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

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 result is an effective instrument for analyzing stochastic approximation schemes in the absence of a stability guarantee, by using it obtain an alternate criteria 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 V.S. Borkar, C. Andrieu and H. F. Chen , by allowing for the presence of set-valued drift functions.