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.

翻译：我们将Borkar-Meyn稳定性定理（BMT）推广至具有任意矩界信息延迟的分布式随机逼近（SAs）。为刻画延迟，我们引入信息年龄过程（AoIPs）：定义在非负整数上、具有单位增长性质的随机过程。我们证明，具有任意矩界的AoIPs不能无限频繁地超过任意时间分数。结合适当选取的步长，该性质足以保证分布式SAs的稳定性。与BMT相比，我们的分析需进行关键修正，并采用新的论证思路来处理由AoI引起的SA误差。在分析中，我们证明这些SA误差满足递推不等式。为评估该递推关系，我们提出一种针对求和下限时变情形的、新的Gronwall型不等式。作为分布式BMT的应用，我们讨论了基于梯度的分布式优化以及一种分析动量SAs的新方法。