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Martin J. Wainwright

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Machine Learning math.ST Statistics Theory Information Theory math.IT math.OC Data Structures and Algorithms Artificial Intelligence Computation Methodology Cryptography and Security Computational Complexity math.PR Discrete Mathematics Systems and Control

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preprint2026arXiv

TILT: Target-induced loss tilting under covariate shift

We introduce and analyze Target-Induced Loss Tilting (TILT) for unsupervised domain adaptation under covariate shift. It is based on a novel objective function that decomposes the source predictor as $f+b$, fits $f+b$ on labeled source data while simultaneously penalizing the auxiliary component $b$ on unlabeled target inputs. The resulting fit $f$ is deployed as the final target predictor. At the population level, we show that this target-side penalty implicitly induces relative importance weighting at the population level, but in terms of an estimand $b^*_f$ that is self-localized to the current error, and remains uniformly bounded for any source-target pair (even those with disjoint supports). We prove a general finite-sample oracle inequality on the excess risk, and use it to give an end-to-end guarantee for training with sparse ReLU networks. Experiments on controlled regression problems and shifted CIFAR-100 distillation show that TILT improves target-domain performance over source-only training, exact importance weighting, and relative density-ratio baselines, with a stable dependence on the regularization parameter.

preprint2023arXiv

Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency

We study optimal procedures for estimating a linear functional based on observational data. In many problems of this kind, a widely used assumption is strict overlap, i.e., uniform boundedness of the importance ratio, which measures how well the observational data covers the directions of interest. When it is violated, the classical semi-parametric efficiency bound can easily become infinite, so that the instance-optimal risk depends on the function class used to model the regression function. For any convex and symmetric function class $\mathcal{F}$, we derive a non-asymptotic local minimax bound on the mean-squared error in estimating a broad class of linear functionals. This lower bound refines the classical semi-parametric one, and makes connections to moduli of continuity in functional estimation. When $\mathcal{F}$ is a reproducing kernel Hilbert space, we prove that this lower bound can be achieved up to a constant factor by analyzing a computationally simple regression estimator. We apply our general results to various families of examples, thereby uncovering a spectrum of rates that interpolate between the classical theories of semi-parametric efficiency (with $\sqrt{n}$-consistency) and the slower minimax rates associated with non-parametric function estimation.

preprint2022arXiv

A Diffusion Process Perspective on Posterior Contraction Rates for Parameters

We analyze the posterior contraction rates of parameters in Bayesian models via the Langevin diffusion process, in particular by controlling moments of the stochastic process and taking limits. Analogous to the non-asymptotic analysis of statistical M-estimators and stochastic optimization algorithms, our contraction rates depend on the structure of the population log-likelihood function, and stochastic perturbation bounds between the population and sample log-likelihood functions. Convergence rates are determined by a non-linear equation that relates the population-level structure to stochastic perturbation terms, along with a term characterizing the diffusive behavior. Based on this technique, we also prove non-asymptotic versions of a Bernstein-von-Mises guarantee for the posterior. We illustrate this general theory by deriving posterior convergence rates for various concrete examples, as well as approximate posterior distributions computed using Langevin sampling procedures.

preprint2022arXiv

A new similarity measure for covariate shift with applications to nonparametric regression

We study covariate shift in the context of nonparametric regression. We introduce a new measure of distribution mismatch between the source and target distributions that is based on the integrated ratio of probabilities of balls at a given radius. We use the scaling of this measure with respect to the radius to characterize the minimax rate of estimation over a family of Hölder continuous functions under covariate shift. In comparison to the recently proposed notion of transfer exponent, this measure leads to a sharper rate of convergence and is more fine-grained. We accompany our theory with concrete instances of covariate shift that illustrate this sharp difference.

preprint2022arXiv

Instability, Computational Efficiency and Statistical Accuracy

Many statistical estimators are defined as the fixed point of a data-dependent operator, with estimators based on minimizing a cost function being an important special case. The limiting performance of such estimators depends on the properties of the population-level operator in the idealized limit of infinitely many samples. We develop a general framework that yields bounds on statistical accuracy based on the interplay between the deterministic convergence rate of the algorithm at the population level, and its degree of (in)stability when applied to an empirical object based on $n$ samples. Using this framework, we analyze both stable forms of gradient descent and some higher-order and unstable algorithms, including Newton's method and its cubic-regularized variant, as well as the EM algorithm. We provide applications of our general results to several concrete classes of models, including Gaussian mixture estimation, non-linear regression models, and informative non-response models. We exhibit cases in which an unstable algorithm can achieve the same statistical accuracy as a stable algorithm in exponentially fewer steps -- namely, with the number of iterations being reduced from polynomial to logarithmic in sample size $n$.

preprint2022arXiv

Instance-Dependent Confidence and Early Stopping for Reinforcement Learning

Various algorithms for reinforcement learning (RL) exhibit dramatic variation in their convergence rates as a function of problem structure. Such problem-dependent behavior is not captured by worst-case analyses and has accordingly inspired a growing effort in obtaining instance-dependent guarantees and deriving instance-optimal algorithms for RL problems. This research has been carried out, however, primarily within the confines of theory, providing guarantees that explain \textit{ex post} the performance differences observed. A natural next step is to convert these theoretical guarantees into guidelines that are useful in practice. We address the problem of obtaining sharp instance-dependent confidence regions for the policy evaluation problem and the optimal value estimation problem of an MDP, given access to an instance-optimal algorithm. As a consequence, we propose a data-dependent stopping rule for instance-optimal algorithms. The proposed stopping rule adapts to the instance-specific difficulty of the problem and allows for early termination for problems with favorable structure.

preprint2022arXiv

Stabilizing Q-learning with Linear Architectures for Provably Efficient Learning

The $Q$-learning algorithm is a simple and widely-used stochastic approximation scheme for reinforcement learning, but the basic protocol can exhibit instability in conjunction with function approximation. Such instability can be observed even with linear function approximation. In practice, tools such as target networks and experience replay appear to be essential, but the individual contribution of each of these mechanisms is not well understood theoretically. This work proposes an exploration variant of the basic $Q$-learning protocol with linear function approximation. Our modular analysis illustrates the role played by each algorithmic tool that we adopt: a second order update rule, a set of target networks, and a mechanism akin to experience replay. Together, they enable state of the art regret bounds on linear MDPs while preserving the most prominent feature of the algorithm, namely a space complexity independent of the number of step elapsed. We show that the performance of the algorithm degrades very gracefully under a novel and more permissive notion of approximation error. The algorithm also exhibits a form of instance-dependence, in that its performance depends on the "effective" feature dimension.

preprint2021arXiv

A Permutation-based Model for Crowd Labeling: Optimal Estimation and Robustness

The task of aggregating and denoising crowd-labeled data has gained increased significance with the advent of crowdsourcing platforms and massive datasets. We propose a permutation-based model for crowd labeled data that is a significant generalization of the classical Dawid-Skene model, and introduce a new error metric by which to compare different estimators. We derive global minimax rates for the permutation-based model that are sharp up to logarithmic factors, and match the minimax lower bounds derived under the simpler Dawid-Skene model. We then design two computationally-efficient estimators: the WAN estimator for the setting where the ordering of workers in terms of their abilities is approximately known, and the OBI-WAN estimator where that is not known. For each of these estimators, we provide non-asymptotic bounds on their performance. We conduct synthetic simulations and experiments on real-world crowdsourcing data, and the experimental results corroborate our theoretical findings.

preprint2021arXiv

Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients

Hamiltonian Monte Carlo (HMC) is a state-of-the-art Markov chain Monte Carlo sampling algorithm for drawing samples from smooth probability densities over continuous spaces. We study the variant most widely used in practice, Metropolized HMC with the Störmer-Verlet or leapfrog integrator, and make two primary contributions. First, we provide a non-asymptotic upper bound on the mixing time of the Metropolized HMC with explicit choices of step-size and number of leapfrog steps. This bound gives a precise quantification of the faster convergence of Metropolized HMC relative to simpler MCMC algorithms such as the Metropolized random walk, or Metropolized Langevin algorithm. Second, we provide a general framework for sharpening mixing time bounds of Markov chains initialized at a substantial distance from the target distribution over continuous spaces. We apply this sharpening device to the Metropolized random walk and Langevin algorithms, thereby obtaining improved mixing time bounds from a non-warm initial distribution.

preprint2021arXiv

Minimax Off-Policy Evaluation for Multi-Armed Bandits

We study the problem of off-policy evaluation in the multi-armed bandit model with bounded rewards, and develop minimax rate-optimal procedures under three settings. First, when the behavior policy is known, we show that the Switch estimator, a method that alternates between the plug-in and importance sampling estimators, is minimax rate-optimal for all sample sizes. Second, when the behavior policy is unknown, we analyze performance in terms of the competitive ratio, thereby revealing a fundamental gap between the settings of known and unknown behavior policies. When the behavior policy is unknown, any estimator must have mean-squared error larger -- relative to the oracle estimator equipped with the knowledge of the behavior policy -- by a multiplicative factor proportional to the support size of the target policy. Moreover, we demonstrate that the plug-in approach achieves this worst-case competitive ratio up to a logarithmic factor. Third, we initiate the study of the partial knowledge setting in which it is assumed that the minimum probability taken by the behavior policy is known. We show that the plug-in estimator is optimal for relatively large values of the minimum probability, but is sub-optimal when the minimum probability is low. In order to remedy this gap, we propose a new estimator based on approximation by Chebyshev polynomials that provably achieves the optimal estimation error. Numerical experiments on both simulated and real data corroborate our theoretical findings.

