Dynamic Regret and Tightened Constraint Violation for Communication-Efficient Distributed Online Bilevel Optimization

Abstract

This paper investigates the distributed online bilevel optimization problem with time-varying inequality constraints. In this setting, a multi-agent network is confronted with a hierarchical decision-making task characterized by coupling between the inner and outer optimization levels. We first propose a communication-efficient single-loop distributed online primal--dual algorithm based on communication compression, which tracks both the inner-level solution and the Hessian-inverse-vector product using only one update per-round. We show that this algorithm achieves a dynamic network regret bound of $\mathcal{O}(T^{\max \{{\theta }_1,1-{\theta }_1\}}+T^{{\theta }_1}{P}_T+H_{1,T}+E_{1,T})$, where~${\theta }_1\in (0,1)$, $P_T$ is the outer-level path variation, $H_{1,T}$ is the inner-level path variation, and $E_{1,T}$ is the gradient path variation alongside a network cumulative constraint violation bound of $\mathcal{O}(T^{1-{\theta }_1/2})$, which is further reduced to $\mathcal{O}(T^{1-{\theta }_1})$ under Slater's condition. Furthermore, to eliminate the dependence of regret on the inner-level and gradient path variation, we propose a novel adaptive multi-loop distributed online algorithm to control the inner-level approximation accuracy at each round. By dynamically adjusting the number of inner-level updates based on a gradient-norm tolerance condition, this algorithm yields an improved dynamic network regret bound of $\mathcal{O}(T^{\max \{{\theta }_1,1-{\theta }_1\}}+T^{{\theta }_1}{P}_T)$ which depends solely on the outer-level path variation while maintaining the same network cumulative constraint violation bound. Finally, numerical simulations verify the effectiveness of the proposed algorithms.