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DOI: 10.4230/LIPIcs.ITP.2022.31

Formalization of a Stochastic Approximation Theorem

Authors: Koundinya Vajjha, Barry Trager, Avraham Shinnar, and Vasily Pestun

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

Abstract

Stochastic approximation algorithms are iterative procedures which are used to approximate a target value in an environment where the target is unknown and direct observations are corrupted by noise. These algorithms are useful, for instance, for root-finding and function minimization when the target function or model is not directly known. Originally introduced in a 1951 paper by Robbins and Monro, the field of Stochastic approximation has grown enormously and has come to influence application domains from adaptive signal processing to artificial intelligence. As an example, the Stochastic Gradient Descent algorithm which is ubiquitous in various subdomains of Machine Learning is based on stochastic approximation theory. In this paper, we give a formal proof (in the Coq proof assistant) of a general convergence theorem due to Aryeh Dvoretzky [Dvoretzky, 1956] (proven in 1956) which implies the convergence of important classical methods such as the Robbins-Monro and the Kiefer-Wolfowitz algorithms. In the process, we build a comprehensive Coq library of measure-theoretic probability theory and stochastic processes.

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Koundinya Vajjha, Barry Trager, Avraham Shinnar, and Vasily Pestun. Formalization of a Stochastic Approximation Theorem. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 31:1-31:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{vajjha_et_al:LIPIcs.ITP.2022.31,
  author =	{Vajjha, Koundinya and Trager, Barry and Shinnar, Avraham and Pestun, Vasily},
  title =	{{Formalization of a Stochastic Approximation Theorem}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{31:1--31:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.31},
  URN =		{urn:nbn:de:0030-drops-167402},
  doi =		{10.4230/LIPIcs.ITP.2022.31},
  annote =	{Keywords: Formal Verification, Stochastic Approximation, Stochastic Processes, Probability Theory, Optimization Algorithms}
}

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Formalization of a Stochastic Approximation Theorem

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