preprint2020arXiv

Derivative-Free Methods for Policy Optimization: Guarantees for Linear Quadratic Systems

We study derivative-free methods for policy optimization over the class of linear policies. We focus on characterizing the convergence rate of these methods when applied to linear-quadratic systems, and study various settings of driving noise and reward feedback. We show that these methods provably converge to within any pre-specified tolerance of the optimal policy with a number of zero-order evaluations that is an explicit polynomial of the error tolerance, dimension, and curvature properties of the problem. Our analysis reveals some interesting differences between the settings of additive driving noise and random initialization, as well as the settings of one-point and two-point reward feedback. Our theory is corroborated by extensive simulations of derivative-free methods on these systems. Along the way, we derive convergence rates for stochastic zero-order optimization algorithms when applied to a certain class of non-convex problems.

preprint2020arXiv

FedSplit: An algorithmic framework for fast federated optimization

Motivated by federated learning, we consider the hub-and-spoke model of distributed optimization in which a central authority coordinates the computation of a solution among many agents while limiting communication. We first study some past procedures for federated optimization, and show that their fixed points need not correspond to stationary points of the original optimization problem, even in simple convex settings with deterministic updates. In order to remedy these issues, we introduce FedSplit, a class of algorithms based on operator splitting procedures for solving distributed convex minimization with additive structure. We prove that these procedures have the correct fixed points, corresponding to optima of the original optimization problem, and we characterize their convergence rates under different settings. Our theory shows that these methods are provably robust to inexact computation of intermediate local quantities. We complement our theory with some simple experiments that demonstrate the benefits of our methods in practice.

preprint2020arXiv

High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm

We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of $d$-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most $\varepsilon > 0$ in Wasserstein distance from the target distribution in $O\left(\frac{d^{1/4}}{ \varepsilon^{1/2}} \right)$ steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with $α$-th order smoothness, we prove that the mixing time scales as $O \left(\frac{d^{1/4}}{\varepsilon^{1/2}} + \frac{d^{1/2}}{\varepsilon^{1/(α- 1)}} \right)$.

preprint2020arXiv

HopSkipJumpAttack: A Query-Efficient Decision-Based Attack

The goal of a decision-based adversarial attack on a trained model is to generate adversarial examples based solely on observing output labels returned by the targeted model. We develop HopSkipJumpAttack, a family of algorithms based on a novel estimate of the gradient direction using binary information at the decision boundary. The proposed family includes both untargeted and targeted attacks optimized for $\ell_2$ and $\ell_\infty$ similarity metrics respectively. Theoretical analysis is provided for the proposed algorithms and the gradient direction estimate. Experiments show HopSkipJumpAttack requires significantly fewer model queries than Boundary Attack. It also achieves competitive performance in attacking several widely-used defense mechanisms. (HopSkipJumpAttack was named Boundary Attack++ in a previous version of the preprint.)

preprint2020arXiv

Instance-dependent $\ell_\infty$-bounds for policy evaluation in tabular reinforcement learning

Markov reward processes (MRPs) are used to model stochastic phenomena arising in operations research, control engineering, robotics, and artificial intelligence, as well as communication and transportation networks. In many of these cases, such as in the policy evaluation problem encountered in reinforcement learning, the goal is to estimate the long-term value function of such a process without access to the underlying population transition and reward functions. Working with samples generated under the synchronous model, we study the problem of estimating the value function of an infinite-horizon, discounted MRP on finitely many states in the $\ell_\infty$-norm. We analyze both the standard plug-in approach to this problem and a more robust variant, and establish non-asymptotic bounds that depend on the (unknown) problem instance, as well as data-dependent bounds that can be evaluated based on the observations of state-transitions and rewards. We show that these approaches are minimax-optimal up to constant factors over natural sub-classes of MRPs. Our analysis makes use of a leave-one-out decoupling argument tailored to the policy evaluation problem, one which may be of independent interest.

preprint2020arXiv

Is Temporal Difference Learning Optimal? An Instance-Dependent Analysis

We address the problem of policy evaluation in discounted Markov decision processes, and provide instance-dependent guarantees on the $\ell_\infty$-error under a generative model. We establish both asymptotic and non-asymptotic versions of local minimax lower bounds for policy evaluation, thereby providing an instance-dependent baseline by which to compare algorithms. Theory-inspired simulations show that the widely-used temporal difference (TD) algorithm is strictly suboptimal when evaluated in a non-asymptotic setting, even when combined with Polyak-Ruppert iterate averaging. We remedy this issue by introducing and analyzing variance-reduced forms of stochastic approximation, showing that they achieve non-asymptotic, instance-dependent optimality up to logarithmic factors.

preprint2020arXiv

Lower bounds in multiple testing: A framework based on derandomized proxies

The large bulk of work in multiple testing has focused on specifying procedures that control the false discovery rate (FDR), with relatively less attention being paid to the corresponding Type II error known as the false non-discovery rate (FNR). A line of more recent work in multiple testing has begun to investigate the tradeoffs between the FDR and FNR and to provide lower bounds on the performance of procedures that depend on the model structure. Lacking thus far, however, has been a general approach to obtaining lower bounds for a broad class of models. This paper introduces an analysis strategy based on derandomization, illustrated by applications to various concrete models. Our main result is meta-theorem that gives a general recipe for obtaining lower bounds on the combination of FDR and FNR. We illustrate this meta-theorem by deriving explicit bounds for several models, including instances with dependence, scale-transformed alternatives, and non-Gaussian-like distributions. We provide numerical simulations of some of these lower bounds, and show a close relation to the actual performance of the Benjamini-Hochberg (BH) algorithm.

preprint2020arXiv

On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration

We undertake a precise study of the asymptotic and non-asymptotic properties of stochastic approximation procedures with Polyak-Ruppert averaging for solving a linear system $\bar{A} θ= \bar{b}$. When the matrix $\bar{A}$ is Hurwitz, we prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity. The CLT characterizes the exact asymptotic covariance matrix, which is the sum of the classical Polyak-Ruppert covariance and a correction term that scales with the step size. Under assumptions on the tail of the noise distribution, we prove a non-asymptotic concentration inequality whose main term matches the covariance in CLT in any direction, up to universal constants. When the matrix $\bar{A}$ is not Hurwitz but only has non-negative real parts in its eigenvalues, we prove that the averaged LSA procedure actually achieves an $O(1/T)$ rate in mean-squared error. Our results provide a more refined understanding of linear stochastic approximation in both the asymptotic and non-asymptotic settings. We also show various applications of the main results, including the study of momentum-based stochastic gradient methods as well as temporal difference algorithms in reinforcement learning.

preprint2020arXiv

Singularity, Misspecification, and the Convergence Rate of EM

A line of recent work has analyzed the behavior of the Expectation-Maximization (EM) algorithm in the well-specified setting, in which the population likelihood is locally strongly concave around its maximizing argument. Examples include suitably separated Gaussian mixture models and mixtures of linear regressions. We consider over-specified settings in which the number of fitted components is larger than the number of components in the true distribution. Such misspecified settings can lead to singularity in the Fisher information matrix, and moreover, the maximum likelihood estimator based on $n$ i.i.d. samples in $d$ dimensions can have a non-standard $\mathcal{O}((d/n)^{\frac{1}{4}})$ rate of convergence. Focusing on the simple setting of two-component mixtures fit to a $d$-dimensional Gaussian distribution, we study the behavior of the EM algorithm both when the mixture weights are different (unbalanced case), and are equal (balanced case). Our analysis reveals a sharp distinction between these two cases: in the former, the EM algorithm converges geometrically to a point at Euclidean distance of $\mathcal{O}((d/n)^{\frac{1}{2}})$ from the true parameter, whereas in the latter case, the convergence rate is exponentially slower, and the fixed point has a much lower $\mathcal{O}((d/n)^{\frac{1}{4}})$ accuracy. Analysis of this singular case requires the introduction of some novel techniques: in particular, we make use of a careful form of localization in the associated empirical process, and develop a recursive argument to progressively sharpen the statistical rate.

preprint2016arXiv

Active Ranking from Pairwise Comparisons and when Parametric Assumptions Don't Help

