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, [Formula: see text]. It provides the first proof that when the operator [Formula: see text] is linear, symmetric, and contractive, the windowed AA, which uses a sliding window of prior iterates, improves the root-linear convergence factor over the fixed-point iterations. When [Formula: see text] is nonlinear, yet has a symmetric, non-scalar, and locally contractive 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.