Competition-Based Resilience in Distributed Quadratic Optimization

TL;DR

This work tackles resilience of distributed quadratic optimization over networks under misbehaving agents by introducing competition-based resilience through Friedkin-Johnsen dynamics with a tunable parameter $\lambda$. The authors analytically and numerically show a nontrivial trade-off between collaboration and competition, with an interior optimal $\lambda^*\in(0,1)$ that increases with attack intensity and number of malicious agents, and tendencies for $\lambda^*$ to approach 1 under severe attacks. Numerical results demonstrate that modest competition (small $\lambda$) can outperform traditional methods like MSR in sparse networks, while maintaining practical error bounds. Overall, the approach offers a computationally light, tunable mechanism to enhance resilience without requiring robustness guarantees, making it appealing for resource-constrained networked control applications.

Abstract

This paper proposes a novel approach to resilient distributed optimization with quadratic costs in a networked control system (e.g., wireless sensor network, power grid, robotic team) prone to external attacks (e.g., hacking, power outage) that cause agents to misbehave. Departing from classical filtering strategies proposed in literature, we draw inspiration from a game-theoretic formulation of the consensus problem and argue that adding competition to the mix can enhance resilience in the presence of malicious agents. Our intuition is corroborated by analytical and numerical results showing that i) our strategy highlights the presence of a nontrivial tradeoff between blind collaboration and full competition, and ii) such competition-based approach can outperform state-of-the-art algorithms based on Mean Subsequence Reduced.

Figures (4)

• Figure 1: Consensus error of FJ dynamics with $3$-regular graph, $N = 100$, one malicious agent, and $d\in\{0,10,\dots,100\}$. The arrow in the left box shows how the error curve varies as the outlier noise intensity $d$ increases.
• Figure 2: Consensus error of FJ dynamics with $3$-regular graph, $d = 10$, $N = 100$, and $M\in\{1,\dots,10\}$ . The arrow in the left box shows how the error varies as more regular agents turn malicious.
• Figure 3: Consensus error of FJ dynamics with $3$-regular graph, $d = 10$, $R = 100$, and $M\in\{1,\dots,10\}$. The arrow in the left box shows how the error varies as more malicious agents $M$ are added to the network.
• Figure 4: Comparison between standard consensus, FJ dynamics (best $\lambda$), and W-MSR with $3$-regular communication graph attacked by $2$ malicious agents.

Theorems & Definitions (19)

• Proposition 1: Consensus protocol vs. FJ dynamics with outliers
• proof
• Remark 1: Limit behavior of FJ dynamics
• Remark 2: Malicious agents disrupt optimization
• Proposition 2: Consensus protocol vs. FJ dynamics with malicious agents
• proof
• Lemma 1
• proof
• Lemma 2
• proof