Cubic Regularization with Momentum for Nonconvex Optimization
Cubic Regularization with Momentum for Nonconvex Optimization
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发表时间：
2018-10
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通讯作者：
Zhe Wang;Yi Zhou;Yingbin Liang;Guanghui Lan
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作者：
Zhe Wang;Yi Zhou;Yingbin Liang;Guanghui Lan
动量是一种在实际训练中用于加速收敛的常用技术，其对一阶算法收敛保证的影响已得到充分研究。然而，这样一个成功的加速技术还没有被提出的二阶算法在nonconvex optimization.In本文中，我们应用动量计划的立方正则化（CR）牛顿的方法，并探讨潜在的加速。我们对各种非凸优化问题的数值实验表明，动量格式可以大大促进三次正则化的收敛，甚至比Nesterov的CR加速方案更好。理论上，我们证明了CR动量下实现了最好的可能收敛速度到一个非凸优化的二阶稳定点。此外，我们研究了所提出的算法求解问题满足误差界条件，并建立了一个局部二次收敛速度。然后，特别是有限和问题，我们表明，该算法可以允许计算不精确，降低了整体样本的复杂性，而不会降低收敛速度。
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.