A Poisson Jump-driven SDE Approach to Distributed Gradient Descent with Sparse Communication

TL;DR

This work introduces a Poisson Jump-driven SDE framework to embed asynchronous, sparse communication directly into distributed optimization dynamics, bridging the gap between idealized and real networked settings. By modeling inter-agent channels as stochastic jumps, the authors derive a distributed gradient flow whose updates occur at Poisson times, and they establish stability and convergence guarantees for unconstrained quadratic objectives. The main theoretical contributions are explicit conditions on channel rates and drift bounds that ensure mean-square stability and, at higher but sparse rates, convergence up to the nominal gradient flow rate. The framework is general, enabling analysis and design of energy-efficient, event-driven communication for a broad class of distributed algorithms and hardware architectures, with demonstrated validation on a three-agent quadratic example.

Abstract

To bridge the gap between idealised communication models and the stochastic reality of networked systems, we introduce a framework for embedding asynchronous communication directly into algorithm dynamics using stochastic differential equations (SDE) driven by Poisson Jumps. We apply this communication-aware design to the continuous-time gradient flow, yielding a distributed algorithm where updates occur via sparse Poisson events. Our analysis establishes communication rate bounds for asymptotic stability and, crucially, a higher, yet sparse, rate that provably any desired exponential convergence performance slower than the nominal, centralized flow. These theoretical results, shown for unconstrained quadratic optimisation, are validated by a numerical simulation.

Figures (6)

• Figure 1: Sample path of the Poisson jump-driven SDE, showing the continuous evolution of $\vec{x}(t, \omega_0)$ interrupted by discrete jumps.
• Figure 2: Illustration of two nodes within the communication graph and their directed communication channels: solid lines represent continuous signals, while dashed lines represent discrete communication events. Channel and node dynamics are continuous-time dynamics.
• Figure 3: Sample paths with different channel dynamics, showing the continuous evolution of ($\vec{x}_1(t, \omega)$,\ref{['plot:figure:samplepath-nodrift:x']}) and ($\vec{z}_{(1, 2)}(t, \omega)$,\ref{['plot:figure:samplepath-nodrift:z']}) interrupted by discrete communication events $\odif{N}_{(1,2)}(t, \omega)$.
• Figure 4: Switched RC circuit modelling a leaky integrator channel, where input $x_j$ charges the capacitor at Poisson jumps, and output $\vec{z}_{(j, i)}$ leaks through $R$.
• Figure 5: Comparison of Lyapunov-function evolutions with zero channel drift rates ($\mat{a} = \mat{0}$). Depicted are $\bar{V}(t)$ for diffent communication rates: ($\lambda_1 = 10$,\ref{['plot:figure:lyapunov-function-evolution:A:mean']}), ($\lambda_2 = 27$,\ref{['plot:figure:lyapunov-function-evolution:B:mean']}) and ($\lambda_3 = 50$,\ref{['plot:figure:lyapunov-function-evolution:C:mean']}) alongside their respective confidence intervals, as well as a reference exponential function with decay rate ($2\eigmin{Q}$,\ref{['plot:figure:lyapunov-function-evolution:reference']}) and a single sample path (\ref{['plot:figure:lyapunov-function-evolution:B:V50']}).

Theorems & Definitions (14)

• definition 1: Global Practical Uniform Exponential Stability in mean-square sense
• lemma 1: Global Practical Uniform Exponential Stability in mean-square sense
• definition 2: Quadratic Optimization Problem
• definition 3: Gradient Descent Flow
• definition 4
• lemma 2: Error Dynamics
• theorem 1
• remark 1
• remark 2
• corollary 1