Cubic Regularization with Momentum for Nonconvex Optimization
This work addresses a bottleneck in nonconvex optimization for researchers and practitioners by accelerating second-order methods, though it is incremental as it extends momentum to an existing framework.
The paper tackles the lack of momentum acceleration for second-order algorithms in nonconvex optimization by applying momentum to cubic regularized Newton's method, achieving the best possible convergence rate to a second-order stationary point and demonstrating substantial convergence facilitation in experiments.
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.