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

• URL: http://arxiv.org/abs/2311.02490v2
• Date: Fri, 8 Mar 2024 15:15:11 GMT
• Title: Improved Convergence Rates of Windowed Anderson Acceleration for Symmetric Fixed-Point Iterations
• Abstract summary: This paper studies the commonly utilized windowed Anderson acceleration (AA) algorithm for fixed-point methods. Experiments with different data models demonstrate AA is significantly superior to the standard fixed-point methods for Tyler's M-estimation.
• Score: 15.420927564123193
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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