Stochastic Majorization-Minimization Algorithms for Large-Scale Optimization

TL;DR

This paper introduces a scalable stochastic majorization-minimization framework for large-scale optimization, achieving fast convergence rates for convex problems and almost sure convergence for non-convex problems, with applications in logistic regression, sparse estimation, and matrix factorization.

Contribution

It presents a novel stochastic majorization-minimization scheme that extends classical methods to large-scale and possibly non-convex problems, with proven convergence guarantees.

Findings

Achieves $O(1/ abla{n})$ convergence rate for convex problems.

Almost sure convergence to stationary points for non-convex problems.

Experimental results match state-of-the-art solvers in large-scale logistic regression.

Abstract

Majorization-minimization algorithms consist of iteratively minimizing a majorizing surrogate of an objective function. Because of its simplicity and its wide applicability, this principle has been very popular in statistics and in signal processing. In this paper, we intend to make this principle scalable. We introduce a stochastic majorization-minimization scheme which is able to deal with large-scale or possibly infinite data sets. When applied to convex optimization problems under suitable assumptions, we show that it achieves an expected convergence rate of $O(1/\sqrt{n})$ after $n$ iterations, and of $O(1/n)$ for strongly convex functions. Equally important, our scheme almost surely converges to stationary points for a large class of non-convex problems. We develop several efficient algorithms based on our framework. First, we propose a new stochastic proximal gradient method, which experimentally matches state-of-the-art solvers for large-scale $\ell_1$-logistic regression. Second, we develop an online DC programming algorithm for non-convex sparse estimation. Finally, we demonstrate the effectiveness of our approach for solving large-scale structured matrix factorization problems.