Diffusion Stochastic Learning Over Adaptive Competing Networks

TL;DR

This paper addresses stochastic, decentralized two-network games where competing teams minimize distinct objectives $J^{(1)}(x,y)$ and $J^{(2)}(x,y)$. It proposes diffusion-based algorithms, ATC-ITC for weak cross-team connectivity and ATC-C for strong cross-team connectivity, enabling within-team cooperation while inferring or observing adversaries’ actions with constant step sizes. The authors prove existence and uniqueness of a Nash equilibrium and establish mean-square convergence to a neighborhood of the equilibrium (scaling as $O(mu)$) with network-centroid dynamics guiding the analysis. They validate the approach through Cournot team competition and decentralized Wasserstein GAN training, showing faster convergence and broader stability ranges than baselines, highlighting practical impact for distributed economic and machine-learning systems. Overall, the work advances adaptive, scalable diffusion strategies for competitive networks without requiring bipartite interconnections or convexity of local costs, enabling robust, decentralized strategic learning.

Abstract

This paper studies a stochastic dynamic game between two competing teams, each consisting of a network of collaborating agents. Unlike fully cooperative settings, where all agents share a common objective, each team in this game aims to minimize its own distinct objective. In the adversarial setting, their objectives could be conflicting as in zero-sum games. Throughout the competition, agents share strategic information within their own team while simultaneously inferring and adapting to the strategies of the opposing team. We propose diffusion learning algorithms to address two important classes of this network game: i) a zero-sum game characterized by weak cross-team subgraph interactions, and ii) a general non-zero-sum game exhibiting strong cross-team subgraph interactions. We analyze the stability performance of the proposed algorithms under reasonable assumptions and illustrate the theoretical results through experiments on Cournot team competition and decentralized GAN training.

Figures (4)

• Figure 1: Illustration of within-team subgraphs, cross-team subgraphs, and associated combination and inference matrices.
• Figure 2: Cournot cross-team information flow with two representatives in each team. The top subfigure corresponds to ATC-C, while the bottom corresponds to ATC-ITC.
• Figure 3: Performance of ATC-ITC (with Assumption \ref{['ass:add_grad']}), CD, ATC-C in Cournot team-competition
• Figure 4: Evolution of gradient norm and mean square error distance between the true values (i.e. mean $\pi$, standard deviation $\sigma$) and estimated ones (i.e. $\hat{\pi}_{k,i}$, $\hat{\sigma}_{k,i}$): In (a), (b), (c), and (d), the true model is given by $\pi = 0, \sigma = 0.01$.

Theorems & Definitions (25)

• Definition 1: Strong cross-team subgraph
• Definition 2: Weak cross-team subgraph
• Remark
• Definition 3: Nash equilibrium
• Lemma 1
• Lemma 2: Within-team and cross-team consensus
• proof
• Lemma 3: Learning dynamics
• Theorem 1: Mean-square-error stability
• Remark