arXiv subjects

Martin J. Wainwright

Publications and source records attributed to Martin J. Wainwright.

At least 145 records · Page 8Linked to original sources

arXiv2012

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.

cs.IT↗

arXiv2012

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.

math.OC↗

arXiv2012

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.

stat.ML↗

arXiv2011

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.

math.ST↗

arXiv2011

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.

stat.ML↗

arXiv2011

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.

math.ST↗

arXiv2011

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.

cs.IT↗

arXiv2011

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.

stat.ML↗

arXiv2010

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.

math.ST↗

arXiv2009

Estimation of (near) low-rank matrices with noise and high-dimensional scaling

High-dimensional inference refers to problems of statistical estimation in which the ambient dimension of the data may be comparable to or possibly even larger than the sample size. We study an instance of high-dimensional inference in which the goal is to estimate a matrix $Θ^* \in \real^{k \times p}$ on the basis of $N$ noisy observations, and the unknown matrix $Θ^*$ is assumed to be either exactly low rank, or ``near'' low-rank, meaning that it can be well-approximated by a matrix with low rank. We consider an $M$-estimator based on regularization by the trace or nuclear norm over matrices, and analyze its performance under high-dimensional scaling. We provide non-asymptotic bounds on the Frobenius norm error that hold for a general class of noisy observation models, and then illustrate their consequences for a number of specific matrix models, including low-rank multivariate or multi-task regression, system identification in vector autoregressive processes, and recovery of low-rank matrices from random projections. Simulation results show excellent agreement with the high-dimensional scaling of the error predicted by our theory.

math.ST↗

arXiv2009

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.

math.ST↗

arXiv2009

High-dimensional analysis of semidefinite relaxations for sparse principal components

Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the ``large $p$, small $n$'' setting, in which the problem dimension $p$ is comparable to or larger than the sample size $n$. This paper studies PCA in this high-dimensional regime, but under the additional assumption that the maximal eigenvector is sparse, say, with at most $k$ nonzero components. We consider a spiked covariance model in which a base matrix is perturbed by adding a $k$-sparse maximal eigenvector, and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector, as follows: (a) a simple diagonal thresholding method, which transitions from success to failure as a function of the rescaled sample size $θ_{\mathrm{dia}}(n,p,k)=n/[k^2\log(p-k)]$; and (b) a more sophisticated semidefinite programming (SDP) relaxation, which succeeds once the rescaled sample size $θ_{\mathrm{sdp}}(n,p,k)=n/[k\log(p-k)]$ is larger than a critical threshold. In addition, we prove that no method, including the best method which has exponential-time complexity, can succeed in recovering the support if the order parameter $θ_{\mathrm{sdp}}(n,p,k)$ is below a threshold. Our results thus highlight an interesting trade-off between computational and statistical efficiency in high-dimensional inference.

math.ST↗

arXiv2009

Information-theoretic limits of selecting binary graphical models in high dimensions

The problem of graphical model selection is to correctly estimate the graph structure of a Markov random field given samples from the underlying distribution. We analyze the information-theoretic limitations of the problem of graph selection for binary Markov random fields under high-dimensional scaling, in which the graph size $p$ and the number of edges $k$, and/or the maximal node degree $d$ are allowed to increase to infinity as a function of the sample size $n$. For pairwise binary Markov random fields, we derive both necessary and sufficient conditions for correct graph selection over the class $\mathcal{G}_{p,k}$ of graphs on $p$ vertices with at most $k$ edges, and over the class $\mathcal{G}_{p,d}$ of graphs on $p$ vertices with maximum degree at most $d$. For the class $\mathcal{G}_{p, k}$, we establish the existence of constants $c$ and $c'$ such that if $\numobs < c k \log p$, any method has error probability at least 1/2 uniformly over the family, and we demonstrate a graph decoder that succeeds with high probability uniformly over the family for sample sizes $\numobs > c' k^2 \log p$. Similarly, for the class $\mathcal{G}_{p,d}$, we exhibit constants $c$ and $c'$ such that for $n < c d^2 \log p$, any method fails with probability at least 1/2, and we demonstrate a graph decoder that succeeds with high probability for $n > c' d^3 \log p$.

cs.IT↗

arXiv2009

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.

