Globally Convergent Type-I Anderson Acceleration for Non-Smooth Fixed-Point Iterations

TL;DR

This paper introduces a globally convergent variant of Type-I Anderson acceleration for non-smooth fixed-point problems, improving convergence of first-order algorithms with practical implementation in convex optimization solvers.

Contribution

It develops the first globally convergent Anderson acceleration method for non-expansive fixed-point iterations, incorporating safeguarding, regularization, and restart strategies.

Findings

Enhanced convergence in various first-order algorithms

Effective acceleration demonstrated through extensive numerical experiments

Implementation in SCS 2.0 improves practical solver performance

Abstract

We consider the application of the type-I Anderson acceleration to solving general non-smooth fixed-point problems. By interleaving with safe-guarding steps, and employing a Powell-type regularization and a re-start checking for strong linear independence of the updates, we propose the first globally convergent variant of Anderson acceleration assuming only that the fixed-point iteration is non-expansive. We show by extensive numerical experiments that many first order algorithms can be improved, especially in their terminal convergence, with the proposed algorithm. Our proposed method of acceleration is being implemented in SCS 2.0, one of the default solvers used in the convex optimization parser-solver CVXPY 1.0.