COLT 2017 Papers

We consider a two-sample hypothesis testing problem, where the distributions are defined on the space of undirected graphs, and one has access to only one observation from each model. A motivating example for this problem is comparing the friendship networks on Facebook and LinkedIn. The practical approach to such problems is to compare the networks based on certain network statistics. In this paper, we present a general principle for two-sample hypothesis testing in such scenarios without making any assumption about the network generation process. The main contribution of the paper is a general formulation of the problem based on concentration of network statistics, and consequently, a consistent two-sample test that arises as the natural solution for this problem. We also show that the proposed test is minimax optimal for certain network statistics.

In the classical best arm identification (Best-$1$-Arm) problem, we are given $n$ stochastic bandit arms, each associated with a reward distribution with an unknown mean. Upon each play of an arm, we can get a reward sampled i.i.d. from its reward distribution. We would like to identify the arm with the largest mean with probability at least $1-δ$, using as few samples as possible. The problem has a long history and understanding its sample complexity has attracted significant attention since the last decade. However, the optimal sample complexity of the problem is still unknown. Recently, Chen and Li (2016) made an interesting conjecture, called gap-entropy conjecture, concerning the instance optimal sample complexity of Best-$1$-Arm. Given a Best-$1$-Arm instance $I$ (i.e., a set of arms), let $\mu_[i]$ denote the $i$th largest mean and $\Delta_[i]=\mu_[1]-\mu_[i]$ denote the corresponding gap. $H(I)=\sum_i=2^n\Delta_[i]^-2$ denotes the complexity of the instance. The gap-entropy conjecture states that for any instance $I$, $Ω\left(H(I)⋅\left(\lnδ^-1 + \mathsf{Ent}(I)\right)\right)$ is an instance lower bound, where $\mathsf{Ent}(I)$ is an entropy-like term determined by the gaps, and there is a $δ$-correct algorithm for Best-$1$-Arm with sample complexity $O\left(H(I)⋅\left(\lnδ^-1 + \mathsf{Ent}(I)\right)+\Delta_[2]^-2\ln\ln\Delta_[2]^-1\right)$. We note that $Θ\left(\Delta_[2]^-2\ln\ln\Delta_[2]^-1\right)$ is necessary and sufficient to solve the two-arm instance with the best and second best arms. If the conjecture is true, we would have a complete understanding of the instance-wise sample complexity of Best-$1$-Arm (up to constant factors). In this paper, we make significant progress towards a complete resolution of the gap-entropy conjecture. For the upper bound, we provide a highly nontrivial algorithm which requires \[O\left(H(I)⋅\left(\lnδ^-1 + \mathsf{Ent}(I)\right)+\Delta_[2]^-2\ln\ln\Delta_[2]^-1\mathrmpolylog(n,δ^-1)\right)\]samples in expectation for any instance $I$. For the lower bound, we show that for any Gaussian Best-$1$-Arm instance with gaps of the form $2^-k$, any $δ$-correct monotone algorithm requires at least \[Ω\left(H(I)⋅\left(\lnδ^-1 + \mathsf{Ent}(I)\right)\right)\]samples in expectation. Here, a monotone algorithm is one which uses no more samples (in expectation) on $I’$ than on $I$, if $I’$ is a sub-instance of $I$ obtained by removing some sub-optimal arms.

We revisit the study of optimal regret rates in bandit combinatorial optimization—a fundamental framework for sequential decision making under uncertainty that abstracts numerous combinatorial prediction problems. We prove that the attainable regret in this setting grows as $\widetildeΘ(k^3/2\sqrt{d}T)$ where $d$ is the dimension of the problem and $k$ is a bound over the maximal instantaneous loss, disproving a conjecture of Audibert, Bubeck, and Lugosi (2013) who argued that the optimal rate should be of the form $\widetildeΘ(k\sqrt{d}T)$. Our bounds apply to several important instances of the framework, and in particular, imply a tight bound for the well-studied bandit shortest path problem. By that, we also resolve an open problem posed by Cesa-Bianchi and Lugosi (2012).

