Adaptive Restart of the Optimized Gradient Method for Convex Optimization.

Abstract

First-order methods with momentum such as Nesterov's fast gradient method are very useful for convex optimization problems, but can exhibit undesirable oscillations yielding slow convergence for some applications. An adaptive restarting scheme can improve the convergence rate of the fast gradient methd, when the parameter of a strongly convex cost function is unknown or when the iterates of the algorithm enter a locally well-conditioned region. Recently, we introduced an optimized gradient method, a first-order algorithm that has an inexpensive per-iteration computational cost similar to that of the fast gradient method, yet has a worst-case cost function convergence bound that is twice smaller than that of the fast gradient method and that is optimal for large-dimensional smooth convex problems. Building upon the success of accelerating the fast gradient method using adaptive restart, this paper investigates similar heuristic acceleration of the optimized gradient method. We first derive new step coefficients of the optimized gradient method for a strongly convex quadratic problem with known function parameters, yielding a convergence rate that is faster than that of the analogous version of the fast gradient method. We then provide a heuristic analysis and numerical experiments that illustrate that adaptive restart can accelerate the convergence of the optimized gradient method. Numerical results also illustrate that adaptive restart is helpful for a proximal version of the optimized gradient method for nonsmooth composite convex functions.