Improved Convergence Factor of Windowed Anderson Acceleration for Symmetric Fixed-Point Iterations

This paper studies the commonly utilized windowed Anderson acceleration (AA) algorithm for fixed-point methods, x(k+1)=q(x(k)). It provides the first proof that when the operator q is linear and symmetric the windowed AA, which uses a sliding window of prior iterates, improves the root-linear convergence factor over the fixed-point iterations. When q is nonlinear, yet has a symmetric Jacobian at a fixed point, a slightly modified AA algorithm is proved to have an analogous root-linear convergence factor improvement over fixed-point iterations. Simulations verify our observations. Furthermore, experiments with different data models demonstrate AA is significantly superior to the standard fixed-point methods for Tyler's M-estimation.