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.

中文翻译：

---

无稳定性保证情况下集值图随机逼近方案分析及其稳定性


在本文中，我们分析了在缺乏稳定性保证的情况下具有集值映射的随机逼近方案的行为。我们证明，经过大量迭代后，如果随机逼近过程进入吸引集的吸引域，则它以很高的概率锁定在吸引集中。我们证明，上述结果是在没有稳定性保证的情况下分析随机逼近方案的有效工具，方法是在存在平均场局部吸引集的情况下使用它来获得收敛的替代标准，并使用它表明，当平均场具有全局吸引集时，涉及定期重置迭代的反馈机制可以稳定该方案，从而保证收敛。本文的结果建立在 Borkar、Andrieu 等人和 Chen 等人的工作基础上，允许存在集值漂移函数。