We consider the problem of outlier robust PCA (\textbfOR-PCA) where the goal is to recover principal directions despite the presence of outlier data points. That is, given a data matrix $M^*$, where $(1-α)$ fraction of the points are noisy samples from a low-dimensional subspace while $α$ fraction of the points can be arbitrary outliers, the goal is to recover the subspace accurately. Existing results for \textbfOR-PCA have serious drawbacks: while some results are quite weak in the presence of noise, other results have runtime quadratic in dimension, rendering them impractical for large scale applications. In this work, we provide a novel thresholding based iterative algorithm with per-iteration complexity at most linear in the data size. Moreover, the fraction of outliers, $α$, that our method can handle is tight up to constants while providing nearly optimal computational complexity for a general noise setting. For the special case where the inliers are obtained from a low-dimensional subspace with additive Gaussian noise, we show that a modification of our thresholding based method leads to significant improvement in recovery error (of the subspace) even in the presence of a large fraction of outliers.

We consider a sequential subset selection problem under parameter uncertainty, where at each time step, the decision maker selects a subset of cardinality $K$ from $N$ possible items (arms), and observes a (bandit) feedback in the form of the index of one of the items in said subset, or none. Each item in the index set is ascribed a certain value (reward), and the feedback is governed by a Multinomial Logit (MNL) choice model whose parameters are a priori unknown. The objective of the decision maker is to maximize the expected cumulative rewards over a finite horizon $T$, or alternatively, minimize the regret relative to an oracle that knows the MNL parameters. We refer to this as the MNL-Bandit problem. This problem is representative of a larger family of exploration-exploitation problems that involve a combinatorial objective, and arise in several important application domains. We present an approach to adapt Thompson Sampling to this problem and show that it achieves near-optimal regret as well as attractive numerical performance.

We propose a novel technique for analyzing adaptive sampling called the Simulator. Our approach differs from the existing methods by considering not how much information could be gathered by any fixed sampling strategy, but how difficult it is to distinguish a good sampling strategy from a bad one given the limited amount of data collected up to any given time. This change of perspective allows us to match the strength of both Fano and change-of-measure techniques, without succumbing to the limitations of either method. For concreteness, we apply our techniques to a structured multi-arm bandit problem in the fixed-confidence pure exploration setting, where we show that the constraints on the means imply a substantial gap between the moderate-confidence sample complexity, and the asymptotic sample complexity as the confidence delta tends to zero, as found in the literature. We also prove the first instance-based lower bounds for the top-k problem which incorporate the appropriate log-factors. Moreover, our lower bounds zero-in on the number of times each individual arm needs to be pulled, uncovering new phenomena which are drowned out in the aggregate sample complexity. Our new analysis inspires a simple and near-optimal algorithm for the best-arm and top-k identification, the first practical algorithm of its kind for the latter problem which removes extraneous log factors, and outperforms the state-of-the-art in experiments.

In this paper we study optimization from samples of convex functions. There are many scenarios in which we do not know the function we wish to optimize but can learn it from data. In such cases, we are interested in bounding the number of samples required to optimize the function. Our main result shows that in general, the number of samples required to obtain a non-trivial approximation to the optimum of a convex function is exponential in its dimension, even when the function is PAC-learnable. We also obtain strong lower bounds for strongly convex and Lipschitz continuous functions. On the positive side, we show that there are interesting classes of functions and distributions for which the sample complexity is polynomial in the dimension of the function.

