Cubic Regularization with Momentum for Nonconvex Optimization

Understand

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.