Optimal Anytime Algorithms for Online Convex Optimization with Adversarial Constraints

2025-10-26

AI Summary

This paper studies online convex optimization (OCO) in adversarial environments, where both the sequence of convex cost functions and online convex constraints are subject to adversarial perturbations, and no prior knowledge of the time horizon is available. We propose a novel algorithm based on a time-varying Lyapunov function, eliminating the need for conventional doubling tricks. Our approach achieves, for the first time, tight joint bounds at any time $t$: dynamic regret and cumulative constraint violation are both $O(sqrt{t})$ (the latter up to a logarithmic factor), matching the optimal asymptotic rate. By moving beyond fixed Lyapunov function analysis, our framework naturally accommodates optimistic predictions and is sensitive to prediction errors, while extending seamlessly to dynamic regret settings. Experiments on the online shortest path problem validate both the tightness of our theoretical bounds and the practical superiority of the proposed method.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Online Learning & Bandits
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $ ilde{O}(sqrt{t})$ cumulative constraint violation bounds for any $tgeq 1$. 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.

Problem

• Learning adversarial convex cost functions while satisfying constraints
• Achieving optimal performance without knowing time horizon length
• Overcoming limitations of doubling trick with time-varying Lyapunov functions

Innovation

• Anytime online algorithm without doubling trick
• Time-varying Lyapunov functions for constraints
• Achieves O(sqrt(t)) regret and constraint violation