Convergence rates of the heavy ball method for quasi-strongly convex optimization

In this paper, we study the behavior of solutions of the ODE associated to the heavy ball method. Since the pioneering work of B. T. Polyak in 1964, it has been well known that such a scheme is very efficient for C-2 strongly convex functions with Lipschitz gradient. But much less is known when the C-2 assumption is dropped. Depending on the geometry of the function to minimize, we obtain optimal convergence rates for the class of convex functions with some additional regularity such as quasi-strong convexity or strong convexity. We perform this analysis in continuous time for the ODE, and then we transpose these results for discrete optimization schemes. In particular, we propose a variant of the heavy ball algorithm which has the best state of the art convergence rate for first-order methods to minimize strongly composite nonsmooth convex functions.