Decentralized Nonconvex Robust Optimization over Unsafe Multiagent Systems: System Modeling, Utility, Resilience, and Privacy Analysis

TL;DR

The paper addresses decentralized nonconvex optimization in unsafe multi-agent systems subject to privacy leakage and Byzantine faults under the Polyak-Łojasiewicz condition. It introduces DP-SCC-PL, a gradient-masking plus resilient-aggregation algorithm that achieves differential privacy and Byzantine resilience without requiring strong convexity or bounded gradients. A contraction-based theoretical framework yields consensus and convergence guarantees, highlighting a trilemma among utility, resilience, and privacy, with sublinear convergence under decaying steps and a preserved exact convergence when privacy/Byzantine issues are absent. Numerical experiments validate utility, resilience, and privacy under diverse Byzantine attacks on a nonconvex problem satisfying the P-Ł condition. The work advances practical private and robust decentralized optimization for complex MASs and paves the way for asynchronous extensions and broader nonconvex settings.

Abstract

Privacy leakage and Byzantine failures are two adverse factors to the intelligent decision-making process of multi-agent systems (MASs). Considering the presence of these two issues, this paper targets the resolution of a class of nonconvex optimization problems under the Polyak-Łojasiewicz (P-Ł) condition. To address this problem, we first identify and construct the adversary system model. To enhance the robustness of stochastic gradient descent methods, we mask the local gradients with Gaussian noises and adopt a resilient aggregation method self-centered clipping (SCC) to design a differentially private (DP) decentralized Byzantine-resilient algorithm, namely DP-SCC-PL, which simultaneously achieves differential privacy and Byzantine resilience. The convergence analysis of DP-SCC-PL is challenging since the convergence error can be contributed jointly by privacy-preserving and Byzantine-resilient mechanisms, as well as the nonconvex relaxation, which is addressed via seeking the contraction relationships among the disagreement measure of reliable agents before and after aggregation, together with the optimal gap. Theoretical results reveal that DP-SCC-PL achieves consensus among all reliable agents and sublinear (inexact) convergence with well-designed step-sizes. It has also been proved that if there are no privacy issues and Byzantine agents, then the asymptotic exact convergence can be recovered. Numerical experiments verify the utility, resilience, and differential privacy of DP-SCC-PL by tackling a nonconvex optimization problem satisfying the P-Ł condition under various Byzantine attacks.

Figures (5)

• Figure 1: A structural example of an unsafe networked system.
• Figure 2: Performance comparison under sign-flipping attacks with injected noises ${{\tilde{n}}_{i,k}} \sim N\left( {0,0.001^2} \right)$.
• Figure 3: Performance comparison under A-Little-Is-Enough attacks with injected noises ${{\tilde{n}}_{i,k}} \sim N\left( {0,0.01^2} \right)$.
• Figure 4: Performance comparison under dissensus attacks with injected noises ${{\tilde{n}}_{i,k}} \sim N\left( {0,0.1^2} \right)$.
• Figure 5: Performance across varying Gaussian noise variance levels.

Theorems & Definitions (20)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Definition 1
• Corollary 1
• Remark 5
• Lemma 1
• Theorem 1
• Remark 6