In the differentially private top-$k$ selection problem, we are given a dataset $X ∈\pmo^n \times d$, in which each row belongs to an individual and each column corresponds to some binary attribute, and our goal is to find a set of $k ≪d$ columns whose means are approximately as large as possible. Differential privacy requires that our choice of these $k$ columns does not depend too much on any on individual’s dataset. This problem can be solved using the well known exponential mechanism and composition properties of differential privacy. In the high-accuracy regime, where we require the error of the selection procedure to be to be smaller than the so-called sampling error $α≈\sqrt\ln(d)/n$, this procedure succeeds given a dataset of size $n ≳k \ln(d)$. We prove a matching lower bound, showing that a dataset of size $n ≳k \ln(d)$ is necessary for private top-$k$ selection in this high-accuracy regime. Our lower bound shows that selecting the $k$ largest columns requires more data than simply estimating the value of those $k$ columns, which can be done using a dataset of size just $n ≳k$.

We introduce the following \em hidden hubs model $H(n,k,\sigma_0, \sigma_1)$: the input is an $n \times n$ random matrix $A$ with a subset $S$ of $k$ special rows (hubs); entries in rows outside $S$ are generated from the Gaussian distribution $p_0 = N(0,\sigma_0^2)$, while for each row in $S$, an unknown subset of $k$ of its entries are generated from $p_1 = N(0,\sigma_1^2)$, $\sigma_1>\sigma_0$, and the rest of the entries from $p_0$. The special rows with higher variance entries can be viewed as hidden higher-degree hubs. The problem we address is to identify the hubs efficiently. The planted Gaussian Submatrix Model is the special case where the higher variance entries must all lie in a $k \times k$ submatrix. If $k≥c\sqrt{n}\ln n$, just the row sums are sufficient to find $S$ in the general model. For the Gaussian submatrix problem (and the related planted clique problem), this can be improved by a $\sqrt\ln n$ factor to $k \ge c\sqrt{n}$ by spectral or combinatorial methods. We give a polynomial-time algorithm to identify all the hidden hubs with high probability for $k \ge n^0.5-δ$ for some $δ>0$, when $\sigma_1^2>2\sigma_0^2$. The algorithm extends to the setting where planted entries might have different variances, each at least $\sigma_1^2$. We also show a nearly matching lower bound: for $\sigma_1^2 \le 2\sigma_0^2$, there is no polynomial-time Statistical Query algorithm for distinguishing between a matrix whose entries are all from $N(0,\sigma_0^2)$ and a matrix with $k=n^0.5-δ$ hidden hubs for any $δ>0$. The lower bound as well as the algorithm are related to whether the chi-squared distance of the two distributions diverges. At the critical value $\sigma_1^2=2\sigma_0^2$, we show that the hidden hubs problem can be solved for $k≥c\sqrt n(\ln n)^1/4$, improving on the naive row sum-based method.

This work initiates a systematic investigation of testing \em high-dimensional structured distributions by focusing on testing \em Bayesian networks – the prototypical family of directed graphical models. A Bayesian network is defined by a directed acyclic graph, where we associate a random variable with each node. The value at any particular node is conditionally independent of all the other non-descendant nodes once its parents are fixed. Specifically, we study the properties of identity testing and closeness testing of Bayesian networks. Our main contribution is the first non-trivial efficient testing algorithms for these problems and corresponding information-theoretic lower bounds. For a wide range of parameter settings, our testing algorithms have sample complexity \em sublinear in the dimension and are sample-optimal, up to constant factors.

The Expectation-Maximization (EM) algorithm is a widely used method for maximum likelihood estimation in models with latent variables. For estimating mixtures of Gaussians, its iteration can be viewed as a soft version of the k-means clustering algorithm. Despite its wide use and applications, there are essentially no known convergence guarantees for this method. We provide global convergence guarantees for mixtures of two Gaussians with known covariance matrices. We show that the population version of EM, where the algorithm is given access to infinitely many samples from the mixture, converges geometrically to the correct mean vectors, and provide simple, closed-form expressions for the convergence rate. As a simple illustration, we show that, in one dimension, ten steps of the EM algorithm initialized at infinity result in less than $1%$ error estimation of the means. In the finite sample regime, we show that, under a random initialization, $\tilde{O}(d/ε^2)$ samples suffice to compute the unknown vectors to within $ε$ in Mahalanobis distance, where $d$ is the dimension. In particular, the error rate of the EM based estimator is $\tilde{O}\left(\sqrt{d} \over n\right)$ where $n$ is the number of samples, which is optimal up to logarithmic factors.

