Online Convex Optimization Achieves Bounds Without Slater's Condition.

Constrained online convex optimization (OCO) is a critical area for decision-making under uncertainty, but existing algorithms often rely on restrictive assumptions like Slater's condition for stochastic constraints, or use limited comparators for adversarial constraints. This research presents a novel anytime primal-dual framework that overcomes these limitations. The proposed algorithm integrates an adaptive regularizer into the dual update mechanism, which effectively stabilizes the dual process without needing the negative drift typically induced by Slater's condition. For stochastic constraints and convex losses, this framework achieves nearly optimal expected regret and cumulative constraint violation bounds, and also provides high-probability bounds of the same order. With a minor adjustment, the framework also extends to adversarial constraints, offering guarantees for hard constraint violation, bridging a significant gap in OCO theory.