Differentially Private Online Distributed Aggregative Games With Time-Varying and Non-Identical Communication and Feedback Delays

TL;DR

The paper tackles online distributed aggregative games where cost functions vary in time and agents communicate over time-varying, unbalanced graphs while exposing privacy-sensitive information. It introduces an online distributed dual averaging algorithm enhanced with Laplace-based differential privacy and delay-tolerant updates, including left-eigenvector compensation to address graph unbalance and delayed gradient handling. The authors prove a sublinear dynamic regret bound that captures the coupling between privacy noise and communication/feedback delays, and they provide privacy guarantees at both per-step and horizon levels. Simulations on a time-varying Nash-Cournot setting with five firms validate convergence of the running-average actions and illustrate the trade-offs between privacy, delays, and learning performance, highlighting practical impact for privacy-preserving, robust distributed game learning in dynamic networks.

Abstract

This paper investigates online distributed aggregative games with time-varying cost functions, where agents are interconnected through an unbalanced communication graph. Due to the distributed and noncooperative nature of the game, some curious agents may wish to steal sensitive information from neighboring agents during parameter exchanges. Additionally, communication delays arising from network congestion, particularly in wireless settings, as well as feedback delays, can hinder the convergence of agents to a Nash equilibrium. Although a recent work addressed both communication and feedback delays in aggregative games, it is based on the unrealistic assumption that the delays are fixed over time and identical across agents. Hence, the case of time-varying and non-identical delays across agents has never been considered in aggregative games. In this work, we address the combined challenges of privacy leakage with time-varying and non-identical communication and feedback delays for the first time. We propose an online distributed dual averaging algorithm that simultaneously tackles these challenges while achieving a provably low regret bound. Our simulation result shows that the running average of each client's local action converges over time.

Figures (8)

• Figure 1: Time-varying unbalanced graph networks with a broken path at odd iterations.
• Figure 2: Trajectories of $x_i(t)$ (agents' actions) and $\frac{1}{t}\sum_{i=1}^t F_{i,t}(x_i(t),\psi(t))$ (average loss) with no communication and feedback delays, $\epsilon = 0.2$, $\eta_t = \frac{\gamma}{\sqrt{t+1}}$.
• Figure 3: (effect of increasing learning rate) Trajectories of $x_i(t)$ (agents' actions) and $\frac{1}{t}\sum_{i=1}^t F_{i,t}(x_i(t),\psi(t))$ (average loss) with no communication and feedback delays, $\epsilon = 0.2$, and $\eta_t = 10*\gamma/\sqrt{t+1}$.
• Figure 4: (effect of increasing privacy) Trajectories of $x_i(t)$ (agents' actions) and $\frac{1}{t}\sum_{i=1}^t F_{i,t}(x_i(t),\psi(t))$ (average loss) with no communication and feedback delays, and $\epsilon = 0.1$, and $\eta_t = \gamma/\sqrt{t+1}$.
• Figure 5: (effect of fixed communication delays with no privacy) Trajectories of $x_i(t)$ (agents' actions) and $\frac{1}{t}\sum_{i=1}^t F_{i,t}(x_i(t),\psi(t))$ (average loss) with $\tau_i(t) = 0 \quad\forall i$ and $\tau_{42}(t)=\tau_{42} = 2$ and no privacy.