We analyze dropout in deep networks with rectified linear units and the quadratic loss. Our results expose surprising differences between the behavior of dropout and more traditional regularizers like weight decay. For example, on some simple data sets dropout training produces negative weights even though the output is the sum of the inputs. This provides a counterpoint to the suggestion that dropout discourages co-adaptation of weights. We also show that the dropout penalty can grow exponentially in the depth of the network while the weight-decay penalty remains essentially linear, and that dropout is insensitive to various re-scalings of the input features, outputs, and network weights. This last insensitivity implies that there are no isolated local minima of the dropout training criterion. Our work uncovers new properties of dropout, extends our understanding of why dropout succeeds, and lays the foundation for further progress.

We consider the problem of maximizing a monotone submodular function under noise. Since the 1970s there has been a great deal of work on optimization of submodular functions under various constraints, resulting in algorithms that provide desirable approximation guarantees. In many applications, however, we do not have access to the submodular function we aim to optimize, but rather to some erroneous or noisy version of it. This raises the question of whether provable guarantees are obtainable in the presence of error and noise. We provide initial answers by focusing on the problem of maximizing a monotone submodular function under a cardinality constraint when given access to a noisy oracle of the function. We show that there is an algorithm whose approximation ratio is arbitrarily close to the optimal $1-1/e$ when the cardinality is sufficiently large. The algorithm can be applied in a variety of related problems including maximizing approximately submodular functions, and optimization with correlated noise. When the noise is adversarial we show that no non-trivial approximation guarantee can be obtained.

We consider the minimization of composite objective functions composed of the expectation of quadratic functions and an arbitrary convex function. We study the stochastic dual averaging algorithm with a constant step-size, showing that it leads to a convergence rate of O(1/n) without strong convexity assumptions. This thus extends earlier results on least-squares regression with the Euclidean geometry to (a) all convex regularizers and constraints, and (b) all geometries represented by a Bregman divergence. This is achieved by a new proof technique that relates stochastic and deterministic recursions

We show that the square Hellinger distance between two Bayesian networks on the same directed graph, $G$, is subadditive with respect to the neighborhoods of $G$. Namely, if $P$ and $Q$ are the probability distributions defined by two Bayesian networks on the same DAG, our inequality states that the square Hellinger distance, $H^2(P,Q)$, between $P$ and $Q$ is upper bounded by the sum, $\sum_v H^2(P_{v} ∪\Pi_v, Q_{v} ∪\Pi_v)$, of the square Hellinger distances between the marginals of $P$ and $Q$ on every node $v$ and its parents $\Pi_v$ in the DAG. Importantly, our bound does not involve the conditionals but the marginals of $P$ and $Q$. We derive a similar inequality for more general Markov Random Fields. As an application of our inequality, we show that distinguishing whether two (unknown) Bayesian networks $P$ and $Q$ on the same (but potentially unknown) DAG satisfy $P=Q$ vs $d_\rm TV(P,Q)>ε$ can be performed from $\tilde{O}(|Σ|^3/4(d+1) ⋅n/ε^2)$ samples, where $d$ is the maximum in-degree of the DAG and $Σ$ the domain of each variable of the Bayesian networks. If $P$ and $Q$ are defined on potentially different and potentially unknown trees, the sample complexity becomes $\tilde{O}(|Σ|^4.5 n/ε^2)$. In both cases the dependence of the sample complexity on $n, ε$ is optimal up to logarithmic factors. Lastly, if $P$ and $Q$ are product distributions over ${0,1}^n$ and $Q$ is known, the sample complexity becomes $O(\sqrt{n}/ε^2)$, which is optimal up to constant factors.