We consider sequential or active ranking of a set of n items based on noisy pairwise comparisons. Items are ranked according to the probability that a given item beats a randomly chosen item, and ranking refers to partitioning the items into sets of pre-specified sizes according to their scores. This notion of ranking includes as special cases the identification of the top-k items and the total ordering of the items. We first analyze a sequential ranking algorithm that counts the number of comparisons won, and uses these counts to decide whether to stop, or to compare another pair of items, chosen based on confidence intervals specified by the data collected up to that point. We prove that this algorithm succeeds in recovering the ranking using a number of comparisons that is optimal up to logarithmic factors. This guarantee does not require any structural properties of the underlying pairwise probability matrix, unlike a significant body of past work on pairwise ranking based on parametric models such as the Thurstone or Bradley-Terry-Luce models. It has been a long-standing open question as to whether or not imposing these parametric assumptions allows for improved ranking algorithms. For stochastic comparison models, in which the pairwise probabilities are bounded away from zero, our second contribution is to resolve this issue by proving a lower bound for parametric models. This shows, perhaps surprisingly, that these popular parametric modeling choices offer at most logarithmic gains for stochastic comparisons.

preprint2016arXiv

Asymptotic behavior of $\ell_p$-based Laplacian regularization in semi-supervised learning

Given a weighted graph with $N$ vertices, consider a real-valued regression problem in a semi-supervised setting, where one observes $n$ labeled vertices, and the task is to label the remaining ones. We present a theoretical study of $\ell_p$-based Laplacian regularization under a $d$-dimensional geometric random graph model. We provide a variational characterization of the performance of this regularized learner as $N$ grows to infinity while $n$ stays constant, the associated optimality conditions lead to a partial differential equation that must be satisfied by the associated function estimate $\hat{f}$. From this formulation we derive several predictions on the limiting behavior the $d$-dimensional function $\hat{f}$, including (a) a phase transition in its smoothness at the threshold $p = d + 1$, and (b) a tradeoff between smoothness and sensitivity to the underlying unlabeled data distribution $P$. Thus, over the range $p \leq d$, the function estimate $\hat{f}$ is degenerate and "spiky," whereas for $p\geq d+1$, the function estimate $\hat{f}$ is smooth. We show that the effect of the underlying density vanishes monotonically with $p$, such that in the limit $p = \infty$, corresponding to the so-called Absolutely Minimal Lipschitz Extension, the estimate $\hat{f}$ is independent of the distribution $P$. Under the assumption of semi-supervised smoothness, ignoring $P$ can lead to poor statistical performance, in particular, we construct a specific example for $d=1$ to demonstrate that $p=2$ has lower risk than $p=\infty$ due to the former penalty adapting to $P$ and the latter ignoring it. We also provide simulations that verify the accuracy of our predictions for finite sample sizes. Together, these properties show that $p = d+1$ is an optimal choice, yielding a function estimate $\hat{f}$ that is both smooth and non-degenerate, while remaining maximally sensitive to $P$.

preprint2016arXiv

Convexified Convolutional Neural Networks

We describe the class of convexified convolutional neural networks (CCNNs), which capture the parameter sharing of convolutional neural networks in a convex manner. By representing the nonlinear convolutional filters as vectors in a reproducing kernel Hilbert space, the CNN parameters can be represented as a low-rank matrix, which can be relaxed to obtain a convex optimization problem. For learning two-layer convolutional neural networks, we prove that the generalization error obtained by a convexified CNN converges to that of the best possible CNN. For learning deeper networks, we train CCNNs in a layer-wise manner. Empirically, CCNNs achieve performance competitive with CNNs trained by backpropagation, SVMs, fully-connected neural networks, stacked denoising auto-encoders, and other baseline methods.

preprint2016arXiv

Feeling the Bern: Adaptive Estimators for Bernoulli Probabilities of Pairwise Comparisons

We study methods for aggregating pairwise comparison data in order to estimate outcome probabilities for future comparisons among a collection of n items. Working within a flexible framework that imposes only a form of strong stochastic transitivity (SST), we introduce an adaptivity index defined by the indifference sets of the pairwise comparison probabilities. In addition to measuring the usual worst-case risk of an estimator, this adaptivity index also captures the extent to which the estimator adapts to instance-specific difficulty relative to an oracle estimator. We prove three main results that involve this adaptivity index and different algorithms. First, we propose a three-step estimator termed Count-Randomize-Least squares (CRL), and show that it has adaptivity index upper bounded as $\sqrt{n}$ up to logarithmic factors. We then show that that conditional on the hardness of planted clique, no computationally efficient estimator can achieve an adaptivity index smaller than $\sqrt{n}$. Second, we show that a regularized least squares estimator can achieve a poly-logarithmic adaptivity index, thereby demonstrating a $\sqrt{n}$-gap between optimal and computationally achievable adaptivity. Finally, we prove that the standard least squares estimator, which is known to be optimally adaptive in several closely related problems, fails to adapt in the context of estimating pairwise probabilities.

preprint2016arXiv

Function-Specific Mixing Times and Concentration Away from Equilibrium

Slow mixing is the central hurdle when working with Markov chains, especially those used for Monte Carlo approximations (MCMC). In many applications, it is only of interest to estimate the stationary expectations of a small set of functions, and so the usual definition of mixing based on total variation convergence may be too conservative. Accordingly, we introduce function-specific analogs of mixing times and spectral gaps, and use them to prove Hoeffding-like function-specific concentration inequalities. These results show that it is possible for empirical expectations of functions to concentrate long before the underlying chain has mixed in the classical sense, and we show that the concentration rates we achieve are optimal up to constants. We use our techniques to derive confidence intervals that are sharper than those implied by both classical Markov chain Hoeffding bounds and Berry-Esseen-corrected CLT bounds. For applications that require testing, rather than point estimation, we show similar improvements over recent sequential testing results for MCMC. We conclude by applying our framework to real data examples of MCMC, providing evidence that our theory is both accurate and relevant to practice.

preprint2016arXiv

Linear Regression with an Unknown Permutation: Statistical and Computational Limits

Consider a noisy linear observation model with an unknown permutation, based on observing $y = Π^* A x^* + w$, where $x^* \in \mathbb{R}^d$ is an unknown vector, $Π^*$ is an unknown $n \times n$ permutation matrix, and $w \in \mathbb{R}^n$ is additive Gaussian noise. We analyze the problem of permutation recovery in a random design setting in which the entries of the matrix $A$ are drawn i.i.d. from a standard Gaussian distribution, and establish sharp conditions on the SNR, sample size $n$, and dimension $d$ under which $Π^*$ is exactly and approximately recoverable. On the computational front, we show that the maximum likelihood estimate of $Π^*$ is NP-hard to compute, while also providing a polynomial time algorithm when $d =1$.

preprint2016arXiv

Local Maxima in the Likelihood of Gaussian Mixture Models: Structural Results and Algorithmic Consequences

We provide two fundamental results on the population (infinite-sample) likelihood function of Gaussian mixture models with $M \geq 3$ components. Our first main result shows that the population likelihood function has bad local maxima even in the special case of equally-weighted mixtures of well-separated and spherical Gaussians. We prove that the log-likelihood value of these bad local maxima can be arbitrarily worse than that of any global optimum, thereby resolving an open question of Srebro (2007). Our second main result shows that the EM algorithm (or a first-order variant of it) with random initialization will converge to bad critical points with probability at least $1-e^{-Ω(M)}$. We further establish that a first-order variant of EM will not converge to strict saddle points almost surely, indicating that the poor performance of the first-order method can be attributed to the existence of bad local maxima rather than bad saddle points. Overall, our results highlight the necessity of careful initialization when using the EM algorithm in practice, even when applied in highly favorable settings.

preprint2016arXiv

Simple, Robust and Optimal Ranking from Pairwise Comparisons

We consider data in the form of pairwise comparisons of n items, with the goal of precisely identifying the top k items for some value of k < n, or alternatively, recovering a ranking of all the items. We analyze the Copeland counting algorithm that ranks the items in order of the number of pairwise comparisons won, and show it has three attractive features: (a) its computational efficiency leads to speed-ups of several orders of magnitude in computation time as compared to prior work; (b) it is robust in that theoretical guarantees impose no conditions on the underlying matrix of pairwise-comparison probabilities, in contrast to some prior work that applies only to the BTL parametric model; and (c) it is an optimal method up to constant factors, meaning that it achieves the information-theoretic limits for recovering the top k-subset. We extend our results to obtain sharp guarantees for approximate recovery under the Hamming distortion metric, and more generally, to any arbitrary error requirement that satisfies a simple and natural monotonicity condition.

preprint2016arXiv

Stochastically Transitive Models for Pairwise Comparisons: Statistical and Computational Issues

There are various parametric models for analyzing pairwise comparison data, including the Bradley-Terry-Luce (BTL) and Thurstone models, but their reliance on strong parametric assumptions is limiting. In this work, we study a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity. This class includes parametric models including the BTL and Thurstone models as special cases, but is considerably more general. We provide various examples of models in this broader stochastically transitive class for which classical parametric models provide poor fits. Despite this greater flexibility, we show that the matrix of probabilities can be estimated at the same rate as in standard parametric models. On the other hand, unlike in the BTL and Thurstone models, computing the minimax-optimal estimator in the stochastically transitive model is non-trivial, and we explore various computationally tractable alternatives. We show that a simple singular value thresholding algorithm is statistically consistent but does not achieve the minimax rate. We then propose and study algorithms that achieve the minimax rate over interesting sub-classes of the full stochastically transitive class. We complement our theoretical results with thorough numerical simulations.