math.ST↗

arXiv2009

On surrogate loss functions and $f$-divergences

The goal of binary classification is to estimate a discriminant function $γ$ from observations of covariate vectors and corresponding binary labels. We consider an elaboration of this problem in which the covariates are not available directly but are transformed by a dimensionality-reducing quantizer $Q$. We present conditions on loss functions such that empirical risk minimization yields Bayes consistency when both the discriminant function and the quantizer are estimated. These conditions are stated in terms of a general correspondence between loss functions and a class of functionals known as Ali-Silvey or $f$-divergence functionals. Whereas this correspondence was established by Blackwell [Proc. 2nd Berkeley Symp. Probab. Statist. 1 (1951) 93--102. Univ. California Press, Berkeley] for the 0--1 loss, we extend the correspondence to the broader class of surrogate loss functions that play a key role in the general theory of Bayes consistency for binary classification. Our result makes it possible to pick out the (strict) subset of surrogate loss functions that yield Bayes consistency for joint estimation of the discriminant function and the quantizer.

math.ST↗

arXiv2008

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.

math.ST↗

arXiv2008

High-dimensional covariance estimation by minimizing $\ell_1$-penalized log-determinant divergence

Given i.i.d. observations of a random vector $X \in \mathbb{R}^p$, we study the problem of estimating both its covariance matrix $Σ^*$, and its inverse covariance or concentration matrix {$Θ^* = (Σ^*)^{-1}$.} We estimate $Θ^*$ by minimizing an $\ell_1$-penalized log-determinant Bregman divergence; in the multivariate Gaussian case, this approach corresponds to $\ell_1$-penalized maximum likelihood, and the structure of $Θ^*$ is specified by the graph of an associated Gaussian Markov random field. We analyze the performance of this estimator under high-dimensional scaling, in which the number of nodes in the graph $p$, the number of edges $s$ and the maximum node degree $d$, are allowed to grow as a function of the sample size $n$. In addition to the parameters $(p,s,d)$, our analysis identifies other key quantities covariance matrix $Σ^*$; and (b) the $\ell_\infty$ operator norm of the sub-matrix $Γ^*_{S S}$, where $S$ indexes the graph edges, and $Γ^* = (Θ^*)^{-1} \otimes (Θ^*)^{-1}$; and (c) a mutual incoherence or irrepresentability measure on the matrix $Γ^*$ and (d) the rate of decay $1/f(n,δ)$ on the probabilities $ \{|\hatΣ^n_{ij}- Σ^*_{ij}| > δ\}$, where $\hatΣ^n$ is the sample covariance based on $n$ samples. Our first result establishes consistency of our estimate $\hatΘ$ in the elementwise maximum-norm. This in turn allows us to derive convergence rates in Frobenius and spectral norms, with improvements upon existing results for graphs with maximum node degrees $d = o(\sqrt{s})$. In our second result, we show that with probability converging to one, the estimate $\hatΘ$ correctly specifies the zero pattern of the concentration matrix $Θ^*$.

stat.ML↗

arXiv2008

Information-theoretic limits on sparse signal recovery: Dense versus sparse measurement matrices

We study the information-theoretic limits of exactly recovering the support of a sparse signal using noisy projections defined by various classes of measurement matrices. Our analysis is high-dimensional in nature, in which the number of observations $n$, the ambient signal dimension $p$, and the signal sparsity $k$ are all allowed to tend to infinity in a general manner. This paper makes two novel contributions. First, we provide sharper necessary conditions for exact support recovery using general (non-Gaussian) dense measurement matrices. Combined with previously known sufficient conditions, this result yields sharp characterizations of when the optimal decoder can recover a signal for various scalings of the sparsity $k$ and sample size $n$, including the important special case of linear sparsity ($k = Θ(p)$) using a linear scaling of observations ($n = Θ(p)$). Our second contribution is to prove necessary conditions on the number of observations $n$ required for asymptotically reliable recovery using a class of $γ$-sparsified measurement matrices, where the measurement sparsity $γ(n, p, k) \in (0,1]$ corresponds to the fraction of non-zero entries per row. Our analysis allows general scaling of the quadruplet $(n, p, k, γ)$, and reveals three different regimes, corresponding to whether measurement sparsity has no effect, a minor effect, or a dramatic effect on the information-theoretic limits of the subset recovery problem.

math.ST↗