We explain how Anderson Acceleration (AA) speeds up the Alternating Direction Method of Multipliers (ADMM), for the case where ADMM by itself converges linearly. We do so by considering the spectral properties of the Jacobians of ADMM and a stationary version of AA evaluated at the fixed point, where the coefficients of the stationary version are computed such that its asymptotic linear convergence factor is optimal. Numerical tests show that this allows us to quantify the improved linear asymptotic convergence speed of AA-ADMM as compared to the convergence factor of ADMM used by itself. This way of estimating AA-ADMM convergence speed is useful because there are no known convergence bounds for AA with finite window size that would allow quantification of this improvement in asymptotic convergence speed.

中文翻译：

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量化应用于 ADMM 的 Anderson Acceleration 的渐近线性收敛速度

我们解释了 Anderson Acceleration (AA) 如何加速乘法器交替方向法 (ADMM)，在 ADMM 本身线性收敛的情况下。我们通过考虑 ADMM 的雅可比矩阵的频谱特性和在固定点评估的 AA 的平稳版本来做到这一点，其中计算平稳版本的系数，使其渐近线性收敛因子是最佳的。数值测试表明，与自身使用的 ADMM 收敛因子相比，这使我们能够量化 AA-ADMM 改进的线性渐近收敛速度。这种估计 AA-ADMM 收敛速度的方法很有用，因为对于具有有限窗口大小的 AA 来说，没有已知的收敛界限，可以量化渐近收敛速度的这种改进。