preprint2015arXiv

A More Powerful Two-Sample Test in High Dimensions using Random Projection

We consider the hypothesis testing problem of detecting a shift between the means of two multivariate normal distributions in the high-dimensional setting, allowing for the data dimension p to exceed the sample size n. Specifically, we propose a new test statistic for the two-sample test of means that integrates a random projection with the classical Hotelling T^2 statistic. Working under a high-dimensional framework with (p,n) tending to infinity, we first derive an asymptotic power function for our test, and then provide sufficient conditions for it to achieve greater power than other state-of-the-art tests. Using ROC curves generated from synthetic data, we demonstrate superior performance against competing tests in the parameter regimes anticipated by our theoretical results. Lastly, we illustrate an advantage of our procedure's false positive rate with comparisons on high-dimensional gene expression data involving the discrimination of different types of cancer.

preprint2015arXiv

Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds

We study the following generalized matrix rank estimation problem: given an $n \times n$ matrix and a constant $c \geq 0$, estimate the number of eigenvalues that are greater than $c$. In the distributed setting, the matrix of interest is the sum of $m$ matrices held by separate machines. We show that any deterministic algorithm solving this problem must communicate $Ω(n^2)$ bits, which is order-equivalent to transmitting the whole matrix. In contrast, we propose a randomized algorithm that communicates only $\widetilde O(n)$ bits. The upper bound is matched by an $Ω(n)$ lower bound on the randomized communication complexity. We demonstrate the practical effectiveness of the proposed algorithm with some numerical experiments.

preprint2015arXiv

Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence

Data in the form of pairwise comparisons arises in many domains, including preference elicitation, sporting competitions, and peer grading among others. We consider parametric ordinal models for such pairwise comparison data involving a latent vector $w^* \in \mathbb{R}^d$ that represents the "qualities" of the $d$ items being compared; this class of models includes the two most widely used parametric models--the Bradley-Terry-Luce (BTL) and the Thurstone models. Working within a standard minimax framework, we provide tight upper and lower bounds on the optimal error in estimating the quality score vector $w^*$ under this class of models. The bounds depend on the topology of the comparison graph induced by the subset of pairs being compared via its Laplacian spectrum. Thus, in settings where the subset of pairs may be chosen, our results provide principled guidelines for making this choice. Finally, we compare these error rates to those under cardinal measurement models and show that the error rates in the ordinal and cardinal settings have identical scalings apart from constant pre-factors.

preprint2015arXiv

Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees

Optimization problems with rank constraints arise in many applications, including matrix regression, structured PCA, matrix completion and matrix decomposition problems. An attractive heuristic for solving such problems is to factorize the low-rank matrix, and to run projected gradient descent on the nonconvex factorized optimization problem. The goal of this problem is to provide a general theoretical framework for understanding when such methods work well, and to characterize the nature of the resulting fixed point. We provide a simple set of conditions under which projected gradient descent, when given a suitable initialization, converges geometrically to a statistically useful solution. Our results are applicable even when the initial solution is outside any region of local convexity, and even when the problem is globally concave. Working in a non-asymptotic framework, we show that our conditions are satisfied for a wide range of concrete models, including matrix regression, structured PCA, matrix completion with real and quantized observations, matrix decomposition, and graph clustering problems. Simulation results show excellent agreement with the theoretical predictions.

preprint2015arXiv

Learning Halfspaces and Neural Networks with Random Initialization

We study non-convex empirical risk minimization for learning halfspaces and neural networks. For loss functions that are $L$-Lipschitz continuous, we present algorithms to learn halfspaces and multi-layer neural networks that achieve arbitrarily small excess risk $ε>0$. The time complexity is polynomial in the input dimension $d$ and the sample size $n$, but exponential in the quantity $(L/ε^2)\log(L/ε)$. These algorithms run multiple rounds of random initialization followed by arbitrary optimization steps. We further show that if the data is separable by some neural network with constant margin $γ>0$, then there is a polynomial-time algorithm for learning a neural network that separates the training data with margin $Ω(γ)$. As a consequence, the algorithm achieves arbitrary generalization error $ε>0$ with ${\rm poly}(d,1/ε)$ sample and time complexity. We establish the same learnability result when the labels are randomly flipped with probability $η<1/2$.

preprint2015arXiv

Newton Sketch: A Linear-time Optimization Algorithm with Linear-Quadratic Convergence

We propose a randomized second-order method for optimization known as the Newton Sketch: it is based on performing an approximate Newton step using a randomly projected or sub-sampled Hessian. For self-concordant functions, we prove that the algorithm has super-linear convergence with exponentially high probability, with convergence and complexity guarantees that are independent of condition numbers and related problem-dependent quantities. Given a suitable initialization, similar guarantees also hold for strongly convex and smooth objectives without self-concordance. When implemented using randomized projections based on a sub-sampled Hadamard basis, the algorithm typically has substantially lower complexity than Newton's method. We also describe extensions of our methods to programs involving convex constraints that are equipped with self-concordant barriers. We discuss and illustrate applications to linear programs, quadratic programs with convex constraints, logistic regression and other generalized linear models, as well as semidefinite programs.

preprint2015arXiv

On the Computational Complexity of High-Dimensional Bayesian Variable Selection

We study the computational complexity of Markov chain Monte Carlo (MCMC) methods for high-dimensional Bayesian linear regression under sparsity constraints. We first show that a Bayesian approach can achieve variable-selection consistency under relatively mild conditions on the design matrix. We then demonstrate that the statistical criterion of posterior concentration need not imply the computational desideratum of rapid mixing of the MCMC algorithm. By introducing a truncated sparsity prior for variable selection, we provide a set of conditions that guarantee both variable-selection consistency and rapid mixing of a particular Metropolis-Hastings algorithm. The mixing time is linear in the number of covariates up to a logarithmic factor. Our proof controls the spectral gap of the Markov chain by constructing a canonical path ensemble that is inspired by the steps taken by greedy algorithms for variable selection.

preprint2015arXiv

Optimal prediction for sparse linear models? Lower bounds for coordinate-separable M-estimators

For the problem of high-dimensional sparse linear regression, it is known that an $\ell_0$-based estimator can achieve a $1/n$ "fast" rate on the prediction error without any conditions on the design matrix, whereas in absence of restrictive conditions on the design matrix, popular polynomial-time methods only guarantee the $1/\sqrt{n}$ "slow" rate. In this paper, we show that the slow rate is intrinsic to a broad class of M-estimators. In particular, for estimators based on minimizing a least-squares cost function together with a (possibly non-convex) coordinate-wise separable regularizer, there is always a "bad" local optimum such that the associated prediction error is lower bounded by a constant multiple of $1/\sqrt{n}$. For convex regularizers, this lower bound applies to all global optima. The theory is applicable to many popular estimators, including convex $\ell_1$-based methods as well as M-estimators based on nonconvex regularizers, including the SCAD penalty or the MCP regularizer. In addition, for a broad class of nonconvex regularizers, we show that the bad local optima are very common, in that a broad class of local minimization algorithms with random initialization will typically converge to a bad solution.

preprint2015arXiv

Randomized sketches for kernels: Fast and optimal non-parametric regression

Kernel ridge regression (KRR) is a standard method for performing non-parametric regression over reproducing kernel Hilbert spaces. Given $n$ samples, the time and space complexity of computing the KRR estimate scale as $\mathcal{O}(n^3)$ and $\mathcal{O}(n^2)$ respectively, and so is prohibitive in many cases. We propose approximations of KRR based on $m$-dimensional randomized sketches of the kernel matrix, and study how small the projection dimension $m$ can be chosen while still preserving minimax optimality of the approximate KRR estimate. For various classes of randomized sketches, including those based on Gaussian and randomized Hadamard matrices, we prove that it suffices to choose the sketch dimension $m$ proportional to the statistical dimension (modulo logarithmic factors). Thus, we obtain fast and minimax optimal approximations to the KRR estimate for non-parametric regression.

preprint2015arXiv

Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima

We provide novel theoretical results regarding local optima of regularized $M$-estimators, allowing for nonconvexity in both loss and penalty functions. Under restricted strong convexity on the loss and suitable regularity conditions on the penalty, we prove that \emph{any stationary point} of the composite objective function will lie within statistical precision of the underlying parameter vector. Our theory covers many nonconvex objective functions of interest, including the corrected Lasso for errors-in-variables linear models; regression for generalized linear models with nonconvex penalties such as SCAD, MCP, and capped-$\ell_1$; and high-dimensional graphical model estimation. We quantify statistical accuracy by providing bounds on the $\ell_1$-, $\ell_2$-, and prediction error between stationary points and the population-level optimum. We also propose a simple modification of composite gradient descent that may be used to obtain a near-global optimum within statistical precision $ε$ in $\log(1/ε)$ steps, which is the fastest possible rate of any first-order method. We provide simulation studies illustrating the sharpness of our theoretical results.

