Universal subgradient and proximal bundle methods for convex and strongly convex hybrid composite optimization

TL;DR

This paper introduces two universal, parameter-free optimization methods for convex and strongly convex hybrid composite problems, providing complexity bounds without needing problem-specific parameters or restarts.

Contribution

The paper presents the first universal, parameter-free subgradient and bundle methods that adapt to strong convexity without restarts or multiple threads.

Findings

Established functional complexity bounds for both methods.

Methods are universal, requiring no prior knowledge of problem parameters.

No restarts or multiple threads needed for convergence.

Abstract

This paper develops two parameter-free methods for solving convex and strongly convex hybrid composite optimization problems, namely, a composite subgradient type method and a proximal bundle type method. Functional complexity bounds for the two methods are established in terms of the unknown strong convexity parameter. The two proposed methods are universal with respect to all problem parameters, including the strong convexity one, and require no knowledge of the optimal value. Moreover, in contrast to previous works, they do not restart nor use multiple threads.