Accelerated Primal-Dual Proximal Block Coordinate Updating Methods for Constrained Convex Optimization

TL;DR

This paper introduces an accelerated primal-dual block coordinate update method for large-scale constrained convex optimization, achieving faster convergence rates and improved stability over previous methods, especially under strong convexity or independence conditions.

Contribution

It proposes a novel accelerated primal-dual BCU algorithm with proven $O(1/t^2)$ convergence and linear convergence under specific conditions, enhancing efficiency for large-scale problems.

Findings

Achieves $O(1/t^2)$ convergence rate under weak convexity.

Attains linear convergence when one block is independent.

Demonstrates stable performance with minimal parameter tuning.

Abstract

Block Coordinate Update (BCU) methods enjoy low per-update computational complexity because every time only one or a few block variables would need to be updated among possibly a large number of blocks. They are also easily parallelized and thus have been particularly popular for solving problems involving large-scale dataset and/or variables. In this paper, we propose a primal-dual BCU method for solving linearly constrained convex program in multi-block variables. The method is an accelerated version of a primal-dual algorithm proposed by the authors, which applies randomization in selecting block variables to update and establishes an $O(1/t)$ convergence rate under weak convexity assumption. We show that the rate can be accelerated to $O(1/t^2)$ if the objective is strongly convex. In addition, if one block variable is independent of the others in the objective, we then show that the algorithm can be modified to achieve a linear rate of convergence. The numerical experiments show that the accelerated method performs stably with a single set of parameters while the original method needs to tune the parameters for different datasets in order to achieve a comparable level of performance.