preprint2015arXiv

Statistical and Computational Guarantees for the Baum-Welch Algorithm

The Hidden Markov Model (HMM) is one of the mainstays of statistical modeling of discrete time series, with applications including speech recognition, computational biology, computer vision and econometrics. Estimating an HMM from its observation process is often addressed via the Baum-Welch algorithm, which is known to be susceptible to local optima. In this paper, we first give a general characterization of the basin of attraction associated with any global optimum of the population likelihood. By exploiting this characterization, we provide non-asymptotic finite sample guarantees on the Baum-Welch updates, guaranteeing geometric convergence to a small ball of radius on the order of the minimax rate around a global optimum. As a concrete example, we prove a linear rate of convergence for a hidden Markov mixture of two isotropic Gaussians given a suitable mean separation and an initialization within a ball of large radius around (one of) the true parameters. To our knowledge, these are the first rigorous local convergence guarantees to global optima for the Baum-Welch algorithm in a setting where the likelihood function is nonconvex. We complement our theoretical results with thorough numerical simulations studying the convergence of the Baum-Welch algorithm and illustrating the accuracy of our predictions.

preprint2015arXiv

The geometry of kernelized spectral clustering

Clustering of data sets is a standard problem in many areas of science and engineering. The method of spectral clustering is based on embedding the data set using a kernel function, and using the top eigenvectors of the normalized Laplacian to recover the connected components. We study the performance of spectral clustering in recovering the latent labels of i.i.d. samples from a finite mixture of nonparametric distributions. The difficulty of this label recovery problem depends on the overlap between mixture components and how easily a mixture component is divided into two nonoverlapping components. When the overlap is small compared to the indivisibility of the mixture components, the principal eigenspace of the population-level normalized Laplacian operator is approximately spanned by the square-root kernelized component densities. In the finite sample setting, and under the same assumption, embedded samples from different components are approximately orthogonal with high probability when the sample size is large. As a corollary we control the fraction of samples mislabeled by spectral clustering under finite mixtures with nonparametric components.

preprint2014arXiv

Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates

We establish optimal convergence rates for a decomposition-based scalable approach to kernel ridge regression. The method is simple to describe: it randomly partitions a dataset of size N into m subsets of equal size, computes an independent kernel ridge regression estimator for each subset, then averages the local solutions into a global predictor. This partitioning leads to a substantial reduction in computation time versus the standard approach of performing kernel ridge regression on all N samples. Our two main theorems establish that despite the computational speed-up, statistical optimality is retained: as long as m is not too large, the partition-based estimator achieves the statistical minimax rate over all estimators using the set of N samples. As concrete examples, our theory guarantees that the number of processors m may grow nearly linearly for finite-rank kernels and Gaussian kernels and polynomially in N for Sobolev spaces, which in turn allows for substantial reductions in computational cost. We conclude with experiments on both simulated data and a music-prediction task that complement our theoretical results, exhibiting the computational and statistical benefits of our approach.

preprint2014arXiv

Iterative Hessian sketch: Fast and accurate solution approximation for constrained least-squares

We study randomized sketching methods for approximately solving least-squares problem with a general convex constraint. The quality of a least-squares approximation can be assessed in different ways: either in terms of the value of the quadratic objective function (cost approximation), or in terms of some distance measure between the approximate minimizer and the true minimizer (solution approximation). Focusing on the latter criterion, our first main result provides a general lower bound on any randomized method that sketches both the data matrix and vector in a least-squares problem; as a surprising consequence, the most widely used least-squares sketch is sub-optimal for solution approximation. We then present a new method known as the iterative Hessian sketch, and show that it can be used to obtain approximations to the original least-squares problem using a projection dimension proportional to the statistical complexity of the least-squares minimizer, and a logarithmic number of iterations. We illustrate our general theory with simulations for both unconstrained and constrained versions of least-squares, including $\ell_1$-regularization and nuclear norm constraints. We also numerically demonstrate the practicality of our approach in a real face expression classification experiment.

preprint2014arXiv

Local Privacy, Data Processing Inequalities, and Statistical Minimax Rates

Working under a model of privacy in which data remains private even from the statistician, we study the tradeoff between privacy guarantees and the utility of the resulting statistical estimators. We prove bounds on information-theoretic quantities, including mutual information and Kullback-Leibler divergence, that depend on the privacy guarantees. When combined with standard minimax techniques, including the Le Cam, Fano, and Assouad methods, these inequalities allow for a precise characterization of statistical rates under local privacy constraints. We provide a treatment of several canonical families of problems: mean estimation, parameter estimation in fixed-design regression, multinomial probability estimation, and nonparametric density estimation. For all of these families, we provide lower and upper bounds that match up to constant factors, and exhibit new (optimal) privacy-preserving mechanisms and computationally efficient estimators that achieve the bounds.

preprint2014arXiv

Lower bounds on the performance of polynomial-time algorithms for sparse linear regression

Under a standard assumption in complexity theory (NP not in P/poly), we demonstrate a gap between the minimax prediction risk for sparse linear regression that can be achieved by polynomial-time algorithms, and that achieved by optimal algorithms. In particular, when the design matrix is ill-conditioned, the minimax prediction loss achievable by polynomial-time algorithms can be substantially greater than that of an optimal algorithm. This result is the first known gap between polynomial and optimal algorithms for sparse linear regression, and does not depend on conjectures in average-case complexity.

preprint2014arXiv

Optimal rates for zero-order convex optimization: the power of two function evaluations

We consider derivative-free algorithms for stochastic and non-stochastic convex optimization problems that use only function values rather than gradients. Focusing on non-asymptotic bounds on convergence rates, we show that if pairs of function values are available, algorithms for $d$-dimensional optimization that use gradient estimates based on random perturbations suffer a factor of at most $\sqrt{d}$ in convergence rate over traditional stochastic gradient methods. We establish such results for both smooth and non-smooth cases, sharpening previous analyses that suggested a worse dimension dependence, and extend our results to the case of multiple ($m \ge 2$) evaluations. We complement our algorithmic development with information-theoretic lower bounds on the minimax convergence rate of such problems, establishing the sharpness of our achievable results up to constant (sometimes logarithmic) factors.

preprint2014arXiv

Optimality guarantees for distributed statistical estimation

Large data sets often require performing distributed statistical estimation, with a full data set split across multiple machines and limited communication between machines. To study such scenarios, we define and study some refinements of the classical minimax risk that apply to distributed settings, comparing to the performance of estimators with access to the entire data. Lower bounds on these quantities provide a precise characterization of the minimum amount of communication required to achieve the centralized minimax risk. We study two classes of distributed protocols: one in which machines send messages independently over channels without feedback, and a second allowing for interactive communication, in which a central server broadcasts the messages from a given machine to all other machines. We establish lower bounds for a variety of problems, including location estimation in several families and parameter estimation in different types of regression models. Our results include a novel class of quantitative data-processing inequalities used to characterize the effects of limited communication.

preprint2014arXiv

Randomized Sketches of Convex Programs with Sharp Guarantees

Random projection (RP) is a classical technique for reducing storage and computational costs. We analyze RP-based approximations of convex programs, in which the original optimization problem is approximated by the solution of a lower-dimensional problem. Such dimensionality reduction is essential in computation-limited settings, since the complexity of general convex programming can be quite high (e.g., cubic for quadratic programs, and substantially higher for semidefinite programs). In addition to computational savings, random projection is also useful for reducing memory usage, and has useful properties for privacy-sensitive optimization. We prove that the approximation ratio of this procedure can be bounded in terms of the geometry of constraint set. For a broad class of random projections, including those based on various sub-Gaussian distributions as well as randomized Hadamard and Fourier transforms, the data matrix defining the cost function can be projected down to the statistical dimension of the tangent cone of the constraints at the original solution, which is often substantially smaller than the original dimension. We illustrate consequences of our theory for various cases, including unconstrained and $\ell_1$-constrained least squares, support vector machines, low-rank matrix estimation, and discuss implications on privacy-sensitive optimization and some connections with de-noising and compressed sensing.

preprint2014arXiv

Statistical guarantees for the EM algorithm: From population to sample-based analysis

We develop a general framework for proving rigorous guarantees on the performance of the EM algorithm and a variant known as gradient EM. Our analysis is divided into two parts: a treatment of these algorithms at the population level (in the limit of infinite data), followed by results that apply to updates based on a finite set of samples. First, we characterize the domain of attraction of any global maximizer of the population likelihood. This characterization is based on a novel view of the EM updates as a perturbed form of likelihood ascent, or in parallel, of the gradient EM updates as a perturbed form of standard gradient ascent. Leveraging this characterization, we then provide non-asymptotic guarantees on the EM and gradient EM algorithms when applied to a finite set of samples. We develop consequences of our general theory for three canonical examples of incomplete-data problems: mixture of Gaussians, mixture of regressions, and linear regression with covariates missing completely at random. In each case, our theory guarantees that with a suitable initialization, a relatively small number of EM (or gradient EM) steps will yield (with high probability) an estimate that is within statistical error of the MLE. We provide simulations to confirm this theoretically predicted behavior.

