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Trimmed Statistical Estimation via Variance Reduction

Author

Listed:
  • Aleksandr Aravkin

    (Department of Applied Mathematics, University of Washington, Seattle, Washington 98195)

  • Damek Davis

    (School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14850)

Abstract

In this paper, we show how to transform any optimization problem that arises from fitting a machine learning model into one that (1) detects and removes contaminated data from the training set while (2) simultaneously fitting the trimmed model on the uncontaminated data that remains. To solve the resulting nonconvex optimization problem, we introduce a fast stochastic proximal-gradient algorithm that incorporates prior knowledge through nonsmooth regularization. For data sets of size n , our approach requires O ( n 2/3 / ℇ ) gradient evaluations to reach ℇ -accuracy, and when a certain error bound holds, the complexity improves to O ( κn 2/3 log(1/ ℇ )), where κ is a “condition number.” These rates are n 1/3 times better than those achieved by typical, nonstochastic methods.

Suggested Citation

  • Aleksandr Aravkin & Damek Davis, 2020. "Trimmed Statistical Estimation via Variance Reduction," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 292-322, February.
    • Handle: RePEc:inm:ormoor:v:45:y:2020:i:1:p:292-322
    DOI: 10.1287/moor.2019.0992
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    References listed on IDEAS

    as
    1. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    2. Dmitriy Drusvyatskiy & Adrian S. Lewis, 2018. "Error Bounds, Quadratic Growth, and Linear Convergence of Proximal Methods," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 919-948, August.
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