Randomized subspace correction methods for convex optimization
This work provides a generalized approach for solving convex optimization problems in fields like PDEs and data science, though it is incremental in extending prior methods.
The paper introduces a unified framework for randomized subspace correction methods in convex optimization, extending existing algorithms to handle arbitrary decompositions, inexact solvers, and weaker assumptions, with convergence rates analyzed under various conditions.
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.