Cost-Aware Distributed Online Learning with Strict Rejection Behavior against Adversarial Agents

Distributed online learning in multi-agent systems(MASs) is highly vulnerable to adversarial influence, especially when malicious agents cannot be fully isolated during the transient stage. While existing studies mainly pursue resilient consensus or secure fusion, they pay much less attention to the learning inefficiency and extra evolution cost accumulated during the defense process. This paper addresses this gap by developing a cost-aware distributed online learning framework with strict rejection behavior against adversarial agents. Under this mechanism, the state evolution cost of online adaptation is formulated and the cost amplification effect caused by adversarial interactions is theoretically characterized. To balance robustness, convergence efficiency, and long-term cost, we propose an adaptive adjustment mechanism for the state-evolution rate. The resulting outer-layer update can be equivalently viewed as a constrained online optimization problem. We further establish the well-posedness and regularity of the associated periodic Riccati layer, and show that the outer-layer update ensures feasibility and controlled variation. Based on these properties, closed-loop practical stability is rigorously established via a two-time-scale Lyapunov framework. Simulations demonstrate that the proposed method achieves robust and low-cost convergence under adversarial disturbances. Furthermore, a scenario involving a satellite-assisted IoT monitoring network for target tracking further validates the practical effectiveness of the strict rejection behavior.