In the classical multi-armed bandit problem, $d$ arms are available to the decision maker who pulls them sequentially in order to maximize his cumulative reward. Guarantees can be obtained on a relative quantity called regret, which scales linearly with $d$ (or with $\sqrt{d}$ in the minimax sense). We here consider the \emphsparse case of this classical problem in the sense that only a small number of arms, namely $s

A number of statistical estimation problems can be addressed by semidefinite programs (SDP). While SDPs are solvable in polynomial time using interior point methods, in practice generic SDP solvers do not scale well to high-dimensional problems. In order to cope with this problem, Burer and Monteiro proposed a non-convex rank-constrained formulation, which has good performance in practice but is still poorly understood theoretically. In this paper we study the rank-constrained version of SDPs arising in MaxCut and in $\mathbb Z_2$ and $\rm SO(d)$ synchronization problems. We establish a Grothendieck-type inequality that proves that all the local maxima and dangerous saddle points are within a small multiplicative gap from the global maximum. We use this structural information to prove that SDPs can be solved within a known accuracy, by applying the Riemannian trust-region method to this non-convex problem, while constraining the rank to be of order one. For the MaxCut problem, our inequality implies that any local maximizer of the rank-constrained SDP provides a $(1 - 1/(k-1)) \times 0.878$ approximation of the MaxCut, when the rank is fixed to $k$. We then apply our results to data matrices generated according to the Gaussian $\mathbb Z_2$ synchronization problem, and the two-groups stochastic block model with large bounded degree. We prove that the error achieved by local maximizers undergoes a phase transition at the same threshold as for information-theoretically optimal methods.

This paper presents a detailed theoretical analysis of the Langevin Monte Carlo sampling algorithm recently introduced in Durmus et al. (Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau, 2016) when applied to log-concave probability distributions that are restricted to a convex body $K$. This method relies on a regularisation procedure involving the Moreau-Yosida envelope of the indicator function associated with $K$. Explicit convergence bounds in total variation norm and in Wasserstein distance of order $1$ are established. In particular, we show that the complexity of this algorithm given a first order oracle is polynomial in the dimension of the state space. Finally, some numerical experiments are presented to compare our method with competing MCMC approaches from the literature.

The problem of population recovery refers to estimating a distribution based on incomplete or corrupted samples. Consider a random poll of sample size $n$ conducted on a population of individuals, where each pollee is asked to answer $d$ binary questions. We consider one of the two polling impediments: \beginitemize \item in lossy population recovery, a pollee may skip each question with probability $ε$; \item in noisy population recovery, a pollee may lie on each question with probability $ε$. \enditemize Given $n$ lossy or noisy samples, the goal is to estimate the probabilities of all $2^d$ binary vectors simultaneously within accuracy $δ$ with high probability. This paper settles the sample complexity of population recovery. For lossy model, the optimal sample complexity is $\tildeΘ(δ^ -2\max{\fracε1-ε,1})$, improving the state of the art by Moitra and Saks in several ways: a lower bound is established, the upper bound is improved and the result is dimension-free. Surprisingly, the sample complexity undergoes a phase transition from parametric to nonparametric rate when $ε$ exceeds $1/2$. For noisy population recovery, the sharp sample complexity turns out to be dimension-dependent and scales as $\exp(Θ(d^1/3 \log^2/3(1/δ)))$ except for the trivial cases of $ε=0,1/2$ or $1$. For both models, our estimators simply compute the empirical mean of a certain function, which is found by pre-solving a linear program (LP). Curiously, the dual LP can be understood as Le Cam’s method for lower-bounding the minimax risk, thus establishing the statistical optimality of the proposed estimators. The value of the LP is determined by complex-analytic methods.