Convergence rates of the Heavy-Ball method with Lojasiewicz property

In this paper, a joint study of the behavior of solutions of the Heavy Ball ODE and Heavy Ball type algorithms is given. Since the pioneering work of B.T. Polyak [38], it is well known that such a scheme is very efficient for C 2 strongly convex functions with Lipschitz gradient. But much less is known when only growth conditions are considered. Depending on the geometry of the function to minimize, convergence rates for convex functions, with some additional regularity such as quasi-strong convexity, or strong convexity, were recently obtained in [12]. Convergence results with much weaker assumptions are given in the present paper: namely, linear convergence rates when assuming a growth condition (which amounts to a Lojasiewicz property in the convex case). This analysis is firstly performed in continuous time for the ODE, and then transposed for discrete optimization schemes. In particular, a variant of the Heavy Ball algorithm is proposed, which converges geometrically whatever the parameters choice, and which has the best state of the art convergence rate for first order methods to minimize composite non smooth convex functions satisfying a Lojasiewicz property

FISTA is an automatic geometrically optimized algorithm for strongly convex functions

In this work, we are interested in the famous FISTA algorithm. We show that FISTA is an automatic geometrically optimized algorithm for functions satisfying a quadratic growth assumption. This explains why FISTA works better than the standard Forward-Backward algorithm (FB) in such a case, although FISTA is known to have a polynomial asymptotical convergence rate while FB is exponential. We provide a simple rule to tune the α parameter within the FISTA algorithm to reach an ε-solution with an optimal number of iterations. These new results highlight the efficiency of FISTA algorithms, and they rely on new non asymptotic bounds for FISTA

High Prevalence of Personality Disorders in Skin-restricted Lupus Patients

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