Adaptive Stochastic Optimisation of Nonconvex Composite Objectives

TL;DR

This paper introduces adaptive stochastic mirror descent algorithms for nonconvex composite optimization, achieving efficient convergence and low-dimensional exploitation, especially in high-dimensional zeroth-order problems.

Contribution

It presents a new family of adaptive stochastic composite mirror descent algorithms with entropy-like updates and variance reduction, improving convergence guarantees without prior problem knowledge.

Findings

Algorithms converge without prior knowledge of the problem.

Achieve logarithmic complexity dependence on dimensionality.

Effective in high-dimensional zeroth-order optimization.

Abstract

In this paper, we propose and analyse a family of generalised stochastic composite mirror descent algorithms. With adaptive step sizes, the proposed algorithms converge without requiring prior knowledge of the problem. Combined with an entropy-like update-generating function, these algorithms perform gradient descent in the space equipped with the maximum norm, which allows us to exploit the low-dimensional structure of the decision sets for high-dimensional problems. Together with a sampling method based on the Rademacher distribution and variance reduction techniques, the proposed algorithms guarantee a logarithmic complexity dependence on dimensionality for zeroth-order optimisation problems.