Sparse recovery by reduced variance stochastic approximation

TL;DR

This paper introduces a multistage stochastic optimization algorithm for sparse signal recovery that achieves linear convergence and improved reliability, applicable to noisy observations in generalized linear regression.

Contribution

Develops a novel multistage stochastic mirror descent algorithm with linear convergence for sparse recovery, incorporating median-of-means techniques for enhanced reliability.

Findings

Algorithm achieves linear convergence in early phases.

Performance bounds are close to best known accuracy under weak assumptions.

Effective in recovering sparse and low-rank signals in noisy settings.

Abstract

In this paper, we discuss application of iterative Stochastic Optimization routines to the problem of sparse signal recovery from noisy observation. Using Stochastic Mirror Descent algorithm as a building block, we develop a multistage procedure for recovery of sparse solutions to Stochastic Optimization problem under assumption of smoothness and quadratic minoration on the expected objective. An interesting feature of the proposed algorithm is linear convergence of the approximate solution during the preliminary phase of the routine when the component of stochastic error in the gradient observation which is due to bad initial approximation of the optimal solution is larger than the "ideal" asymptotic error component owing to observation noise "at the optimal solution." We also show how one can straightforwardly enhance reliability of the corresponding solution by using Median-of-Means like techniques. We illustrate the performance of the proposed algorithms in application to classical problems of recovery of sparse and low rank signals in generalized linear regression framework. We show, under rather weak assumption on the regressor and noise distributions, how they lead to parameter estimates which obey (up to factors which are logarithmic in problem dimension and confidence level) the best known to us accuracy bounds.