Optimal Anytime Algorithms for Online Convex Optimization with Adversarial Constraints
Dhruv Sarkar, Abhishek Sinha
TL;DR
This work tackles COCO, aiming to minimize cumulative cost while keeping constraint violations sublinear against adversarial, time-varying constraints, without horizon knowledge. It introduces time-varying Lyapunov functions with a decreasing $\lambda_t$ and a multiplicative queue update to preserve a key monotonicity property, enabling an anytime $O(\sqrt{t})$ regret and $\tilde{O}(\sqrt{t})$ CCV. The framework extends to dynamic regret and optimistic settings, yielding adaptive guarantees that depend on path length $\mathcal{P}_t$ or prediction error $\mathcal{E}_t$, without the impractical doubling trick. Empirical results on constrained online shortest path demonstrate practical advantages, including stability and superior performance relative to horizon-fixed or restart-based baselines, highlighting the method's relevance for safety-constrained online decision making.
Abstract
We propose an anytime online algorithm for the problem of learning a sequence of adversarial convex cost functions while approximately satisfying another sequence of adversarial online convex constraints. A sequential algorithm is called \emph{anytime} if it provides a non-trivial performance guarantee for any intermediate timestep $t$ without requiring prior knowledge of the length of the entire time horizon $T$. Our proposed algorithm achieves optimal performance bounds without resorting to the standard doubling trick, which has poor practical performance due to multiple restarts. Our core technical contribution is the use of time-varying Lyapunov functions to keep track of constraint violations. This must be contrasted with prior works that used a fixed Lyapunov function tuned to the known horizon length $T$. The use of time-varying Lyapunov function poses unique analytical challenges as properties, such as \emph{monotonicity}, on which the prior proofs rest, no longer hold. By introducing a new analytical technique, we show that our algorithm achieves $O(\sqrt{t})$ regret and $\tilde{O}(\sqrt{t})$ cumulative constraint violation bounds for any $t\geq 1$. We extend our results to the dynamic regret setting, achieving bounds that adapt to the path length of the comparator sequence without prior knowledge of its total length. We also present an adaptive algorithm in the optimistic setting, whose performance gracefully scales with the cumulative prediction error. We demonstrate the practical utility of our algorithm through numerical experiments involving the online shortest path problem.