Convergence Analysis of Mixed Timescale Cross-Layer Stochastic Optimization

TL;DR

This paper develops a convergence analysis framework for mixed timescale stochastic algorithms in wireless networks, modeling their dynamics with stochastic differential equations and establishing stability and tracking bounds.

Contribution

It introduces a novel convergence analysis framework for multi-timescale stochastic algorithms using SDE modeling and Lyapunov stability, with an adaptive compensation method.

Findings

Established a sufficient stability condition for the algorithm.

Derived a tracking error bound based on process parameters.

Validated the framework with a wireless network example.

Abstract

This paper considers a cross-layer optimization problem driven by multi-timescale stochastic exogenous processes in wireless communication networks. Due to the hierarchical information structure in a wireless network, a mixed timescale stochastic iterative algorithm is proposed to track the time-varying optimal solution of the cross-layer optimization problem, where the variables are partitioned into short-term controls updated in a faster timescale, and long-term controls updated in a slower timescale. We focus on establishing a convergence analysis framework for such multi-timescale algorithms, which is difficult due to the timescale separation of the algorithm and the time-varying nature of the exogenous processes. To cope with this challenge, we model the algorithm dynamics using stochastic differential equations (SDEs) and show that the study of the algorithm convergence is equivalent to the study of the stochastic stability of a virtual stochastic dynamic system (VSDS). Leveraging the techniques of Lyapunov stability, we derive a sufficient condition for the algorithm stability and a tracking error bound in terms of the parameters of the multi-timescale exogenous processes. Based on these results, an adaptive compensation algorithm is proposed to enhance the tracking performance. Finally, we illustrate the framework by an application example in wireless heterogeneous network.