arXiv subjects

Martin J. Wainwright

Publications and source records attributed to Martin J. Wainwright.

At least 109 records · Page 6Linked to original sources

arXiv2016

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.

cs.LG↗

arXiv2016

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.

cs.LG↗

arXiv2016

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 $P$.

cs.LG↗

arXiv2015

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.

stat.ML↗

arXiv2015

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.

math.ST↗

arXiv2015

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 $η<1/2$.

cs.LG↗

arXiv2015

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.

math.ST↗

arXiv2015

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.

math.ST↗

arXiv2015

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.

math.ST↗

arXiv2015

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.

math.OC↗

arXiv2015

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.

cs.LG↗

arXiv2015

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.

math.ST↗

arXiv2015

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.

cs.DS↗

arXiv2015

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.

stat.ML↗

arXiv2015

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.

math.ST↗

arXiv2014

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.

math.ST↗

arXiv2014

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.

math.OC↗

arXiv2014

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.

math.ST↗