Adaptive Proximal Gradient Method for Convex Optimization

TL;DR

This paper introduces adaptive versions of gradient descent and proximal gradient methods for convex optimization that utilize local curvature information, enabling larger stepsizes and convergence guarantees under minimal assumptions.

Contribution

The paper presents novel adaptive algorithms for GD and ProxGD that do not increase computational costs and are proven to converge using only local Lipschitz conditions.

Findings

Adaptive GD and ProxGD outperform standard methods in step size flexibility.

Convergence is guaranteed under local Lipschitzness of the gradient.

Larger stepsizes are achievable compared to previous approaches.

Abstract

In this paper, we explore two fundamental first-order algorithms in convex optimization, namely, gradient descent (GD) and proximal gradient method (ProxGD). Our focus is on making these algorithms entirely adaptive by leveraging local curvature information of smooth functions. We propose adaptive versions of GD and ProxGD that are based on observed gradient differences and, thus, have no added computational costs. Moreover, we prove convergence of our methods assuming only local Lipschitzness of the gradient. In addition, the proposed versions allow for even larger stepsizes than those initially suggested in [MM20].