Linear Asymptotic Convergence of Anderson Acceleration: Fixed-Point Analysis

We study the asymptotic convergence of AA(m), i.e., Anderson acceleration with window size m for accelerating fixed-point methods xk+1 = q(xk), xk ∈ Rn. Convergence acceleration by AA(m) has been widely observed but is not well understood. We consider the case where the fixed-point iteration function q(x) is differentiable and the convergence of the fixed-point method itself is root-linear. We identify numerically several conspicuous properties of AA(m) convergence: First, AA(m) sequences {xk} converge root-linearly but the root-linear convergence factor depends strongly on the initial condition. Second, the AA(m) acceleration coefficients β do not converge but oscillate as {xk} converges to x∗. To shed light on these observations, we write the AA(m) iteration as an augmented fixed-point iteration zk+1 = Ψ(zk), zk ∈ Rn(m+1) and analyze the continuity and differentiability properties of Ψ(z) and β(z). We find that the vector of acceleration coefficients β(z) is not continuous at the fixed point z∗. However, we show that, despite the discontinuity of β(z), the iteration function Ψ(z) is Lipschitz continuous and directionally differentiable at z∗ for AA(1), and we generalize this to AA(m) with m > 1 for most cases. Furthermore, we find that Ψ(z) is not differentiable at z∗. We then discuss how these theoretical findings relate to the observed convergence behaviour of AA(m). The discontinuity of β(z) at z∗ allows β to oscillate as {xk} converges to x∗, and the non-differentiability of Ψ(z) allows AA(m) sequences to converge with root-linear convergence factors that strongly depend on the initial condition. Additional numerical results illustrate our findings for several linear and nonlinear fixed-point iterations xk+1 = q(xk) and for various values of the window size m.