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.

翻译：本文研究固定点方法$x^{(k+1)}=q(x^{(k)})$中常用的窗口化Anderson加速（AA）算法。首次证明：当算子$q$为线性对称时，使用滑动窗口历史迭代的窗口化AA算法相较不动点迭代可改善根线性收敛因子。对于非线性算子$q$（但在不动点处具有对称雅可比矩阵），本文证明经略微修改的AA算法同样能实现与不动点迭代类似的根线性收敛因子提升。数值仿真验证了上述结论。此外，基于不同数据模型的实验表明，在Tyler's M估计中，AA算法显著优于标准不动点方法。