails of Y-m(X) are sub-Gaussian, we prove nonasymptotic bounds improving upon those obtained for alternative estimation methods. Beyond the novel theoretical results established, several illustrative numerical experiments have been carried out. The latter provide strong empirical evidence that the estimation method proposed works very well for various statistical problems involving gradient estimation, namely dimensionality reduction, stochastic gradient descent optimization and quantifying disentanglement.

Yangyi Lu, Amirhossein Meisami, Ambuj Tewari

In a low-rank linear bandit problem, the reward of an action (represented by a matrix of size $d_1\times d_2$) is the inner product between the action and an unknown low-rank matrix ${\Theta }^{\ast }$. We propose an algorithm based on a novel combination of online-to-confidence-set conversion \citep{abbasi2012online} and the exponentially weighted average forecaster constructed by a covering of low-rank matrices. In $T$ rounds, our algorithm achieves $\tilde{O}((d_1+d_2{)}^{3/2}\sqrt{rT})$ regret that improves upon the standard linear bandit regret bound of $\tilde{O}(d_1{d}_2\sqrt{T})$ when the rank of ${\Theta }^{\ast }$: $r\ll \min \{d_1,d_2\}$. We also extend our algorithmic approach to the generalized linear setting to get an algorithm which enjoys a similar bound under regularity conditions on the link function. To get around the computational intractability of covering based approaches, we propose an efficient algorithm by extending the "Explore-Subspace-Then-Refine" algorithm of \citet{jun2019bilinear}. Our efficient algorithm achieves $\tilde{O}((d_1+d_2{)}^{3/2}\sqrt{rT})$ regret under a mild condition on the action set $\mathcal{X}$ and the $r$-th singular value of ${\Theta }^{\ast }$. Our upper bounds match the conjectured lower bound of \cite{jun2019bilinear} for a subclass of low-rank linear bandit problems. Further, we show that existing lower bounds for the sparse linear bandit problem strongly suggest that our regret bounds are unimprovable. To complement our theoretical contributions, we also conduct experiments to demonstrate that our algorithm can greatly outperform the performance of the standard linear bandit approach when ${\Theta }^{\ast }$ is low-rank.

Yue Xing, Qifan Song, Guang Cheng

Modern machine learning and deep learning models are shown to be vulnerable when testing data are slightly perturbed. Theoretical studies of adversarial training algorithms mostly focus on their adversarial training losses or local convergence properties. In contrast, this paper studies the generalization performance of a generic adversarial training algorithm. Specifically, we consider linear regression models and two-layer neural networks (with lazy training) using squared loss under low-dimensional regime and high-dimensional regime. In the former regime, after overcoming the non-smoothness of adversarial training, the adversarial risk of the trained models will converge to the minimal adversarial risk. In the latter regime, we discover that data interpolation prevents the adversarial robust estimator from being consistent (i.e. converge in probability). Therefore, inspired by successes of the least absolute shrinkage and selection operator (LASSO), we incorporate the ${\mathcal{L}}_1$ penalty in the high dimensional adversarial learning, and show that it leads to consistent adversarial robust estimation. A series of numerical studies are conducted to demonstrate that how the smoothness and ${\mathcal{L}}_1$ penalization help to improve the adversarial robustness of DNN models.

Yue Xing, Ruizhi Zhang, Guang Cheng

Adversarial robust learning aims to design algorithms that are robust to small adversarial perturbations on input variables. Beyond the existing studies on the predictive performance to adversarial samples, our goal is to understand statistical properties of adversarial robust estimates and analyze adversarial risk in the setup of linear regression models. By discovering the statistical minimax rate of convergence of adversarial robust estimators, we emphasize the importance of incorporating model information, e.g., sparsity, in adversarial robust learning. Further, we reveal an explicit connection of adversarial and standard estimates, and propose a straightforward two-stage adversarial training framework, which facilitates to utilize model structure information to improve adversarial robustness. In theory, the consistency of the adversarial robust estimator is proven and its Bahadur representation is also developed for the statistical inference purpose. The proposed estimator converges in a sharp rate under either low-dimensional or sparse scenario. Moreover, our theory confirms two phenomena in adversarial robust learning: adversarial robustness hurts generalization, and unlabeled data help improve the generalization. In the end, we conduct numerical simulations to verify our theory.

Botao Hao, Nevena Lazic, Yasin Abbasi-Yadkori, Pooria Joulani, Csaba Szepesvari

Model-free reinforcement learning algorithms combined with value function approximation have recently achieved impressive performance in a variety of application domains. However, the theoretical understanding of such algorithms is limited, and existing results are largely focused on episodic or discounted Markov decision processes (MDPs). In this work, we present adaptive approximate policy iteration (AAPI), a learning scheme which enjoys a O(T^{2/3}) regret bound for undiscounted, continuing learning in uniformly ergodic MDPs. This is an improvement over the best existing bound of O(T^{3/4}) for the average-reward case with function approximation. Our algorithm and analysis rely on online learning techniques, where value functions are treated as losses. The main technical novelty is the use of a data-dependent adaptive learning rate coupled with a so-called optimistic prediction of upcoming losses. In addition to theoretical guarantees, we demonstrate the advantages of our approach empirically on several environments.