On solving large scale polynomial convex problems by randomized first-order algorithms

TL;DR

This paper introduces a randomized first-order saddle point method for large-scale polynomial convex problems, replacing expensive exact gradients with cheaper unbiased estimates, leading to accelerated solutions.

Contribution

It extends randomized acceleration techniques from bilinear to polynomial saddle point problems, improving efficiency in large-scale convex optimization.

Findings

Randomized gradients reduce computational cost.

Acceleration increases with problem size.

Numerical experiments confirm theoretical advantages.

Abstract

One of the most attractive recent approaches to processing well-structured large-scale convex optimization problems is based on smooth convex-concave saddle point reformu-lation of the problem of interest and solving the resulting problem by a fast First Order saddle point method utilizing smoothness of the saddle point cost function. In this paper, we demonstrate that when the saddle point cost function is polynomial, the precise gra-dients of the cost function required by deterministic First Order saddle point algorithms and becoming prohibitively computationally expensive in the extremely large-scale case, can be replaced with incomparably cheaper computationally unbiased random estimates of the gradients. We show that for large-scale problems with favourable geometry, this randomization accelerates, progressively as the sizes of the problem grow, the solution process. This extends significantly previous results on acceleration by randomization, which, to the best of our knowledge, dealt solely with bilinear saddle point problems. We illustrate our theoretical findings by instructive and encouraging numerical experiments.