Generalized Framework for Nonlinear Acceleration

Nonlinear acceleration algorithms improve the performance of iterative methods, such as gradient descent, using the information contained in past iterates. However, their efficiency is still not entirely understood even in the quadratic case. In this paper, we clarify the convergence analysis by giving general properties that share several classes of nonlinear acceleration: Anderson acceleration (and variants), quasi-Newton methods (such as Broyden Type-I or Type-II, SR1, DFP, and BFGS) and Krylov methods (Conjugate Gradient, MINRES, GMRES). In particular, we propose a generic family of algorithms that contains all the previous methods and prove its optimal rate of convergence when minimizing quadratic functions. We also propose multi-secants updates for the quasi-Newton methods listed above. We provide a Matlab code implementing the algorithm.