Beyond Peak Backlog: Conditional Energy and Temporal Geometry in Capacity-Constrained Delayed Bandit Optimization
What is the right delay complexity when a learner can track only $C$ pending feedback items and discarded feedback is permanently lost? Existing one-point bandit convex optimization guarantees in this model pay $\sqrt{T\sigma_{\max}}$, where $\sigma_{\max}$ is the peak backlog, although unlimited tracking admits the sharper $\sqrt{d_{\mathrm{tot}}}$ dependence on total delay. We introduce a scheduler-side conditional-energy interface that separates rate adaptation from the one-point perturbation filtration and handles the dependent importance weights created by randomized admission. Under the same semi-clairvoyant oracle and pathwise hard-capacity contract, this yields an untuned learner whose delay term scales as $O(\sqrt{E_C d_{\mathrm{tot}}})$, with only an explicit restart factor $E_C$; a public constant-factor peak bound removes this factor while $d_{\mathrm{tot}}$ remains unknown. Under strong convexity, the same interface yields the temporal cost $H_A(d)=\sum_t \sigma_t/(A+t)$. Two delay vectors with identical delay multisets, $d_{\mathrm{tot}}$, $\sigma_{\max}$, and capacity can nevertheless have polynomially different minimax regret, showing that timing matters under curvature even when aggregate delay summaries agree. Finally, a continuous hard family converts tracking capacity into a zeroth-order query budget and gives a complementary capacity-starvation lower endpoint. The upper bounds require $C\ge \ln T+1$ and do not constitute a complete capacity minimax characterization.