Analysis of Stochastic Approximation Schemes With Set-Valued Maps in the Absence of a Stability Guarantee and Their Stabilization

Yaji, Vinayaka G. ; Bhatnagar, Shalabh (2020) Analysis of Stochastic Approximation Schemes With Set-Valued Maps in the Absence of a Stability Guarantee and Their Stabilization IEEE Transactions on Automatic Control, 65 (3). pp. 1100-1115. ISSN 0018-9286

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Official URL: http://doi.org/10.1109/TAC.2019.2916688

Related URL: http://dx.doi.org/10.1109/TAC.2019.2916688

Abstract

In this paper, we analyze the behavior of stochastic approximation schemes with set-valued maps in the absence of a stability guarantee. We prove that after a large number of iterations, if the stochastic approximation process enters the domain of attraction of an attracting set, it gets locked into the attracting set with high probability. We demonstrate that the above-mentioned result is an effective instrument for analyzing stochastic approximation schemes in the absence of a stability guarantee, by using it to obtain an alternate criterion for convergence in the presence of a locally attracting set for the mean field and by using it to show that a feedback mechanism, which involves resetting the iterates at regular time intervals, stabilizes the scheme when the mean field possesses a globally attracting set, thereby guaranteeing convergence. The results in this paper build on the works of Borkar, Andrieu et al., and Chen et al., by allowing for the presence of set-valued drift functions.

Item Type:Article
Source:Copyright of this article belongs to Institute of Electrical and Electronics Engineers.
Keywords:Stochastic Approximation; Set-Valued Maps; Lock-In Probability; Differential Inclusions; Stabilization Via Resetting.
ID Code:116438
Deposited On:12 Apr 2021 05:54
Last Modified:12 Apr 2021 05:54

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