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

Abstract

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.

Figures (4)

• Figure 1: Cumulative constraint violation on synthetic OCO ($T = 10{,}000$, $d = 10$) across nine constraint variation configurations. Percentage labels indicate SA-PD's reduction relative to PD-Fixed. SA-PD achieves 53--69% lower violation across all classes. Note the log scale.
• Figure 2: Violation scaling with horizon $T$ under (a) smooth, (b) periodic, and (c) sparse-switching constraints. Dashed lines show $O(T^{3/4})$ and $O(T^{1/2})$ reference rates. SA-PD empirically tracks a rate closer to $O(T^{1/2})$ on these instances, reflecting the small $\delta_c$ and $\bar{\delta}_P$ values in our experimental configurations (see Remark \ref{['rem:periodic_regime']} for the theoretical regime analysis). This plot illustrates instance-dependent scaling and does not contradict the worst-case linear growth permitted by \ref{['eq:periodic_violation']} for fixed $P$.
• Figure 3: Cumulative violation over time on real datasets. (a) Electricity scheduling with periodic capacity: SA-PD grows 53% slower than PD-Fixed. (b) ETT load management with 8 maintenance windows (dashed vertical lines): SA-PD's staircase has smaller step heights at each change point.
• Figure 4: Ablation study. Each panel removes one SA-PD mechanism. Under smooth constraints, the adaptive $\beta$ accounts for the majority of improvement. Under sparse-switching, the change-point reset is the dominant factor.

Theorems & Definitions (28)

• Example 1: Parameterized Constraints
• Definition 1: Smooth Class
• Definition 2: Periodic Class
• Definition 3: Sparse-Switching Class
• Remark 1: On the uniform Slater condition
• Definition 4: Smooth constraint variation
• Definition 5: Periodic constraint variation
• Definition 6: Sparse-switching constraint variation
• Theorem 1: Bounds under smooth variation
• Theorem 2: Bounds under periodic variation