A Generalization of the Borkar-Meyn Theorem for Stochastic Recursive Inclusions
This work provides a theoretical generalization for stability analysis in stochastic optimization, which is incremental but useful for researchers in control theory and machine learning.
The paper extends the Borkar-Meyn stability theorem to stochastic recursive inclusions with differential inclusion mean fields, presenting two sets of sufficient conditions for stability and convergence, and applies this to solve the approximate drift problem.
In this paper the stability theorem of Borkar and Meyn is extended to include the case when the mean field is a differential inclusion. Two different sets of sufficient conditions are presented that guarantee the stability and convergence of stochastic recursive inclusions. Our work builds on the works of Benaim, Hofbauer and Sorin as well as Borkar and Meyn. As a corollary to one of the main theorems, a natural generalization of the Borkar and Meyn Theorem follows. In addition, the original theorem of Borkar and Meyn is shown to hold under slightly relaxed assumptions. Finally, as an application to one of the main theorems we discuss a solution to the approximate drift problem.