On the Asymptotic Linear Convergence Speed of Anderson Acceleration, Nesterov Acceleration, and Nonlinear GMRES
This work addresses a theoretical gap in understanding the convergence behavior of acceleration methods for non-convex optimization, which is incremental as it builds on existing methods but provides new quantification.
The paper tackles the problem of quantifying the asymptotic linear convergence speed improvement provided by nonlinear convergence acceleration methods like Anderson acceleration and nonlinear GMRES for fixed-point iterations, and it derives optimal convergence factors and bounds under simplified conditions, with numerical tests on tensor decomposition problems showing that these approaches can estimate convergence speed.
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 $ρ<1$ 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.