preprint2014arXiv

Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses

We investigate the relationship between the structure of a discrete graphical model and the support of the inverse of a generalized covariance matrix. We show that for certain graph structures, the support of the inverse covariance matrix of indicator variables on the vertices of a graph reflects the conditional independence structure of the graph. Our work extends results that have previously been established only in the context of multivariate Gaussian graphical models, thereby addressing an open question about the significance of the inverse covariance matrix of a non-Gaussian distribution. The proof exploits a combination of ideas from the geometry of exponential families, junction tree theory and convex analysis. These population-level results have various consequences for graph selection methods, both known and novel, including a novel method for structure estimation for missing or corrupted observations. We provide nonasymptotic guarantees for such methods and illustrate the sharpness of these predictions via simulations.

preprint2014arXiv

Support recovery without incoherence: A case for nonconvex regularization

We demonstrate that the primal-dual witness proof method may be used to establish variable selection consistency and $\ell_\infty$-bounds for sparse regression problems, even when the loss function and/or regularizer are nonconvex. Using this method, we derive two theorems concerning support recovery and $\ell_\infty$-guarantees for the regression estimator in a general setting. Our results provide rigorous theoretical justification for the use of nonconvex regularization: For certain nonconvex regularizers with vanishing derivative away from the origin, support recovery consistency may be guaranteed without requiring the typical incoherence conditions present in $\ell_1$-based methods. We then derive several corollaries that illustrate the wide applicability of our method to analyzing composite objective functions involving losses such as least squares, nonconvex modified least squares for errors-in variables linear regression, the negative log likelihood for generalized linear models, and the graphical Lasso. We conclude with empirical studies to corroborate our theoretical predictions.

preprint2013arXiv

A Unified Framework for High-Dimensional Analysis of M-Estimators with Decomposable Regularizers

High-dimensional statistical inference deals with models in which the the number of parameters p is comparable to or larger than the sample size n. Since it is usually impossible to obtain consistent procedures unless $p/n\rightarrow0$, a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and structured matrices, low-rank matrices and combinations thereof. In such settings, a general approach to estimation is to solve a regularized optimization problem, which combines a loss function measuring how well the model fits the data with some regularization function that encourages the assumed structure. This paper provides a unified framework for establishing consistency and convergence rates for such regularized M-estimators under high-dimensional scaling. We state one main theorem and show how it can be used to re-derive some existing results, and also to obtain a number of new results on consistency and convergence rates, in both $\ell_2$-error and related norms. Our analysis also identifies two key properties of loss and regularization functions, referred to as restricted strong convexity and decomposability, that ensure corresponding regularized M-estimators have fast convergence rates and which are optimal in many well-studied cases.

preprint2013arXiv

Distance-based and continuum Fano inequalities with applications to statistical estimation

In this technical note, we give two extensions of the classical Fano inequality in information theory. The first extends Fano's inequality to the setting of estimation, providing lower bounds on the probability that an estimator of a discrete quantity is within some distance $t$ of the quantity. The second inequality extends our bound to a continuum setting and provides a volume-based bound. We illustrate how these inequalities lead to direct and simple proofs of several statistical minimax lower bounds.

preprint2013arXiv

Early stopping and non-parametric regression: An optimal data-dependent stopping rule

The strategy of early stopping is a regularization technique based on choosing a stopping time for an iterative algorithm. Focusing on non-parametric regression in a reproducing kernel Hilbert space, we analyze the early stopping strategy for a form of gradient-descent applied to the least-squares loss function. We propose a data-dependent stopping rule that does not involve hold-out or cross-validation data, and we prove upper bounds on the squared error of the resulting function estimate, measured in either the $L^2(P)$ and $L^2(P_n)$ norm. These upper bounds lead to minimax-optimal rates for various kernel classes, including Sobolev smoothness classes and other forms of reproducing kernel Hilbert spaces. We show through simulation that our stopping rule compares favorably to two other stopping rules, one based on hold-out data and the other based on Stein's unbiased risk estimate. We also establish a tight connection between our early stopping strategy and the solution path of a kernel ridge regression estimator.

preprint2013arXiv

Local Privacy and Minimax Bounds: Sharp Rates for Probability Estimation

We provide a detailed study of the estimation of probability distributions---discrete and continuous---in a stringent setting in which data is kept private even from the statistician. We give sharp minimax rates of convergence for estimation in these locally private settings, exhibiting fundamental tradeoffs between privacy and convergence rate, as well as providing tools to allow movement along the privacy-statistical efficiency continuum. One of the consequences of our results is that Warner's classical work on randomized response is an optimal way to perform survey sampling while maintaining privacy of the respondents.

preprint2013arXiv

Privacy Aware Learning

We study statistical risk minimization problems under a privacy model in which the data is kept confidential even from the learner. In this local privacy framework, we establish sharp upper and lower bounds on the convergence rates of statistical estimation procedures. As a consequence, we exhibit a precise tradeoff between the amount of privacy the data preserves and the utility, as measured by convergence rate, of any statistical estimator or learning procedure.

preprint2013arXiv

Sampled forms of functional PCA in reproducing kernel Hilbert spaces

We consider the sampling problem for functional PCA (fPCA), where the simplest example is the case of taking time samples of the underlying functional components. More generally, we model the sampling operation as a continuous linear map from $\mathcal{H}$ to $\mathbb{R}^m$, where the functional components to lie in some Hilbert subspace $\mathcal{H}$ of $L^2$, such as a reproducing kernel Hilbert space of smooth functions. This model includes time and frequency sampling as special cases. In contrast to classical approach in fPCA in which access to entire functions is assumed, having a limited number m of functional samples places limitations on the performance of statistical procedures. We study these effects by analyzing the rate of convergence of an M-estimator for the subspace spanned by the leading components in a multi-spiked covariance model. The estimator takes the form of regularized PCA, and hence is computationally attractive. We analyze the behavior of this estimator within a nonasymptotic framework, and provide bounds that hold with high probability as a function of the number of statistical samples n and the number of functional samples m. We also derive lower bounds showing that the rates obtained are minimax optimal.

preprint2012arXiv

Belief Propagation for Continuous State Spaces: Stochastic Message-Passing with Quantitative Guarantees

The sum-product or belief propagation (BP) algorithm is a widely used message-passing technique for computing approximate marginals in graphical models. We introduce a new technique, called stochastic orthogonal series message-passing (SOSMP), for computing the BP fixed point in models with continuous random variables. It is based on a deterministic approximation of the messages via orthogonal series expansion, and a stochastic approximation via Monte Carlo estimates of the integral updates of the basis coefficients. We prove that the SOSMP iterates converge to a δ-neighborhood of the unique BP fixed point for any tree-structured graph, and for any graphs with cycles in which the BP updates satisfy a contractivity condition. In addition, we demonstrate how to choose the number of basis coefficients as a function of the desired approximation accuracy δand smoothness of the compatibility functions. We illustrate our theory with both simulated examples and in application to optical flow estimation.

preprint2012arXiv

Discussion: Latent variable graphical model selection via convex optimization

Discussion of "Latent variable graphical model selection via convex optimization" by Venkat Chandrasekaran, Pablo A. Parrilo and Alan S. Willsky [arXiv:1008.1290].

preprint2012arXiv

Fast global convergence of gradient methods for high-dimensional statistical recovery

Many statistical $M$-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-dimensional framework that allows the data dimension $\pdim$ to grow with (and possibly exceed) the sample size $\numobs$. This high-dimensional structure precludes the usual global assumptions---namely, strong convexity and smoothness conditions---that underlie much of classical optimization analysis. We define appropriately restricted versions of these conditions, and show that they are satisfied with high probability for various statistical models. Under these conditions, our theory guarantees that projected gradient descent has a globally geometric rate of convergence up to the \emph{statistical precision} of the model, meaning the typical distance between the true unknown parameter $θ^*$ and an optimal solution $\hatθ$. This result is substantially sharper than previous convergence results, which yielded sublinear convergence, or linear convergence only up to the noise level. Our analysis applies to a wide range of $M$-estimators and statistical models, including sparse linear regression using Lasso ($\ell_1$-regularized regression); group Lasso for block sparsity; log-linear models with regularization; low-rank matrix recovery using nuclear norm regularization; and matrix decomposition. Overall, our analysis reveals interesting connections between statistical precision and computational efficiency in high-dimensional estimation.

preprint2012arXiv

High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity

Although the standard formulations of prediction problems involve fully-observed and noiseless data drawn in an i.i.d. manner, many applications involve noisy and/or missing data, possibly involving dependence, as well. We study these issues in the context of high-dimensional sparse linear regression, and propose novel estimators for the cases of noisy, missing and/or dependent data. Many standard approaches to noisy or missing data, such as those using the EM algorithm, lead to optimization problems that are inherently nonconvex, and it is difficult to establish theoretical guarantees on practical algorithms. While our approach also involves optimizing nonconvex programs, we are able to both analyze the statistical error associated with any global optimum, and more surprisingly, to prove that a simple algorithm based on projected gradient descent will converge in polynomial time to a small neighborhood of the set of all global minimizers. On the statistical side, we provide nonasymptotic bounds that hold with high probability for the cases of noisy, missing and/or dependent data. On the computational side, we prove that under the same types of conditions required for statistical consistency, the projected gradient descent algorithm is guaranteed to converge at a geometric rate to a near-global minimizer. We illustrate these theoretical predictions with simulations, showing close agreement with the predicted scalings.

