Cubic Regularization with Momentum for Nonconvex Optimization

Abstract

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.

References in corpus (9)

• Accelerated Methods for Non-Convex Optimization
• Newton-Type Methods for Non-Convex Optimization Under Inexact Hessian Information
• Accelerated Gradient Descent Escapes Saddle Points Faster than Gradient Descent
• Convergence Analysis of Proximal Gradient with Momentum for Nonconvex Optimization
• On Noisy Negative Curvature Descent: Competing with Gradient Descent for Faster Non-convex Optimization
• A Unified Scheme to Accelerate Adaptive Cubic Regularization and Gradient Methods for Convex Optimization
• NEON+: Accelerated Gradient Methods for Extracting Negative Curvature for Non-Convex Optimization
• A Family of Inexact SQA Methods for Non-Smooth Convex Minimization with Provable Convergence Guarantees Based on the Luo-Tseng Error Bound Property
• On the Quadratic Convergence of the Cubic Regularization Method under a Local Error Bound Condition

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• Faster Stochastic Quasi-Newton Methods