Structure-Dependent Regret and Constraint Violation Bounds for Online Convex Optimization with Time-Varying Constraints

Online convex optimization (OCO) with time-varying constraints is a critical framework for sequential decision-making in dynamic networked systems, where learners must minimize cumulative loss while satisfying regions of feasibility that shift across rounds. Existing theoretical analyses typically treat constraint variation as a monolithic adversarial process, resulting in joint regret and violation bounds that are overly conservative for real-world network dynamics. In this paper, we introduce a structured characterization of constraint variation - smooth drift, periodic cycles, and sparse switching - mapping these classes to common network phenomena such as slow channel fading, diurnal traffic patterns, and discrete maintenance windows. We derive structure-dependent joint bounds that strictly improve upon adversarial rates when the constraint process exhibits regularity. To realize these gains, we propose the Structure-Adaptive Primal-Dual (SA-PD) algorithm, which utilizes observable constraint signals to detect environmental structure online and adapt dual update strategies accordingly. Extensive experiments on synthetic benchmarks and real-world datasets - including online electricity scheduling and transformer load management - demonstrate that SA-PD reduces cumulative constraint violation by up to 53% relative to structure-agnostic baselines while maintaining competitive utility. This work serves as a comprehensive guide for exploiting temporal regularity in constrained online learning for robust network engineering.