preprint2012arXiv

Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions

We analyze a class of estimators based on convex relaxation for solving high-dimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation $\mathfrak{X}$ of the sum of an approximately) low rank matrix $Θ^\star$ with a second matrix $Γ^\star$ endowed with a complementary form of low-dimensional structure; this set-up includes many statistical models of interest, including factor analysis, multi-task regression, and robust covariance estimation. We derive a general theorem that bounds the Frobenius norm error for an estimate of the pair $(Θ^\star, Γ^\star)$ obtained by solving a convex optimization problem that combines the nuclear norm with a general decomposable regularizer. Our results utilize a "spikiness" condition that is related to but milder than singular vector incoherence. We specialize our general result to two cases that have been studied in past work: low rank plus an entrywise sparse matrix, and low rank plus a columnwise sparse matrix. For both models, our theory yields non-asymptotic Frobenius error bounds for both deterministic and stochastic noise matrices, and applies to matrices $Θ^\star$ that can be exactly or approximately low rank, and matrices $Γ^\star$ that can be exactly or approximately sparse. Moreover, for the case of stochastic noise matrices and the identity observation operator, we establish matching lower bounds on the minimax error. The sharpness of our predictions is confirmed by numerical simulations.

preprint2012arXiv

Randomized Smoothing for Stochastic Optimization

We analyze convergence rates of stochastic optimization procedures for non-smooth convex optimization problems. By combining randomized smoothing techniques with accelerated gradient methods, we obtain convergence rates of stochastic optimization procedures, both in expectation and with high probability, that have optimal dependence on the variance of the gradient estimates. To the best of our knowledge, these are the first variance-based rates for non-smooth optimization. We give several applications of our results to statistical estimation problems, and provide experimental results that demonstrate the effectiveness of the proposed algorithms. We also describe how a combination of our algorithm with recent work on decentralized optimization yields a distributed stochastic optimization algorithm that is order-optimal.

preprint2012arXiv

Stochastic Belief Propagation: A Low-Complexity Alternative to the Sum-Product Algorithm

The sum-product or belief propagation (BP) algorithm is a widely-used message-passing algorithm for computing marginal distributions in graphical models with discrete variables. At the core of the BP message updates, when applied to a graphical model with pairwise interactions, lies a matrix-vector product with complexity that is quadratic in the state dimension $d$, and requires transmission of a $(d-1)$-dimensional vector of real numbers (messages) to its neighbors. Since various applications involve very large state dimensions, such computation and communication complexities can be prohibitively complex. In this paper, we propose a low-complexity variant of BP, referred to as stochastic belief propagation (SBP). As suggested by the name, it is an adaptively randomized version of the BP message updates in which each node passes randomly chosen information to each of its neighbors. The SBP message updates reduce the computational complexity (per iteration) from quadratic to linear in $d$, without assuming any particular structure of the potentials, and also reduce the communication complexity significantly, requiring only $\log{d}$ bits transmission per edge. Moreover, we establish a number of theoretical guarantees for the performance of SBP, showing that it converges almost surely to the BP fixed point for any tree-structured graph, and for graphs with cycles satisfying a contractivity condition. In addition, for these graphical models, we provide non-asymptotic upper bounds on the convergence rate, showing that the $\ell_{\infty}$ norm of the error vector decays no slower than $O(1/\sqrt{t})$ with the number of iterations $t$ on trees and the mean square error decays as $O(1/t)$ for general graphs. These analysis show that SBP can provably yield reductions in computational and communication complexities for various classes of graphical models.

preprint2012arXiv

Stochastic optimization and sparse statistical recovery: An optimal algorithm for high dimensions

We develop and analyze stochastic optimization algorithms for problems in which the expected loss is strongly convex, and the optimum is (approximately) sparse. Previous approaches are able to exploit only one of these two structures, yielding an $\order(\pdim/T)$ convergence rate for strongly convex objectives in $\pdim$ dimensions, and an $\order(\sqrt{(\spindex \log \pdim)/T})$ convergence rate when the optimum is $\spindex$-sparse. Our algorithm is based on successively solving a series of $\ell_1$-regularized optimization problems using Nesterov's dual averaging algorithm. We establish that the error of our solution after $T$ iterations is at most $\order((\spindex \log\pdim)/T)$, with natural extensions to approximate sparsity. Our results apply to locally Lipschitz losses including the logistic, exponential, hinge and least-squares losses. By recourse to statistical minimax results, we show that our convergence rates are optimal up to multiplicative constant factors. The effectiveness of our approach is also confirmed in numerical simulations, in which we compare to several baselines on a least-squares regression problem.

preprint2011arXiv

Approximation properties of certain operator-induced norms on Hilbert spaces

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L2 norm, and study their approximation properties over Hilbert subspaces of L2 . The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in reproducing kernel Hilbert spaces (RKHS). Our results have implications to the analysis of M-estimators in models based on finite-dimensional linear approximation of functions, and also to some related packing problems.

preprint2011arXiv

Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization

Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardness of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We improve upon known results and obtain tight minimax complexity estimates for various function classes.

preprint2011arXiv

Minimax-optimal rates for sparse additive models over kernel classes via convex programming

Sparse additive models are families of $d$-variate functions that have the additive decomposition $f^* = \sum_{j \in S} f^*_j$, where $S$ is an unknown subset of cardinality $s \ll d$. In this paper, we consider the case where each univariate component function $f^*_j$ lies in a reproducing kernel Hilbert space (RKHS), and analyze a method for estimating the unknown function $f^*$ based on kernels combined with $\ell_1$-type convex regularization. Working within a high-dimensional framework that allows both the dimension $d$ and sparsity $s$ to increase with $n$, we derive convergence rates (upper bounds) in the $L^2(\mathbb{P})$ and $L^2(\mathbb{P}_n)$ norms over the class $\MyBigClass$ of sparse additive models with each univariate function $f^*_j$ in the unit ball of a univariate RKHS with bounded kernel function. We complement our upper bounds by deriving minimax lower bounds on the $L^2(\mathbb{P})$ error, thereby showing the optimality of our method. Thus, we obtain optimal minimax rates for many interesting classes of sparse additive models, including polynomials, splines, and Sobolev classes. We also show that if, in contrast to our univariate conditions, the multivariate function class is assumed to be globally bounded, then much faster estimation rates are possible for any sparsity $s = Ω(\sqrt{n})$, showing that global boundedness is a significant restriction in the high-dimensional setting.

preprint2011arXiv

Restricted strong convexity and weighted matrix completion: Optimal bounds with noise

We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong convexity with respect to weighted Frobenius norm. Using this property, we obtain as corollaries a number of error bounds on matrix completion in the weighted Frobenius norm under noisy sampling and for both exact and near low-rank matrices. Our results are based on measures of the "spikiness" and "low-rankness" of matrices that are less restrictive than the incoherence conditions imposed in previous work. Our technique involves an $M$-estimator that includes controls on both the rank and spikiness of the solution, and we establish non-asymptotic error bounds in weighted Frobenius norm for recovering matrices lying with $\ell_q$-"balls" of bounded spikiness. Using information-theoretic methods, we show that no algorithm can achieve better estimates (up to a logarithmic factor) over these same sets, showing that our conditions on matrices and associated rates are essentially optimal.

preprint2011arXiv

Support union recovery in high-dimensional multivariate regression

In multivariate regression, a $K$-dimensional response vector is regressed upon a common set of $p$ covariates, with a matrix $B^*\in\mathbb{R}^{p\times K}$ of regression coefficients. We study the behavior of the multivariate group Lasso, in which block regularization based on the $\ell_1/\ell_2$ norm is used for support union recovery, or recovery of the set of $s$ rows for which $B^*$ is nonzero. Under high-dimensional scaling, we show that the multivariate group Lasso exhibits a threshold for the recovery of the exact row pattern with high probability over the random design and noise that is specified by the sample complexity parameter $θ(n,p,s):=n/[2ψ(B^*)\log(p-s)]$. Here $n$ is the sample size, and $ψ(B^*)$ is a sparsity-overlap function measuring a combination of the sparsities and overlaps of the $K$-regression coefficient vectors that constitute the model. We prove that the multivariate group Lasso succeeds for problem sequences $(n,p,s)$ such that $θ(n,p,s)$ exceeds a critical level $θ_u$, and fails for sequences such that $θ(n,p,s)$ lies below a critical level $θ_{\ell}$. For the special case of the standard Gaussian ensemble, we show that $θ_{\ell}=θ_u$ so that the characterization is sharp. The sparsity-overlap function $ψ(B^*)$ reveals that, if the design is uncorrelated on the active rows, $\ell_1/\ell_2$ regularization for multivariate regression never harms performance relative to an ordinary Lasso approach and can yield substantial improvements in sample complexity (up to a factor of $K$) when the coefficient vectors are suitably orthogonal. For more general designs, it is possible for the ordinary Lasso to outperform the multivariate group Lasso. We complement our analysis with simulations that demonstrate the sharpness of our theoretical results, even for relatively small problems.

preprint2010arXiv

High-dimensional Ising model selection using ${\ell_1}$-regularized logistic regression

