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In this paper, we analyze the behavior of stochastic approximation schemeswith set-valued maps in the absence of a stability guarantee. We prove thatafter a large number of iterations if the stochastic approximation processenters the domain of attraction of an attracting set it gets locked into theattracting set with high probability. We demonstrate that the above result isan effective instrument for analyzing stochastic approximation schemes in theabsence of a stability guarantee, by using it obtain an alternate criteria forconvergence in the presence of a locally attracting set for the mean field andby using it to show that a feedback mechanism, which involves resetting theiterates at regular time intervals, stabilizes the scheme when the mean fieldpossesses a globally attracting set, thereby guaranteeing convergence. Theresults in this paper build on the works of V.S. Borkar, C. Andrieu and H. F.Chen , by allowing for the presence of set-valued drift functions.

Analysis of stochastic approximation schemes with set-valued maps in the absence of a stability guarantee and their stabilization