Accelerating Proximal Gradient-type Algorithms using Damped Anderson Acceleration with Restarts and Nesterov Initialization

TL;DR

This paper introduces a two-phase acceleration scheme for proximal gradient algorithms that combines Nesterov's momentum and Anderson acceleration with restarts, significantly improving convergence speed in large-scale optimization.

Contribution

The paper proposes a novel two-phase scheme integrating Nesterov acceleration and Anderson acceleration with restarts for proximal gradient methods, enhancing convergence efficiency.

Findings

Substantial convergence improvements over existing methods.

Effective phase switching based on restart schemes.

Enhanced performance in sparse optimization problems.

Abstract

Despite their frequent slow convergence, proximal gradient schemes are widely used in large-scale optimization tasks due to their tremendous stability, scalability, and ease of computation. In this paper, we develop and investigate a general two-phase scheme for accelerating the convergence of proximal gradient algorithms. By using Nesterov's momentum method in an initialization phase, our procedure delivers fast initial descent that is robust to the choice of starting value. Once iterates are much closer to the solution after the first phase, we utilize a variation of Anderson acceleration to deliver more rapid local convergence in the second phase. Drawing upon restarting schemes developed for Nesterov acceleration, we can readily identify points where it is advantageous to switch from the first to the second phase, which enables use of the procedure without requiring one to specify the number of iterations used in each phase. For the second phase, we adapt and extend a version of Anderson acceleration with algorithm restarts, and we introduce a subsetted version of this procedure that improves performance in problems with substantial sparsity. Through simulation studies involving four representative optimization problems, we show that our proposed algorithm can generate substantial improvements over competing acceleration methods.