Universal Online Convex Optimization With Delayed Feedback

Abstract

We study universal online convex optimization with delayed feedback. We develop a single algorithm that simultaneously adapts to general convex, strongly convex, and exp-concave losses without prior knowledge of the curvature class or parameters. Our regret bounds match the best known dependence on the time horizon $T$ and delays up to doubly logarithmic factors: roughly $\sqrt{T + d_{\mathrm{tot}}}$ for convex losses and $σ_{\max}\log T$ rates with curved losses, where $d_{\mathrm{tot}}$ is the total cumulative delay and $σ_{\max}$ is the maximum number of simultaneously missing observations. Our main technical contribution is a delayed meta-learner with a second-order excess-loss guarantee, which enables curvature-adaptive aggregation of delayed base learners. In addition, our new exp-concave base learner yields a sharper regret bound which improves the state-of-the-art delay dependence by replacing the maximum delay with the potentially much smaller $σ_{\max}$, while retaining the complementary $\sqrt{d_{\mathrm{tot}}}$ dependence.