Stochastic Mirror Descent for Large-Scale Sparse Recovery

TL;DR

This paper introduces a stochastic mirror descent method for high-dimensional sparse recovery, providing theoretical convergence guarantees and demonstrating its effectiveness on sparse regression problems with numerical experiments.

Contribution

It develops a multistage stochastic optimization approach using Composite Stochastic Mirror Descent for sparse parameter estimation with proven convergence rates.

Findings

Achieves linear convergence in the initial phase and sublinear in the asymptotic phase.

Attains optimal convergence rates under weak assumptions in sparse regression.

Numerical results confirm the method's effectiveness on high-dimensional data.

Abstract

In this paper we discuss an application of Stochastic Approximation to statistical estimation of high-dimensional sparse parameters. The proposed solution reduces to resolving a penalized stochastic optimization problem on each stage of a multistage algorithm; each problem being solved to a prescribed accuracy by the non-Euclidean Composite Stochastic Mirror Descent (CSMD) algorithm. Assuming that the problem objective is smooth and quadratically minorated and stochastic perturbations are sub-Gaussian, our analysis prescribes the method parameters which ensure fast convergence of the estimation error (the radius of a confidence ball of a given norm around the approximate solution). This convergence is linear during the first "preliminary" phase of the routine and is sublinear during the second "asymptotic" phase. We consider an application of the proposed approach to sparse Generalized Linear Regression problem. In this setting, we show that the proposed algorithm attains the optimal convergence of the estimation error under weak assumptions on the regressor distribution. We also present a numerical study illustrating the performance of the algorithm on high-dimensional simulation data.