A Proof That Anderson Acceleration Improves the Convergence Rate in Linearly Converging Fixed-Point Methods (But Not in Those Converging Quadratically)

Abstract

This paper provides the first proof that Anderson acceleration (AA) improves the convergence rate of general fixed point iterations. AA has been used for decades to speed up nonlinear solvers in many applications, however a rigorous mathematical justification of the improved convergence rate has remained lacking. The key ideas of the analysis presented here are relating the difference of consecutive iterates to residuals based on performing the inner-optimization in a Hilbert space setting, and explicitly defining the gain in the optimization stage to be the ratio of improvement over a step of the unaccelerated fixed point iteration. The main result we prove is that AA improves the convergence rate of a fixed point iteration to first order by a factor of the gain at each step. In addition to improving the convergence rate, our results indicate that AA increases the radius of convergence. Lastly, our estimate shows that while the linear convergence rate is improved, additional quadratic terms arise in the estimate, which shows why AA does not typically improve convergence in quadratically converging fixed point iterations. Results of several numerical tests are given which illustrate the theory.

Benchmarking results for the Newton–Anderson method

“…In each of the above cases, Newton-Anderson converged faster than accelerated-Newton. In agreement with Table 1: Results for problems from [10] where Newton-Anderson(1) converges the results of [1,2], these results indicate that Anderson slows the convergence of quadratically-converging iterations, and should not be used on problems for which Newton is known to converge robustly and quadratically. For the Brown almost-linear function (14), Newton-Anderson converged in 1.5 times as long as Newton (6 additional iterations) with n = 5 and no damping, but 0.15 times as long as Newton (316 fewer iterations), for the problem with dimension n = 20 with a damping factor of 0.8.…”

On Quasi‐Newton methods in fast Fourier transform‐based micromechanics

“…Eyert and Fang and Saad pointed out the relation of Anderson acceleration to Quasi‐Newton schemes and identified it as a generalized form of Broyden's second method. Recently, Evans et al provided a proof that Anderson acceleration improves the convergence rate of linearly converging fixed‐point methods.…”

“…More precisely, it was identified as a generalized multisecant form of the second Broyden method (or “bad Broyden method”) which approximates the Hessian in terms of a number m (called depth) of past iterates and gradients. Recently, Evans et al proved that Anderson acceleration improved the first‐order convergence rate for fixed‐point iterations. Pollock and Rebholz extended the analysis to the noncontractive setting and provided sharper residual bounds.…”

Accelerating ADMM for efficient simulation and optimization

“…Our experiments show that Anderson acceleration is effective in reducing the number of iterations, but we do not have a theoretical guarantee for such property. This is still an open research problem, and the only existing result we are aware of is [Evans et al 2018], which proves that Anderson acceleration improves the convergence rate for linearly converging fixed-point methods if a set of strong assumptions is satisfied. Further theoretical analysis of our method is needed to understand and guarantee its performance.…”