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The proximal Robbins–Monro method

Author

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  • Panos Toulis
  • Thibaut Horel
  • Edoardo M. Airoldi

Abstract

The need for statistical estimation with large data sets has reinvigorated interest in iterative procedures and stochastic optimization. Stochastic approximations are at the forefront of this recent development as they yield procedures that are simple, general and fast. However, standard stochastic approximations are often numerically unstable. Deterministic optimization, in contrast, increasingly uses proximal updates to achieve numerical stability in a principled manner. A theoretical gap has thus emerged. While standard stochastic approximations are subsumed by the framework Robbins and Monro (The annals of mathematical statistics, 1951, pp. 400–407), there is no such framework for stochastic approximations with proximal updates. In this paper, we conceptualize a proximal version of the classical Robbins–Monro procedure. Our theoretical analysis demonstrates that the proposed procedure has important stability benefits over the classical Robbins–Monro procedure, while it retains the best known convergence rates. Exact implementations of the proximal Robbins–Monro procedure are challenging, but we show that approximate implementations lead to procedures that are easy to implement, and still dominate standard procedures by achieving numerical stability, practically without trade‐offs. Moreover, approximate proximal Robbins–Monro procedures can be applied even when the objective cannot be calculated analytically, and so they generalize stochastic proximal procedures currently in use.

Suggested Citation

  • Panos Toulis & Thibaut Horel & Edoardo M. Airoldi, 2021. "The proximal Robbins–Monro method," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 188-212, February.
  • Handle: RePEc:bla:jorssb:v:83:y:2021:i:1:p:188-212
    DOI: 10.1111/rssb.12405
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    References listed on IDEAS

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    1. Borkar,Vivek S., 2008. "Stochastic Approximation," Cambridge Books, Cambridge University Press, number 9780521515924, October.
    2. Pascal Bianchi & Walid Hachem, 2016. "Dynamical Behavior of a Stochastic Forward–Backward Algorithm Using Random Monotone Operators," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 90-120, October.
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    Cited by:

    1. Rutger-Jan Lange & Bram van Os & Dick van Dijk, 2022. "Implicit score-driven filters for time-varying parameter models," Tinbergen Institute Discussion Papers 22-066/III, Tinbergen Institute, revised 01 Jun 2024.
    2. Bram van Os, 2023. "Information-Theoretic Time-Varying Density Modeling," Tinbergen Institute Discussion Papers 23-037/III, Tinbergen Institute.

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