Capacity-Constrained Online Convex Optimization with Delayed Feedback

Online learning with delayed feedback typically assumes that the learner can track all pending rounds until their feedback arrives. In practice, tracking resources are finite, and feedback from untracked rounds is permanently lost. In this paper, we study delayed online convex optimization (OCO) under a hard capacity constraint, where at most $C$ pending rounds can be tracked at any time. To model delay information, we introduce a semi-clairvoyant model that refines the clairvoyant assumption from prior work: rather than requiring delays to be known at prediction time, the learner observes delay expirations online, consistent with the classical unconstrained delayed setting. Our approach proceeds via a reduction to a novel ``delayed and weighted'' OCO problem, using a scheduler that randomizes tracking decisions and importance-weights the resulting observations. For this base problem, we propose and analyze Delayed-Weighted FTRL and its bandit analogue, establishing regret bounds that explicitly characterize the interaction between time-varying weights and delayed feedback. Combining these base learners with our schedulers yields the first regret guarantees for capacity-constrained OCO under convex and strongly convex losses, for both first-order and bandit feedback. For first-order feedback, capacity $C = Ω(\log T)$ suffices to recover standard delayed OCO rates up to logarithmic factors. For bandit feedback, the regret rates are modulated by powers of $(1 + σ_{\text{max}}/C)$, where $σ_{\text{max}}$ is the maximum number of pending observations at any time. This allows the regret bound to degrade gracefully when $C < σ_{\text{max}}$, while remaining sublinear.

Noise-Adaptive High-Probability Regret Bounds for Online Convex Optimization

We study high-probability regret bounds for online convex optimization (OCO) with strongly convex losses and establish three results that resolve open questions at the intersection of noise adaptivity, feedback structure, and constraint satisfaction. For the full-information setting with sub-Gaussian stochastic gradients, we prove a noise-adaptive high-probability regret bound in which the martingale deviation term scales with the noise level $σ$ rather than the gradient bound $G$, yielding a multiplicative improvement of $G/σ$ over the classical Azuma-Hoeffding baseline. Our analysis introduces an exponential supermartingale argument that bypasses the bounded-difference requirement of Freedman's inequality, enabling direct treatment of unbounded sub-Gaussian noise without truncation artifacts. For bandit feedback, we prove a minimax lower bound: the high-probability regret scales linearly in $\log(1/δ)$, in contrast to the $\sqrt{\log(1/δ)}$ confidence cost under full information. This constitutes a formal separation in the confidence cost of strongly convex OCO across feedback models. Regarding constrained OCO with stochastic constraints satisfying a Slater condition, we provide simultaneous high-probability guarantees for both cumulative regret and long-run constraint violation, achieving $\mathcal{O}(\sqrt{T\log(m/δ)})$ regret and $\mathcal{O}(\sqrt{T}/(ζδ) + m\sqrt{T\log(m/δ)})$ violation. Synthetic experiments corroborate all theoretical predictions.