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Combining Stochastic Adaptive Cubic Regularization with Negative Curvature for Nonconvex Optimization

Author

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  • Seonho Park

    (University of Florida)

  • Seung Hyun Jung

    (Korea Institute of Industrial Technology (KITECH))

  • Panos M. Pardalos

    (University of Florida)

Abstract

We focus on minimizing nonconvex finite-sum functions that typically arise in machine learning problems. In an attempt to solve this problem, the adaptive cubic-regularized Newton method has shown its strong global convergence guarantees and the ability to escape from strict saddle points. In this paper, we expand this algorithm to incorporating the negative curvature method to update even at unsuccessful iterations. We call this new method Stochastic Adaptive cubic regularization with Negative Curvature (SANC). Unlike the previous method, in order to attain stochastic gradient and Hessian estimators, the SANC algorithm uses independent sets of data points of consistent size over all iterations. It makes the SANC algorithm more practical to apply for solving large-scale machine learning problems. To the best of our knowledge, this is the first approach that combines the negative curvature method with the adaptive cubic-regularized Newton method. Finally, we provide experimental results, including neural networks problems supporting the efficiency of our method.

Suggested Citation

  • Seonho Park & Seung Hyun Jung & Panos M. Pardalos, 2020. "Combining Stochastic Adaptive Cubic Regularization with Negative Curvature for Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 953-971, March.
    • Handle: RePEc:spr:joptap:v:184:y:2020:i:3:d:10.1007_s10957-019-01624-6
    DOI: 10.1007/s10957-019-01624-6
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    References listed on IDEAS

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    1. El Houcine Bergou & Youssef Diouane & Serge Gratton, 2018. "A Line-Search Algorithm Inspired by the Adaptive Cubic Regularization Framework and Complexity Analysis," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 885-913, September.
    2. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Javier Cano & Javier M. Moguerza & Francisco J. Prieto, 2017. "Using Improved Directions of Negative Curvature for the Solution of Bound-Constrained Nonconvex Problems," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 474-499, August.
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    Cited by:

    1. Yonggang Pei & Shaofang Song & Detong Zhu, 2023. "A sequential adaptive regularisation using cubics algorithm for solving nonlinear equality constrained optimization," Computational Optimization and Applications, Springer, vol. 84(3), pages 1005-1033, April.

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