Parallel Direction Method of Multipliers

TL;DR

This paper introduces PDMM, a parallel randomized block coordinate method for multi-block convex optimization with linear constraints, demonstrating convergence and superior performance over existing methods.

Contribution

The paper proposes PDMM, a novel parallel randomized block coordinate method for multi-block convex problems, with proven convergence and applicability to overlapping blocks.

Findings

PDMM converges globally with constant step size.

PDMM outperforms state-of-the-art methods in experiments.

Effective for overlapping block structures.

Abstract

We consider the problem of minimizing block-separable convex functions subject to linear constraints. While the Alternating Direction Method of Multipliers (ADMM) for two-block linear constraints has been intensively studied both theoretically and empirically, in spite of some preliminary work, effective generalizations of ADMM to multiple blocks is still unclear. In this paper, we propose a randomized block coordinate method named Parallel Direction Method of Multipliers (PDMM) to solve the optimization problems with multi-block linear constraints. PDMM randomly updates some primal and dual blocks in parallel, behaving like parallel randomized block coordinate descent. We establish the global convergence and the iteration complexity for PDMM with constant step size. We also show that PDMM can do randomized block coordinate descent on overlapping blocks. Experimental results show that PDMM performs better than state-of-the-arts methods in two applications, robust principal component analysis and overlapping group lasso.