Parallel Successive Convex Approximation for Nonsmooth Nonconvex Optimization

TL;DR

This paper introduces a parallel block coordinate descent method for nonsmooth nonconvex optimization, enabling simultaneous updates of variable blocks using convex approximations, with proven convergence and practical efficiency.

Contribution

It proposes an inexact parallel BCD algorithm that allows multiple variable blocks to be updated simultaneously, extending convergence analysis to both convex and non-convex cases.

Findings

Cyclic block selection can outperform randomized selection in Lasso problems.

The method converges for both convex and non-convex objectives.

Numerical experiments demonstrate practical efficiency of the proposed approach.

Abstract

Consider the problem of minimizing the sum of a smooth (possibly non-convex) and a convex (possibly nonsmooth) function involving a large number of variables. A popular approach to solve this problem is the block coordinate descent (BCD) method whereby at each iteration only one variable block is updated while the remaining variables are held fixed. With the recent advances in the developments of the multi-core parallel processing technology, it is desirable to parallelize the BCD method by allowing multiple blocks to be updated simultaneously at each iteration of the algorithm. In this work, we propose an inexact parallel BCD approach where at each iteration, a subset of the variables is updated in parallel by minimizing convex approximations of the original objective function. We investigate the convergence of this parallel BCD method for both randomized and cyclic variable selection rules. We analyze the asymptotic and non-asymptotic convergence behavior of the algorithm for both convex and non-convex objective functions. The numerical experiments suggest that for a special case of Lasso minimization problem, the cyclic block selection rule can outperform the randomized rule.