Efficiency of Coordinate Descent Methods on Huge-Scale Optimization Problems

TLDR

Large-scale optimization problems make even basic full-dimensional vector operations prohibitively expensive. The authors aim to develop coordinate descent methods that use random partial updates to efficiently solve huge-scale optimization problems. They implement random partial update coordinate descent with both constrained and unconstrained formulations, including an accelerated variant. The methods achieve provable convergence rates that surpass standard deterministic bounds for some objective classes, and numerical tests confirm high efficiency on very large problems.

Abstract

In this paper we propose new methods for solving huge-scale optimization problems. For problems of this size, even the simplest full-dimensional vector operations are very expensive. Hence, we propose to apply an optimization technique based on random partial update of decision variables. For these methods, we prove the global estimates for the rate of convergence. Surprisingly, for certain classes of objective functions, our results are better than the standard worst-case bounds for deterministic algorithms. We present constrained and unconstrained versions of the method and its accelerated variant. Our numerical test confirms a high efficiency of this technique on problems of very big size.