We consider nonlinear convergence acceleration methods for fixed-point iteration $x_{k+1}=q(x_k)$, including Anderson acceleration (AA), nonlinear GMRES (NGMRES), and Nesterov-type acceleration (corresponding to AA with window size one). We focus on fixed-point methods that converge asymptotically linearly with convergence factor $\rho<1$ and that solve an underlying fully smooth and non-convex optimization problem. It is often observed that AA and NGMRES substantially improve the asymptotic convergence behavior of the fixed-point iteration, but this improvement has not been quantified theoretically. We investigate this problem under simplified conditions. First, we consider stationary versions of AA and NGMRES, and determine coefficients that result in optimal asymptotic convergence factors, given knowledge of the spectrum of $q'(x)$ at the fixed point $x^*$. This allows us to understand and quantify the asymptotic convergence improvement that can be provided by nonlinear convergence acceleration, viewing $x_{k+1}=q(x_k)$ as a nonlinear preconditioner for AA and NGMRES. Second, for the case of infinite window size, we consider linear asymptotic convergence bounds for GMRES applied to the fixed-point iteration linearized about $x^*$. Since AA and NGMRES are equivalent to GMRES in the linear case, one may expect the GMRES convergence factors to be relevant for AA and NGMRES as $x_k \rightarrow x^*$. Our results are illustrated numerically for a class of test problems from canonical tensor decomposition, comparing steepest descent and alternating least squares (ALS) as the fixed-point iterations that are accelerated by AA and NGMRES. Our numerical tests show that both approaches allow us to estimate asymptotic convergence speed for nonstationary AA and NGMRES with finite window size.
翻译:我们考虑非线性趋同加速方法,用于固定点变异 $x ⁇ k+1 ⁇ q(x_k), 包括 Anderson 加速(AA)、 非线性 GMRES(NGMRES) 和 Nesterov 类型的加速(对AA 的响应,窗口大小为1) 。我们关注固定点的加速方法,这些方法在线性趋同系数中与 $\rho <1美元相趋同,并解决一个完全平稳和非对调的优化问题。人们经常看到, AAAA和NGMRES 大大改进了固定点变异性趋同行为,但这一改进没有在理论上量化。首先,我们考虑AA和NGMRES 的固定版本,根据对 美元(x) 数字趋异性变异性变异性变异性变异性变异性变异性变异性变异性变异。