Nonlinear Acceleration of Primal-Dual Algorithms

We describe a convergence acceleration scheme for multi-step optimization algorithms. The extrapolated solution is written as a nonlinear average of the iterates produced by the original optimization algorithm. Our scheme does not need the underlying fixed-point operator to be symmetric, hence handles e.g. algorithms with momentum terms such as Nesterov’s accelerated method, or primal-dual methods such as Chambolle-Pock. The weights are computed via a simple linear system and we analyze performance in both online and offline modes. We use Crouzeix’s conjecture to show that acceleration is controlled by the solution of a Chebyshev problem on the numerical range of a nonsymmetric operator modelling the behavior of iterates near the optimum. Numerical experiments are detailed on image processing and logistic regression problems.