On the constrained online convex optimization with feedback delay

We investigate the problem of online convex optimization (OCO) under feedback delay, where feedback for a decision is received after a delay, and long-term constraints, where constraints can be violated in intermediate iterations but must be satisfied over the long run. Existing approaches are primarily limited to fixed delay settings and general convex loss functions. In this paper, we employ a stricter metric based on cumulative constraint violations. We first propose a novel algorithm tailored for the fixed $d$-slot delay setting, achieving a regret bound of $O\left(\sqrt{dT}\right)$ and a cumulative constraint violation of $O$ ($T^{\frac{1}{4}}$), demonstrating superior performance compared to existing results. Moreover, when the loss functions are strongly convex, the regret and violation bounds can be further reduced to $O$ ($dlnT$) and $O$ ($\sqrt{d}lnT$), respectively. Additionally, we extend our algorithm to the more realistic re-indexed delay setting, achieving $O\left(\sqrt{dT}\right)$ regret and $O\left(T^{\frac{1}{4}}\right)$ cumulative constraint violation. Under strong convexity, these bounds are further improved to $O(\hat{d}lnT)$ and $O(\sqrt{\hat{d}}lnT)$, where $\hat{d}={max}_{t\in \left[T\right]}{d}_t$ denotes the maximum delay.