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Stochastic Approximation

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  • Borkar,Vivek S.

Abstract

This simple, compact toolkit for designing and analyzing stochastic approximation algorithms requires only basic literacy in probability and differential equations. Yet these algorithms have powerful applications in control and communications engineering, artificial intelligence and economic modelling. The dynamical systems viewpoint treats an algorithm as a noisy discretization of a limiting differential equation and argues that, under reasonable hypotheses, it tracks the asymptotic behaviour of the differential equation with probability one. The differential equation, which can usually be obtained by inspection, is easier to analyze. Novel topics include finite-time behaviour, multiple timescales and asynchronous implementation. There is a useful taxonomy of applications, with concrete examples from engineering and economics. Notably it covers variants of stochastic gradient-based optimization schemes, fixed-point solvers, which are commonplace in learning algorithms for approximate dynamic programming, and some models of collective behaviour. Three appendices give background on differential equations and probability.

Suggested Citation

  • Borkar,Vivek S., 2008. "Stochastic Approximation," Cambridge Books, Cambridge University Press, number 9780521515924.
    • Handle: RePEc:cup:cbooks:9780521515924
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    Cited by:

    1. 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.
    2. Tadić, Vladislav B., 2015. "Convergence and convergence rate of stochastic gradient search in the case of multiple and non-isolated extrema," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 1715-1755.
    3. Li Cai, 2010. "High-dimensional Exploratory Item Factor Analysis by A Metropolis–Hastings Robbins–Monro Algorithm," Psychometrika, Springer;The Psychometric Society, vol. 75(1), pages 33-57, March.
    4. Helin Zhu & Joshua Hale & Enlu Zhou, 2018. "Simulation optimization of risk measures with adaptive risk levels," Journal of Global Optimization, Springer, vol. 70(4), pages 783-809, April.
    5. Georgios Chasparis & Jeff Shamma, 2012. "Distributed Dynamic Reinforcement of Efficient Outcomes in Multiagent Coordination and Network Formation," Dynamic Games and Applications, Springer, vol. 2(1), pages 18-50, March.

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