Stability and Convergence of Distributed Stochastic Approximations with large Unbounded Stochastic Information Delays

We generalize the Borkar-Meyn stability Theorem (BMT) to distributed stochastic approximations (SAs) with information delays that possess an arbitrary moment bound. To model the delays, we introduce Age of Information Processes (AoIPs): stochastic processes on the non-negative integers with a unit growth property. We show that AoIPs with an arbitrary moment bound cannot exceed any fraction of time infinitely often. In combination with a suitably chosen stepsize, this property turns out to be sufficient for the stability of distributed SAs. Compared to the BMT, our analysis requires crucial modifications and a new line of argument to handle the SA errors caused by AoI. In our analysis, we show that these SA errors satisfy a recursive inequality. To evaluate this recursion, we propose a new Gronwall-type inequality for time-varying lower limits of summations. As applications to our distributed BMT, we discuss distributed gradient-based optimization and a new approach to analyzing SAs with momentum.