Fast Gradient Methods for Uniformly Convex and Weakly Smooth Problems

TL;DR

This paper introduces modified fast gradient methods that adapt to uniformly convex and weakly smooth problems without restarting, demonstrating improved theoretical and numerical performance.

Contribution

It proposes a novel momentum-based acceleration technique for uniformly convex and weakly smooth problems, avoiding restarting procedures used in prior methods.

Findings

Proposed methods outperform existing algorithms in numerical experiments.

Theoretical analysis confirms optimal convergence rates.

Methods effectively handle weak smoothness and uniform convexity.

Abstract

In this paper, acceleration of gradient methods for convex optimization problems with weak levels of convexity and smoothness is considered. Starting from the universal fast gradient method which was designed to be an optimal method for weakly smooth problems whose gradients are H\"older continuous, its momentum is modified appropriately so that it can also accommodate uniformly convex and weakly smooth problems. Different from the existing works, fast gradient methods proposed in this paper do not use the restarting technique but use momentums that are suitably designed to reflect both the uniform convexity and the weak smoothness information of the target energy function. Both theoretical and numerical results that support the superiority of proposed methods are presented.