We consider the problem of estimating the graph associated with a binary Ising Markov random field. We describe a method based on $\ell_1$-regularized logistic regression, in which the neighborhood of any given node is estimated by performing logistic regression subject to an $\ell_1$-constraint. The method is analyzed under high-dimensional scaling in which both the number of nodes $p$ and maximum neighborhood size $d$ are allowed to grow as a function of the number of observations $n$. Our main results provide sufficient conditions on the triple $(n,p,d)$ and the model parameters for the method to succeed in consistently estimating the neighborhood of every node in the graph simultaneously. With coherence conditions imposed on the population Fisher information matrix, we prove that consistent neighborhood selection can be obtained for sample sizes $n=Ω(d^3\log p)$ with exponentially decaying error. When these same conditions are imposed directly on the sample matrices, we show that a reduced sample size of $n=Ω(d^2\log p)$ suffices for the method to estimate neighborhoods consistently. Although this paper focuses on the binary graphical models, we indicate how a generalization of the method of the paper would apply to general discrete Markov random fields.

preprint2009arXiv

Estimating divergence functionals and the likelihood ratio by convex risk minimization

We develop and analyze $M$-estimation methods for divergence functionals and the likelihood ratios of two probability distributions. Our method is based on a non-asymptotic variational characterization of $f$-divergences, which allows the problem of estimating divergences to be tackled via convex empirical risk optimization. The resulting estimators are simple to implement, requiring only the solution of standard convex programs. We present an analysis of consistency and convergence for these estimators. Given conditions only on the ratios of densities, we show that our estimators can achieve optimal minimax rates for the likelihood ratio and the divergence functionals in certain regimes. We derive an efficient optimization algorithm for computing our estimates, and illustrate their convergence behavior and practical viability by simulations.

preprint2009arXiv

Minimax rates of estimation for high-dimensional linear regression over $\ell_q$-balls

Consider the standard linear regression model $\y = \Xmat \betastar + w$, where $\y \in \real^\numobs$ is an observation vector, $\Xmat \in \real^{\numobs \times \pdim}$ is a design matrix, $\betastar \in \real^\pdim$ is the unknown regression vector, and $w \sim \mathcal{N}(0, σ^2 I)$ is additive Gaussian noise. This paper studies the minimax rates of convergence for estimation of $\betastar$ for $\ell_\rpar$-losses and in the $\ell_2$-prediction loss, assuming that $\betastar$ belongs to an $\ell_{\qpar}$-ball $\Ballq(\myrad)$ for some $\qpar \in [0,1]$. We show that under suitable regularity conditions on the design matrix $\Xmat$, the minimax error in $\ell_2$-loss and $\ell_2$-prediction loss scales as $\Rq \big(\frac{\log \pdim}{n}\big)^{1-\frac{\qpar}{2}}$. In addition, we provide lower bounds on minimax risks in $\ell_{\rpar}$-norms, for all $\rpar \in [1, +\infty], \rpar \neq \qpar$. Our proofs of the lower bounds are information-theoretic in nature, based on Fano's inequality and results on the metric entropy of the balls $\Ballq(\myrad)$, whereas our proofs of the upper bounds are direct and constructive, involving direct analysis of least-squares over $\ell_{\qpar}$-balls. For the special case $q = 0$, a comparison with $\ell_2$-risks achieved by computationally efficient $\ell_1$-relaxations reveals that although such methods can achieve the minimax rates up to constant factors, they require slightly stronger assumptions on the design matrix $\Xmat$ than algorithms involving least-squares over the $\ell_0$-ball.

preprint2008arXiv

On optimal quantization rules for some problems in sequential decentralized detection

We consider the design of systems for sequential decentralized detection, a problem that entails several interdependent choices: the choice of a stopping rule (specifying the sample size), a global decision function (a choice between two competing hypotheses), and a set of quantization rules (the local decisions on the basis of which the global decision is made). This paper addresses an open problem of whether in the Bayesian formulation of sequential decentralized detection, optimal local decision functions can be found within the class of stationary rules. We develop an asymptotic approximation to the optimal cost of stationary quantization rules and exploit this approximation to show that stationary quantizers are not optimal in a broad class of settings. We also consider the class of blockwise stationary quantizers, and show that asymptotically optimal quantizers are likelihood-based threshold rules.

preprint2008arXiv

Probabilistic Analysis of Linear Programming Decoding

We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous non-asymptotic results for LDPC codes, and in particular exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a generalized matching. An interesting by-product of our analysis is to establish the existence of ``probabilistic expansion'' in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst-case setting.

Martin J. Wainwright

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74 published item(s)

TILT: Target-induced loss tilting under covariate shift

Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency

A Diffusion Process Perspective on Posterior Contraction Rates for Parameters

A new similarity measure for covariate shift with applications to nonparametric regression

Instability, Computational Efficiency and Statistical Accuracy

Instance-Dependent Confidence and Early Stopping for Reinforcement Learning

Stabilizing Q-learning with Linear Architectures for Provably Efficient Learning

A Permutation-based Model for Crowd Labeling: Optimal Estimation and Robustness

Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients

Minimax Off-Policy Evaluation for Multi-Armed Bandits

Derivative-Free Methods for Policy Optimization: Guarantees for Linear Quadratic Systems

FedSplit: An algorithmic framework for fast federated optimization

High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm

HopSkipJumpAttack: A Query-Efficient Decision-Based Attack

Instance-dependent $\ell_\infty$-bounds for policy evaluation in tabular reinforcement learning

Is Temporal Difference Learning Optimal? An Instance-Dependent Analysis

Lower bounds in multiple testing: A framework based on derandomized proxies

On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration

Singularity, Misspecification, and the Convergence Rate of EM

Active Ranking from Pairwise Comparisons and when Parametric Assumptions Don't Help

Asymptotic behavior of $\ell_p$-based Laplacian regularization in semi-supervised learning

Convexified Convolutional Neural Networks

Feeling the Bern: Adaptive Estimators for Bernoulli Probabilities of Pairwise Comparisons

Function-Specific Mixing Times and Concentration Away from Equilibrium

Linear Regression with an Unknown Permutation: Statistical and Computational Limits

Local Maxima in the Likelihood of Gaussian Mixture Models: Structural Results and Algorithmic Consequences

Simple, Robust and Optimal Ranking from Pairwise Comparisons

Stochastically Transitive Models for Pairwise Comparisons: Statistical and Computational Issues

A More Powerful Two-Sample Test in High Dimensions using Random Projection

Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds

Estimation from Pairwise Comparisons: Sharp Minimax Bounds with Topology Dependence

Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees

Learning Halfspaces and Neural Networks with Random Initialization

Newton Sketch: A Linear-time Optimization Algorithm with Linear-Quadratic Convergence

On the Computational Complexity of High-Dimensional Bayesian Variable Selection

Optimal prediction for sparse linear models? Lower bounds for coordinate-separable M-estimators

Randomized sketches for kernels: Fast and optimal non-parametric regression

Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima

Statistical and Computational Guarantees for the Baum-Welch Algorithm

The geometry of kernelized spectral clustering

Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates

Iterative Hessian sketch: Fast and accurate solution approximation for constrained least-squares

Local Privacy, Data Processing Inequalities, and Statistical Minimax Rates

Lower bounds on the performance of polynomial-time algorithms for sparse linear regression

Optimal rates for zero-order convex optimization: the power of two function evaluations

Optimality guarantees for distributed statistical estimation

Randomized Sketches of Convex Programs with Sharp Guarantees

Statistical guarantees for the EM algorithm: From population to sample-based analysis

Structure estimation for discrete graphical models: Generalized covariance matrices and their inverses

Support recovery without incoherence: A case for nonconvex regularization

A Unified Framework for High-Dimensional Analysis of M-Estimators with Decomposable Regularizers

Distance-based and continuum Fano inequalities with applications to statistical estimation

Early stopping and non-parametric regression: An optimal data-dependent stopping rule

Local Privacy and Minimax Bounds: Sharp Rates for Probability Estimation

Privacy Aware Learning

Sampled forms of functional PCA in reproducing kernel Hilbert spaces

Belief Propagation for Continuous State Spaces: Stochastic Message-Passing with Quantitative Guarantees

Discussion: Latent variable graphical model selection via convex optimization

Fast global convergence of gradient methods for high-dimensional statistical recovery

High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity

Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions

Randomized Smoothing for Stochastic Optimization

Stochastic Belief Propagation: A Low-Complexity Alternative to the Sum-Product Algorithm

Stochastic optimization and sparse statistical recovery: An optimal algorithm for high dimensions

Approximation properties of certain operator-induced norms on Hilbert spaces

Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization

Minimax-optimal rates for sparse additive models over kernel classes via convex programming

Restricted strong convexity and weighted matrix completion: Optimal bounds with noise

Support union recovery in high-dimensional multivariate regression

High-dimensional Ising model selection using ${\ell_1}$-regularized logistic regression

Estimating divergence functionals and the likelihood ratio by convex risk minimization

Minimax rates of estimation for high-dimensional linear regression over $\ell_q$-balls

On optimal quantization rules for some problems in sequential decentralized detection

Probabilistic Analysis of Linear Programming Decoding