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Stochastic inexact augmented Lagrangian method for nonconvex expectation constrained optimization

Author

Listed:
  • Zichong Li

    (Rensselaer Polytechnic Institute)

  • Pin-Yu Chen

    (Thomas J. Watson Research Center)

  • Sijia Liu

    (Michigan State University)

  • Songtao Lu

    (Thomas J. Watson Research Center)

  • Yangyang Xu

    (Rensselaer Polytechnic Institute)

Abstract

Many real-world problems not only have complicated nonconvex functional constraints but also use a large number of data points. This motivates the design of efficient stochastic methods on finite-sum or expectation constrained problems. In this paper, we design and analyze stochastic inexact augmented Lagrangian methods (Stoc-iALM) to solve problems involving a nonconvex composite (i.e. smooth + nonsmooth) objective and nonconvex smooth functional constraints. We adopt the standard iALM framework and design a subroutine by using the momentum-based variance-reduced proximal stochastic gradient method (PStorm) and a postprocessing step. Under certain regularity conditions (assumed also in existing works), to reach an $$\varepsilon $$ ε -KKT point in expectation, we establish an oracle complexity result of $$O(\varepsilon ^{-5})$$ O ( ε - 5 ) , which is better than the best-known $$O(\varepsilon ^{-6})$$ O ( ε - 6 ) result. Numerical experiments on the fairness constrained problem and the Neyman–Pearson classification problem with real data demonstrate that our proposed method outperforms an existing method with the previously best-known complexity result.

Suggested Citation

  • Zichong Li & Pin-Yu Chen & Sijia Liu & Songtao Lu & Yangyang Xu, 2024. "Stochastic inexact augmented Lagrangian method for nonconvex expectation constrained optimization," Computational Optimization and Applications, Springer, vol. 87(1), pages 117-147, January.
    • Handle: RePEc:spr:coopap:v:87:y:2024:i:1:d:10.1007_s10589-023-00521-z
    DOI: 10.1007/s10589-023-00521-z
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    References listed on IDEAS

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    1. repec:inm:orijoo:v:3:y:2021:i:4:p:373-397 is not listed on IDEAS
    2. Yangyang Xu & Yibo Xu, 2023. "Momentum-Based Variance-Reduced Proximal Stochastic Gradient Method for Composite Nonconvex Stochastic Optimization," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 266-297, January.
    3. Qihang Lin & Runchao Ma & Yangyang Xu, 2022. "Complexity of an inexact proximal-point penalty method for constrained smooth non-convex optimization," Computational Optimization and Applications, Springer, vol. 82(1), pages 175-224, May.
    4. Guanghui Lan & Zhiqiang Zhou, 2020. "Algorithms for stochastic optimization with function or expectation constraints," Computational Optimization and Applications, Springer, vol. 76(2), pages 461-498, June.
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