On the Linear Convergence of the Approximate Proximal Splitting Method for Non-Smooth Convex Optimization

TL;DR

This paper introduces a general class of approximate proximal splitting methods for convex optimization, proving their linear convergence under certain conditions, which applies to several well-known algorithms and problems like compressive sensing.

Contribution

The paper establishes the linear convergence of a broad class of APS methods under a local error bound, extending convergence results to algorithms like BCD without strong convexity.

Findings

APS methods include PSM, BCD, and gradient projection.

Linear convergence is proven under local error bounds.

Results apply to compressive sensing and sparse group LASSO.

Abstract

Consider the problem of minimizing the sum of two convex functions, one being smooth and the other non-smooth. In this paper, we introduce a general class of approximate proximal splitting (APS) methods for solving such minimization problems. Methods in the APS class include many well-known algorithms such as the proximal splitting method (PSM), the block coordinate descent method (BCD) and the approximate gradient projection methods for smooth convex optimization. We establish the linear convergence of APS methods under a local error bound assumption. Since the latter is known to hold for compressive sensing and sparse group LASSO problems, our analysis implies the linear convergence of the BCD method for these problems without strong convexity assumption.