Stochastic Coordinate Minimization with Progressive Precision for Stochastic Convex Optimization

TL;DR

This paper introduces a novel stochastic coordinate minimization framework with adaptive precision control, achieving optimal regret for convex functions and enabling scalable, distributed large-scale optimization.

Contribution

It develops an optimal control strategy for precision in coordinate minimization, extending low-dimensional routines to high-dimensional stochastic convex optimization.

Findings

Achieves order-optimal regret for strongly convex functions.

Framework is independent of specific coordinate minimization routines.

Suitable for online, distributed, large-scale optimization.

Abstract

A framework based on iterative coordinate minimization (CM) is developed for stochastic convex optimization. Given that exact coordinate minimization is impossible due to the unknown stochastic nature of the objective function, the crux of the proposed optimization algorithm is an optimal control of the minimization precision in each iteration. We establish the optimal precision control and the resulting order-optimal regret performance for strongly convex and separably nonsmooth functions. An interesting finding is that the optimal progression of precision across iterations is independent of the low-dimensional CM routine employed, suggesting a general framework for extending low-dimensional optimization routines to high-dimensional problems. The proposed algorithm is amenable to online implementation and inherits the scalability and parallelizability properties of CM for large-scale optimization. Requiring only a sublinear order of message exchanges, it also lends itself well to distributed computing as compared with the alternative approach of coordinate gradient descent.