Provable non-accelerations of the heavy-ball method

In this work, we show that the heavy-ball (HB) method provably does not reach an accelerated convergence rate on smooth strongly convex problems. More specifically, we show that for any condition number and any choice of algorithmic parameters, either the worst-case convergence rate of HB on the class of L-smooth and μ-strongly convex quadratic functions is not accelerated (that is, slower than 1-O(κ)), or there exists an L-smooth μ-strongly convex function and an initialization such that the method does not converge. To the best of our knowledge, this result closes a simple yet open question on one of the most used and iconic first-order optimization techniques. Our approach builds on finding functions for which HB fails to converge and instead cycles over finitely many iterates. We analytically describe all parametrizations of HB that exhibit this cycling behavior on a particular cycle shape, whose choice is supported by a systematic and constructive approach to the study of cycling behaviors of first-order methods. We show the robustness of our results to perturbations of the cycle, and extend them to classes of functions that also satisfy higher-order regularity conditions.