Randomized subspace correction methods for convex optimization

TL;DR

This paper presents a unified abstract framework for randomized subspace correction methods applicable to convex optimization, encompassing various existing algorithms and extending to more general problem settings.

Contribution

It introduces a general framework that unifies and extends existing subspace correction algorithms for convex optimization, including inexact solvers and weaker assumptions.

Findings

Provides convergence rate analysis under minimal and practical assumptions.

Extends applicability to arbitrary space decompositions and weaker convexity conditions.

Applicable to nonlinear PDEs, imaging, and data science problems.

Abstract

This paper introduces an abstract framework for randomized subspace correction methods for convex optimization, which unifies and generalizes a broad class of existing algorithms, including domain decomposition, multigrid, and block coordinate descent methods. We provide a convergence rate analysis ranging from minimal assumptions to more practical settings, such as sharpness and strong convexity. While most existing studies on block coordinate descent methods focus on nonoverlapping decompositions and smooth or strongly convex problems, our framework extends to more general settings involving arbitrary space decompositions, inexact local solvers, and problems with weaker smoothness or convexity assumptions. The proposed framework is broadly applicable to convex optimization problems arising in areas such as nonlinear partial differential equations, imaging, and data science.