On the convergence of restarted Anderson acceleration for symmetric linear systems

Anderson acceleration (AA) is a technique for accelerating the convergence of an underlying fixed-point iteration. AA is widely used within computational science, with applications ranging from electronic structure calculation to the training of neural networks. Despite AA's widespread use, relatively little is understood about it theoretically. An important and unanswered question in this context is: To what extent can AA actually accelerate convergence of the underlying fixed-point iteration? While simple enough to state, this question appears rather difficult to answer. For example, it is unanswered even in the simplest (non-trivial) case where the underlying fixed-point iteration consists of applying a two-dimensional affine function. In this note we consider a restarted variant of AA applied to solve symmetric linear systems with restart window of size one. Several results are derived from the analytical solution of a nonlinear eigenvalue problem characterizing residual propagation of the AA iteration. This includes a complete characterization of the method to solve $2 \times 2$ linear systems, rigorously quantifying how the asymptotic convergence factor depends on the initial iterate, and quantifying by how much AA accelerates the underlying fixed-point iteration. We also prove that even if the underlying fixed-point iteration diverges, the associated AA iteration may still converge.