  Momentum is a popular technique to accelerate the convergence in practical
training, and its impact on convergence guarantee has been well-studied for
first-order algorithms. However, such a successful acceleration technique has
not yet been proposed for second-order algorithms in nonconvex optimization.In
this paper, we apply the momentum scheme to cubic regularized (CR) Newton's
method and explore the potential for acceleration. Our numerical experiments on
various nonconvex optimization problems demonstrate that the momentum scheme
can substantially facilitate the convergence of cubic regularization, and
perform even better than the Nesterov's acceleration scheme for CR.
Theoretically, we prove that CR under momentum achieves the best possible
convergence rate to a second-order stationary point for nonconvex optimization.
Moreover, we study the proposed algorithm for solving problems satisfying an
error bound condition and establish a local quadratic convergence rate. Then,
particularly for finite-sum problems, we show that the proposed algorithm can
allow computational inexactness that reduces the overall sample complexity
without degrading the convergence rate.
