Accelerated Methods for $\alpha$-Weakly-Quasi-Convex Problems

TL;DR

This paper explores $\alpha$-weakly-quasi-convex optimization problems, demonstrating that existing methods like Sequential Subspace Optimization and Nemirovski's conjugate gradients retain optimal convergence rates under these broader conditions.

Contribution

It extends the understanding of convergence rates for optimization algorithms to the class of $\alpha$-weakly-quasi-convex problems, generalizing previous results.

Findings

Sequential Subspace Optimization remains optimal for smooth $\alpha$-weakly-quasi-convex objectives.

Nemirovski's conjugate gradients method achieves optimal rates under weaker $\alpha$-weak-quasi-convexity.

Results generalize known convergence rates beyond the special case of 1-weak-quasi-convexity.

Abstract

We provide a quick overview of the class of $\alpha$-weakly-quasi-convex problems and its relationships with other problem classes. We show that the previously known Sequential Subspace Optimization method retains its optimal convergence rate when applied to minimization problems with smooth $\alpha$-weakly-quasi-convex objectives. We also show that Nemirovski's conjugate gradients method of strongly convex minimization achieves its optimal convergence rate under weaker conditions of $\alpha$-weak-quasi-convexity and quad\-ratic growth. Previously known results only capture the special case of 1-weak-quasi-convexity or give convergence rates with worse dependence on the parameter $\alpha$.