Near-Optimal Regret Bounds for Monotone Stochastic Optimization under Semi-Bandit Feedback

Abstract

Stochastic optimization models sequential decision-making under uncertainty, typically assuming known input distributions. When these distributions are unknown, \citet{agarwal2026semibandit} give an online-to-offline reduction for monotone stochastic optimization under semi-bandit feedback, where only probed variables are observed. Its expected regret relative to the offline guarantee is $\tilde{O}(n\sqrt{T})$, where $n$ is the number of variables and $T$ the horizon. For optimal stopping and Pandora's box, \citet{liu2025improved} achieve $\tilde{O}(\sqrt{nT})$ using problem-specific algorithms. \emph{Can a general online-to-offline reduction achieve this improved rate?} We give such a reduction for strongly monotone stochastic optimization, answering the corresponding open question. Strong monotonicity is defined relative to the chosen offline algorithm. Our oracle-efficient reduction achieves $\tilde{O}(\sqrt{nT})$ expected regret while preserving approximation guarantees for oracles satisfying the same condition. It combines Bernstein-type optimistic distribution estimates with a Hellinger stability bound for adaptive policy transcripts, aggregating squared marginal errors weighted by the executed policy's probing probabilities. The rate is optimal up to logarithmic factors for $T\ge n$, even under full feedback. Under monotonicity and uniformly bounded coordinate-wise variation, the same algorithm achieves $\tilde{O}(n^{3/4}\sqrt{T})$ expected approximation regret, improving the previous linear dependence on $n$. The exponent $3/4$ is sharp for its ordinary regret over instances and exact oracle choices. Our framework applies to prophet inequality, Pandora's box, best-of-$K$ selection, stochastic knapsack, stochastic orienteering, and stochastic submodular maximization.

arXiv comment68 pages, 20 figures, 1 table