Can Competition Outperform Collaboration? The Role of Misbehaving Agents

TL;DR

The paper addresses resilience in distributed quadratic optimization when some agents misbehave. It introduces a competition-based protocol built on Friedkin-Johnsen dynamics with a tunable parameter $\lambda$ to balance collaboration and competition. A key contribution is the competition-collaboration trade-off, with an explicit decomposition of the steady-state consensus error and an interior optimal $\lambda^*$ under mild conditions, supported by numerical experiments that show performance gains over MSR-based methods. The work also analyzes how network topology and connectivity influence resilience and discusses future directions, including heterogeneous parameter design and extensions to distributed optimization under adversarial settings.

Abstract

We investigate a novel approach to resilient distributed optimization with quadratic costs in a multi-agent system prone to unexpected events that make some agents misbehave. In contrast to commonly adopted filtering strategies, we draw inspiration from phenomena modeled through the Friedkin-Johnsen dynamics and argue that adding competition to the mix can improve resilience in the presence of misbehaving agents. Our intuition is corroborated by analytical and numerical results showing that (i) there exists a nontrivial trade-off between full collaboration and full competition and (ii) our competition-based approach can outperform state-of-the-art algorithms based on Weighted Mean Subsequence Reduced. We also study impact of communication topology and connectivity on resilience, pointing out insights to robust network design.

Figures (20)

• Figure 1: Competition vs. collaboration in distributed quadratic optimization. The global cost $e_{\text{tot}}$ is the sum of two contributions that reflect two contrasting attitudes of regular agents: $e_{\text{deception}}$ is caused by (erroneously) trusting misbehaving agents, which makes them drift away from the nominal average, while $e_{\text{consensus}}$ is due to the competition among regular agents, which mitigates misbehaviors but also prevents regular agents from reaching a consensus. The tunable parameter $\lambda \in [0,1]$ allows regular agents to smoothly transition from full collaboration ($\lambda=0$), where they fully trust all agents in the network, to full competition ($\lambda=1$), where they trust no other agent, producing a rich range of behaviors at local and global scale.
• Figure 2: FJ dynamics consensus error with $3$-regular graph, exponential decay of observation covariances, and one misbehaving agent. The arrow shows how the error curve varies as the intensity $d$ of the deception bias increases.
• Figure 3: FJ dynamics consensus error with $3$-regular graph, diagonal prior covariance matrix $\Sigma$, and one misbehaving agent.
• Figure 4: FJ dynamics consensus error with $3$-regular graph and diagonal prior covariance matrix $\Sigma$. The arrow on the left box shows how the error varies as the number of misbehaving nodes $M$ increases (with $R = 100$).
• Figure 5: Optimal $\lambda$ as a function of $M$ with $d = 10$. Each pair of misbehaving agents affects the same regular agent (e.g., the first two belong to $\mathcal{N}_{1}$).

Theorems & Definitions (28)

• Remark 1: Misbehavior vs. intelligent attacks
• Remark 2: Connections with game theory and opinion dynamics
• Remark 3: Competition for resilience
• Remark 4: Heterogeneous competition
• Lemma 1: Drift vs. competition
• proof
• Proposition 1: Full competition vs. full collaboration
• proof : Sketch of proof
• Theorem 1: Competition-collaboration trade-off.
• proof : Sketch of proof