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Graphical Convergence of Subgradients in Nonconvex Optimization and Learning

Author

Listed:
  • Damek Davis

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

  • Dmitriy Drusvyatskiy

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

Abstract

We investigate the stochastic optimization problem of minimizing population risk, where the loss defining the risk is assumed to be weakly convex. Compositions of Lipschitz convex functions with smooth maps are the primary examples of such losses. We analyze the estimation quality of such nonsmooth and nonconvex problems by their sample average approximations. Our main results establish dimension-dependent rates on subgradient estimation in full generality and dimension-independent rates when the loss is a generalized linear model. As an application of the developed techniques, we analyze the nonsmooth landscape of a robust nonlinear regression problem.

Suggested Citation

  • Damek Davis & Dmitriy Drusvyatskiy, 2022. "Graphical Convergence of Subgradients in Nonconvex Optimization and Learning," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 209-231, February.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:1:p:209-231
    DOI: 10.1287/moor.2021.1126
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    References listed on IDEAS

    as
    1. Stephen M. Robinson, 1996. "Analysis of Sample-Path Optimization," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 513-528, August.
    2. Damek Davis & Dmitriy Drusvyatskiy & Kellie J. MacPhee & Courtney Paquette, 2018. "Subgradient Methods for Sharp Weakly Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 962-982, December.
    3. Alan J. King & R. Tyrrell Rockafellar, 1993. "Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 148-162, February.
    4. Daniel Ralph & Huifu Xu, 2011. "Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 568-592, August.
    Full references (including those not matched with items on IDEAS)

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