  The Heavy Ball Method, proposed by Polyak over five decades ago, is a
first-order method for optimizing continuous functions. While its stochastic
counterpart has proven extremely popular in training deep networks, there are
almost no known functions where deterministic Heavy Ball is provably faster
than the simple and classical gradient descent algorithm in non-convex
optimization. The success of Heavy Ball has thus far eluded theoretical
understanding. Our goal is to address this gap, and in the present work we
identify two non-convex problems where we provably show that the Heavy Ball
momentum helps the iterate to enter a benign region that contains a global
optimal point faster. We show that Heavy Ball exhibits simple dynamics that
clearly reveal the benefit of using a larger value of momentum parameter for
the problems. The first of these optimization problems is the phase retrieval
problem, which has useful applications in physical science. The second of these
optimization problems is the cubic-regularized minimization, a critical
subroutine required by Nesterov-Polyak cubic-regularized method to find
second-order stationary points in general smooth non-convex problems.
