Beyond Peak Backlog: Conditional Energy and Temporal Geometry in Capacity-Constrained Delayed Bandit Optimization

2026-08-17

AI Summary

This study addresses the dependence of regret bounds on peak backlog in capacity-constrained multi-armed bandits with delayed feedback. We propose a scheduler-side conditional energy interface that decouples rate adaptation from disturbance filtering to optimize delay complexity. By revealing how temporal geometry influences regret, we demonstrate that minimax regret can differ polynomially under identical delay statistics, thereby overcoming the limitations of aggregated metrics. Leveraging semi-transparent oracles and strong convexity analysis, we derive a parameter-free regret bound with a delay term of o(√(e_c d_tot)) and establish a lower bound for capacity-scarce regimes. These contributions significantly refine the theoretical granularity of regret analysis in delayed feedback settings beyond conventional aggregate measures.

Abstract

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 $\sqrt{Tσ_{\max}}$, where $σ_{\max}$ 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 $H_A(d)=\sum_t σ_t/(A+t)$. Two delay vectors with identical delay multisets, $d_{\mathrm{tot}}$, $σ_{\max}$, 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.

Problem

• Capacity-Constrained Delayed Bandit
• Delay Complexity
• Peak Backlog
• Temporal Geometry
• Minimax Regret

Innovation

• Conditional-Energy Interface
• Capacity-Constrained Delayed Bandit
• Temporal Geometry
• Importance Weighting
• Delay Complexity