Optimistic Safety for Linearly-Constrained Online Convex Optimization

The setting of online convex optimization (OCO) under unknown constraints has garnered significant attention in recent years. In this work, we consider a version of this problem with static linear constraints that the player receives noisy feedback of and must always satisfy. By leveraging our novel design paradigm of optimistic safety, we give an algorithm for this problem that enjoys $\tilde{\mathcal{O}}(\sqrt{T})$ regret. This improves on the previous best regret bound of $\tilde{\mathcal{O}}(T^{2/3})$ while using only slightly stronger assumptions of independent noise and an oblivious adversary. Then, by recasting this problem as OCO under time-varying stochastic linear constraints, we show that our algorithm enjoys the same regret guarantees in such a setting and never violates the constraints in expectation. This contributes to the literature on OCO under time-varying stochastic constraints, where the state-of-the-art algorithms enjoy $\tilde{\mathcal{O}}(\sqrt{T})$ regret and $\tilde{\mathcal{O}}(\sqrt{T})$ violation when the constraints are convex and the player receives full feedback. Additionally, we provide a version of our algorithm that is more computationally efficient and give numerical experiments comparing it with benchmark algorithms.