Constrained Online Convex Optimization without Slater's Condition
It solves a long-standing gap in constrained online learning by removing the need for Slater's condition, which is often unrealistic in practice, thus broadening applicability.
This paper proposes an anytime primal-dual framework for constrained online convex optimization that achieves O(√T) regret and O(√T log T) constraint violation for stochastic constraints without requiring Slater's condition, and extends to adversarial constraints with hard violation guarantees.
We study constrained online convex optimization with adversarial losses and stochastic or adversarial constraints. For stochastic constraints, existing algorithms that achieve nearly optimal regret and constraint violation bounds typically rely on regularity assumptions such as Slater's condition, while adversarial-constraint algorithms avoid these assumptions by using a rather restrictive round-wise feasible comparator. We bridge this gap with an anytime primal-dual framework that incorporates an adaptive regularizer into the dual update. The regularizer stabilizes the dual process without relying on the negative drift induced by Slater's condition. For stochastic constraints and convex losses, our algorithm achieves $O(\sqrt{T})$ expected regret and $O(\sqrt{T}\log T)$ expected cumulative constraint violation. Furthermore, we show that our algorithm also admits high-probability bounds of the same order on regret and constraint violation. For strongly convex losses, the regret bound improves to $O(\log T)$ with a violation bound of the same order. With a minor modification, the framework also applies to adversarial constraints and provides guarantees for hard constraint violation.