  Anderson Acceleration (AA) is a popular acceleration technique to enhance the
convergence of fixed-point iterations. The analysis of AA approaches typically
focuses on the convergence behavior of a corresponding fixed-point residual,
while the behavior of the underlying objective function values along the
accelerated iterates is currently not well understood. In this paper, we
investigate local properties of AA with restarting applied to a basic gradient
scheme in terms of function values. Specifically, we show that AA with
restarting is a local descent method and that it can decrease the objective
function faster than the gradient method. These new results theoretically
support the good numerical performance of AA when heuristic descent conditions
are used for globalization and they provide a novel perspective on the
convergence analysis of AA that is more amenable to nonconvex optimization
problems. Numerical experiments are conducted to illustrate our theoretical
findings.
