A Variant of Anderson Mixing with Minimal Memory Size

Anderson mixing (AM) is a useful method that can accelerate fixed-point iterations by exploring the information from historical iterations. Despite its numerical success in various applications, the memory requirement in AM remains a bottleneck when solving large-scale optimization problems in a resource-limited machine. To address this problem, we propose a novel variant of AM method, called Min-AM, by storing only one vector pair, that is the minimal memory size requirement in AM. Our method forms a symmetric approximation to the inverse Hessian matrix and is proved to be equivalent to the full-memory Type-I AM for solving strongly convex quadratic optimization. Moreover, for general nonlinear optimization problems, we establish the convergence properties of Min-AM under reasonable assumptions and show that the mixing parameters can be adaptively chosen by estimating the eigenvalues of the Hessian.