Lipschitz-free Projected Subgradient Method with Time-varying Step-size

TL;DR

This paper proposes a new family of time-varying step-sizes for the projected subgradient method, achieving optimal convergence without requiring the objective function to be Lipschitz continuous, thus extending its applicability.

Contribution

It introduces a novel class of step-sizes that ensure optimal ergodic convergence for the projected subgradient method without Lipschitz assumptions.

Findings

Achieves optimal ergodic convergence rates.

Extends convergence guarantees to non-Lipschitz functions.

Does not depend on Lipschitz continuity of the objective.

Abstract

We introduce a novel family of time-varying step-sizes for the classical projected subgradient method, offering optimal ergodic convergence. Importantly, this approach does not depend on the Lipschitz assumption of the objective function, thereby broadening the convergence result of projected subgradient method to non-Lipschitz case.