Sparse recovery by reduced variance stochastic approximation

Author:

Juditsky Anatoli1,Kulunchakov Andrei2,Tsyntseus Hlib3

Affiliation:

1. LJK, Université Grenoble Alpes, 700 Avenue Centrale , 38401 Domaine Universitaire de Saint-Martin-d’Héres, France

2. Université Grenoble Alpes , Inria, CNRS, Grenoble INP, LJK, Grenoble 38000, France

3. Google France , 75009 Paris, France

Abstract

AbstractIn 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 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 the 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 accuracy bounds.

Funder

Multidisciplinary Institute in Artificial intelligence MIAI @ Grenoble Alpes

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Theory and Mathematics,Numerical Analysis,Statistics and Probability,Analysis

Reference59 articles.

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