On the Global Convergence of Continuous–Time Stochastic Heavy–Ball Method for Nonconvex Optimization

We study the convergence behavior of a stochastic heavy-ball method with a small stepsize. Under a change of time scale, we approximate the discrete scheme by a stochastic differential equation that models small random perturbations of a coupled system of nonlinear oscillators. We rigorously show that the perturbed system converges to a local minimum in a logarithmic time. This indicates that for the diffusion process that approximates the stochastic heavy-ball method, it takes (up to a logarithmic factor) only a linear time of the square root of the inverse stepsize to escape from all saddle points. This results may suggest a fast convergence of its discrete-time counterpart. Our theoretical results are validated by numerical experiments.